Can a photon have a stable orbit around a black hole?

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Can a black hole free from an accretion disk collect photons in the photon sphere in a prolonged stable orbit?

Could enough photons be in orbit to shade the actual black hole with light?

Would a rogue black hole appear to have an atmosphere made of photons from the light emitted from other distant stars entrapped and released?

For any massive object the gravitational potential energy is given by Newton's law:

$$V(r) = -frac{GMm}{r}$$

The gravitational potential energy is due to the attractive gravitational force, but for an orbiting object there is also a (fictitious) centrifugal force pushing it outwards. If we calculate the potential energy due to the centrifugal force and add it to the gravitational potential energy we get an effective potential energy:

$$V_mathrm{eff}(r) = -frac{GMm}{r} + frac{L^2}{2mr^2} ag{1}$$

where $$L$$ is the angular momentum, which is a constant for an orbiting object (because angular momentum is conserved in a central field). If we plot this graph for the Earth-Moon system we get a graph like this:

(this comes from my answer to the question Could we send a man safely to the Moon in a rocket without knowledge of general relativity? on the Physics SE)

Note that there is a minimum in the potential energy curve, and this minimum gives the radius of the stable orbit. Note also that because it's a minimum if we displace the object away from the minimum it will fall back towards the minimum again i.e. this is a stable orbit.

Now, for light we cannot simply use Newtonian mechanics because light is massless, but we can do the calculation using general relativity. The details are a bit intimidating, but we end up with an effective potential just as described above. For light the effective potential turns out to be:

$$V_mathrm{eff}(r) = sqrt{1 - frac{2GM}{c^2r}}frac{L}{r} ag{2}$$

And if we graph this it looks like this:

This looks very different from our first graph, and it's different because light is massless and only ever travels at the same speed of $$c$$. The graph of $$V_mathrm{eff}$$ for light has a maximum not a minimum. The maximum corresponds to the position of a circular orbit, just like the minimum in the first graph, and we find the radius of this circular orbit is given by:

$$frac{r}{r_mathrm s} = 1.5$$

where $$r_mathrm s$$ is the Schwarzschild radius. This radius gives the position of the notorious photon sphere.

But for light $$V_mathrm{eff}$$ has a maximum not a minimum. That means if we displace the light by even the tiniest distance from this maximum it will lower its potential energy by moving either inwards or outwards. The orbit at $$1.5r_mathrm s$$ is unstable and the tiniest perturbation will cause the light to spiral into the black hole or away from it. This means we cannot accumulate light in the photon sphere as the question asks. Any attempt to put light into this orbit is doomed to failure as even the tiniest perturbation (e.g. from other objects orbiting the black hole, or even from passing gravitational waves) will destabilise the orbit and the light will be lost.

Photon sphere

A photon sphere [1] or photon circle [2] is an area or region of space where gravity is so strong that photons are forced to travel in orbits. (It is sometimes called the last photon orbit.) [3] The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole:

where G is the gravitational constant, M is the black hole mass, and c is the speed of light in vacuum and rs is the Schwarzschild radius (the radius of the event horizon) - see below for a derivation of this result.

This equation entails that photon spheres can only exist in the space surrounding an extremely compact object (a black hole or possibly an "ultracompact" neutron star [4] ).

The photon sphere is located farther from the center of a black hole than the event horizon. Within a photon sphere, it is possible to imagine a photon that's emitted from the back of one's head, orbiting the black hole, only then to be intercepted by the person's eyes, allowing one to see the back of the head. For non-rotating black holes, the photon sphere is a sphere of radius 3/2 rs. There are no stable free fall orbits that exist within or cross the photon sphere. Any free fall orbit that crosses it from the outside spirals into the black hole. Any orbit that crosses it from the inside escapes to infinity or falls back in and spirals into the black hole. No unaccelerated orbit with a semi-major axis less than this distance is possible, but within the photon sphere, a constant acceleration will allow a spacecraft or probe to hover above the event horizon.

Another property of the photon sphere is centrifugal force (note: not centripetal) reversal. [5] Outside the photon sphere, the faster one orbits the greater the outward force one feels. Centrifugal force falls to zero at the photon sphere, including non-freefall orbits at any speed, i.e. you weigh the same no matter how fast you orbit, and becomes negative inside it. Inside the photon sphere the faster you orbit the greater your felt weight or inward force. This has serious ramifications for the fluid dynamics of inward fluid flow.

