Black Hole Light

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According to Einstein’s classical theory of gravity, a black hole is a region of spacetime with an extremely strong gravitational field from which nothing can escape (not even light). As a result, black holes can only absorb matter. A consequence of this property is that the size of black holes never decreases. However, when quantum effects are taken into account, this scenario changes drastically. As this article will show, due to vacuum fluctuations of quantum fields in the vicinity of its surface (the so-called event horizon), the black hole emits particles, such as photons (particles of light), neutrinos and others. Black holes, therefore, are not completely black! The author of this groundbreaking discovery was the famous British physicist Stephen Hawking.
The prediction that particles are emitted by static black holes surprised (to put it mildly) the physics community. Previous work, also by Hawking, showed that rotating black holes create particles. But contrary to his expectation that, in the absence of rotation, no particles would be produced, he found that even static black holes create particles.
In this article, I will follow the derivation in Mukhanov and Winitzki of the Hawking temperature for a massless scalar field in 2-dimensional spacetime (the derivation in 4-dimensional spacetime can be found in Hawking’s original paper).
The standard explanation of the origin of the Hawking radiation is the following. Quantum fluctuations are, in general, characterized by the constant production of virtual particle-antiparticle pairs. What often happens in the vicinity of the black hole’s event horizon is that one of the two particles crosses the horizon while the other particle escapes as Hawking radiation.
Though this explanation is ubiquitous, it does not correspond precisely to the actual computation, which will be explained here. For a more detailed discussion of this issue, see Schmelzer and Baez.
The Schwarzschild black hole is nonrotating and spherically symmetric. It is the simplest black hole and contains only one parameter, its mass M. The 2D Schwarzschild line element in spherical coordinates reads:
For a detailed explanation of the expression above see the following article.
An Apparent Accident May Contain a Great Mystery in Physics
Eq. 1 seems to indicate that there are two singularities (locations in spacetime where the gravitational field becomes infinite) in the Schwarzschild metric, one at r=0 and another at r= 2M, the so-called Schwarzschild radius. More specifically, substituting r= 2M, the two components of the line element become:
However, as will be shown in the next section, the only physical singularity of the Schwarzschild black hole is at r=0. The apparent singularity at r = 2M is, in fact, merely a coordinate singularity that can be removed by changing to a new system of coordinates. To show that, we will introduce the so-called Kruskal–Szekeres coordinates.
The Kruskal–Szekeres coordinates are obtained as follows. First, one defines the “tortoise coordinate” r*(r) and the corresponding tortoise lightcone coordinates:
Notice that r* is singular at r = 2M. This implies that the tortoise lightcone coordinates, which are also singular at r = 2M, do not cover the region where r < 2M, which is defined as the interior of the black hole. To cover the full Schwarzschild black hole spacetime, we need to change our coordinate systems again. This new coordinate system will be the analytical extension of the Kruskal–Szekeres lightcone coordinates which is defined as follows:
Now compare it with Eq. 1. We see that with this choice of coordinate system, the singularity at the Schwarzschild radius r=2M vanishes. Hence, as previously mentioned, r = 2M is a coordinate singularity (an artifice of the coordinate system) and not a physical singularity. This means that when a free-falling observer crosses the radius r = 2M, he will not feel anything unusual.
Now, note that the Kruskal–Szekeres lightcone coordinates have the following domain of validity: -∞< u <0 and 0 < v < ∞. These coordinates do not cover the region r < 2M. In other words, they are valid only outside the black hole, for r >2M. Note, however, that the line element Eq. 5 is well-defined for all -∞< u <∞ and -∞ < v < ∞, covering all Schwarzschild spacetime. We must then analytically extend the coordinates u and v to include values u > 0 and v < 0.
Following Mukhanov and Winitzki we can express the new, Kruskal–Szekeres lightcone coordinates (u, v) in terms of the original (t, r) using Eqs. 3, 4, and 5. We obtain the following two relations:
Provided we analytically continue u and v, these relations will be valid for all Schwarzschild black hole spacetime.
To obtain the well-known Kruskal-Szekeres diagram for the Schwarzschild black hole we introduce two new coordinates, namely
Now, to cover the full diagram we have to extend the coordinates (u, v). Expressing the extended variables (u, v) in terms of the original coordinates (t, r) we obtain two sets of coordinates, namely:
