# Black Holes

# BLACK HOLES

A black hole (the term was coined by John Archibald Wheeler in 1967) is a closed surface through which gravity prevents light from propagating. Insofar as relativity prohibits anything from traveling faster than light, it follows that nothing can escape through the surface of a relativistic black hole. That said, in general relativity the notion of energy is problematic, and energy and hence mass can be extracted by classical and quantum processes. Classically the interior of a black hole contains a *singularity* : Along certain paths physical quantities become ill-behaved (e.g., the gravitational field may become infinite). While nothing can pass back through the surface of a black hole, it is possible in certain models to travel into other universes. All of these properties have philosophically disturbing implications that have strongly influenced the development of physics, especially since there are solid theoretical and experimental reasons to believe that black holes are not merely hypothetical, but actually exist.

## History

Black holes (henceforth BHs) arise in the general theory of relativity (GTR). However something similar is possible in Newtonian physics. John Michell (1784) and Pierre Simon Laplace (1796) pointed out that, as a ball thrown upwards with insufficient speed will eventually fall back to Earth, if a star of a given mass were smaller than a certain size (in modern parlance, its *critical radius* ) then even light corpuscles, emitted from the surface at the speed of light, would eventually be pulled back to its surface. If such a star were a sufficient distance away it would not be directly visible (though faster or accelerating bodies could escape).

Karl Schwarzschild discovered the first exact solution of GTR in 1916, before the Einstein field equations of the theory were cast in their final form. The model has a point mass *M* at its center and in radial co-ordinates (two angles, θ and Φ, giving the latitude and longitude of a point, its distance *r* from the center and time *t* ) the line element (the distance between two infinitesimally separated points) is:

where *G* is Newton's gravitational constant and *c* the speed of light. The idea in GTR is that *M* determines *ds* ^{2}, which determines the geometry of spacetime, which explains the effect of gravity. Inspection of the second and first terms reveals *ds* ^{2} diverges at *r* = 2GM/*c* ^{2}—the "Schwarzschild radius"—and *r* = 0—the location of the point mass, respectively. The singularity at *r* = 0 is genuine, though one would suspect (wrongly) that it would not occur if it were not for the idealization of a point mass—see below. The divergence at *r* = 2GM/c^{2} is not physical, but merely an artifact of the co-ordinates used to describe the solution, a point that was not properly appreciated until the late 1930s. By way of analogy, if we used *x′* =1/(*x* −1), *y′* =*y* as co-ordinates for the x-y plane, then d*s* ^{2}=d *x′* ^{2} *x′* ^{4} + d*y′* ^{2}. Along *x* =1 the plane is perfectly smooth but *ds* ^{2} is singular since *x′* =∞, a reflection of the "poor" choice of co-ordinates.

In the Schwarzschild solution the singularity reflects not a geometric irregularity but the existence of a sphere of radius 2GM/c^{2} (named the "horizon" by Wolfgang Rindler in the 1950s) from which no light can escape (a point first made by Johannes Droste in 1916). Clearly if a body is smaller than 2GM/c^{2} then light cannot escape its horizon, so a star's Schwarzschild radius is its critical radius: The horizon forms a BH around any star smaller than its Schwartzschild radius. Other solutions for BHs were discovered by Hans Reissner (1916) and Gunnar Nordström (1918) for a charged BH, by Roy Kerr (1964) for a spinning BH, and by Ted Newman (1965) for a charged and spinning BH. It is important to emphasize that the nature of the horizon and hence of the BH (and hence of the early solutions) was not properly understood until the mid-1960s. Remarkably a so-called "No Hair Theorem" shows that the exterior (but not the interior) of any BH is completely characterized by its mass, charge, and spin and not on any other details of its composition or formation: The exterior of every possible BH is described by one of the four models mentioned here.

