Black Holes

Black holes are some of the most fascinating and mind-bending objects in the cosmos. The very thing that characterizes a black hole also makes it hard to study: its intense gravity. All the mass in a black hole is concentrated in a tiny region, surrounded by a boundary called the “event horizon”. Nothing that crosses that boundary can return to the outside universe, not even light. A black hole itself is invisible.

But astronomers can still observe black holes indirectly by the way their gravity affects stars and pulls matter into orbit. As gas flows around a black hole, it heats up, paradoxically making these invisible objects into some of the brightest things in the entire universe. As a result, we can see some black holes from billions of light-years away. For one large black hole in a nearby galaxy, astronomers even managed to see a ring of light around the event horizon, using a globe-spanning array of powerful telescopes.

Center for Astrophysics | Harvard & Smithsonian scientists participate in many black hole-related projects:

Using the Event Horizon Telescope (EHT) to capture the first image of a black hole’s “shadow”: the absence of light that marks where the event horizon is located. The EHT is composed of many telescopes working together to create one Earth-sized observatory , all monitoring the supermassive black hole at the center of the galaxy M87, leading to the first image ever captured of a black hole. CfA Plays Central Role In Capturing Landmark Black Hole Image

Observing supermassive black holes in other galaxies to understand how they evolve and shape their host galaxies. CfA astronomers use telescopes across the entire spectrum of light, from radio waves to X-rays to gamma rays. A Surprising Blazar Connection Revealed

Studying the infall of matter — called “accretion” — onto black holes, using NASA’s Chandra X-ray Observatory and other telescopes. In addition, CfA researchers use cutting-edge supercomputers to create theoretical models for the disks and jets of matter that black holes create around themselves. Supermassive Black Hole Spins Super-Fast

Hunting for black hole interactions with other astronomical objects. That includes “disruption” events, where black holes tear stars or other objects apart, creating bursts of intense light. Black Hole Meal Sets Record for Length and Size

Observing clusters of stars to find intermediate mass black holes, and modeling how they shape their environments. A Middleweight Black Hole is Hiding at the Center of a Giant Star Cluster

Hunting for and characterizing stellar mass black holes, which can include information about their birth process and evolution. NASA's Chandra Adds to Black Hole Birth Announcement

The Varieties of Black Holes

Black holes come in three categories:

Stellar Mass Black Holes are born from the death of stars much more massive than the Sun. When some of these stars run out of the nuclear fuel that makes them shine, their cores collapse into black holes under their own gravity. Other stellar mass black holes form from the collision of neutron stars , such as the ones first detected by LIGO and Virgo in 2017. These are probably the most common black holes in the cosmos, but are hard to detect unless they have an ordinary star for a companion. When that happens, the black hole can strip material from the star, causing the gas to heat up and glow brightly in X-rays.

Supermassive Black Holes are the monsters of the universe, living at the centers of nearly every galaxy. They range in mass from 100,000 to billions of times the mass of the Sun, far too massive to be born from a single star. The Milky Way’s black hole is about 4 million times the Sun’s mass, putting it in the middle of the pack. In the form of quasars and other “active” galaxies , these black holes can shine brightly enough to be seen from billions of light-years away. Understanding when these black holes formed and how they grow is a major area of research. Center for Astrophysics | Harvard & Smithsonian scientists are part of the Event Horizon Telescope (EHT) collaboration, which captured the first-ever image of the black hole: the supermassive black hole at the center of the galaxy M87.

Intermediate Mass Black Holes are the most mysterious, since we’ve hardly seen any of them yet. They weigh 100 to 10,000 times the mass of the Sun, putting them between stellar and supermassive black holes. We don’t know exactly how many of these are, and like supermassive black holes, we don’t fully understand how they’re born or grow. However, studying them could tell us a lot about how the most supermassive black holes came to be.

Black holes can seem bizarre and incomprehensible, but in truth they’re remarkably understandable. Despite not being able to see black holes directly, we know quite a bit about them. They are …

Simple . All three black hole types can be described by just two observable quantities: their mass and how fast they spin. That’s much simpler than a star, for example, which in addition to mass is a product of its unique history and evolution , including its chemical makeup. Mass and spin tell us everything we need to know about a black hole: it “forgets” everything that went into making it. Those two quantities determine how big the event horizon is, and the way gravity affects any matter falling onto the black hole.

Compact . Black holes are tiny compared to their mass. The event horizon of a black hole the mass of the Sun would be no more than 6 kilometers across, and the faster it spins, the smaller that size is. Even a supermassive black hole would fit easily inside our Solar System.

Powerful . The combination of large mass and small size results in very strong gravity. This gravity is strong enough to pull a star apart if it gets too close, producing powerful bursts of light. A supermassive black hole heats gas falling onto it to temperatures of millions of degrees, making it glow brightly enough in X-rays and other types of radiation to be seen across the universe.

Very common . From theoretical calculations based on observations, astronomers think the Milky Way might have as many as a hundred million black holes, most of which are stellar mass. And with at least one supermassive black hole in most galaxies, there could be hundreds of billions of supermassive black holes in the observable universe.

Very important . Black holes have a reputation for eating everything that comes by, but they turn out to be messy eaters. A lot of stuff that falls toward a black hole gets jetted away, thanks to the complicated churning of gas near the event horizon. These jets and outflows of gas called “winds” spread atoms throughout the galaxy, and can either boost or throttle the birth of new stars, depending on other factors. That means supermassive black holes play an important role in the life of galaxies, even far beyond the black hole’s gravitational pull.

And yes, mysterious . Along with astronomers, physicists are interested in black holes because they’re a laboratory for “quantum gravity”. Black holes are described by Albert Einstein’s general relativity, which is our modern theory of gravity, but the other forces of nature are described by quantum physics. So far, nobody has developed a complete quantum gravity theory, but we already know black holes will be an important test of any proposed theory.

The first image of a black hole

The first image of a black hole in human history, captured by the Event Horizon Telescope, showing light emitted by matter as it swirls under the influence of intense gravity. This black hole is 6.5 billion times the mass of the Sun and resides at the center of the galaxy M87.

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Scientists have found a black hole so large it eats the equivalent of one sun per day

Joe Hernandez

research paper in black hole

This illustration provided by the European Southern Observatory this month depicts the record-breaking quasar J059-4351, the bright core of a distant galaxy that is powered by a supermassive black hole. M. Kornmesser/AP hide caption

This illustration provided by the European Southern Observatory this month depicts the record-breaking quasar J059-4351, the bright core of a distant galaxy that is powered by a supermassive black hole.

Picture a black hole so powerful that it swallows the equivalent of one sun every day.

Now imagine that black hole also has a mass that's 17 billion times larger than our sun.

Scientists in Australia have just found exactly that.

New research that was published in the journal Nature Astronomy on Monday lays out the discovery of a distant quasar containing what scientists say is the fastest-growing black hole ever recorded.

"It's a surprise it remained undetected until now, given what we know about many other, less impressive black holes," co-author Christopher Onken, a researcher at Australian National University, said in a university news release . "It was hiding in plain sight."

James Webb Telescope detects earliest known black hole — it's really big for its age

James Webb Telescope detects earliest known black hole — it's really big for its age

Researchers first detected the black hole using a telescope at the university's Siding Spring Observatory in New South Wales and then confirmed it with the European Southern Observatory's Very Large Telescope, one of the largest telescopes in the world.

Though black holes themselves don't emit any light, very large ones form bright objects called quasars . Located in the middle of galaxies, quasars are illuminated by all the matter that heats up as it gets pulled in.

Australian National University professor Christian Wolf, the study's lead author, said in the news release that the quasar is now the brightest known object in the universe, shining 500 trillion times brighter than the sun.

"The incredible rate of growth also means a huge release of light and heat," Wolf said.

Black holes are surrounded by an accretion disk, a swirling zone where material is first dragged in before being consumed. This black hole's accretion disk is 7 light-years wide.

"It looks like a gigantic and magnetic storm cell with temperatures of 10,000 degrees Celsius, lightning everywhere and winds blowing so fast they would go around Earth in a second," Wolf added.

This particular quasar, J059-4351, is located 12 billion light-years away.

Researchers discover fastest-growing black hole that consumes the mass of 'the Sun and all the planets' combined — every day

The fastest-growing black hole ever recorded — increasing the equivalent of one sun every day — has been discovered by a team of international researchers led by experts at the Australian National University (ANU).

The black hole's mass is roughly 17 billion times that of our solar system's Sun, and exists within the brightest currently known thing being continually powered in the universe.

It exists in a quasar — a swirling storm surrounding an active supermassive black hole at the centre of a galaxy.

Lead author and ANU associate professor, Christian Wolf, said the black hole was indeed creating matter from its environment, and consuming a lot to do so.

"This black hole eats as much mass in a single day as there is in our entire solar system – the Sun and all the planets combined," he said.

A man with grey hair and glasses smiles in front of an artist rendering of a supermassive black hole.

"The accretion disk [the holding pattern for all the material waiting to be devoured] is so massive and dense and hot that it starts glowing brightly, and that's the light that we see.

"It's a lot of light that comes out of that accretion disk, about 500 trillion times the amount of light that our Sun emits, or about 20,000 times the amount of light that our entire Milky Way galaxy – with all its billions of stars – emits."

Dr Wolf said the quasar's accretion disk was incredibly big, being seven light-years in diameter — 1.5 times the distance from our solar system to the next star in the sky, Alpha Centauri.

An artist rendering of a black hole inside a mass of fiery red and orange material.

The quasar, known as J0529-435, was first detected using a 2.3 metre telescope at the ANU Siding Spring Observatory near Coonabarabran in NSW.

The research team then turned to one of the world's largest telescopes — the European Southern Observatory's Very Large Telescope — to confirm the full nature of the black hole and measure its mass.

"Finding it wasn't easy because … if you look at the sky and you see lots and lots of stars, some of them are actually fast-growing black holes, but you don't know which is which," Dr Wolf said.

"They are in the minority – there's lots of stars out there, very few of them are black holes."

Though very far away, the quasar can be seen from Earth with a backyard telescope with a lens at least the size of a basketball.

It cannot be seen by the naked eye, but given its distance from Earth, Dr Wolf said it's impressive it can be seen with such little work.

Space with a small square blown up to show the location of a black hole.

"It's very far away, the light has been travelling for 12 billion years to reach us, so evidently it has to be very luminous for us to see it from here," he said.

"And indeed, it's the most luminous object that we now know in the universe.

"All this light comes from the accretion disk, which is a giant storm cell — a storm cell with seven light years of diameter, and with material on the outside that moves at wind speeds of a few thousand kilometres per second."

The scientific community's understanding of black holes is quite limited, with nothing known about what is inside one.

"Physicists describe black holes as entities that only have two properties: mass and rotation," Dr Wolf said.

"Mass and rotation are the only things we can possibly measure — everything else is unknown.

"What's inside, it doesn't show in any way."

A man smiling in the camera with a blue light cast over his face.

ANU astrophysicist and former vice-chancellor, Brian Schmidt, said the "quite remarkable" discovery was significant for expanding our understanding of black holes.

"I think the reason this is mind blowing is just the fact that the thing exists at all," he said.

"It is not something I would've thought would have been in the universe, it just seems too big and too active.

"But it'll be a great laboratory to figure out what's going on with black holes, because there are a lot of mysteries about why there are so many — and so many that are big — in the universe."

The authors of the research paper suggested the black hole at the centre of quasar J0529-435 was amassing near the Eddington limit — the proposed upper limit to the mass of a star or an accretion disk.

But to know for sure whether that is the case further research and observations will be needed to explore its growth rate.

The researchers' findings are published in Nature Astronomy .

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research paper in black hole

New paper demonstrates link between black holes and early star formation

By ZACHARY BAHAR | February 19, 2024

artists-rendering-ulas-j1120-0641

ESO/M. KORNMESSER / CC BY 4.0

Cosmologist Joseph Silk explores the role of supermassive black holes in the early universe in a recent paper.

Supermassive black holes have long fascinated physicists and astronomers. Almost every large galaxy has a supermassive black hole located at its center, and with solar masses ranging from 100,000 to billions or even hundreds of billions, these structures bind galaxies. As gas falls onto its accretion disks , it heats up and releases powerful waves of electromagnetic energy. How do these cosmic maelstroms emerge? What could enable their formation?

An international group of cosmologists led by Homewood Professor of Physics and Astronomy Joseph Silk proposed a novel solution to this question in a paper recently published in the Astrophysical Journal Letters . Silk’s team proposes that not only did supermassive black holes dominate the composition of the early universe, but they also seeded stars and developed galaxies. 

Recent data from the James Webb Space Telescope (JWST), capable of looking farther into the early universe than any of its predecessors due to infrared sensors, left scientists confused: The early universe was filled with both far more supermassive black holes and luminous galaxies than expected. Silk’s team analyzed this data and concluded that these two factors are directly related: The existence of supermassive black holes leads directly to luminous galaxies. 

Silk discussed these findings in an interview with The News-Letter .

“What we have discovered — it’s a new idea — is that the presence of black holes is often correlated with those same galaxies which contain very luminous and very large star formations. We think it's extremely unlikely these things are independent of each other,” he said. “As you go back in time, the galaxies that surrounded black holes have relatively fewer stars than today. This suggested to us two things: One is that the black holes most likely came first. Second, if that were the case, then the energy they produce when they collect gas from their surroundings, stimulated large amounts of star formation.”

Black holes form when massive stars implode on themselves. The team hypothesized that the first black holes were concentrated in the center of early galaxies, where dense gas clouds enabled rapid star formation. As these first stars died and created black holes, they were drawn gravitationally towards one another, coalescing in the galactic center and merging. This process continues leading to increasingly massive black holes. Gas from the galactic nucleus also falls onto the developing black hole, giving it even more mass.

As these supermassive black holes form, they begin pulling the gas around them into an accretion disk , a region of hot gas circling the black hole and releasing jets of ionized particles. These jets spur the creation of new stars.

“When the black holes form, they're uprooting lots of gas from their surroundings, and they act like these incredible factories of energy. Gas falls on the black hole and gets converted into a high energy outflow of very, very hot gas,” Silk explained. “This happens because the black hole acts like a gigantic furnace with directionality attached to it because it's spinning very fast. That tends to jerk stuff out. The outflows overwhelm the gas clouds nearby, crush them and turn them into stars.”

This process continues until the black holes reach a critical mass at which point their energetic outflows are so overwhelming that, rather than crush surrounding gas clouds into stars, they clear them out, dispelling their contents. At this point, the black hole prompts star formation to cease.