A rotating black hole has two photon spheres. As a black hole rotates, it drags space with it. The photon sphere that is closer to the black hole is moving in the same direction as the rotation, whereas the photon sphere further away is moving against it. The greater the angular velocity of the rotation of a black hole, the greater the distance between the two photon spheres. Since the black hole has an axis of rotation, this only holds true if approaching the black hole in the direction of the equator. If approaching at a different angle, such as one from the poles of the black hole to the equator, there is only one photon sphere. This is because approaching at this angle the possibility of traveling with or against the rotation does not exist.

Conceptually yes. There is an orbit called the photon sphere with a 1.5 gravitational radius. A photon can circle around a black hole on this orbit indefinitely without losing energy. In reality however this is not possible, because the orbit is unstable. Any deviation of the direction of light from this exact orbit (e.g. due to dispersion) would send the photon either to the black hole or away from it. So in reality, light may make a number of circles, but eventually will all disperse away from the photon sphere.

The wavelength is not important, but it cannot be longer than the circumference of the orbit (very long radio waves).

A photon, being a quantum of EM radiation, is not only a particle but also a wave. Basic solutions of wave equation around the black hole are quasinormal modes, which are exponentially decaying in time. They correspond to a wave either being absorbed by a black hole horizon or escaping to infinity. So no, a photon could not be orbiting black hole indefinitely even in principle. For a given value of angular momentum number $ell$ there would be a quasinormal mode with the lowest imaginary part of an eigenvalue (this is a generalization of a circular orbit on a photon sphere), but this imaginary part (which is a decay constant of that mode) would approach a constant value for a large $ell$ .

Additionally, if a photon has high enough energy, its back-reaction on the gravitational field would cause it to lose energy via emission of gravitons. Conceptually the process is similar to an electron in excited state in an atom losing energy via emission of a photon.

Can you be in stable orbit inside a blcak hole?

Thanks for your reply. I read the 'photon sphere' entry in Wikipiedia with interest, however I am not convinced by this. I have two issues.

Firstly the issue of a rotating or non-rotating black hole. It is said that a rotating singularity 'drags' spacetime around with it. If this were the case any rotating body such as the Earth or the Sun would do the same - and even though the effects would be orders of magnitude smaller, this could be measured. Has it been?

It seems to me that spacetime dragging would require quantum gravity with carrier 'particles' and I am far from convinced of this concept. It starts by failing Occam's Razor by appearing to be completely un-nessesary. But more empirically there is a measurable effect that I think contradicts this notion. It takes 500 seconds for light to reach the Earth from the Sun and thus we see the Sun subtend an angle that is 500 seconds out of date. But satelites orbiting Earth act (so I am told) as though in a gravity field that is a superimpostion of that from the Earth, the moon and the Sun subtending the angle of its actual position without the 500 second delay.

Some interpret this as 'evidence' that gravity travells faster than light. I interpret it as evidence that gravity is an omni-present field due to mass (non-quantized) and that only changes in mass and thus gravity propogate at the speed of light. If the Sun was sudendly converted to light it would take us 500 seconds to notice and 500 seconds to exit the current orbit and fly off at a tangent.

The second issue is one of energy and the application of the Lorentz contraction. Has the said contraction been properly applied?

If we fire accelerated protons for example such that thier path will go through the event horizon but miss the center. If the protons are accelerated to a million times thier mass (easily achived by cosmic rays) time for the protons goes a million times slower and to the protons the black hole looks a million times smaller - We see them go through the horizon but actually they don't go anywhere near it as far as they are concened.

If Photons Have Zero Mass, How Can Black Holes Pull Them In? [Video]

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In the limit that the wavelength of an electromagnetic wave is similar to the size of the black hole, we can no longer use the approximation that electromagnetic waves travel along geodesics. Instead one has to study the solutions of Maxwell's equations on the black hole background.

The solution that is proposed in the question would be a solution (with no waves coming in from infinity or the past horizon) to the vacuum Maxwell equations that does not decay overtime. One can prove that no such solutions exist. (At least, for some known types of black holes. arXiv:1910.02854 proves this statement for Kerr black holes and massless fields of arbitrary spin.)

