# Black Holes

**Black Holes** are regions of space in which gravitational fields are so
strong that no particle or signal can escape the pull of gravity.
The boundary of this no-escape region is called the event horizon, since
distant observers outside the black hole cannot see (cannot get light from)
events inside.

Although the fundamental possibility of such an object exists within Newton's classical theory of gravitation, Einstein's theory of gravity makes black holes inevitable under some circumstances. Prior to the early 1960s, black holes seemed to be only an interesting theoretical concept with no astrophysical plausibility, but with the discovery of quasars in 1963 it became clear that very exotic astrophysical objects could exist. Nowadays it is taken for granted that black holes do exist in at least two different forms. Stellar mass black holes are the endpoint of the death of some stars, and supermassive black holes are the result of coalescences in the centers of most galaxies, including our own.

No signal can propagate from inside a black hole, but the
gravitational influence of a black hole is always present. (This
influence does not *propagate* out of the hole; it is permanently
present outside, and depends only on the total amount of mass, angular momentum,
and electric charge that have gone into forming the hole.) Black holes can be detected through the influence of
this strong gravity on the surroundings just outside the hole. In this
way, stellar mass holes produce detectable X-rays, supermassive black
holes produce a wide spectrum of electromagnetic signals, and both
types can be inferred from the orbital motion of luminous stars and matter around
them. Phenomena involving black holes of any mass can produce strong
gravitational waves, and are of interest as sources for present and
future gravitational wave detectors.

## Classical vs. relativistic black holes

Something like a black hole exists within Newton's classical theory of gravity. In that theory, an energy argument tells us that there is an escape velocity \(v_{\rm esc} =\sqrt{2GM/R}\) from the surface of any spherical object of mass \(M\) and radius \(R\ .\) If this velocity is greater than the speed of light \(c\) then light from this object cannot escape to infinity. Thus the condition for such an "unseeable" object is

\[R<2GM/c^2\ .\]

In the classical theory, a particle could overcome this gravity with strong enough engines to provide the energy needed for escape. This is not so in general relativity, Einstein's theory of gravitation. In that theory, escaping the black hole is equivalent to moving faster than light, an impossibility in relativity.

To understand the relativistic black hole it is useful to think of space being dragged inward towards a gravitational center, at a faster rate near the center than far from it. The distance at which space is moving inward at the speed of light represents the location of the event horizon, since no signal can progress outward through space faster than \(c\ .\) This comparison is more than a metaphor; black hole analog experiments with accelerating gas flows and other phenomena are being designed.

An important difference from Newton's theory is that Einstein's, and other relativistic theories of gravitation are nonlinear in the sense that gravitation (as well as mass) can be a source of gravity. Thus when a massive object collapses small enough, the tendency to continue the collapse and form a black hole can become unstoppable.

## Stationary black holes

In the Newtonian theory, gravity is described by the potential \(\Phi\ .\) Inside a spherical object the form of \(\Phi(r)\) depends on the interior structure, but in the vacuum outside matter the potential \(-GM/r\) depends only on the interior mass. Similarly, in Einstein's theory the stationary (time-independent) spherically symmetric exterior solution, called the Schwarzschild spacetime, depends only on the mass of the interior object. If the interior object is small enough, then the Schwarzschild exterior extends to small enough radius that there is a horizon, a surface across which light cannot move outward. This horizon radius \(R_H =2GM/c^2\) is, coincidentally, the same as the critical radius for "unseeable" objects in Newton's theory. (The meaning of "radius" as distance to the center is not straightforward for the Schwarzschild solution. Radius \(R_H \)here actually means that the area of the event horizon is \(4\pi R_H ^2\ .\))

In Einstein's theory, the "exterior" solution can be taken to apply with no interior solution. In this case it is gravity itself, rather than matter, that acts as the source of gravity. The inward-extended exterior solution does not reach a center, but rather is connected via a spacetime bridge to another universe, or another section of our own. For an astrophysical black hole, formed from the collapse of matter, a physical solution for the matter distribution replaces the pure vacuum Schwarzschild solution in the interior of the black hole. This physical solution lacks the spacetime bridge of ideal mathematical black holes, but contains a central "singularity" where matter is compressed to infinite density. Very close to this singularity it is expected that the laws of general relativity will no longer apply, and as-yet unknown laws of quantum gravity are needed.

A more general stationary black hole solution of Einstein's theory is the Kerr solution, a vacuum spacetime with both mass and angular momentum, and taken to represent a rotating black hole. In its pure mathematical form the Kerr hole contains a spacetime bridge, but as in the case of the Schwarzschild black hole this bridge is absent in realistic black holes that form by the collapse of matter.