Silk emphasized that this theory, like many in astronomy, is driven by observation. New theories emerge hand in hand with new telescopic technology — such as JWST — that enable more accurate observations.

“We've had a very difficult time extrapolating from what we observe with the Hubble [Space] Telescope to these further away, earlier-in-time regions of the universe. We had various theories interpreting what [the Hubble Space Telescope] found, and they failed to predict this new phenomenon,” Silk said. “In astronomy, we play catch up: We find these amazing things with our new telescopes and then interpret them, model them and do our best, only to find out that — when you take another step back in time — it’s very hard to predict what you’ll see.”

Over his career, Silk has authored over 700 papers and earned countless laurels in the astronomy community. He recently released Back to the Moon: The Next Giant Leap for Humankind , which offers an impassioned argument in favor of lunar science. His current project investigating the development of early black holes has been ongoing for over a decade, but now the tools available are capable of surveying the past unlike ever before. Silk’s recent paper concluded with a set of observations to be taken in the coming years to further validate this theory.

“This particular project dealing with how you make black holes and how they interact is something I've been working on for at least 10 years now. It’s only with the [JWST], that we finally have the data to really test these theories and refine these theories,” Silk said.

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Study: Without more data, a black hole’s origins can be “spun” in any direction

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2 black holes next to each other, with pink and purple cloudy light touching and spinning around them.

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2 black holes next to each other, with pink and purple cloudy light touching and spinning around them.

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Clues to a black hole’s origins can be found in the way it spins. This is especially true for binaries, in which two black holes circle close together before merging. The spin and tilt of the respective black holes just before they merge can reveal whether the invisible giants arose from a quiet galactic disk or a more dynamic cluster of stars.

Astronomers are hoping to tease out which of these origin stories is more likely by analyzing the 69 confirmed binaries detected to date. But a new study finds that for now, the current catalog of binaries is not enough to reveal anything fundamental about how black holes form.

In a study appearing today in the journal Astronomy and Astrophysics Letters, MIT physicists show that when all the known binaries and their spins are worked into models of black hole formation, the conclusions can look very different, depending on the particular model used to interpret the data. 

A black hole’s origins can therefore be “spun” in different ways, depending on a model’s assumptions of how the universe works.

“When you change the model and make it more flexible or make different assumptions, you get a different answer about how black holes formed in the universe,” says study co-author Sylvia Biscoveanu, an MIT graduate student working in the LIGO Laboratory. “We show that people need to be careful because we are not yet at the stage with our data where we can believe what the model tells us.”

The study’s co-authors include Colm Talbot, an MIT postdoc; and Salvatore Vitale, an associate professor of physics and a member of the Kavli Institute of Astrophysics and Space Research at MIT.

A tale of two origins

Black holes in binary systems are thought to arise via one of two paths. The first is through “field binary evolution,” in which two stars evolve together and eventually explode in supernovae, leaving behind two black holes that continue circling in a binary system. In this scenario, the black holes should have relatively aligned spins, as they would have had time — first as stars, then black holes — to pull and tug each other into similar orientations. If a binary’s black holes have roughly the same spin, scientists believe they must have evolved in a relatively quiet environment, such as a galactic disk.

Black hole binaries can also form through “dynamical assembly,” where two black holes evolve separately, each with its own distinct tilt and spin. By some extreme astrophysical processes, the black holes are eventually brought together, close enough to form a binary system. Such a dynamical pairing would likely occur not in a quiet galactic disk, but in a more dense environment, such as a globular cluster, where the interaction of thousands of stars can knock two black holes together. If a binary’s black holes have randomly oriented spins, they likely formed in a globular cluster.

But what fraction of binaries form through one channel versus the other? The answer, astronomers believe, should lie in data, and particularly, measurements of black hole spins.

To date, astronomers have derived the spins of black holes in 69 binaries, which have been discovered by a network of gravitational-wave detectors including LIGO in the U.S., and its Italian counterpart Virgo. Each detector listens for signs of gravitational waves — very subtle reverberations through space-time that are left over from extreme, astrophysical events such as the merging of massive black holes.

With each binary detection, astronomers have estimated the respective black hole’s properties, including their mass and spin. They have worked the spin measurements into a generally accepted model of black hole formation, and found signs that binaries could have both a preferred, aligned spin, as well as random spins. That is, the universe could produce binaries in both galactic disks and globular clusters.

“But we wanted to know, do we have enough data to make this distinction?” Biscoveanu says. “And it turns out, things are messy and uncertain, and it’s harder than it looks.”

Spinning the data

In their new study, the MIT team tested whether the same data would yield the same conclusions when worked into slightly different theoretical models of how black holes form.

The team first reproduced LIGO’s spin measurements in a widely used model of black hole formation. This model assumes that a fraction of binaries in the universe prefer to produce black holes with aligned spins, where the rest of the binaries have random spins. They found that the data appeared to agree with this model’s assumptions and showed a peak where the model predicted there should be more black holes with similar spins.

They then tweaked the model slightly, altering its assumptions such that it predicted a slightly different orientation of preferred black hole spins. When they worked the same data into this tweaked model, they found the data shifted to line up with the new predictions. The data also made similar shifts in 10 other models, each with a different assumption of how black holes prefer to spin.

“Our paper shows that your result depends entirely on how you model your astrophysics, rather than the data itself,” Biscoveanu says.

“We need more data than we thought, if we want to make a claim that is independent of the astrophysical assumptions we make,” Vitale adds.

Just how much more data will astronomers need? Vitale estimates that once the LIGO network starts back up in early 2023, the instruments will detect one new black hole binary every few days. Over the next year, that could add up to hundreds more measurements to add to the data.

“The measurements of the spins we have now are very uncertain,” Vitale says. “But as we build up a lot of them, we can gain better information. Then we can say, no matter the detail of my model, the data always tells me the same story — a story that we could then believe.”

This research was supported in part by the National Science Foundation.

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Scientists have found a black hole so large it eats the equivalent of one sun per day

This illustration provided by the European Southern Observatory in February 2024 depicts the record-breaking quasar J059-4351, the bright core of a distant galaxy that is powered by a supermassive black hole.

This illustration provided by the European Southern Observatory in February 2024 depicts the record-breaking quasar J059-4351, the bright core of a distant galaxy that is powered by a supermassive black hole.

M. Kornmesser / AP

Picture a black hole so powerful that it swallows the equivalent of one sun every day.

Now imagine that black hole also has a mass that's 17 billion times larger than our sun.

Scientists in Australia have just found exactly that.

New research that was published in the journal Nature Astronomy on Monday lays out the discovery of a distant quasar containing what scientists say is the fastest-growing black hole ever recorded.

"It's a surprise it remained undetected until now, given what we know about many other, less impressive black holes," co-author Christopher Onken, a researcher at Australian National University, said in a university press release . "It was hiding in plain sight."

Researchers first detected the black hole using a telescope at the university's Siding Spring Observatory in New South Wales, and then confirmed it with the European Southern Observatory's Very Large Telescope, one of the largest telescopes in the world.

Though black holes themselves don't emit any light, very large ones form bright objects called quasars . Located in the middle of galaxies, quasars are illuminated by all the matter that heats up as it gets pulled in.

Australian National University professor Christian Wolf, the study's lead author, said in a statement that the quasar is now the brightest known object in the universe, shining 500 trillion times brighter than the sun.

"The incredible rate of growth also means a huge release of light and heat," Wolf said.

Black holes are surrounded by an accretion disk, a swirling zone where material is first dragged in before being consumed. This black hole's accretion disk is seven light-years wide.

"It looks like a gigantic and magnetic storm cell with temperatures of 10,000 degrees Celsius, lightning everywhere and winds blowing so fast they would go around Earth in a second," Wolf added.

This particular quasar, J059-4351, is located 12 billion light-years away.

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Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar

Oliver James 3,1 , Eugénie von Tunzelmann 1 , Paul Franklin 1 and Kip S Thorne 2

Published 13 February 2015 • © 2015 IOP Publishing Ltd Classical and Quantum Gravity , Volume 32 , Number 6 Citation Oliver James et al 2015 Class. Quantum Grav. 32 065001 DOI 10.1088/0264-9381/32/6/065001

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1 Double Negative Ltd., 160 Great Portland Street, London W1W 5QA, UK

2 Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

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3 Author to whom any correspondence should be addressed.

  • Received 27 November 2014
  • Revised 12 January 2015
  • Accepted 13 January 2015
  • Published 13 February 2015

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Method : Single-anonymous Revisions: 1 Screened for originality? No

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Interstellar is the first Hollywood movie to attempt depicting a black hole as it would actually be seen by somebody nearby. For this, our team at Double Negative Visual Effects , in collaboration with physicist Kip Thorne, developed a code called Double Negative Gravitational Renderer (DNGR) to solve the equations for ray-bundle (light-beam) propagation through the curved spacetime of a spinning (Kerr) black hole, and to render IMAX-quality, rapidly changing images. Our ray-bundle techniques were crucial for achieving IMAX-quality smoothness without flickering; and they differ from physicists' image-generation techniques (which generally rely on individual light rays rather than ray bundles), and also differ from techniques previously used in the film industry's CGI community. This paper has four purposes: (i) to describe DNGR for physicists and CGI practitioners, who may find interesting and useful some of our unconventional techniques. (ii) To present the equations we use, when the camera is in arbitrary motion at an arbitrary location near a Kerr black hole, for mapping light sources to camera images via elliptical ray bundles. (iii) To describe new insights, from DNGR, into gravitational lensing when the camera is near the spinning black hole, rather than far away as in almost all prior studies; we focus on the shapes, sizes and influence of caustics and critical curves, the creation and annihilation of stellar images, the pattern of multiple images, and the influence of almost-trapped light rays, and we find similar results to the more familiar case of a camera far from the hole. (iv) To describe how the images of the black hole Gargantua and its accretion disk, in the movie Interstellar , were generated with DNGR—including, especially, the influences of (a) colour changes due to doppler and gravitational frequency shifts, (b) intensity changes due to the frequency shifts, (c) simulated camera lens flare, and (d) decisions that the film makers made about these influences and about the Gargantua's spin, with the goal of producing images understandable for a mass audience. There are no new astrophysical insights in this accretion-disk section of the paper, but disk novices may find it pedagogically interesting, and movie buffs may find its discussions of Interstellar interesting.

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1. Introduction and overview

1.1. previous research and visualizations.

At a summer school in Les Houches, France, in summer 1972, Bardeen [ 1 ], building on earlier work of Carter [ 2 ], initiated research on gravitational lensing by spinning black holes. Bardeen gave a thorough analytical analysis of null geodesics (light-ray propagation) around a spinning black hole; and, as part of his analysis, he computed how a black hole's spin affects the shape of the shadow that the hole casts on light from a distant star field. The shadow bulges out on the side of the hole moving away from the observer, and squeezes inward and flattens on the side moving toward the observer. The result, for a maximally spinning hole viewed from afar, is a D-shaped shadow; cf figure 4 . (When viewed up close, the shadow's flat edge has a shallow notch cut out of it, as hinted by figure 8 .)

Despite this early work, gravitational lensing by black holes remained a backwater of physics research until decades later, when the prospect for actual observations brought it to the fore.

There were, we think, two especially memorable accomplishments in the backwater era. The first was a 1978 simulation of what a camera sees as it orbits a non-spinning black hole, with a star field in the background. This simulation was carried out by Palmer et al [ 3 ] on an Evans and Sutherland vector graphics display at Simon Fraser University. Palmer et al did not publish their simulation, but they showed a film clip from it in a number of lectures in that era. The nicest modern-era film clip of this same sort that we know of is by Riazuelo (contained in his DVD [ 4 ] and available on the web at [ 5 ]); see figure 3 and associated discussion. And see [ 6 ] for an online application by Müller and Weiskopf for generating similar film clips. Also of much interest in our modern era are film clips by Hamilton [ 7 ] of what a camera sees when falling into a nonspinning black hole; these have been shown at many planetariums, and elsewhere.

The other most memorable backwater-era accomplishment was a black and white simulation by Luminet [ 8 ] of what a thin accretion disk, gravitationally lensed by a nonspinning black hole, would look like as seen from far away but close enough to resolve the image. In figure 15 (c), we show a modern-era colour version of this, with the camera close to a fast-spinning black hole.

Gravitational lensing by black holes began to be observationally important in the 1990s. Rauch and Blandford [ 9 ] recognized that, when a hot spot, in a black hole's accretion disk or jet, passes through caustics of the Earth's past light cone (caustics produced by the hole's spacetime curvature), the brightness of the hot spot's x-rays will undergo sharp oscillations with informative shapes. This has motivated a number of quantitative studies of the Kerr metric's caustics; see, especially [ 9 – 11 ] and references therein.

In the 1990s astrophysicists began to envision an era in which very long baseline interferometry would make possible the imaging of black holes—specifically, their shadows and their accretion disks. This motivated visualizations, with ever increasing sophistication, of accretion disks around black holes: modern variants of Luminet's [ 8 ] pioneering work. See, especially, Fukue and Yokoyama [ 13 ], who added colours to the disk; Viergutz [ 14 ], who made his black hole spin, treated thick disks, and produced particularly nice and interesting coloured images and included the disk's secondary image which wraps under the black hole; Marck [ 15 ], who laid the foundations for a lovely movie now available on the web [ 16 ] with the camera moving around close to the disk, and who also included higher-order images, as did Fanton et al [ 17 ] and Beckwith and Done [ 18 ]. See also papers cited in these articles.

In the 2000s astrophysicists have focused on perfecting the mm-interferometer imaging of black-hole shadows and disks, particularly the black hole at the centre of our own Milky Way Galaxy (Sgr A*). See, e.g., the 2000 feasibility study by Falcke et al [ 19 ]. See also references on the development and exploitation of general relativistic magnetohydrodynamical (GRMHD) simulation codes for modelling accretion disks like that in Sgr A* [ 20 – 22 ]; and references on detailed GRMHD models of Sgr A* and the models' comparison with observations [ 23 – 26 ]. This is culminating in a mm interferometric system called the Event Horizon Telescope [ 27 ], which is beginning to yield interesting observational results though not yet full images of the shadow and disk in Sgr A*.

All the astrophysical visualizations of gravitational lensing and accretion disks described above, and all others that we are aware of, are based on tracing huge numbers of light rays through curved spacetime. A primary goal of today's state-of-the-art, astrophysical ray-tracing codes (e.g., the Chan et al massively parallel, GPU-based code GRay [ 28 ]) is very fast throughput, measured, e.g., in integration steps per second; the spatial smoothness of images has been only a secondary concern. For our Interstellar work, by contrast, a primary goal is smoothness of the images, so flickering is minimized when objects move rapidly across an IMAX screen; fast throughput has been only a secondary concern.