You can find solutions that correspond (roughly) to waves going around the photon sphere, but these solutions decay exponentially with time. Such solutions are known as the (electromagnetic) quasi-normal modes of the black hole. They occur only at specific frequencies of the electromagnetic field, determined by the mass, angular momentum (and charge) of the black hole.

Introduction

In classical general relativity, singularity is one of the most fundamental questions. The first version of singularity theorem is proposed by Penrose [1], which states that the formation of singularities in spacetime is inevitable assuming the weak energy condition and global hyperbolicity. The singularity theorem that people often refer to is the version presented and proved by Hawking and Penrose [2], which says a spacetime (mathcal ) cannot satisfy causal geodesic completeness provided that Einstein’s equations and some assumptions hold. In contrast to the other singularity theorems, the conditions required by the Hawking–Penrose singularity theorem are the easiest to implement physically and cover a wide range of areas. Up to now, many attempts have been made to eliminate the singularity, which includes and is not limited to considering quantum corrections [3,4,5] and alternative gravities [6]. Recently, a novel 4D Einstein–Gauss–Bonnet (EGB) gravity was formulated by D. Glavan and Lin [7]. By focusing on the positive GB coupling constant, they discovered a static and spherically symmetric black hole solution, which is practically free from the singularity problem. It’s interesting to note that the same solution was already found before, initially in the gravity with a conformal anomaly [8] and then in gravity with quantum corrections [3, 4]. In contrast, in [7] the GB action should be considered as a classical modified gravity theory, so the theory is on an equal footing with general relativity.

However, since the publication of the paper [7], there have appeared several works [9,10,11,12,13,14,15] debating that the procedure of taking (D ightarrow 4) limit in [7] may not be consistent. For example, in [14] by studying tree-level graviton scattering amplitudes it was shown that in four dimensions there are no new scattering amplitudes than those of the general gravity. On the other hand, some proposals have been raised to circumvent the issues of the novel 4D EGB gravity. These proposals can be divided into two classes. One is adding an extra degree of freedom to the theory. For example, [16, 17] considered using the Kaluza–Klein approach of the (D ightarrow 4) limit to obtain a well-defined theory that belongs to the Horndeski class [18]. The same theory can also be deduced by introducing a counter term into the action [19, 20]. The other proposal is to keep the two dynamical degrees of freedom unchanged at the price of breaking the temporal diffeomorphism invariance [21]. In summary, the novel 4D EGB gravity formulated in [7] may run into trouble at the level of action or equations of motion. Nevertheless, the spherically symmetric black hole solution derived in [7] and in early literatures [3, 4, 8] can be successfully reproduced in those consistent theories of 4D EGB gravity, which is a little bit surprise. Therefore, the spherically symmetric black hole solution itself is meaningful and worthy of study.

It can be expected that due to the publication of [7], lots of works concerning every aspect of the spherically symmetric 4D EGB black hole solution will emerge, including theoretical study and the viability of the solution in the real world. In astronomical survey, the singularities of black holes cannot be directly observed, since they are always inside the event horizons of the black holes. In fact, the event horizon cannot be directly observed by astronomical telescopes. However, the emergence of black hole photograph shows, the black hole shadow and the orbit of the light emitter around the black hole can be seen by the Event Horizon Telescope (EHT), and thus the parameters of a black hole can be identified based on the black hole model [22, 23]. On the other hand, the first detection of gravitational waves from a binary black hole merger by the LIGO/Virgo Collaborations [24] opened a new window to probe gravity in the strong field regime, which then enables us to test gravity theories alternative to general relativity [25]. The progress in both areas may help us to distinguish Schwarzschild black hole from other black hole models, including the 4D EGB black hole, in the near future.

Based on these, we would like to investigate the geodesic motions of both timelike and null particles in the background of the 4D EGB black hole, by focusing on the innermost stable circular orbit (ISCO) of the timelike particle, the unstable photon sphere and the associated shadow of the black hole. The ISCO plays an important role in the study of realistic astrophysics and gravitational wave physics. For example, in the Novikov–Thorne accretion disk model [26], the inner edge of the disk is at the ISCO. Moreover, according to the Buonanno–Kidder–Lehner approach [27], one can estimate the final black hole spin of a binary black hole coalescence with arbitrary initial masses and spins. The key point is that one may approximate the merger process as a test timelike particle orbiting at the ISCO around a Kerr black hole. On the other hand, for the motion of the null particles, besides the observable black hole shadow, the photon sphere (or the light ring) provides information on the quasinormal modes of the final black hole in the ringdown phase of the black hole merger [28] (see however [29]). From the theoretical point of view, a sequence of inequalities were proposed recently in [30], which involve the radii of the event horizon, the photon sphere and the shadow. It would be interesting to verify the conjecture for the 4D EGB black hole.