Unlike the Schwarzschild spacetime, the Kerr solution is *not*
the exterior spacetime of a material object with angular momentum. (In
fact no realistic solution has been found to join a Kerr exterior to a
material interior.) The Kerr solution only becomes the exterior
spacetime asymptotically at very late times after the collapse of an
object.

Two other exact mathematical black hole solutions are the Reissner-Nordström spacetime, representing a hole with mass and electrical charge, and the Kerr-Newman spacetime, representing a hole with mass, electrical charge, and angular momentum. These spacetimes are not astrophysically relevant, since astrophysical bodies have negligible net electrical charge. [For a detailed description of these spacetimes see, e.g., Misner et al. (1973), Part VII; or Wald (1984), Chap. 12.]

All of these spacetimes, including the ones with angular momentum, are stationary: that is, they are independent of time. But in relativity there is no unique meaning to time, so an important question is: "Just what 'time' is it of which the stationary black holes are independent?" The answer lies in the fact that one can assign every spacetime point four coordinates, four labels that uniquely identify the location of each point. One of these coordinates is called the "coordinate time." Spacetimes that are said to be stationary, like the spacetime of a Kerr hole, have a special property: the time coordinate may be chosen so that the spacetime geometry is the same at any moment of this time coordinate.

At large distances from a stationary black hole, where spacetime curvatures are weak, this stationary time coordinate can be chosen also to have another important property: to agree with the "proper time," or ordinary clock time, of an observer at rest with respect to the hole. Since we ourselves are more-or-less at rest (or are at nonrelativistic velocities) very far from black holes, this kind of stationary coordinate time is the time used in astronomical observations. For observers near the hole, however, proper time and stationary coordinate time will not agree. An interval of proper time between two events is shorter (near the horizon, much shorter) than the interval of stationary coordinate time between those two events. The relationship between stationary coordinate and proper time is further complicated by special relativistic time dilation for observers who are rapidly moving. Figure 2 illustrates this by showing the coordinate time vs. radius for a particle falling into a black hole, and comparing it with the proper time measured by an observer riding along with the infalling particle. The progress as measured with proper time is in no way special at the horizon: falling observers will not notice anything strange as they pass the point of no return. But as described in coordinate time, the observer (and likewise, the surface of a collapsing star) takes an infinite amount of time to reach the horizon. Since coordinate time is the proper time of distant observers, astronomers will see the particle reach the horizon only in the infinite future.

The two (or more) types of time are sometimes a source of confusion in discussions of black hole phenomena, since they often give totally different answers to the question "how long does it take?"

## Black hole parameters

Astrophysical black holes are characterized by two parameters:
their mass and their angular momentum (or *spin*). The
mass parameter \(M\) is equivalent to a characteristic length
\(GM/c^2=1.48\,{\rm km}(M/M_{\rm o})\ ,\) or a characteristic timescale
\(GM/c^3=4.93\times10^{-6} \,{\rm sec}(M/M_{\rm o})\ ,\) where \(M_{\rm o}\)
denotes the mass of the Sun. These scales, for
example, give the order of magnitude of the radii and periods of
near-hole orbits. The timescale also applies to the process in which a
developing horizon settles into its asymptotically stationary
form. For a stellar mass hole this is of order \(10^{-5} \,{\rm sec} \ ,\)
while for a supermassive hole of \(10^8M_{\rm o}\ ,\) it is thousands of
seconds.

For Schwarzschild holes, and approximately for Kerr holes, the horizon
is at radius \(R_H= 2GM/c^2 \ .\) At the horizon, the "acceleration of
gravity" has no meaning, since a falling observer cannot stop at the
horizon to be weighed. What is relevant at the horizon is the tidal
stresses that stretch and distort the falling observer. This tidal
stretching is given by the same expression, the gradient of the
gravitational acceleration, as in Newtonian theory\[2GM/R_H^3=c^6/(4G^2M^2)\ .\] In the case of a solar mass black hole the
tidal stress (acceleration per unit length) is enormous at the
horizon, on the order of \(3\times10^9(M_{\rm o}/M)^2\mathrm{s}^{-2}\ :\)
that is, a person would experience a
differential gravitational field of about \(10^9\) Earth gravities,
enough to rip apart ordinary materials. For a supermassive hole, by
contrast, the tidal force at the horizon is smaller by a typical
factor \(10^{10\mbox{--}16}\) and would be easily survivable. However,
at the central singularity, deep inside the event horizon, the tidal
stress is *infinite*.

In addition to its mass \(M\ ,\) the Kerr spacetime is described with a spin parameter \(a\) defined by the dimensionless expression

\[\frac{a}{M}=\frac{cJ}{GM^2}\,,\]

where \(J\) is the angular momentum of the hole. For the Sun (based on surface rotation) this number is about 0.2, and is much larger for many stars. Since angular momentum is ubiquitous in astrophysics, and since it is expected to be approximately conserved during collapse and black hole formation, astrophysical holes are expected to have significant values of \(a/M\ ,\) from several tenths up to and approaching unity.