With these different primary goals, in our own code, called Double Negative Gravitational Renderer (DNGR), we have been driven to employ a different set of visualization techniques from those of the astrophysics community—techniques based on propagation of ray bundles (light beams) instead of discrete light rays, and on carefully designed spatial filtering to smooth the overlaps of neighbouring beams; see section 2 and appendix A . Although, at Double Negative, we have many GPU-based workstations, the bulk of our computational work is done on a large compute cluster (the Double Negative render-farm) that does not employ GPUs.

In appendix A.7 we shall give additional comparisons of our DNGR code with astrophysical codes and with other film-industry CGI codes.

1.2. This paper

Our work on gravitational lensing by black holes began in May 2013, when Christopher Nolan asked us to collaborate on building realistic images of a spinning black hole and its disk, with IMAX resolution, for his science-fiction movie Interstellar . We saw this not only as an opportunity to bring realistic black holes into the Hollywood arena, but also an opportunity to create a simulation code capable of exploring a black hole's lensing with a level of image smoothness and dynamics not previously available.

To achieve IMAX quality (with 23 million pixels per image and adequately smooth transitions between pixels), our code needed to integrate not only rays (photon trajectories) from the light source to the simulated camera, but also bundles of rays (light beams) with filtering to smooth the beams' overlap; see section 2 , appendices A.2 , and A.3 . And because the camera would sometimes be moving with speeds that are a substantial fraction of the speed of light, our code needed to incorporate relativistic aberration as well as Doppler shifts and gravitational redshifts.

Thorne, having had a bit of experience with this kind of stuff, put together a step-by-step prescription for how to map a light ray and ray bundle from the light source (the celestial sphere or an accretion disk) to the camera's local sky; see appendices A.1 and A.2 . He implemented his prescription in Mathematica to be sure it produced images in accord with others' prior simulations and his own intuition. He then turned his prescription over to our Double Negative team, who created the fast, high-resolution code DNGR that we describe in section 2 and appendix A , and created the images to be lensed: fields of stars and in some cases also dust clouds, nebulae, and the accretion disk around Interstellar 's black hole, Gargantua.

Finally, in section 5 we summarize and we point to our use of DNGR to produce images of gravitational lensing by wormholes.

Throughout we use geometrized units in which G (Newton's gravitation constant) and c (the speed of light) are set to unity, and we use the MTW sign conventions [ 33 ].

2. DNGR: our Double Negative Gravitational Renderer code

Our computer code for making images of what a camera would see in the vicinity of a black hole or wormhole is called the Double Negative Gravitational Renderer, or DNGR—which obviously can also be interpreted as the Double Negative General Relativistic code.

2.1. Ray tracing

The ray tracing part of DNGR produces a map from the celestial sphere (or the surface of an accretion disk) to the camera's local sky. More specifically (see appendix A.1 for details and figure 1 for the ray-tracing geometry):

  • (vi)   If the ray originates on the surface of an accretion disk, we integrate the null geodesic equation backward from the camera until it hits the disk's surface, and thereby deduce the map from a point on the disk's surface to one on the camera's sky. For more details on this case, see appendix A.6 .
  • (vii)   We also compute, using the relevant Doppler shift and gravitational redshift, the net frequency shift from the ray's source to the camera, and the corresponding net change in light intensity.

Figure 1.

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2.2. Ray-bundle (light-beam) propagation

DNGR achieves its IMAX-quality images by integrating a bundle of light rays (a light beam) backward along the null geodesic from the camera to the celestial sphere using a slightly modified variant of a procedure formulated in the 1970s by Pineault and Roeder [ 35 , 36 ]. This procedure is based on the equation of geodesic deviation and is equivalent to the optical scalar equations [ 37 ] that have been widely used by astrophysicists in analytical (but not numerical) studies of gravitational lensing; see references in section 2.3 of [ 38 ]. Our procedure, in brief outline, is this (see figure 1 ); for full details, see appendix A.2 .

  • (i)   In DNGR, we begin with an initially circular (or sometimes initially elliptical) bundle of rays, with very small opening angle, centred on a pixel on the camera's sky.
  • (iii)   We then add up the spectrum and intensity of all the light emitted from within that ellipse; and thence, using the frequency and intensity shifts that were computed by ray tracing, we deduce the spectrum and intensity of the light arriving in the chosen camera pixel.

2.3. Filtering, implementation, and code characteristics

Novel types of filtering are key to generating our IMAX-quality images for movies. In DNGR we use spatial filtering to smooth the interfaces between beams (ray bundles), and temporal filtering to make dynamical images look like they were filmed with a movie camera. For details, see appendix A.3 .

In appendix A.4 we describe some details of our DNGR implementation of the ray-tracing, ray-bundle, and filtering equations; in appendix A.5 we describe some characteristics of our code and of Double Negative's Linux-based render-farm on which we do our computations; in appendix A.6 we describe our DNGR modelling of accretion disks; and in appendix A.7 we briefly compare DNGR with other film-industry CGI codes and state-of-the-art astrophysical simulation codes.

3. Lensing of a star field as seen by a moving camera near a black hole

3.1. nonspinning black hole.

In this subsection we review well known features of gravitational lensing by a nonspinning (Schwarzschild) black hole, in preparation for discussing the same things for a fast-spinning hole.

Figure 2.

Images outside the Einstein ring (the violet circle) move rightward and deflect away from the ring. These are called primary images . Images inside the Einstein ring ( secondary images ) appear, in the film clip, to emerge from the edge of the black hole's shadow, loop leftward around the hole, and descend back into the shadow. However, closer inspection with higher resolution reveals that their tracks actually close up along the shadow's edge as shown in the figure; the close-up is not seen in the film clip because the images are so very dim along the inner leg of their tracks. At all times, each star's two images are on opposite sides of the shadow's centre.

This behaviour is generic. Every star (if idealized as a point source of light), except a set of measure zero, has two images that behave in the same manner as the red and yellow ones. Outside the Einstein ring, the entire primary star field flows rightward, deflecting around the ring; inside the ring, the entire secondary star field loops leftward, confined by the ring then back rightward along the shadow's edge. (There actually are more, unseen, images of the star field, even closer to the shadow's edge, that we shall discuss in section 3.2 .)

As is well known, this behaviour is easily understood by tracing light rays from the camera to the celestial sphere; see figure 3 .

The Einstein ring is the image, on the camera's sky, of a point source that is on the celestial sphere, diametrically opposite the camera; i.e., at the location indicated by the red dot and labeled 'Caustic' in figure 3 . Light rays from that caustic point generate the purple ray surface that converges on the camera, and the Einstein ring is the intersection of that ray surface with the camera's local sky.

(The caustic point (red dot) is actually the intersection of the celestial sphere with a caustic line (a one-dimensional sharp edge) on the camera's past light cone. This caustic line extends radially from the black hole's horizon to the caustic point.)

The figure shows a single star (black dot) on the celestial sphere and two light rays that travel from that star to the camera, gravitationally deflecting around opposite sides of the black hole. One of these rays, the primary one, arrives at the camera outside the Einstein ring; the other, secondary ray, arrives inside the Einstein ring.

Because the caustic point and the star on the celestial sphere both have dimension zero, as the camera moves, causing the caustic point to move relative to the star, there is zero probability for it to pass through the star. Therefore, the star's two images will never cross the Einstein ring; one will remain forever outside it and the other inside—and similarly for all other stars in the star field.

However, if a star with finite size passes close to the ring, the gravitational lensing will momentarily stretch its two images into lenticular shapes that hug the Einstein ring and will produce a great, temporary increase in each image's energy flux at the camera due to the temporary increase in the total solid angle subtended by each lenticular image. This increase in flux still occurs when the star's actual size is too small for its images to be resolved, and also in the limit of a point star. For examples, see Riazuelo's film clip [ 5 ].

(Large amplifications of extended images are actually seen in nature, for example in the gravitational lensing of distant galaxies by more nearby galaxies or galaxy clusters; see, e.g. [ 39 ].)

3.2. Fast-spinning black hole: introduction

In the figure we show in violet two critical curves—analogs of the Einstein ring for a nonspinning black hole. These are images, on the camera sky, of two caustic curves that reside on the celestial sphere; see discussion below.

We shall discuss in turn the region outside the secondary (inner) critical curve, and then the region inside.

3.3. Fast-spinning hole: outer region—outside the secondary critical curve

As the camera moves through one full orbit around the hole, the stellar images in the outer region make one full circuit along the red and yellow curves and other curves like them, largely avoiding the two critical curves of figure 4 —particularly the outer (primary) one.

(For a nonspinning black hole (figure 2 ) there are also two critical curves, with the stellar-image motions confined by them: the Einstein ring, and a circular inner critical curve very close to the black hole's shadow, that prevents the inner star tracks from plunging into the shadow and deflects them around the shadow so they close up.)

3.3.1. Primary and secondary critical curves and their caustics

After seeing these stellar-image motions in our simulations, we explored the nature of the critical curves and caustics for a camera near a fast-spinning black hole, and their influence. Our exploration, conceptually, is a rather straightforward generalization of ideas laid out by Rauch and Blandford [ 9 ] and by Bozza [ 10 ]. They studied a camera or observer on the celestial sphere and light sources orbiting a black hole; our case is the inverse: a camera orbiting the hole and light sources on the celestial sphere.

Just as the Einstein ring, for a nonspinning black hole, is the image of a caustic point on the celestial sphere—the intersection of the celestial sphere with a caustic line on the camera's past light cone—so the critical curves for our spinning black hole are also images of the intersection of the celestial sphere with light-cone caustics. But the spinning hole's light-cone caustics generically are two-dimensional (2D) surfaces (folds) in the three-dimensional light cone, so their intersections with the celestial sphere are one-dimensional: they are closed caustic curves in the celestial sphere, rather than caustic points. The hole's rotation breaks spherical symmetry and converts non-generic caustic points into generic caustic curves. (For this reason, theorems about caustics in the Schwarzschild spacetime, which are rather easy to prove, are of minor importance compared to results about generic caustics in the Kerr spacetime.)

Figure 3.

Figure 3.  Light rays around a Schwarzschild black hole: geometric construction for explaining figure 2 .

Figure 4.

3.3.2. Image creations and annihilations on critical curves

Because the spinning hole's caustics have finite cross sections on the celestial sphere, by contrast with the point caustics of a nonspinning black hole, stars, generically, can cross through them; see, e.g., the dashed stellar path in figure 5 . As is well known from the elementary theory of fold caustics (see, e.g., section 7.5 of [ 41 ]), at each crossing two stellar images, on opposite sides of the caustic's critical curve, merge and annihilate; or two are created. And at the moment of creation or annihilation, the images are very bright.

Figure 6.

The primary track (order 0) does not intersect the primary critical curve, so a single primary image travels around it as the camera orbits the black hole. The secondary track (order 1) is the one depicted red in figure 6 and discussed above. It crosses the secondary critical curve twice, so there is a single pair creation event and a single annihilation event; at some times there is a single secondary image on the track, and at others there are three. It is not clear to us whether the red secondary track crosses the tertiary critical curve (not shown); but if it does, there will be no pair creations or annihilations at the crossing points, because the secondary track and the tertiary critical curve are generated by rays with different numbers of poloidal turning points, and so the critical curve is incapable of influencing images on the track. The extension to higher-order tracks and critical curves, all closer to the hole's shadow, should be clear. This pattern is qualitatively the same as when the light source is near the black hole and the camera far away, but in the hole's equatorial plane [ 11 ].

The film clips at stacks.iop.org/cqg/32/065001/mmedia exhibit these tracks and images all together, and show a plethora of image creations and annihilations. Exploring these clips can be fun and informative.

3.4. Fast-spinning hole: inner region—inside the secondary critical curve

The version of DNGR that we used for Interstellar showed a surprisingly complex, fingerprint-like structure of gravitationally lensed stars inside the secondary critical curve, along the left side of the shadow.

We searched for errors that might be responsible for it, and finding none, we thought it real. But Riazuelo [ 42 ] saw nothing like it in his computed images. Making detailed comparisons with Riazuelo, we found a bug in DNGR. When we corrected the bug, the complex pattern went away, and we got excellent agreement with Riazuelo (when using the same coordinate system), and with images produced by Bohn et al using their Cornell/Caltech SXS imaging code [ 43 ]. Since the SXS code is so very different from ours (it is designed to visualize colliding black holes), that agreement gives us high confidence in the results reported below.

Fortunately, the bug we found had no noticeable impact on the images in Interstellar .

With our debugged code, the inner region, inside the secondary critical curve, appears to be a continuation of the pattern seen in the exterior region. There is a third critical curve within the second, and there are signs of higher-order critical curves, all nested inside each other. These are most visible near the flattened edge of the black hole's shadow on the side where the horizon's rotation is toward the camera (the left side in this paper's figures). The dragging of inertial frames moves the critical curves outward from the shadow's flattened edge, enabling us to see things that otherwise could only be seen with a strong zoom-in.

3.4.1. Critical curves and caustics for {{r}_{c}}=2.60M

Figure 8.

Figure 9.  (a) The secondary caustic (red) on the celestial sphere and secondary critical curve (green) on the camera's sky, for the black hole and camera of figure 8 . Points on each curve that are ray-mapped images of each other are marked by letters a, b, c, d. (b) The tertiary caustic and tertiary critical curve.

3.4.2. Multiple images for {{r}_{c}}=2.60M

Returning to the gravitationally lensed star-field image in figure 8 (b): notice the series of images of the galactic plane (fuzzy white curves). Above and below the black hole's shadow there is just one galactic-plane image between the primary and secondary critical curves, and just one between the secondary and tertiary critical curves. This is what we expect from the example of a nonspinning black hole. However, near the fast-spinning hole's left shadow edge, the pattern is very different: three galactic-plane images between the primary and secondary critical curves, and eight between the secondary and tertiary critical curves.

These multiple galactic-plane images are caused by the large sizes of the caustics—particularly their wrapping around the celestial sphere—and the resulting ease with which stars cross them, producing multiple stellar images. An extension of an argument by Bozza [ 10 ] (paragraph preceding his equation (17)) makes this more precise. (This argument will be highly plausible but not fully rigorous because we have not developed a sufficiently complete understanding to make it rigorous.)

When above the caustics, the star produces one image of each order: A primary image (no poloidal turning points) that presumably will remain outside the primary critical curve when the star returns to its equatorial location; a secondary image (one poloidal turning point) that presumably will be between the primary and secondary critical curves when the star returns; a tertiary image between the secondary and tertiary critical curves; etc.