The Kruskal-Szekeres diagram is shown below:
Notice a few important characteristics:
For simplicity, let us write the action of a scalar massless field in curved spacetime with two-dimensions:
The scalar field equation corresponding to S can be written in terms of lightcone tortoise coordinates or Kruskal–Szekeres lightcone coordinates. In both cases, it is a sum of two terms:
This is a consequence of the conformal invariance of S (which is explained in more detail in one of my previous articles). The As and Bs are well-behaved functions (with and without the “~”). A simple example is the following solution:
The Schwarzschild line element written in terms of the lightcone tortoise coordinates is given by:
At very large distances from the black hole, this line element becomes:
We see that very far from the black hole, the proper time of a stationary observer coincides with t in Eq. 15. Hence, this observer associates the modes with positive ωs (the frequencies with respect to t) with particles (see Mukhanov and Winitzki).
As usual in quantum field theory, we expand the field in terms of creation and annihilation operators:
The vacuum corresponding the b annihilation operator is called Boulware vacuum and it contains no particles according to the observer that is located at large distances from the black hole:
Since, as we saw before, the lightcone tortoise coordinates only cover part of the Schwarzschild black hole spacetime (outside the horizon), the Boulware vacuum has singular behavior on the horizon, and therefore it is not physically valid. Furthermore, the corresponding energy density would diverge at the horizon and quantum fluctuations would generate a strong backreaction which would invalidate the underlying hypothesis of a weakly perturbed classical gravitational field.
In contrast with the lightcone tortoise coordinates, the Kruskal–Szekeres lightcone coordinates cover all Schwarzschild black hole spacetime and are well-defined (nonsingular) at the event horizon. In the neighborhood of the horizon the line element in Kruskal–Szekeres coordinates reads:
Hence, an observer that moves across the event horizon is associated with a mode with frequency ω > 0 with respect to T. Again expanding the field as in Eq. 16 we obtain:
The nonsingular vacuum state now has a finite energy density leading to a small backreaction of the quantum fluctuations, keeping the classical gravitational field weekly perturbed as it should. The Kruskal vacuum, which obeys
can, therefore, be identified with the “true” vacuum close to a black hole.
Now, for the distant observer, the Kruskal–Szekeres vacuum contains particles! Their number density can be obtained following precisely the procedure at the end of one of my previous articles, linked below. The only modification we need to do is to substitute the acceleration a by the term 1/4M (the so-called surface gravity).
Connections Between Acceleration and Temperature
We then obtain the following expression for the thermal spectrum measured by the distant observer and the corresponding Hawking temperature:
The Hawking radiation is very dim and is swamped by the radiation emitted by the enormous amounts of hot gas that fall into all black holes we currently identify.
That could be one of the reasons why it hasn’t been observed yet. From Eq. 21, we see that only black holes with tiny mass would emit Hawking radiation with measurable intensities.
My Github and personal website www.marcotavora.me have some other interesting material both about physics and other topics such as math, machine learning, deep learning, and finance. Check it out!
I’m a physicist. For me, nothing is more magical than uncovering connections between domains of knowledge that seem, at first, to be entirely unrelated.
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https://www.cnn.com/2020/06/25/world/black-hole-merger-light-trnd-scn
Перевести · 25.06.2020 · The supermassive black hole at the center of the M87 galaxy, the first to ever be imaged, can now be seen in polarized light. Swirling lines reveal the magnetic field near the …
https://towardsdatascience.com/why-black-holes-emit-light-908cdef80153
Перевести · 04.08.2020 · According to Einstein’s classical theory of gravity, a black hole is a region of spacetime with an e x tremely strong gravitational field from which nothing can escape (not even light). As a result, black holes can only absorb matter. A consequence of this property is that the size of black holes never decreases.
https://en.m.wikipedia.org/wiki/Black_hole
Ориентировочное время чтения: 10 мин
Опубликовано: 20.11.2001
A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. Although it has an enormous effect on the fate and circumstances of an object crossing it, according to general relativity it has no locally detectable features. In many way…
A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. Although it has an enormous effect on the fate and circumstances of an object crossing it, according to general relativity it has no locally detectable features. In many ways, a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe directly.

Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, and its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; it was not until the 1960s that theoretical work showed they were a generic prediction of general relativity. The discovery of neutron stars by Jocelyn Bell Burnell in 1967 sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality. The first black hole known as such was Cygnus X-1, identified by several researchers independently in 1971.

Black holes of stellar mass form when very massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses (M☉) may form. There is consensus that supermassive black holes exist in the centers of most galaxies.

The presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light. Matter that falls onto a black hole can form an external accretion disk heated by friction, forming quasars, some of the brightest objects in the universe. Stars passing too close to a supermassive black hole can be shred into streamers that shine very brightly before being "swallowed." If there are other stars orbiting a black hole, their orbits can be used to determine the black hole's mass and location. Such observations can be used to exclude possible alternatives such as neutron stars. In this way, astronomers have identified numerous stellar black hole candidates in binary systems, and established that the radio source known as Sagittarius A*, at the core of the Milky Way galaxy, contains a supermassive black hole of about 4.3 million solar masses.

On 11 February 2016, the LIGO Scientific Collaboration and the Virgo collaboration announced the first direct detection of gravitational waves, which also represented the first observation of a black hole merger. As of December 2018 , eleven gravitational wave events have been observed that originated from ten merging black holes (along with one binary neutron star merger). On 10 April 2019, the first direct image of a black hole and its vicinity was published, following observations made by the Event Horizon Telescope (EHT) in 2017 of the supermassive black hole in Messier 87's galactic centre. In March 2021, the EHT Collaboration presented, for the first time, a polarized-based image of the black hole which may help better reveal the forces giving rise to quasars.

As of 2021, the nearest known body thought to be a black hole is around 1500 light-years away (see List of nearest black holes). Though only a couple dozen black holes have been found so far in the Milky Way, there are thought to be hundreds of millions, most of which are solitary and do not cause emission of radiation, so would only be detectable by gravitational lensing.
https://owlcation.com/stem/Why-Does-Light-Get-Sucked-Into-Black-Holes
Перевести · 30.10.2011 · When light travels into a black hole it will eventually hit the event horizon, and as spacetime continues to bend into itself; the light will follow. So really, light will never get sucked into black holes. Instead, light is simply following its normal behavior, and traveling straight into black holes on its own!
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https://www.sciencealert.com/visible-light-emitted-from-a-black-hole-has-been-detected...
Перевести · 15.06.2015 · For the first time, astronomers have observed bursts of visible light being released by a black hole as it swallows matter from nearby stars. These flashes of light, which lasted between several minutes to a few hours, were seen coming from a black hole in the Cygnus constellation, located about 7,800 light-years away from Earth.
https://www.newsweek.com/nasa-shows-video-2-black-holes-dance-around-bend-light-off...
Перевести · 16.04.2021 · NASA released a video of a depiction of two massive black holes colliding, which could give the space organization an inner-look at ripple effects of the black holes through the light …
How is light affected by a black hole?
How is light affected by a black hole?
When light passes by black holes, as it shifts in that straight line of space-time, it doesn’t speed up its acceleration, which things with mass would do, because light has a universally constant velocity. However, the frequency of the light is changed by this space-time geometry distortion, which affects the color of light that we can observe.
www.scienceabc.com/nature/universe/if-li…
Can gravity trap light in a black hole?
Can gravity trap light in a black hole?
But contrary to what you may think, gravity is not the key component in a black hole's ability to trap light. The real culprit is the mass of a black hole, and its effects on spacetime. (Also referred to as spacetime or space-time) Everything that has mass causes the spacetime around it to bend. More mass creates a larger bend in spacetime.
owlcation.com/stem/Why-Does-Light-Get …
For the first time, astronomers have observed bursts of visible light being released by a black hole as it swallows matter from nearby stars. These flashes of light, which lasted between several minutes to a few hours, were seen coming from a black hole in the Cygnus constellation, located about 7,800 light-years away from Earth.
www.sciencealert.com/visible-light-emitte…
What kind of gravity does a black hole have?
What kind of gravity does a black hole have?
According to Einstein’s classical theory of gravity, a black hole is a region of spacetime with an e x tremely strong gravitational field from which nothing can escape (not even light). As a result, black holes can only absorb matter.
towardsdatascience.com/why-black-holes …
https://www.scienceabc.com/.../if-light-has-no-mass-why-is-it-affected-by-black-holes.html
The Movement of Light: Einstein’s Theory of Relativity
Black Holes vs. Light
Why Does Light (photons) Feel The Effects of Gravity When It Has No Mass?
References
Now, we have been talking about light and its colors being affected by gravity wells and passing near black holes, but what about the main event? People say that black holes have such powerful gravitation that not even light can escape it, but that seems contrary to everything we’ve just learned. Light can’t change its velocity, so how could it ever be “contained” or “captured” by a black hole? Well, bl…
https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-a-black-hole...
Перевести · 21.05.2015 · A black hole is a place
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Astronomers may have detected light from a black hole ...
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What Is a Black Hole? | NASA
Black Hole Light