The Schwarzschild solution was quickly accepted as the description of gravity outside a (stationary) star, where the mass could be treated as located at the star's center—that is, providing that the star was larger than its Schwarzschild radius (18.5km for the Sun). The early pioneers of GTR did not properly understand the horizon (they worried about its possible singular nature and the fact that bodies approaching it would apparently take bodies an infinite time to reach it) and tried to argue that they could not occur in nature. However the question arose of what would happen to a star after it exhausted its fuel supply and began to cool and contract. By 1925 it was apparent that such stars could shrink under their own gravity to form *white dwarfs* 1,000s of times denser than the Sun, but in the early 1930s Subrahmanyan Chandrasekhar showed that white dwarfs of masses more than 1.5 the mass of the Sun (*M* _{⊙}) were not stable against gravity. In 1933 Walter Baade and Fritz Zwicky proposed that stars could further implode to form *neutron stars* as dense as atomic nuclei (the gravitational energy released by the implosion explaining supernovae), but in 1938 Robert Oppenheimer and George Volkoff argued that neutron stars heavier than a few times *M* _{⊙} (the contemporary value is 2M_{⊙}) would be unable to resist their own gravity, and in 1956 Wheeler and Masami Wakano demonstrated that there were no denser stable objects than neutron stars. In 1939 Oppenheimer and Hartland Snyder provided the first model of a star collapsing through its Schwarzschild radius, a model vindicated in the early 1960s by simulations based on hydrogen bomb research (subsequent models show that stars below around 20M_{⊙} will eject sufficient matter during implosion to avoid complete collapse). In other words, it became clear not only that BHs were quite possible but also that their formation was likely. One philosophically significant aspect of this scientific revolution is how it was affected by physicists' changing intuitions about what mathematical models were physically realistic.

In the first years of the twenty-first century the astronomical evidence for BHs was strong and rapidly evolving; as of this writing, there are some fifty known candidates, half of them strong candidates. One class of candidates occurs in binary systems in which a star orbits a heavy body that strips material from it. The speed of the star can be calculated from the Doppler shift of its emission spectrum and the mass of the heavy body derived from that: If it is above 2M_{⊙} it is too heavy to be a neutron star and is presumably a BH. Typical BHs of this type have 5−15M_{⊙}. A second kind of BH candidate is the *supermassive* BH, thought to occur at the center of galaxies. For instance it is believed that a BH over 3 × 10^{6}*M* _{⊙} lies at the heart of our galaxy in the constellation Sagittarius. Observational work has been done to verify that these candidates are not some unknown, denser objects, for example by looking for nuclear reactions that can occur only on material surfaces and not on horizons.

## Bh Interior

When objections to the physical possibility of a horizon were overcome, the question became whether real BHs, like the known solutions, contained singularities. According to Oppenheimer and Snyder's model, stellar matter collapses to a point to form a singularity, but their work was not definitive because it assumed an unrealistically symmetric distribution of collapsing matter. However in the late 1960s Roger Penrose and then Steven Hawking proved Singularity Theorems showing that singularities must arise under very general conditions, including those believed to hold in BH formation, while in 1969 Vladimir Belinsky, Isaac Khalatnikov, and Evgeny Lifshitz found a singularity that would form if stellar collapse was only approximately symmetric.

Ordinarily a singularity in a function means that it diverges somewhere in its domain. The situation is more complicated in GTR because space has no existence separately from the fields: GTR is the theory of the metric field (the coefficients of *ds* ^{2}), which describes the geometry of space. Intuitively speaking, when the fields of GTR become singular the very notion of spatial points fails, and space can contain a singularity even though the fields do not diverge anywhere. Singularities potentially raise several philosophically interesting issues. One is that singularities are associated with failures of determinism: The problem is roughly analogous to that of calculating the propagation of an electromagnetic wave in a space with a hole removed. Another is that Physicists have thus postulated various forms of *Cosmic Censorship* : That singularities cause at most localized failures of determinism (e.g., only inside a BH).

One may be struck by the similar (mistaken) initial reaction to the horizon (also by the "transcendental" nature of this move—without censorship and determinism, physics of a certain kind is impossible), though censorship has been shown in a range of cases. If uncensored "naked singularities" are possible, then it would be possible to use them to complete "supertasks" in a finite time relative to a distant observer. Important to remember is that our discussion so far has been in the context of classical GTR (utilizing some quantum properties of matter), but around a singularity, quantum gravitational effects likely become important. Physicists generally expect that singularities will not occur in a quantized version of GTR. If so, classical singularities may offer no philosophically important lessons after all. However John Earman's *Bangs, Crunches, Whimpers, and Shrieks* argues that classical singularities may pose no physical problem, so physicists need not demand that quantum gravity banish them.

Everything that enters the Schwarzschild BH eventually reaches its singularity, but in the other models it is possible to avoid the singularity altogether and travel on to a flat region of spacetime: The other BHs contain "worm holes" or "Einstein-Rosen bridges" to a "new" universe or to a region of space far from the BH (in the latter case spacetime would contain "closed timelike curves," paths that allow one to travel to one's past). However even in classical GTR the models assume unrealistic symmetries, and so the interior parts cannot be trusted: As of this writing, while it has not been shown that more realistic classical BHs do not contain worm holes, it is widely assumed that they do not. The situation in quantum gravity is even less clear, though Lee Smolin (1992) has speculated that a new universe is created inside whenever a BH forms, with laws of nature that vary in a small, random way from their parent universe, so that all possible physics eventually comes into existence.