When the star moves downward through the upper left branch of the astroidal primary caustic, it creates two primary images, one on each side of the primary critical curve. When it moves downward through the upper left branch of the secondary caustic (figure 9 (a)), it creates two secondary images, one on each side of the secondary caustic. And when it moves downward through the six sky-wrapped upper left branches of the tertiary caustic (figure 9 (b)), it creates 12 tertiary images, six on each side of the tertiary caustic. And because the upper left branches of all three caustics map onto the left sides of their corresponding critical curves, all the created images will wind up on the left sides of the critical curves and thence the left side of the black hole's shadow. And by symmetry, with the camera and the star both in the equatorial plane, all these images will wind up in the equatorial plane.

So now we can count. In the equatorial plane to the left of the primary critical curve, there are two images: one original primary image, and one caustic-created primary image. These are to the left of the region depicted in figure 8 (a). Between the primary and secondary critical curves there are three images: one original secondary image, one caustic-created primary, and one caustic-created secondary image. These are representative stellar images in the three galactic-plane images between the primary and secondary critical curves of figure 8 . And between the secondary and tertiary critical curves there are eight stellar images: one original tertiary, one caustic-created secondary, and six caustic-created tertiary images. These are representative stellar images in the eight galactic-plane images between the secondary and tertiary critical curves of figure 8 .

3.4.3. Checkerboard to elucidate the multiple-image pattern

Figure 10 is designed to help readers explore this multiple-image phenomenon in greater detail. There we have placed, on the celestial sphere, a checkerboard of paint swatches (figure 10 (a)), with dashed lines running along the constant-latitude spaces between paint swatches, i.e., along the celestial-sphere tracks of stars. In figure 10 (b) we show the gravitationally lensed checkerboard on the camera's entire sky; and in figure 10 (c) we show a blowup of the camera-sky region near the left edge of the black hole's shadow. We have labeled the critical curves 1CC, 2CC and 3CC for primary, secondary, and tertiary.

Figure 10.

The multiple images of lines of constant celestial-sphere longitude show up clearly in the blow-up, between pairs of critical curves; and the figure shows those lines being stretched vertically, enormously, in the vicinity of each critical curve. The dashed lines (star-image tracks) on the camera's sky show the same kind of pattern as we saw in figure 7 .

3.4.4. Multiple images explained by light-ray trapping

The multiple images near the left edge of the shadow can also be understood in terms of the light rays that bring the stellar images to the camera. Those light rays travel from the celestial sphere inward to near the black hole, where they get temporarily trapped, for a few round-trips, on near circular orbits (orbits with nearly constant Boyer–Lindquist radius r ), and then escape to the camera. Each such nearly trapped ray is very close to a truly (but unstably) trapped, constant- r ray such as that shown in figure 11 . These trapped rays (discussed in [ 45 ] and in chapters 6 and 8 of [ 40 ]) wind up and down spherical strips with very shallow pitch angles.

Figure 11.

As the camera makes each additional prograde trip around the black hole, the image carried by each temporarily trapped mapping ray gets wound around the constant- r sphere one more time (i.e., gets stored there for one more circuit), and it comes out to the camera's sky slightly closer to the shadow's edge and slightly higher or lower in latitude. Correspondingly, as the camera moves, the star's image gradually sinks closer to the hole's shadow and gradually changes its latitude—actually moving away from the equator when approaching a critical curve and toward the equator when receding from a critical curve. This behaviour is seen clearly, near the shadow's left edge, in the film clips at stacks.iop.org/cqg/32/065001/mmedia .

3.5. Aberration: influence of the camera's speed

Figure 12.

Despite these huge differences in lensing patterns, the multiplicity of images between critical curves is unchanged: still three images of some near-equator swatches between the primary and secondary critical curves, and eight between the secondary and tertiary critical curves. This is because the caustics in the camera's past light cone depend only on the camera's location and not on its velocity, so a point source's caustic crossings are independent of camera velocity, and the image pair creations and annihilations along critical curves are independent of camera velocity.

4. Lensing of an accretion disk

4.1. effects of lensing, colour shift, and brightness shift: a pedagogical discussion.

We have used our code, DNGR, to construct images of what a thin accretion disk in the equatorial plane of a fast-spinning black hole would look like, seen up close. For our own edification, we explored successively the influence of the bending of light rays (gravitational lensing), the influence of Doppler frequency shifts and gravitational frequency shifts on the disk's colours, the influence of the frequency shifts on the brightness of the disk's light, and the influence of lens flare due to light scattering and diffraction in the lenses of a simulated 65 mm IMAX camera. Although all these issues except lens flare have been explored previously, e.g. in [ 8 , 13 , 14 , 17 , 18 ] and references therein, our images may be of pedagogical interest, so we show them here. We also show them as a foundation for discussing the choices that were made for Interstellar ' s accretion disk.

4.1.1. Gravitational lensing

Figure 13.

In the figure we see three images of the disk. The upper image swings around the front of the black hole's shadow and then, instead of passing behind the shadow, it swings up over the shadow and back down to close on itself. This wrapping over the shadow has a simple physical origin: light rays from the top face of the disk, which is actually behind the hole, pass up over the top of the hole and down to the camera due to gravitational light deflection; see figure 9.8 of [ 40 ]. This entire image comes from light rays emitted by the disk's top face. By looking at the colours, lengths, and widths of the disk's swatches and comparing with those in the inset, one can deduce, in each region of the disk, the details of the gravitational lensing.

In figure 13 , the lower disk image wraps under the black hole's shadow and then swings inward, becoming very thin, then up over the shadow and back down and outward to close on itself. This entire image comes from light rays emitted by the disk's bottom face: the wide bottom portion of the image, from rays that originate behind the hole, and travel under the hole and back upward to the camera; the narrow top portion, from rays that originate on the disk's front underside and travel under the hole, upward on its back side, over its top, and down to the camera—making one full loop around the hole.

There is a third disk image whose bottom portion is barely visible near the shadow's edge. That third image consists of light emitted from the disk's top face, that travels around the hole once for the visible bottom part of the image, and one and a half times for the unresolved top part of the image.

In the remainder of this section 4 we deal with a moderately realistic accretion disk—but a disk created for Interstellar by Double Negative artists rather than created by solving astrophysical equations such as [ 32 ]. In appendix A.6 we give some details of how this and other Double Negative accretion disk images were created. This artists' Interstellar disk was chosen to be very anemic compared to the disks that astronomers see around black holes and that astrophysicists model—so the humans who travel near it will not get fried by x-rays and gamma-rays. It is physically thin and marginally optically thick and lies in the black hole's equatorial plane. It is not currently accreting onto the black hole, and it has cooled to a position-independent temperature T = 4500 K, at which it emits a black-body spectrum.

Figure 14.

4.1.2. Colour and brightness changes due to frequency shifts

The influences of Doppler and gravitational frequency shifts on the appearance of this disk are shown in figures 15 (b) and (c).

In figure 15 (b), we have turned on the colour changes, but not the corresponding brightness changes. As expected, the disk has become blue on the left and red on the right.

In figure 15 (c), we have turned on both the colour and the brightness changes. Notice that the disk's left side, moving toward the camera, has become very bright, while the right side, moving away, has become very dim. This is similar to astrophysically observed jets, emerging from distant galaxies and quasars; one jet, moving toward Earth is typically bright, while the other, moving away, is often too dim to be seen.

4.2. Lens flare and the accretion disk in the movie Interstellar

Christopher Nolan, the director and co-writer of Interstellar , and Paul Franklin, the visual effects supervisor, were committed to make the film as scientifically accurate as possible—within constraints of not confusing his mass audience unduly and using images that are exciting and fresh. A fully realistic accretion disk, figure 15 (c), that is exceedingly lopsided, with the hole's shadow barely discernible, was obviously unacceptable.

The first image in figure 15 , the one without frequency shifts and associated colour and brightness changes, was particularly appealing, but it lacked one element of realism that few astrophysicists would ever think of (though astronomers take it into account when modelling their own optical instruments). Movie audiences are accustomed to seeing scenes filmed through a real camera—a camera whose optics scatter and diffract the incoming light, producing what is called lens flare . As is conventional for movies (so that computer generated images will have visual continuity with images shot by real cameras), Nolan and Franklin asked that simulated lens flare be imposed on the accretion-disk image. The result, for the first image in figure 15 , is figure 16 .

Figure 15.

Figure 16.  The accretion disk of figure 15 (a) (no colour or brightness shifts) with lens flare added—a type of lens flare called a 'veiling flare', which has the look of a soft glow and is very characteristic of IMAX camera lenses. This is a variant of the accretion disk seen in Interstellar . (Figure created by our Double Negative team using DNGR, and TM  & © Warner Bros. Entertainment Inc. (s15)). This image may be used under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 (CC BY-NC-ND 3.0) license. Any further distribution of these images must maintain attribution to the author(s) and the title of the work, journal citation and DOI. You may not use the images for commercial purposes and if you remix, transform or build upon the images, you may not distribute the modified images.

This, with some embellishments, is the accretion disk seen around the black hole Gargantua in Interstellar .

All of the black-hole and accretion-disk images in Interstellar were generated using DNGR, with a single exception: when Cooper (Matthew McConaughey), riding in the Ranger spacecraft, has plunged into the black hole Gargantua, the camera, looking back upward from inside the event horizon, sees the gravitationally distorted external Universe within the accretion disk and the black-hole shadow outside it—as general relativity predicts. Because DNGR uses Boyer–Lindquist coordinates, which do not extend smoothly through the horizon, this exceptional image had to be constructed by Double Negative artists manipulating DNGR images by hand.

4.3. Some details of the DNGR accretion-disk simulations

4.3.1. simulating lens flare.

In 2002 one of our authors (James) formulated and perfected the following (rather obvious) method for applying lens flare to images. The appearance of a distant star on a camera's focal plane is mainly determined by the point spread function of the camera's optics. For Christopher Nolan's films we measure the point spread function by recording with HDR photography (see e.g. [ 46 ]) a point source of light with the full set of 35 and 65 mm lenses typically used in his IMAX and anamorphic cameras, attached to a single lens reflex camera. We apply the camera's lens flare to an image by convolving it with this point spread function. (For these optics concepts see, e.g. [ 47 ].) For the image 15 (a), this produces figure 16 . More recent work [ 48 ] does a more thorough analysis of reflections between the optical elements in a lens, but requires detailed knowledge of each lens' construction, which was not readily available for our Interstellar work.

4.3.2. Modelling the accretion disk for Interstellar

As discussed above, the accretion disk in Interstellar was an artist's conception, informed by images that astrophysicists have produced, rather than computed directly from astrophysicists' accretion-disk equations such as [ 32 ].

In our work on Interstellar , we developed three different types of disk models:

  • an infinitely thin, planar disk, with colour and optical thickness defined by an artist's image;
  • a three-dimensional 'voxel' model;
  • for close-up shots, a disk with detailed texture added by modifying a commercial renderer, Mantra [ 49 ].

We discuss these briefly in appendix A.6 .

5. Conclusion

In this paper we have described the code DNGR, developed at Double Negative Ltd, for creating general relativistically correct images of black holes and their accretion disks. We have described our use of DNGR to generate the accretion-disk images seen in the movie Interstellar , and to explain effects that influence the disk's appearance: light-ray bending, Doppler and gravitational frequency shifts, shift-induced colour and brightness changes, and camera lens flare.

We have also used DNGR to explore, for a camera orbiting a fast spinning black hole, the gravitational lensing of a star field on the celestial sphere—including the lensing's caustics and critical curves, and how they influence the stellar images' pattern on the camera's sky, the creation and annihilation of image pairs, and the image motions.

Elsewhere [ 50 ], we describe our use of DNGR to explore gravitational lensing by hypothetical wormholes; particularly, the influence of a wormhole's length and shape on its lensing of stars and nebulae; and we describe the choices of length and shape that were made for Interstellar 's wormhole and how we generated that movie's wormhole images.

Acknowledgments

For very helpful advice during the development of DNGR and/or during the research with it reported here, we thank C Nolan, A Riazuelo, J-P Luminet, R Blandford, A Bohn, F Hebert, W Throwe, A Broderick and D Psaltis. For contributions to DNGR and its applications, we thank members of the Double Negative R & D team S Dieckmann, S Pabst, S Christopher, P-G Roberts, and D Maupu; and also Double Negative artists F Zangla, S Roth, Z Lord, I di Luigi, F Fan, N New, T Myles, and P Howlett. The construction of DNGR was funded by Warner Bros. Entertainment Inc., for generating visual effects for the movie Interstellar . We thank Warner Bros. for authorizing this code's additional use for scientific research, and in particular the research reported in this paper.

Appendix A.: Some details of DNGR

A.1. ray-tracing equations.

The foundations for our prescription are: (i) the Kerr metric written in Boyer–Lindquist coordinates 4

(ii) the 3 + 1 split of spacetime into space plus time embodied in this form of the metric; and (iii) the family of FIDOs whose world lines are orthogonal to the three-spaces of constant t , and their orthonormal basis vectors that lie in those three-spaces and are depicted in figure 1 ,

We shall also need three functions of r and of a ray's constants of motion b (axial angular momentum) and q (Carter constant), which appear in the ray's evolution equations ( A.15 ) below—i.e., in the equations for a null geodesic:

The prescription that we use, in DNGR, for computing the ray-tracing map and the blue shift is the following concrete embodiment of the discussion in section 2.1 .

A.2. Ray-bundle equations

Our prescription is a concrete embodiment of the discussion in section 2.2 ; it is a variant of the Sachs optical scalar equations [ 37 ]; and it is a slight modification of a prescription developed in the 1970s by Pineault and Roeder [ 35 , 36 ] 6 .

Our prescription relies on the following functions, which are defined along the reference ray. (a) The components of the ray bundle's four-momentum (wave vector) on the FIDO's orthonormal spherical basis

Here Q 1 , Q 2 , w , and S are given by

We shall state our prescription for computing the shape and orientation of the ray bundle on the celestial sphere separately, for two cases: a ray bundle that begins circular at the camera; and one that begins elliptical. Then, we shall briefly sketch the Pineault–Roeder [ 35 , 36 ] foundations for these prescriptions.

A.2.1. Circular ray bundle at camera

A.2.2. elliptical ray bundle at camera.

Our prescription, for a ray bundle that begins elliptical at the camera, is this:

  • (v)   Continue and conclude with steps (ii) and (iii) of appendix A.2.1 .

A.2.3. Foundations for these ray-bundle prescriptions

For completeness we briefly describe the Pineault–Roeder [ 35 , 36 ] foundations for these ray-bundle-evolution prescriptions.