Before we get started, we note in [7] the black hole solution is constrained to the positive GB coupling constant case, i.e. (alpha >0) and leaves a gap for the negative GB coupling constant. Thus, we firstly give a very careful analysis and show the black hole can exist when (alpha <0) . More precisely, we find that when (-8<alpha le 1) there always exists a black hole. In this case the singular behavior of the solution is hidden behind the horizon and outside the horizon the solution is well defined. For the solution, according to the analysis of [8], we would like to stress that the black hole entropy has the logarithmic behavior. Then, for the first time, we calculate the radius of the ISCO and give a numerical result for the full range of (alpha ) Footnote 1 . Also, we obtain an approximate analytical expression when (alpha ) is very small around 0. We find the radius of the ISCO in the novel solution can be bigger or smaller than the one in Schwarzschild black hole depending on the value of (alpha ) . For the photon sphere and the shadow, we find the exact expressions not only for (0<alpha le 1) but also for (-8<alpha <0) . Comparing the result to that of the Schwarzschild black hole, we find the 4D EGB black hole contains more features and information which deserves further study.

The paper is organized as follows. In Sect. 2, we revisit the spherically symmetric 4D EGB black hole solution and determine the full range of (alpha ) when the spacetime contains a black hole. In Sect. 3, we move to the innermost stable circular orbit of the timelike particle. Next, we turn our attention to the photon sphere and shadow in Sect. 4. Finally, in Sect. 5, we summarize the results. In this work, we have set the fundamental constants c and G to unity, and we will work in the convention ((-,+,+,+)) .

The graph of the metric function f(r) with respect to r for two typical values of (alpha )

Mystery About How Particles Behave Outside a Black Hole Photon Sphere Solved With String Theory

An artist’s impression of a “string” passing near a black hole. As the string approaches the black hole, it is gradually stretched. Then, as it moves past the black hole, it begins to vibrate. The image to the left, which was captured by the Event Horizon Telescope, represents the shadow of the supermassive black hole at the center of the galaxy M87, including the ring of light around it. Credit: EHT Collaboration Kavli IPMU (Kavli IPMU modified EHT’s original image))

A paper by the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) Director Ooguri Hirosi and Project Researcher Matthew Dodelson on the string theoretical effects outside the black hole photon sphere has been selected for the “Editors’ Suggestion” of the journal Physical Review D. Their paper was published on March 24, 2021.

In a quantum theory of point particles, a fundamental quantity is the correlation function, which measures the probability for a particle to propagate from one point to another. The correlation function develops singularities when the two points are connected by light-like trajectories. In a flat spacetime, there is such a unique trajectory, but when spacetime is curved, there can be many light-like trajectories connecting two points. This is a result of gravitational lensing, which describes the effect of curved geometry on the propagation of light.

In the case of a black hole spacetime, there are light-like trajectories winding around the black hole several times, resulting in a black hole photon sphere, as seen in the recent images by the Event Horizon Telescope (EHT) of the supermassive black hole at the center of the galaxy M87.

Released on April 10, 2019, the EHT Collaboration’s images captured the shadow of a black hole and its photon sphere, the ring of light surrounding it. A photon sphere can occur in a region of a black hole where light entering in a horizontal direction can be forced by gravity to travel in various orbits. These orbits lead to singularities in the aforementioned correlation function.

However, there are cases when the singularities generated by trajectories winding around a black hole multiple times contradict with physical expectations. Dodelson and Ooguri have shown that such singularities are resolved in string theory.

In string theory, every particle is considered as a particular excited state of a string. When the particle travels along a nearly light-like trajectory around a black hole, the spacetime curvature leads to tidal effects, which stretch the string.

Dodelson and Ooguri showed that, if one takes these effects into account, the singularities disappear consistently with physical expectations. Their result provides evidence that a consistent quantum gravity must contain extended objects such as strings as its degrees of freedom.

Ooguri says, “Our results show how string theoretical effects are enhanced near a black hole. Though the effects we found are not strong enough to have an observable consequence on ETH’s black hole image, further research may show us a way to test string theory using black holes.”