## Blackness

Consider what would happen as an astronaut entered the horizon of a BH while he was watched from the outside. It is useful to think of the region around the BH as analogous to a deep, sloped, hole in the ground: As one gets nearer the center, the distance up the walls and along the ground to the outside grows rapidly. Analogously, as the BH is approached, light has to travel up ever steeper "walls" of curved spacetime to escape: Since the speed of light is constant, it follows that light takes an increasingly long time to reach the outside. Just as showing movie frames at increasing intervals makes a scene appear to slow, so the astronaut will appear to decelerate. In a BH, however, the time for light to reach the outside becomes infinite at the horizon (though space there is perfectly smooth). The effect is that the astronaut appears to decelerate indefinitely and from the outside can never be seen to enter the BH, as if the movie were slowed until frozen on a single frame. It is crucial to appreciate that this phenomenon is entirely optical: The astronaut himself measures only a finite amount of time until he is inside the BH. (That a collapsed massive star would thus be seen frozen at its horizon was an impediment to understanding BH formation.)

Just as it would take light infinitely long to escape from the horizon of a BH, nothing localized inside the horizon can pass through it (ignoring subtleties concerning the speed of light as a cosmic speed limit): The BH is opaque to its exterior. However it is theoretically possible to extract energy and hence mass, via the relativistic equivalence of mass and energy, from a spinning BH by classical processes that slow the spin. It is rather surprising that this extraction is achieved by throwing matter into the hole, but not surprising that a BH stores energy in its rotation. In other words part of the BH's mass arises from its interaction with the spacetime outside it, so no energy has to leave the interior. More surprisingly, Hawking (1974) showed quantum effects mean that even a non-spinning BH would radiate energy (and mass) with a temperature inversely proportional to it mass.

This effect is philosophically important for two reasons. First it confirmed Jacob Bekenstein's (1972) speculation that BHs obeyed the laws of thermodynamics and hence possessed entropy; among other things BHs are relevant to an arrow time based on the Second Law of thermodynamics. Second what happens when the mass of the BH "evaporates" to zero? One issue is the possibility of a naked singularity. Another is the "loss of information paradox": Physical theories typically allow an earlier state to be retrodicted from a later one, so that no information about the earlier state is lost over time. However, in Hawking's calculation, radiation carries no information about how a BH was formed, so that information remains inside the BH and is lost: Once a BH evaporates, retrodiction is impossible. For these reasons BH radiation became an important issue since theories of "quantum gravity" can be judged according to how they treat BH entropy and the loss of information paradox. In particular, BH entropy is connected to the powerful idea of "holography," which connects the physics of any region to the physics of the surface bounding it.

** See also ** Earman, John; Laplace, Pierre Simon de; Philosophy of Physics; Quantum Mechanics; Space; Time.

## Bibliography

Bekenstein, Jacob D. "Blackholes and Information Theory." *Contemporary Physics* 45 (2003): 31–43. Available from http://www.arXiv.org/abs/quant-ph/0311049.

Belot, Gordon, John Earman, and Laura Ruetsche. "The Hawking Information Loss Paradox: The Anatomy of a Controversy." *British Journal for the Philosophy of Science* 50 (1999): 189–229.

Carroll, Sean. *Spacetime and Geometry: An Introduction to General Relativity*. New York: Pearson Addison Wesley, 2003.

Earman, John. *Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes*. Oxford: Oxford University Press, 1995.

Israel, Werner. "Dark Stars: The Evolution of an Idea." In *Three Hundred Years of Gravitation*, edited by Steven S. Hawking and Werner Israel. Cambridge, U.K.: Cambridge University Press, 1987.

Smolin, Lee. *The Life of the Cosmos*. Oxford: Oxford University Press, 1997.

Taylor, Edwin F., and John Archibald Wheeler. *Exploring Black Holes: Introduction to General Relativity*. New York: Benjamin Cummins, 2000.

Thorne, Kip S. *Black Holes & Time Warps: Einstein's Outrageous Legacy*. New York: Norton, 1994.

Unruh, William. "Black Holes, Dumb Holes, and Entropy." In *Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity*, edited by Craig Callender and Nick Huggett. Cambridge, U.K.: Cambridge University Press, 2001.

Ziólkowski, Janusz. "Black Hole Candidates." arXiv.org e-print Archive. Available from http://www.arXiv.org/abs/astroph/0307307.

*Nick Huggett (2005)*

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