To describe the ray bundle and its evolution, (following Pineault and Roeder) we introduce two complex numbers (transverse vectors) ξ and η , whose real and imaginary parts are functions that are evolved by the ray-bundle equations ( A.23 ):

The outer edge of the ray bundle, at location ζ along the reference ray, is described by the complex number

By an argument similar to the first part of the paragraph before last, one can deduce that anywhere along the evolving ray bundle

and correspondingly that the minor angular diameter at the celestial sphere is the second of equations ( A.25 ).

A.3. Filtering

A.3.1. spatial filtering and how we handle point stars.

In DNGR we treat stars as point sources of light with infinitesimal angular size. We trace rays backwards from the camera to the celestial sphere. If we were to treat these rays as infinitely thin, there would be zero probability of any ray intersecting a star; so instead we construct a beam with a narrow but finite angular width, centred on a pixel on the camera's sky and extending over a small number of adjacent pixels, we evolve the shape of this beam along with the ray, and if the beam intercepts the star, we collect the star's light into it. This gives us some important benefits.

  • The images of our unresolved stars remain small: they don't stretch when they get magnified by gravitational lensing.
  • The fractional change in the beam's solid angle is directly related to the optical magnification due to gravitational lensing and hence to the intensity (brightness) change in the image of an unresolved star.
  • When sampling images of accretion discs or extended structures such as interstellar dust clouds, we minimize moiré artefacts by adapting the resampling filter according to the shape of the beam [ 55 ].

Another consequence is that each star contributes intensity to several pixels. The eye is sensitive to the sum of these and as a single star crosses this grid, this sum can vary depending on the phase of the geometric image of the star on the grid 7 . This can cause distracting flickering of stars as they traverse the virtual camera's image plane. We mitigate this effect by modulating the star's intensity across the beam with a truncated Gaussian filter and by setting the beam's initial radius to twice the pixel separation.

With this filter, the sum is brightest when the geometric image of the star falls exactly on a pixel and dimmest when centred between four pixels. The shape and width of the filter are designed to give a maximum 2% difference between these extremes, which we found, empirically, was not noticeable in fast changing Interstellar scenes.

A.3.2. Temporal filtering: motion blur

For moving images, we need to filter over time as well as space. A traditional film camera typically exposes the film for half the time between individual frames of film, so for a typical 24 fps (frame per second) film, the exposure time will be 1/48 s. (This fraction is directly related to the camera's 'shutter angle': the shutters in film cameras are rotating discs with a slice cut out. The film is exposed when the slice is over the emulsion, and the film moves onto the next frame when covered. The most common shape is a semicircular disk which is a '180° shutter'.) Any movement of the object or camera during the exposure leads to motion blur. In our case, if the camera is orbiting a black hole, stellar images close to a critical curve appear to zip around in frantic arcs and would appear as streaks of light on a 1/48 s photograph. We aim to reproduce that effect when creating synthetic images for films.

This motion blur in computer graphics is typically simulated with Monte Carlo methods, computing multiple rays per pixel over the shutter duration. In order to cleanly depict these streaks with Monte Carlo methods, we would need to compute the paths of many additional rays and the computational cost of calculating each ray is very high. Instead, in DNGR we take an analytic approach to motion blur (cf figure A1 ) by calculating how the motion of the camera influences the motion of the deflected beam:

Figure A1.

Figure A1.  Left-to-right: no motion blur; Monte Carlo motion blur with four time samples per-pixel; analytic motion blur. Relative computation time is approximately in the ratio 1:4:2. Lens flare has been added to each image to illustrate how these images would typically be seen in the context of a movie.

The camera's instantaneous position influences the beam's momentary state, and likewise the camera's motion affects the time derivatives of the beam's state. We augment our ray and ray bundle equations ( A.15 ) and ( A.23 ) to track these derivatives throughout the beam's trajectory and end up with a description of an elliptical beam being swept through space during the exposure. The beam takes the form of a swept ellipse when it reaches the celestial sphere, and we integrate the contributions of all stars within this to create the motion-blurred image 8 . These additional calculations approximately double the computation time, but this is considerably faster than a naive Monte Carlo implementation of comparable quality.

A.3.3. Formal description of our analytic approach to motion blur

We can put our analytic method into a more formal setting, using the mathematical description of motion blurred rendering of Sung et al [ 59 ], as follows. We introduce the quantity

Luckily, in Interstellar , the design of the shots meant this rarely happened: the main source of motion blur was the movement of the spaceship around the black hole, and not local camera motion.

To handle any cases where this approximation might cause a problem, we also implemented a hybrid method where we launch a small number of Monte Carlo rays over the shutter duration and each ray sweeps a correspondingly shorter analytic path.

We used a very similar technique for the accretion disk.

A.4. Implementation

DNGR was written in C++ as a command-line application. It takes as input the camera's position, velocity, and field of view, as well as the black hole's location, mass and spin, plus details of any accretion disk, star maps and nebulae.

Each of the 23 million pixels in an IMAX image defines a beam that is evolved as described in appendix A.2.

The beam's evolution is described by the set of coupled first and second order differential equations ( A.15 ) and ( A.23 ) that we put into fully first order form and then numerically integrate backwards in time using a custom implementation of the Runge–Kutta–Fehlberg method (see, e.g., chapter 7 of [ 57 ]). This method gives an estimate of the truncation error at every integration step so we can adapt the step size during integration: we take small steps when the beam is bending sharply relative to our coordinates, and large steps when it is bending least. We use empirically determined tolerances to control this behaviour. Evolving the beam along with its central ray triples the time per integration step, on average, compared to evolving only the central ray.

The beam either travels near the black hole and then heads off to the celestial sphere; or it goes into the black hole and is never seen again; or it strikes the accretion disk, in which case it gets attenuated by the disk's optical thickness to extinction or continues through and beyond the disk to pick up additional data from other light sources, but with attenuated amplitude. We use automatic differentiation [ 58 ] to track the derivatives of the camera motion through the ray equations.

Each pixel can be calculated independently, so we run the calculations in parallel over multiple CPU cores and over multiple computers on our render-farm.

We use the OpenVDB library [ 60 ] to store and navigate volumetric data and Autodesk's Maya [ 61 ] to design the motion of the camera. (The motion is chosen to fit the film's narrative.) A custom plug-in running within Maya creates the command line parameters for each frame. These commands are queued up on our render-farm for off-line processing.

A.5. DNGR code characteristics and the Double Negative render-farm

A typical IMAX image has 23 million pixels, and for Interstellar we had to generate many thousand images, so DNGR had to be very efficient. It has 40 000 lines of C++ code and runs across Double Negative's Linux-based render-farm. Depending on the degree of gravitational lensing in an image 9 , it typically takes from 30 min to several hours running on ten CPU cores to create a single IMAX image. The longest renders were those of the close-up accretion disk when we shoe-horned DNGR into Mantra. For Interstellar , render times were never a serious enough issue to influence shot composition or action.

Our London render-farm comprises 1633 Dell-M620 blade servers; each blade has two ten-core E5-2680 Intel Xeon CPUs with 156 GB RAM. During production of Interstellar , several hundred of these were typically being used by our DNGR code.

A.6. DNGR modelling of accretion disks

Our code DNGR includes three different implementations of an accretion disk.

Thin disk : for this, we adapted our DNGR ray-bundle code so it detects intersections of the beam with the disk. At each intersection, we sample the artist's image in the manner described in section 2.2 and appendix A.3 above, to determine the colour and intensity of the bundle's light and we attenuate the beam by the optical thickness (cf appendix A.4).

Volumetric disk : the volumetric accretion disk was built by an artist using SideFX Houdini software and stored in a volumetric data structure containing roughly 17 million voxels (a fairly typical number). Each voxel describes the optical density and colour of the disk in that region.

We used extinction-based sampling [ 62 ] to build a mipmap volume representation of this data. The length of the ray bundle was split into short piecewise-linear segments which, in turn, were traced through the voxels. We used the length of the major axis of the beam's cross section to select the closest two levels in the mipmap volume; and in each level we sampled the volume data at steps the length of a voxel, picking up contributions from the colour at each voxel and attenuating the beam by the optical thickness. The results from the two mipmap levels were interpolated before moving on to the next line segment.

Close-up disk with procedural textures : Side Effects Software's renderer, Mantra, has a plug-in architecture that lets you modify its operation. We embedded the DNGR ray-tracing code into a plug-in and used it to generate piecewise-linear ray segments which were evaluated through Mantra. This let us take advantage of Mantra's procedural textures 10 and shading language to create a model of the accretion disk with much more detail than was possible with the limited resolution of a voxelized representation. However this method was much slower so was only used when absolutely necessary.

Disk layers close to the camera : the accretion-disk images in Interstellar were generated in layers that were blended together to form the final images. Occasionally a layer only occupied space in the immediate vicinity of the camera, so close that the infuences of spacetime curvature and gravitational redshifts were negligible. These nearby layers were rendered as if they were in flat spacetime, to reduce computation time.

We set the white balance of our virtual camera to render a 6500 K blackbody spectrum with equal red, green and blue pixel values by applying a simple gain to each colour channel. We did not model the complex, nonlinear interaction between the colour-sensitive layers that occurs in real film.

A.7. Comparison of DNGR with astrophysical codes and with film-industry CGI codes

Near the end of section 1.1 , we compared our code DNGR with the state-of-the-art astrophysical visualization code GRay. The most important differences—DNGR's use of light-beam mappings versus GRay's use of individual-ray mappings, and DNGR's use of ordinary processors versus GRay's use of GPUs—were motivated by our different goals: smoothness of images in movies versus fast throughput in astrophysics.

In appendix A.3.1 we have described our methods of imaging star fields, using stars that are point sources of light: we feed each star's light into every light beam that intersects the star's location, with an appropriate weighting.

Other gravitational-lensing codes deal with star fields differently. For example, Müller and Frauendiener [ 29 ] calculate where each point-star ends up in the camera's image plane, effectively producing the inverse of the traditional ray-tracing algorithm. They do this for lensing by a non-spinning black hole (Schwarzchild metric) which has a high degree of symmetry, making this calculation tractable. Doing so for a spinning black hole would be much more challenging.

A common method for rendering a star-field is to create a 2D source picture (environment map) that records the stars' positions as finite-sized dots in the source plane (see e.g. [ 64 ]). This source picture is then sampled using the evolved rays from the camera. This has the disadvantage that stars can get stretched in an unrealistic way in areas of extreme magnification, such as near the critical curves described in section 3.3.1 of this paper. As we discussed in appendix A.3.1 , our DNGR light-beam technique with point stars circumvents this problem—as does the Müller-Frauendiener technique with point stars.

We also have done such simulations but not in enough detail to reveal much new.

Our choice of Boyer–Lindquist coordinates prevents the camera from descending into the black hole. For that, it is not hard to switch to ingoing Kerr coordinates; but if one wants to descend even further, across an inner (Cauchy) horizon of the maximally extended Kerr spacetime, one must then switch again, e.g., to outgoing Kerr coordinates. Riazuelo [ 42 ] has created an implementation of the ray tracing equations that does such multiple switches. As a foundation for Interstellar , Thorne did so but only for the switch to ingoing Kerr coordinates, as Interstellar assumes the inner horizons have been replaced by the Marolf–Ori shock singularity [ 51 ] and the Poisson–Israel mass-inflation singularity [ 52 ]; see chapters 24 and 26 of [ 40 ].

These are the super-Hamiltonian variant of the null geodesic equations; see, e.g., section 33.5 of [ 33 ] or appendix A of [ 54 ].

The same is true, of course, for each bit of a nebula or accretion disk. We discuss some details of how DNGR handles accretion disks in appendix A.6.

We took a similar approach when motion-blurring the accretion disks discussed in appendix A.6.

More specifically: the render time depends mainly on the number of steps in the integration. This, in turn, depends on the truncation errors in the integration scheme. The tolerances are tighter on the position of the ray than its shape, as errors in position give rise to more noticeable artefacts. Empirically, renders of regions close to the black hole's shadow are much slower than any other.

A procedural texture uses an algorithm, such as fractal noise, to generate detail at arbitrary resolution, unlike a texture based on an image file, which has a finite resolution.

Movie 1. (42 MB, MP4) View of a starfield under the influence of gravitational lensing. The camera is at radius r=2.6 GM/c2

Movie 2. (112 MB, MP4) View of a starfield under the influence of gravitational lensing. The camera is at radius r=6.03 GM/c2

Movie 3. (185 MB, MP4) View of a starfield under the influence of gravitational lensing. The camera is at radius r=6.03 GM/c2. The primary and secondary critical curves are overlaid in purple and the path of a star at polar angle 0.608 pi is overlaid in red.

Movie 4. (540 KB, MP4) For a camera at radius rc = 6.03 GM/c2: Animation showing the mapping between points on the primary critical curve in the camera's sky and the primary caustic curve on the celestial sphere.

Movie 5. (456 KB, MP4) For a camera at radius rc = 6.03 GM/c2: Animation showing the mapping between points on the secondary critical curve in the camera's sky and the secondary caustic curve on the celestial sphere.

Movie 6. (948 KB, MP4) For a camera at radius rc = 2.6 GM/c2: Animation showing the mapping between points on the primary critical curve in the camera's sky and the primary caustic curve on the celestial sphere.

Movie 7. (831 KB, MP4) For a camera at radius rc = 2.6 GM/c2: Animation showing the mapping between points on the secondary critical curve in the camera's sky and the secondary caustic curve on the celestial sphere.

Movie 8. (1.53 MB, MP4) For a camera at radius rc = 2.6 GM/c2: Animation showing the mapping between points on the tertiary critical curve in the camera's sky and the tertiary caustic curve on the celestial sphere.

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  • Proc Natl Acad Sci U S A
  • v.98(19); 2001 Sep 11

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Black holes

B. brügmann.

* Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, 14476 Golm, Germany; ‡ University of California, Los Angeles, CA 90095-1562; and § Astrophysical Institute Potsdam, An der Sternwarte 16, 14482, Potsdam, Germany

Recent progress in black hole research is illustrated by three examples. We discuss the observational challenges that were met to show that a supermassive black hole exists at the center of our galaxy. Stellar-size black holes have been studied in x-ray binaries and microquasars. Finally, numerical simulations have become possible for the merger of black hole binaries.