Reference: “Singularities of thermal correlators at strong coupling” by Matthew Dodelson and Hirosi Ooguri, 24 March 2021, Physical Review D.
DOI: 10.1103/PhysRevD.103.066018

Going against the flow around a supermassive black hole

ALMA image showing two disks of gas moving in opposite directions around the black hole in galaxy NGC 1068. The colors in this image represent the motion of the gas: blue is material moving toward us, red is moving away. The white triangles are added to show the accelerated gas that is expelled from the inner disk - forming a thick, obscuring cloud around the black hole. Credit: ALMA (ESO/NAOJ/NRAO), V. Impellizzeri NRAO/AUI/NSF, S. Dagnello

At the center of a galaxy called NGC 1068, a supermassive black hole hides within a thick doughnut-shaped cloud of dust and gas. When astronomers used the Atacama Large Millimeter/submillimeter Array (ALMA) to study this cloud in more detail, they made an unexpected discovery that could explain why supermassive black holes grew so rapidly in the early Universe.

"Thanks to the spectacular resolution of ALMA, we measured the movement of gas in the inner orbits around the black hole," explains Violette Impellizzeri of the National Radio Astronomy Observatory (NRAO), working at ALMA in Chile and lead author on a paper published in the Astrophysical Journal. "Surprisingly, we found two disks of gas rotating in opposite directions."

Supermassive black holes already existed when the Universe was young—just a billion years after the Big Bang. But how these extreme objects, whose masses are up to billions of times the mass of the Sun, had time to grow in such a relatively short timespan, is an outstanding question among astronomers. This new ALMA discovery could provide a clue. "Counter-rotating gas streams are unstable, which means that clouds fall into the black hole faster than they do in a disk with a single rotation direction," said Impellizzeri. "This could be a way in which a black hole can grow rapidly."

NGC 1068 (also known as Messier 77) is a spiral galaxy approximately 47 million light-years from Earth in the direction of the constellation Cetus. At its center is an active galactic nucleus, a supermassive black hole that is actively feeding itself from a thin, rotating disk of gas and dust, also known as an accretion disk.

Previous ALMA observations revealed that the black hole is not only gulping down material, but also spewing out gas at incredibly high speeds—up to 500 kilometers per second (more than one million miles per hour). This gas that gets expelled from the accretion disk likely contributes to hiding the region around the black hole from optical telescopes.

A star chart showing the location of NGC 1068 (also known as Messier 77), a spiral galaxy approximately 47 million light-years from Earth in the direction of the constellation Cetus. Credit: IAU Sky & Telescope magazine NRAO/AUI/NSF, S. Dagnello

Impellizzeri and her team used ALMA's superior zoom lens ability to observe the molecular gas around the black hole. Unexpectedly, they found two counter-rotating disks of gas. The inner disk spans 2-4 light-years and follows the rotation of the galaxy, whereas the outer disk (also known as the torus) spans 4-22 light-years and is rotating the opposite way.

"We did not expect to see this, because gas falling into a black hole would normally spin around it in only one direction," said Impellizzeri. "Something must have disturbed the flow, because it is impossible for a part of the disk to start rotating backward all on its own."

Counter-rotation is not an unusual phenomenon in space. "We see it in galaxies, usually thousands of light-years away from their galactic centers," explained co-author Jack Gallimore from Bucknell University in Lewisburg, Pennsylvania. "The counter-rotation always results from the collision or interaction between two galaxies. What makes this result remarkable is that we see it on a much smaller scale, tens of light-years instead of thousands from the central black hole."

The astronomers think that the backward flow in NGC 1068 might be caused by gas clouds that fell out of the host galaxy, or by a small passing galaxy on a counter-rotating orbit captured in the disk.

At the moment, the outer disk appears to be in a stable orbit around the inner disk. "That will change when the outer disk begins to fall onto the inner disk, which may happen after a few orbits or a few hundred thousand years. The rotating streams of gas will collide and become unstable, and the disks will likely collapse in a luminous event as the molecular gas falls into the black hole. Unfortunately, we will not be there to witness the fireworks," said Gallimore.

Can a photon have a stable orbit around a black hole? - Astronomy

Can a person go in a black hole and come back out alive and in one piece?

The short answer to this is no! I guess you'd probably like more of an explanation though!