Black holes are a striking example of a prediction of Einstein's theory of gravity, general relativity. Although it took many decades before the physical concept of a black hole was fully understood and widely accepted, recent years have seen rapid advances on both the observational and theoretical side, which we want to illustrate in this brief note with three examples. Black holes have become an astrophysical reality. Solid observational evidence exists for black holes in two mass ranges. Supermassive black holes of 10 6 -10 9 solar masses have been observed at the centers of many galaxies, and here we discuss the observational challenges that were met to show that there exists a black hole at the center of our own galaxy. Stellar-size black holes of about 3–20 solar masses have been studied in x-ray binaries and microquasars. Finally, numerical simulations have become possible for the merger of black hole binaries.

Recent high-resolution imaging studies of stars at the center of our Galaxy have produced strong dynamical evidence for a central concentration of dark matter, establishing the Milky Way as the most convincing case of a galaxy containing a central supermassive black hole ( 1 , 2 ). In those experiments, images obtained over 2–6 years at the Keck telescope ( 1 ) and European Southern Observatory's New Technology Telescope ( 2 ) provided measurements of the stars' velocities in the plane of the sky, from which a statistical analysis revealed the existence of 2−3 × 10 6 solar masses of dark matter contained within a radius of 0.015 parsec (1 parsec = 3.09 × 10 16 m), or 2.6 light weeks. At this meeting, new results from the Keck telescope were reported. With this new data set, which triples the number of maps obtained and doubles the time baseline for the Keck experiment, the velocity uncertainties are reduced by a factor of 3 compared with the earlier Keck work ( 1 ), primarily as a result of the increased time baseline and, in the central square arcsecond, by a factor of 6 compared with ref. 2 , due to the higher angular resolution (0."05 vs. 0."15). In addition to simply increasing the time baseline for velocity measurements, the new measurements have advanced this experiment in two significant ways: ( i ) the first Keck adaptive optics (AO) images of the galactic center have been obtained (Fig. ​ (Fig.1), 1 ), allowing a more complete census of stars in this region to be obtained ( 3 ), and ( ii ) the first measurements of stellar accelerations in this field have now been achieved ( 4 ).

An external file that holds a picture, illustration, etc.
Object name is pq2013657001.jpg

A ≈3" × 3" region showing the Sgr A* cluster (the faint stars located just to the right of the center of the field of view). Both images were taken in May 1999 at 2.2 μm (K-band); however, the image on the left was produced by shift-and-adding Keck I speckle data, and the image on the right was obtained with the new Keck II adaptive optics system. The adaptive optics image represents a large improvement.

With longer integration times, AO should probe a yet larger sample of fainter stars, place stringent limits on Sgr A*, and explore the possibility of a gravitational lensing experiment ( 5 ). For the time being, the AO map has increased the number of stars in the proper motion study ( 3 ). With relative positional accuracies of ≈3 milliarcseconds, the motions of stars are now fit with a second-order polynomial as opposed to a simple linear fit, which was done in earlier work. Among the 90 stars in the original Keck proper motion sample ( 1 ), accelerations of 2–5 milliarseconds/yr 2 , or equivalently 3–6 × 10 −6 km/sec 2 , are now detected for three stars, S0–1, S0–2, and S0–4 ( 4 ). These three stars are independently distinguished in this sample as being among the fastest moving stars ( v = 565 to 1,383 km/sec) and among the closest to the nominal position of Sgr A* (< r > = 0.003 to 0.015 parsec). Acceleration vectors, in principle, are more precise tools than velocity vectors for studying the properties of the central dark mass. These acceleration measurements improve the localization of our Galaxy's dynamical center by a factor of 3, which is critical for reliably associating any near-infrared source with the black hole, given the complexity of the region. In addition, these acceleration measurements increase the minimum mass density inferred by a factor of 8 over previous results, thereby strengthening the case for a black hole.

X-Ray Binaries and Microquasars.

In contrast to the need for measuring dozens of stars to determine the mass of the black hole in the Galactic Center, that of black holes in x-ray binary systems can be deduced either from optical/IR measurements of just one star, namely the companion of the stellar-mass black hole, or from x-ray observations of the binary. In x-ray binaries, a black hole of typically 3–10 solar masses and a normal star (1–30 solar masses) orbit each other. Matter is pulled off the companion star and, because of its angular momentum, is forming an accretion disk as it moves toward the black hole. Before finally falling into the black hole, the matter heats up to several million degrees at the inner part of the disk and emits luminous x-ray radiation. Because the whole accretion process is highly variable, numerous such black hole binaries have been found over the last decade, thanks to x-ray detectors on satellites, such as Compton Gamma-Ray Observatory and Rossi X-Ray Timing Explorer, constantly monitoring the whole sky.

Although optical/IR spectroscopic measurements of the velocity of the companion star can readily determine the mass of the black hole, x-ray measurements promise to be a sharper and even more flexible diagnostic tool as they reach down to the inner edge of the accretion disk at a few Schwarzschild radii of the black hole. High time-resolution observations of black hole binaries have revealed quasiperiodic oscillations in x-ray emission at a stable minimum period, e.g., at 67 Hz for GRS 1915 + 105 ( 6 ), which may very well be related to the period of the innermost stable orbit of the accretion disk. The Kerr metric fixes this period as a function of mass and spin of the black hole. Because the maximum temperature of the innermost disk, as discernable from x-ray spectroscopy, is also thought to be just a function of black hole mass and spin, detailed x-ray observations can be used to determine both the mass AND spin of a black hole ( 7 ).

A small fraction of black hole binaries also eject matter at relativistic speeds into two opposite jets that are observable in the radio band as knots moving apart at superluminal speed. Actually, GRS 1915 + 105 is the most famous representative of this class of object, called microquasars ( 8 ). Simultaneous observations of these microquasars in the x-ray, optical/IR, and radio band have for the first time revealed a relation between accretion disk instabilities and jet ejections ( 9 , 10 ). Theorists now face the challenge of modeling the highly dynamical processes of nonsteady accretion and jet formation, acceleration, and collimation, with all of the complications of three-dimensional magnetohydrodynamics and general relativity.

Another example in which the full Einstein equations have to be solved in the highly dynamic and nonlinear regime is the collision and merger of two black holes. In fact, although single black holes are comparatively simple exact solutions of the Einstein equations, the two-body problem of general relativity for black holes, or neutron stars, is unsolved. As opposed to Newtonian theory, where the Kepler ellipses provide an astrophysically relevant example for the analytic solution of the two-body problem, in Einsteinian gravity there are no corresponding exact solutions. The failure of Einstein's theory to lead to stable orbits is due to the fact that, in general, two orbiting bodies will emit gravitational waves that carry away energy and momentum from the system, leading to an inspiral. Of course, this “leak” is not considered detrimental. It is expected that gravitational wave astronomy will open a new window onto the universe ( 11 ), and binary black hole mergers are considered to be among the most likely candidates for first detection.

Numerical relativity is only now approaching a state where the evolution of rather general three-dimensional data sets can be simulated on a computer to solve the Einstein equations (see, e.g., ref. 12 ). After early computations for the axisymmetric head-on collision of two black holes in the 1970s, it was in 1995 that, for the first time, spherically symmetric data for a single Schwarzschild black hole was evolved with a three-dimensional computer code ( 13 ). The first fully three-dimensional binary black hole evolutions, the grazing collision of nearby spinning and moving black holes, is reported in ref. 14 . Fig. ​ Fig.2 2 shows a visualization of such a black hole merger [M. Alcubierre, W. Benger, B. Brügmann, G. Lanfermann, L. Nerger, E. Seidel & R. Takahashi, R. http://jean-luc.aei.mpg.de/Press/BH1999/ ]. These simulations are still severely limited in achievable evolution time (300 μs for a final black hole of 10 solar masses), i.e., one can evolve through the very last moments of the inspiral when the two black holes merge, but even a single full orbit is not yet possible. Concretely, the computer code crashes when the space–time distortion becomes too severe. The recent computer simulations not only reflect an increase in raw computer power but also are due to theoretical work on how to construct good coordinates dynamically to deal with strong and even singular gravitational fields, and a new way to compute black hole initial data was developed. Work is in progress to obtain at least one orbit and to compute the gravitational waves generated in a black hole merger.

An external file that holds a picture, illustration, etc.
Object name is pq2013657002.jpg

The evolution of the apparent horizon during a grazing black hole collision. Initially there are two separate horizons, which, during the merger, become enclosed by a third one. The coloring represents the curvature of the surface. The black holes appear to grow, because numerical grid points are falling toward and into the black hole.

In conclusion, we believe that black hole physics will be a very dynamic field in the coming years.

Acknowledgments

A.M.G. was supported by the National Science Foundation and the Packard Foundation. J.G. was partly supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF/DLR) under contract 50 QQ 9602 3. The simulations in refs. 14 and 15 were performed at the Albert-Einstein-Institut and at National Center for Supercomputing Applications.

This paper is a summary of a session presented at the sixth annual German–American Frontiers of Science symposium, held June 8–10, 2000, at the Arnold and Mabel Beckman Center of the National Academies of Science and Engineering in Irvine, CA.

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  • Published: 09 July 2021

Divergent reflections around the photon sphere of a black hole

  • Albert Sneppen 1 , 2  

Scientific Reports volume  11 , Article number:  14247 ( 2021 ) Cite this article

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An Author Correction to this article was published on 30 August 2021

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From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor \(e^{2 \pi }\) closer to the black hole’s optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.

Introduction

Black holes are famously objects where the spatial paths of light are drastically bent by the curvature of space-time. While light itself cannot escape the central mass at the event horizon, at further distances light may orbit the black hole. In the generic case of a non-rotating and electrically neutral black hole [i.e. a Schwarzschild black hole 1 ] the event horizon is located at radial coordinate \(R_s=\frac{2GM}{c^2}\) , while photons may follow unstable circular orbits at \(\frac{3}{2} R_s\) , which is the so-called photon-sphere or last photon orbit. Any photon orbiting below this distance will plunge into the black hole, while light that remains further away will spiral out towards infinity.

However, depending on the photon’s proximity to the last photon orbit it may complete several orbits before spiralling into the event horizon or out towards infinity 2 , 3 . As we approach the limit where the photons graze the exact critical orbital radius the photon will orbit an infinite number of times. Inversely, from the perspective of an observer at infinity this implies that light from any point (from the event horizon to the background) may orbit the black hole an arbitrary number of times. For each of these paths the light will reach the observer slightly closer to the edge of the black hole’s shadow 4 . Therefore, the observer will see the entire surface of the event horizon and the entire universe repeating infinitely near the edges of the black hole. This infinite mapping has been extensively studied with the deflection angle diverging logarithmically in the strong field limit 2 , 4 , 5 , 6 , 7 , 8 .

However, we present a methodology, which differs from previous research by reformulating the trajectory of light in terms of a second order differential equation and quantifying its linear stability. In “ Simulated orbits ” we investigate how small deviations away from the optical edge of a black hole behave with a ray-tracing algorithm. We supplement with an analytical derivation in “ Linear stability ”. In both the numerical and analytical case we will show that small perturbations grow exponentially. Inversely, each additional orbit will be mapped to an exponentially thinner ring, with each subsequent image a factor \(e^{2 \pi }\) thinner. Ultimately, this paper investigates a well known problem from a new analytical perspective suggesting not only the deflection angle but the entire trajectories of light near the photon-sphere are prescribed by two duelling exponential functions.

Crucially, this approach is generalizable to any spherically symmetric black hole or even the equatorial plane of a spinning black hole. In “ Generalization to Kerr Metric ”, the exponentials dependence on the spin is derived and illustrated. Here it is proved for the first time that the spatial frequency of prograde and retrograde images will respectively increase and decrease from the Schwarzschild case.

Analytical setup

The Schwarzschild metric has the form in units with the speed of light, \(c=1\) :

Without loss of generality we can set the orbital plane of the light \(\theta =\frac{\pi }{2}\) . Introducing the two conserved quantities, angular momentum and energy, we can for a mass-less particle reduce the equation for the trajectory of light to 3 , 9 :

here b is the constant ratio of a photon’s angular momentum to energy. Rewriting the differential with \(\phi\) is only applicable if there is an angular evolution, so Eq. ( 2 ) does not apply in the limiting and trivial case of light moving radially towards or away from a black hole. Using the substitution \(u = R_s/r\) and differentiating both sides with \(\frac{d}{d\phi }\) yields the simple second order equation:

We can immediately reproduce the stationary orbit of the photon sphere for the Schwarzschild black hole by setting \(\frac{d^2u_{eq}}{d\phi ^2} = 0\) : \(r_{eq} = \frac{R_s}{u_{eq}} = \frac{3}{2} R_s\) . Note, the trivial equilibrium solution for \(u=0\) (i.e. at infinite distances from the black hole) will not be discussed further.

Simulated orbits

Given the differential equation (Eq. 3 ) relating distance to the angular deflection we can numerically integrate Eq. ( 3 ) using quartic Runga-Kutta (see Fig. 1 ). In this approach we are propagating the light from an observer to the black hole or the background universe. This yields the same light-path as the opposite direction, because the solutions of Eq. ( 2 ) are independent of the direction of the light. Integrating Eq. ( 3 ) requires two initial conditions: in Cartesian coordinates we center the black hole at the origin, set the initial direction of light to be \(\hat{v}_0 = (-1, 0)\) and the initial position \(\mathbf {r_0} = (d_0, b_0+\delta _0)\) . Here \(b_0\) (the critical impact parameter) is the distance within which photons are captured and outside which photons are deflected. Here \(d_0\) can be arbitrarily large given \(b_0\) at that distance which in the \(\lim _{d_0 \rightarrow \infty } b_0 = \frac{\sqrt{27}}{2} R_s\) becomes the photon capture radius commonly found in literature 10 . \(\delta _0\) is our initial perturbation which we use to avoid the ambiguity of defining the closest approach for light rays that spiral within the photon sphere. Importantly, \(\delta _0\) is interpretable as how far the observer is looking away from rim of the optical black hole.

figure 1

Simulated rays of light satisfying Eq. ( 3 ) with \(\delta _0 < 0\) (left) and \(\delta _0 > 0\) (right) with coloring indicating magnitude of \(\delta _0\) . The black hole is shaded in grey with the last photon orbit indicated with a dotted grey line. Each successive light-trajectory plotted is a factor of 2 closer to the photon capture radius with the resulting deflection angle increasing just below \(40^{\circ }\) . Thus, the logarithmic scaling towards the photon capture radius maps to a linear evolution in \(\phi\) .

The path for light with varying \(\delta _0\) can be seen in Fig. 1 . Positive and negative perturbations will respectively spiral out to infinity or plunge into the event horizon as expected. As \(\delta _0\) becomes smaller the deflection angle increases. Note, \(\phi\) increases linearly when moving logarithmically closer to the photon capture radius.

In Fig. 2 the results can be seen for positive (where the angle \(\phi\) is the unwrapped deflection angle) and negative perturbations (with the angle \(\phi\) being defined as the angle orbited around the black hole at the time when photons cross the event horizon). For large perturbations ( \(|\delta _0|>10^{-2}\) ) the relationship between angle and distance is not simply exponential. However in the small perturbation regime ( \(|\delta _0| < 10^{-2}\) ) a tight exponential relationship is visible. To determine the exponent in the exponential regime, we can fit, \(\phi = s \ln (\delta _0) + c\) , and with slope: \(s = -1.0000 \pm 0.0001\) . Inverting this expression for \(\delta _0\) implies that to achieve another orbit requires being a factor of \(f = e^{-2 \pi s} = 535.60 \pm 0.45\) closer to the optical edge of the black hole.

figure 2

Angle of rotation for simulated light-rays as a function of deviations from the equilibrium (in dimensionless units with \(\delta _0 = l/R_s\) ) with \(\phi =0\) representing unbent light-rays. For both small positive and negative perturbations a clear exponential relation to \(\phi\) is visible.

Linear stability

To interpret these numerical results we will utilise linear stability analysis by adding small perturbations, \(u \rightarrow u_{eq} + \delta\) to the equilibrium solution of Eq. ( 3 ):

Linearlizing the equation around \(u_{eq} = \frac{2}{3}\) one gets:

which has the solution:

Evidently the first term grows in magnitude while the latter decreases, with the constants \(\delta _1\) and \(\delta _{-1}\) determining in which regime each term dominates. The constants are set by the initial conditions of the trajectory, which will be discussed further in “ Linear stability ”. Note the dual exponential form is to be expected as the equilibrium solution is a saddle point.

Intuition through manifolds

An alternate perspective on these exponential solutions is in the phase-space of Eq. ( 3 ). This is shown in Fig. 3 where for every initial condition ( \(u,\frac{du}{d\phi }\) ) a vector is plotted indicating the angular change in both variables (i.e. \(\frac{d u}{d \phi },\frac{d^2 u}{d \phi ^2}\) ). The trajectories terminating at \(u=1\) (i.e. \(r=R_s\) ) are the rays of light reaching the event horizon, while infinity is at \(u=0\) . Most trajectories will cross the photon-sphere with radial velocities, but if \(\frac{d u}{d \phi }=0\) on the photon-sphere then the photon will stay in its circular orbit indefinitely. Thus, orbits on the photon-sphere represent a fixed point in the phase-space.

figure 3

Entire (left) and zoomed-in (right) phase-space portrait for light trajectories obeying Eq. ( 3 ) with the arrows’ coloring indicating the magnitude of change (brighter hues implies longer vectors). \(u = 1\) is the event-horizon, \(u=0\) represents infinity and \(u=\frac{2}{3}\) is at the photon sphere. If \(u=\frac{2}{3}\) and \(\frac{d u}{d \phi }=0\) the photons are on circular orbit, so this represents a fixed point. Notable, this is not a stable fixed point as deviations will in general grow. The stable and unstable manifolds are drawn which in the enlarged version are approximately linear. The stable manifold evidently represents a separatrix between the initial conditions of trajectories which will cross the event horizon or be ejected to infinity. Thus, the stable manifold is equivalent to the optical rim of the black hole.

The set of initial conditions which converge towards the photon sphere (which is called the stable manifold) is indicated with a blue line. Photons on this trajectory will asymptotically approach the photon sphere. Conversely, the unstable manifold (i.e. the set of initial conditions which reach the fixed point for \(\phi \rightarrow - \infty\) ) is plotted in red. The symmetry between stable and unstable manifolds seen in the phase-space is due to the Schwarzschild metric and therefore Eq. ( 14 ) being independent of the direction of time.

As seen in Fig. 3 (left) for the stable and unstable manifolds \(\frac{\partial u}{\partial \phi }\) is in general not linear in u , but when we are close to the fixed point ( \(\delta ^2 < |\delta |\) , see Eq. ( 5 )), the relationship becomes approximately linear. Importantly, there are two sets of eigenvectors around the photon sphere. The first with an eigenvalue of \(-1\) (the exponentially approaching term) and the unstable manifold with an eigenvalue of \(+1\) (the exponentially diverging term). Thus, the phase space clearly follows the intuition of Eq. ( 6 ).

The different signs of the eigenvalues proves that the fixed point is a saddle-point. A saddle-point is inherently unstable as a perturbation from the photon sphere will generically result in an exponential divergence. Evidently, the positive eigenvalue implies that a trajectory will diverge exponentially from the bound orbit with a factor \(e^{\pi n} = e^{\gamma n}\) for each half orbit n. Here the Lyapunov exponent, \(\gamma\) [following the definition by Johnson 11 ] characterizes the instability of the bound orbit relative to a half-orbit n. Thus, for the Schwarschild case for the photon sphere, \(\gamma =\pi\) .

Lastly, notice the eigenvalues around the fixed point in u are also the eigenvalues for r as \(\frac{\partial u}{\partial \phi } = u \Rightarrow \frac{\partial r}{\partial \phi } = -r\) . Therefore, the \(\pm 1\) eigenvalues in ( \(u,\frac{\partial u}{\partial \phi }\) ) corresponds to the eigenvalues \({\mp } 1\) in ( \(r,\frac{\partial r}{\partial \phi }\) ).

A tail of two exponentials

Given Eq. ( 6 ) we find the linearlized solutions:

When investigating the trajectory of light close to the black hole both exponential terms are needed to cross the equilibrium distance (see Eq. 7 ) or for \(\frac{d u}{d \phi }\) to change sign (as seen in Eq. 8 ). The importance of both exponential terms is also illustrated in Fig. 4 , where the light approaches the photon sphere exponentially (with each rotation bringing it a factor \(e^{2 \pi }\) closer) until at a crossover-angle of \(\phi _c \approx 6 \pi\) . After this the divergent \(e^{\phi }\) dominates and the light is ejected towards infinity. If \(\delta _{1}\) had the opposite sign then \(\frac{du}{d\phi }\) would remain negative so the crossover-angle would be on the last photon orbit, after which the light would diverge exponentially from the photon-sphere towards the black hole. Curiously, this implies the angle swept by the ray around the black hole prior to the photon sphere is similar to the angle swept by the ray from the photon-sphere to the event horizon.

Notably, the light-ray on the trajectory exactly on the rim of the black hole’s shadow (i.e. \(\delta _0=0\) ) is the solution which is exponentially approaching the photon-sphere indefinitely as it neither diverges towards the black hole or the background universe. It follow that the convergent exponential and in extension \(\delta _{-1}\) must be independent of \(\delta _{0}\) . Instead \(\delta _{-1}\) is set by the approximate distance where the linearised expression holds ( \(\delta _{-1} \approx 1\) ). Any deviations from the critical impact parameter, \(\delta _0 \ne 0\) , will grow exponentially, which implies \(\delta _1\) (the divergent exponential) is set by \(\delta _0\) . Thus, the order of magnitude estimates neatly follow the fitted lines in Fig. 4 .

figure 4

The radial distance between a light-ray (with \(\delta _0 = 10^{-15}\) ) and the last photon orbit as a function of deflection angle [in blue]. The predicted analytical combination of an exponentially declining ( \(e^{-\phi }\) ) and exponentially growing ( \(e^{\phi }\) ) term is indicated with a yellow dashed line. Evidently, each term dominates at different angles of \(\phi\) , with fitted lines suggesting \(\delta _1 \approx 10^{-16}\) and \(\delta _{-1} \approx 1\) . For \(u-u_{eq} \approx 1\) , the linearised solution no longer holds.

While the derivation is only applicable in the linearized regime the implications reach beyond the immediate surroundings of the photon-sphere, as the total deflection of light may be dominated by the angular rotation, while the photons are in the linearized regime. When investigating the total deflection angle or angle of rotation for light (as seen “ Simulated orbits ”) we are solving the trajectories for light moving away from \(u_{eq}\) , where the divergent exponential must dominate. Each additional orbit of light will be mapped a factor \(f = e^{2 \pi }\) nearer the rim of the black hole’s shadow, because decreasing \(\delta _0\) by a factor \(e^{2\pi }\) delays the exponentially growing term exactly one orbit. Furthermore, it should be noted that the predicted analytical value, \(f = e^{2 \pi } \approx 535.49\) , is remarkably close to the numerically fitted relationship seen in “ Simulated orbits ”.

Additionally, for deflected light it is noteworthy that the closest approach to the photon-sphere will only decrease by a factor of \(e^{\pi }\) for each additional orbit, because the cross-over angle is set by the intersection of the two exponential terms. Similarly, for light crossing the photon-sphere the angle swept from the event horizon to the photon-sphere is similar to the angle swept from the photon-sphere to the observer, as the cross-over angle is still defined by the intersection.

Generalization to Kerr Metric

It should be emphasised, that the Schwarzschild metric is the limiting case of a non-spinning black hole. Without this requirement one gets the so-called Kerr metric (Here written in Boyer-Lindquist coordinates with \(\Sigma =r^2+a^2 cos(\theta )\) and \(\Delta = r^2 - R_s r + a^2\) ):

here \(0 \le a \le 1\) is the angular momentum factor, so naturally the Kerr metric reduces to the Schwarzschild metric for \(a=0\) . For orbits in the equatorial plane (where a 2-dimensional analysis is still an exhaustive description) we set \(\theta =\frac{\pi }{2}\) . Further deliberation on non-equatorial orbits may be found through elliptic integrals 12 . Introducing the two conserved quantities, angular momentum and energy, the trajectory of photons reduces to 13 :

With b once more being the constant ratio of a photon’s angular momentum to energy. Differentiating both sides with \(\frac{d}{d\phi }\) yields a second order differential equation.

The phase portrait for Eq. ( 11 ) for \(a=0.5\) is illustrated in Fig. 5 . As before, the fixed point is set by the roots of \(\frac{dr^2}{d\phi ^2}\) , which will depend on a . In contrast to the Schwarzschild parameterization, Eq. ( 11 ) depends on b , so the critical impact parameter of prograde and retrograde orbits must be evaluated at any a 13 .

figure 5

Phase-space portrait for light trajectories obeying Eq. ( 11 ) with \(a=0.5\) (left: prograde and right: retrograde) with the arrows’ coloring indicating the magnitude of change (brighter hues imply longer vectors). The stable and unstable manifolds are drawn which behave approximately linear with a flatter and steeper slope than Fig. 3 for respectively the prograde and retrograde orbits. Note the substitution \(u=\frac{1}{r}\) does not remove the critical impact parameter unlike the Schwarzschild case. Therefore the figure remains in r not u like Fig. 3 .

Retrograde and prograde orbits are obtained by evaluating the critical impact parameter with Eq. ( 12 ) and ( 13 ). For any given spin we may then determine the fixed points, \(r_{eq}\) such that \(\frac{\partial r^2}{\partial \phi ^2}=0\) . One root, \(\Delta =0\) , representing the event horizon with the remaining real root describing the photon circle (see Fig. 6 ). Linearizing generically yields:

Importantly, we once more we get a family of two exponentials and for the first time derive the exponential unwinding for the strong field limit of light:

For any spin, a , the exponential coefficient, s is shown in Fig. 6 . Evidently the fixed point is still a saddle point and therefore unstable. Here, the exponential divergent term corresponds to Lyapunov exponent of \(\gamma = \pi \sqrt{s}\) as trajectories will diverge from the photon sphere with a factor of \(e^{\pi \sqrt{s}}\) over a half orbit. Notable, the exponential coefficient s results in a even faster divergence of the logarithmic angle for retrograde orbits. For \(\lim _{a \rightarrow 1}(s_{retrograde})=\frac{27}{16}\) , so another orbit would require being a factor of \(f = e^{2 \pi \sqrt{27/16}} \approx 3500\) closer to the optical edge of the black hole. Conversely, for prograde orbits the exponential function unwinds evermore slowly for larger spins. For a black hole spinning with \(a=0.99\) (as potentially observed 15 ), where \(s=0.012\) , only a factor \(f = e^{2 \pi \sqrt{0.012}} \approx 2\) is required. Here, each repeated image would merely be a factor of 2 closer to the optical edge of the black hole. In the limit of an extreme Kerr field, \(\lim _{a \rightarrow 1}(s_{prograde}) = 0\) , the eigenvectors collapse and the fixed point becomes a degenerate node. Thus, an extremely rotating Kerr black hole has no exponential trajectories for the prograde motion.

figure 6

Radii of photon circle, \(r_{eq}\) (left), and linear coefficient of Taylor-expansion, s (right). Retrograde orbits (in dotted blue) and prograde orbits (in dashed red). The location of the photon circle is in agreement with 14 , while the generic result \(s>0\) for \(0 \le a < 1\) , implies that the fixed point will always be a saddle point with eigenvalues \(\pm \sqrt{s}\) and therefore unstable. Notable at \(a=1\) the two real fixed points of prograde motion bifurcate.

Thus, when viewing the equatorial plane of a spinning black hole both prograde and retrograde reflections display the exponential repetition, but prograde copies of a source will repeat rapidly compared to the retrograde copies. This asymmetry has potentially far-reaching applications to observables as any observational signature is limited by the brightness of subsequent images decreasing sharply 11 . Therefore, the rapid spatial repetition of prograde images will provide the first observational signatures of the exponential repetition within detection capabilities.

Lastly, the mathematical generality of two real eigenvalues existing for all a should not go unstated. Regardless of the spin of the black hole, there will always exist a family of a convergent and divergent exponential. These exponentials prescribe the entire trajectories of light near the photon orbits. Their prescription implies that any source object in the plane be repeated in an exponentially thinner series of copies, with the scale of repetitions set by the spin of the black hole.

This work introduces a family of two distinct exponential solutions which together provide a succinct description of the entire orbital trajectories of light near a Schwarzschild black hole. Thereby we provide analytical insight into the solutions previously developed 2 , 4 , 5 , 6 , 7 , 8 . Our formalism provides a few important interpretations. Firstly, it states that the deflection angle of background light will diverge logarithmically when the trajectory approaches the last photon orbit. Equivalently, from the perspective of a distant observer looking at the optical edge of the black hole (the photon capture radius) the entire background will be mapped to exponentially thinner rings. Secondly, the event horizon of the black hole itself will be mapped repeatedly in exponentially thinner rings just inside the photon capture radius. Therefore, any object accretting onto the black hole may be observed repeatedly nearer and nearer the optical edge. Thirdly, this edge of the black hole is the location of both the stable and unstable manifold.

The proof presented here is immediately generalizable to any spherically symmetric space-time (such as a Reissner–Nordstrøm black hole). Such metrics can similarly be written as a second order differential equation in r with steady state and perturbed solutions. Further work may investigate these exponentials, which will in general be characterised by a constant, \(s \ne 1\) , to be multiplied on \(\phi\) in the exponents of Eq. ( 14 ). Importantly, as seen in “ Generalization to Kerr Metric ”, our methodology may even be applied to non-spherically symmetric black holes, such as the spinning black holes of the Kerr Metric. With increasing spin, the exponential coefficient, s , of prograde trajectories decreases while retrograde conversely increase. Thus, proving that the side of the black hole which rotates towards the observer repeatedly mirrors the universe in wide bands. In the limit of an extremely rotating Kerr Hole, the \(s_{retrograde}=\frac{27}{16}\) and \(s_{prograde}=0\) . Thus, there is no logarithmic divergence for prograde reflections when \(a=1\) , but given any spin \(a<1\) , there exists an exponential family prescribing the trajectories.

Philosophically, there is a mathematical beauty within the dual exponentials of Eqs. ( 7 ) and ( 14 ). The exponentials prescribes, that an observer at infinity will see the entire black hole’s event horizon and anything accreting onto the black hole mapped infinitely when looking closer towards the photon capture radius of the black hole. Just beyond the photon capture radius, the exponentials dictate, that the observer will also see the entire universe mirrored in exponentially smaller slivers until the quantum limit. A divergence which certainly merits further reflection.

Change history

30 august 2021.

A Correction to this paper has been published: https://doi.org/10.1038/s41598-021-97272-w

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Acknowledgements

The author would like to thank Mogens Høgh Jensen, Martin Pessah, Charles Steinhardt and Nikki Arendse for useful deliberations and insightful feedback. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under Grant no. 140.

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A Voracious Black Hole at the Dawn of Time?

Scientists debate whether this object is the brightest in the visible universe, as a new study suggests.

An illustration of a dark black hole at the center of a bright light, pulling in orange and red matter in its orbit.

By Dennis Overbye

Astronomers claimed on Monday that they had discovered what might be the hungriest, most luminous object in the visible universe — a supermassive black hole that was swallowing a star a day. That would be the mass equivalent of 370 suns a year disappearing down a cosmic gullet 11 billion years ago at the dawn of time.

Burp indeed.

In a paper published in Nature Astronomy , Christian Wolf of the Australian National University and his colleagues from Australia and Europe, called the object at the center of a newly discovered quasar known as J0529-4351 “the fastest growing black hole in the universe.”

According to their estimates, this black hole tipped the scales as one of the most massive black holes ever found: 17 billion times as massive as the sun.

But other astrophysicists cast doubt on the result, questioning the methods by which the mass and luminosity of the new quasar had been estimated. They said the calculations were too uncertain to be conclusive. “They may have the right value, but I don’t think other observers would be shocked if it turned out the true mass was somewhat less,” said Daniel Holz, a theoretical astrophysicist at the University of Chicago.

“It does seem like an extreme object,” he said. But, he added, “I would be shocked if this turned out to be the most luminous quasar on the sky.”

Jenny Greene, a professor of astrophysical sciences at Princeton University, called the result “cute.”

“It’s nice to pick out the brightest of something,” she said.

Still, she agreed with Dr. Holz: “I don’t think this luminosity difference between this and other quasars is that big, and given the historical variability of quasars. It is not clear this object even really is more luminous than the others.”

Chung-Pei Ma, an astrophysicist at the University of California, Berkeley, weighed in, saying that estimate of these black hole masses could be off by a factor of two or three, “too large to make me lose sleep over the viability of prevailing cosmological models.”

This is a story of mind-bending big numbers, no matter how it comes out.

“There’s this weird game we play in astronomy where we’re always looking for the biggest, the brightest, the youngest, the oldest, etc.,” Dr. Holz said in an email. “Record-breaking objects are an efficient way to learn about the universe. Extremes help illuminate the contours of a problem, and help push our theories up to (or past) their breaking points.”

So it is with quasars and black holes. Quasars are distant objects that look like stars in the sky. In the 1960s, they were discovered to be emitting improbable torrents of energy, outshining all the stars in the galaxy in which they were embedded.

Astronomers have since concluded that all this energy is produced by matter falling into giant black holes. Just as a bathtub can’t drain in an instant, matter can only disappear down the cosmic drain at a rate, called the Eddington limit, depending on the black hole’s size. The rest is trapped in a sort of turnstile of doom, a swirling, sparking disc radiating energy. Which makes black holes, despite their name, the brightest objects in the universe.

Because they look like stars, quasars are hard to find in the sky. Dr. Wolf, a dedicated quasar hunter, said in an email that he relished the hunt. “It makes me feel like a kid again,” he wrote.

In this case, the quasar was hiding in plain sight in the database of the European Space Agency’s Gaia spacecraft, which has mapped the locations and properties of billions of stars since it was launched in 2013.

Dr. Wolf and his team identified it as a quasar after observing it with a telescope at Siding Spring Observatory in Australia. Follow-up spectrographic measurements with the Very Large Telescope operated by the European Southern Observatory at Paranal in Chile, allowed them to estimate the size of the accretion disc and the speed of the gas within it.

That in turn let them conclude that the black hole was some 17 billion solar masses and was accreting mass as fast as it could, at the Eddington limit, given its size or mass.

“In this process its accretion disc alone releases a radiative energy that is equivalent to the output from between 365 and 640 trillion suns,” the astronomers wrote in their paper. They hope to do better soon with an upgraded version of new high-resolution instrument, called Gravity on the Very Large Telescope, and the upcoming Extremely Large Telescope now under construction in Chile.

Acknowledging that all estimates of these distant early universe black hole masses were indeed uncertain by a large margin, Dr. Wolf said that the new instruments should be able to get a really well-defined image of the rotating storm disc leading to an accurate black hole mass. “This will check the scale that we are using right at the highest and most extreme end, and it may help to settle the debate on all these extrapolations that we currently rely on,” he said. “This will definitely be an important step for cosmology.”

By comparison, the black hole at the center of the Milky Way is only four million times as massive as the sun, and the black hole imaged at the center of the giant galaxy M87 in Virgo is 6.5 billion times as massive as the sun.

The recent detections of supermassive black holes residing in galaxies early in the history of the universe, only a billion or two years after the Big Bang, has spurred debate about how they could have grown so big so fast. Astronomers have long theorized that when the universe was only 100 million years old or so, it was seeded with black holes when the first stars burned out, exploded and collapsed into black holes a few dozen times the mass of the sun. In principle, in cosmic time, they could grow into the monsters found in the centers of almost all galaxies by merging with other black holes, accreting gas and eating the occasional star that wandered too close.

At its observed rate of growth, Dr. Wolf said, the quasar’s black hole would have doubled every 30 million years, which would have allowed the black hole’s mass to have grown to 17 billion suns within three billion years after the Big Bang.

But it was unlikely, he went on to say, that black holes actually grow at their maximum rates all the time. He noted that black holes only intermittently reach their Eddington limits, when a feast presents itself. Even more massive black holes have been discovered in the early times of the universe by telescopes like the James Webb Space Telescope, but none of them are as luminous as J0529-4351.

Which has led some astronomers to speculate that many of these black holes had primordial origins , predating stars and galaxies, and started out very massive.

“I am myself coming around to the idea that black holes formed before the galaxies did, and were the seeds around which galaxies formed rather than the other way around,” Dr. Wolf said.

“This has been proposed decades ago, but was considered too crazy to become mainstream,” he said. But the results from the new James Webb Space Telescope have breathed some life into this idea. “It is a very exciting time,” Dr. Wolf said.

An earlier version of this article misstated the location of the European Southern Observatory. It is situated in Paranal, Chile, not La Silla.

How we handle corrections

Dennis Overbye is the cosmic affairs correspondent for The Times, covering physics and astronomy. More about Dennis Overbye

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Inside the black hole

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Robert W. Brehme; Inside the black hole. Am. J. Phys. 1 May 1977; 45 (5): 423–428. https://doi.org/10.1119/1.10829

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The simplest model of a black hole, the massive point source generating a static spherically symmetric gravitational field, is examined using the Schwarzschild coordinate frame. A brief review is given of this coordinate frame external to the Schwarzschild surface. Greater attention is paid to an interpretation of this frame inside the Schwarzschild surface. Here the roles of space and time are reversed in the sense that the external radial coordinate becomes an internal temporal coordinate, and the external temporal coordinate becomes an internal spatial coordinate. An internal universe is constructed from this frame, and a few simple kinematic phenomena are described in terms of it. The internal and external coordinates are connected graphically by using Kruskal coordinates and physically by considering the world lines of photons and freely moving particles which transit the Schwarzschild surface.

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Webb Finds Evidence for Neutron Star at Heart of Young Supernova Remnant

A three-panel image of a supernova remnant. The left panel is labeled “NIRCam” while the two right panels are labeled “MIRI M R S Argon two” (at top) and “NIRSpec I F U Argon six” (at bottom). At left, a mottled light pinkish-orange oval whose inner edge resembles a string of pearls. Within the oval is a dense blue-green cloud, shaped like a keyhole. Three stars with six-point diffraction patterns surround the oval. Above and below these structures, are very faint orange rings, which form a figure eight pattern. The center of the supernova remnant is surrounded by a white box with lines leading to the upper and lower right of the image, where two stacked panels show a bright orange ring with an orange dot in the middle. The upper panel is fuzzier and more blobby, while the bottom panel has more clearly defined edges around the ring and central dot.

NASA’s James Webb Space Telescope has found the best evidence yet for emission from a neutron star at the site of a recently observed supernova. The supernova, known as SN 1987A, was a core-collapse supernova, meaning the compacted remains at its core formed either a neutron star or a black hole. Evidence for such a compact object has long been sought, and while indirect evidence for the presence of a neutron star has previously been found, this is the first time that the effects of high-energy emission from the probable young neutron star have been detected.

Supernovae – the explosive final death throes of some massive stars – blast out within hours, and the brightness of the explosion peaks within a few months. The remains of the exploding star will continue to evolve at a rapid rate over the following decades, offering a rare opportunity for astronomers to study a key astronomical process in real time.

Supernova 1987A

The supernova SN 1987A occurred 160,000 light-years from Earth in the Large Magellanic Cloud. It was first observed on Earth in February 1987, and its brightness peaked in May of that year. It was the first supernova that could be seen with the naked eye since Kepler's Supernova was observed in 1604.

About two hours prior to the first visible-light observation of SN 1987A, three observatories around the world detected a burst of neutrinos lasting only a few seconds. The two different types of observations were linked to the same supernova event, and provided important evidence to inform the theory of how core-collapse supernovae take place. This theory included the expectation that this type of supernova would form a neutron star or a black hole. Astronomers have searched for evidence for one or the other of these compact objects at the center of the expanding remnant material ever since.

Indirect evidence for the presence of a neutron star at the center of the remnant has been found in the past few years, and observations of much older supernova remnants –such as the Crab Nebula – confirm that neutron stars are found in many supernova remnants. However, no direct evidence of a neutron star in the aftermath of SN 1987A (or any other such recent supernova explosion) had been observed, until now.

Image: Supernova 1987A

Claes Fransson of Stockholm University, and the lead author on this study, explained: “From theoretical models of SN 1987A, the 10-second burst of neutrinos observed just before the supernova implied that a neutron star or black hole was formed in the explosion. But we have not observed any compelling signature of such a newborn object from any supernova explosion. With this observatory, we have now found direct evidence for emission triggered by the newborn compact object, most likely a neutron star.”

Webb’s Observations of SN 1987A

Webb began science observations in July 2022, and the Webb observations behind this work were taken on July 16, making the SN 1987A remnant one of the first objects observed by Webb. The team used the Medium Resolution Spectrograph (MRS) mode of Webb’s MIRI (Mid-Infrared Instrument), which members of the same team helped to develop. The MRS is a type of instrument known as an Integral Field Unit (IFU).

IFUs are able to image an object and take a spectrum of it at the same time. An IFU forms a spectrum at each pixel, allowing observers to see spectroscopic differences across the object. Analysis of the Doppler shift of each spectrum also permits the evaluation of the velocity at each position.

Spectral analysis of the results showed a strong signal due to ionized argon from the center of the ejected material that surrounds the original site of SN 1987A. Subsequent observations using Webb’s NIRSpec (Near-Infrared Spectrograph) IFU at shorter wavelengths found even more heavily ionized chemical elements, particularly five times ionized argon (meaning argon atoms that have lost five of their 18 electrons). Such ions require highly energetic photons to form, and those photons have to come from somewhere.

“To create these ions that we observed in the ejecta, it was clear that there had to be a source of high-energy radiation in the center of the SN 1987A remnant,” Fransson said. “In the paper we discuss different possibilities, finding that only a few scenarios are likely, and all of these involve a newly born neutron star.”

More observations are planned this year, with Webb and ground-based telescopes. The research team hopes ongoing study will provide more clarity about exactly what is happening in the heart of the SN 1987A remnant. These observations will hopefully stimulate the development of more detailed models, ultimately enabling astronomers to better understand not just SN 1987A, but all core-collapse supernovae.

These findings were published in the journal Science.

The James Webb Space Telescope is the world’s premier space science observatory. Webb is solving mysteries in our solar system, looking beyond to distant worlds around other stars, and probing the mysterious structures and origins of our universe and our place in it. Webb is an international program led by NASA with its partners, ESA (European Space Agency) and the Canadian Space Agency.

Right click the images in this article to open a larger version in a new tab/window.

Download full resolution images for this article from the Space Telescope Science Institute.

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  • 1 Introduction
  • 2 The Spacetime Harmonic Function Approach
  • 3 The Spinorial Callias Operator Approach
  • 4 The |$\mu $| -Bubble Approach
  • 5 The Spectral Cube Inequality
  • 6 Black Hole Existence
  • Appendix A. Existence and Regularity of Warped |$\mu $| -Bubbles
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Spectral Torical Band Inequalities and Generalizations of the Schoen–Yau Black Hole Existence Theorem

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Sven Hirsch, Demetre Kazaras, Marcus Khuri, Yiyue Zhang, Spectral Torical Band Inequalities and Generalizations of the Schoen–Yau Black Hole Existence Theorem, International Mathematics Research Notices , Volume 2024, Issue 4, February 2024, Pages 3139–3175, https://doi.org/10.1093/imrn/rnad129

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Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator |$-\Delta +cR$|⁠ , where |$R$| denotes scalar curvature and |$c>0$| is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, |$\mu $| -bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary, we generalize the classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size.

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