The value of \(a/M\) can be unity (an "extreme" Kerr hole), but it cannot be greater than unity. In the mathematics of general relativity, exceeding this limit replaces the event horizon with an inner boundary on the spacetime where tidal forces become infinite. Because this singularity is "visible" to observers, rather than hidden behind a horizon, as in a black hole, it is called a naked singularity. Toy models and heuristic arguments suggest that as \(a/M\) approaches unity it becomes more and more difficult to add angular momentum. The conjecture that such mechanisms will always keep \(a/M\) below unity is called cosmic censorship.

The inclusion of angular momentum changes details of the description of the horizon, so that, for example, the horizon area becomes

\[\mbox{Horizon area}=(4\pi G^2/c^4)\left[ \left(M+\sqrt{M^2-a^2 \;}\ \right)^2+a^2\right]\,.\]

This modification of the Schwarzschild (\(a=0 \)) result is not significant until \(a/M\) becomes very close to unity. For this reason, good estimates can be made in many astrophysical scenarios with \(a \) ignored.

## Dynamical black holes

The event horizon is defined as the outer boundary of the region from which there is no escape. For stationary black holes this surface is at a fixed location in space, but more generally the horizon is dynamical; it can grow, change shape, oscillate. In particular, a horizon can be born.

The birth of a horizon is a change first studied by Oppenheimer and Snyder for the collapse of spherical pressureless fluid. The general scenario is for a horizon to be born deep inside the collapsing matter and to spread outward.

Small changes in a horizon can be treated as perturbations, greatly simplifying the mathematics of Einstein's equations. In this way, it has been found that black holes have characteristic patterns of oscillations, called quasinormal modes. These modes are like mechanical resonances, but are highly damped by the emission of gravitational waves, and have both periods and damping times on the order of the characteristic time \(GM/c^3\ .\)

For large nonspherical changes in a dynamical horizon, only supercomputer simulations can give quantitative answers. The focus of such work has been the merger of two black holes in binary orbit around each other, a scenario of special interest as a source of strong gravitational wave emission. Only in 2005 were technical problems first overcome [Pretorius (2005) (See Fig. 3.), Campanelli et al. (2006), Baker et al. (2006)] so that an accurate picture could emerge of how two horizons join to become a single final horizon.

There is one change that a horizon cannot make according to classical (nonquantum) general relativity: it cannot decrease its area. But considerations by Stephen Hawking and others of quantum effects in black hole spacetimes suggest that radiation arising in the close exterior of the black hole can carry off energy, and decreases the mass (and hence horizon area) of the hole. Although no quantum theory of relativistic gravitation currently exists, it is generally accepted that this Hawking radiation will be a feature of any such theory. The radiation behaves as if the horizon were a blackbody (perfect thermal emitter) at a temperature \(6\times10^{-8}(M_{\rm o}/M)\)K. Thus for astrophysical black holes, mass loss by Hawking radiation is much less important than the mass increase due to absorption of the 3K cosmic microwave background, and that mass increase is itself negligible.

## Astrophysical black holes

Black holes in our Universe can be grouped as: primordial black holes, stellar-mass black holes, and supermassive black holes. The first is highly speculative; the second and third are broadly accepted.

Primordial black holes of all masses are postulated to have formed from quantum fluctuations in the early Universe, but those with mass less than around \(10^{13}\)kg would have already evaporated due to Hawking radiation. No significant observational evidence has yet been found of the existence of these objects [MacGibbon and Carr (1991)].

Stellar-mass black holes, ranging from a few to a few tens of solar masses, are normal but rare endpoints in the evolution of massive stars. When a star exhausts its nuclear fuel and cools, it must collapse unless it is supported by nonthermal forces. It is known that such nonthermal forces cannot resist gravitational compression for masses greater than around 1.5 to 3\(M_{\rm o}\ .\) (The uncertainty is due to uncertainty in our knowledge of nuclear physics at high densities.) There are many stars far more massive than this, but in their death throes massive stars expel a great deal of their mass in supernova explosions. Even stars initially as massive as 20-30\(M_{\rm o}\) may blow off enough mass to leave a remnant neutron star smaller than 1.5\(M_{\rm o}\ .\) Stars more massive than 20-30\(M_{\rm o}\) may form a neutron star core, only to have it collapse due to fallback of material from the stellar mantle. Still more massive stars (above \(\sim40M_{\rm o}\)) may form a black hole directly in collapse, with or without a supernova explosion. These masses are rather uncertain at present, as our understanding of supernova explosion mechanisms is still evolving [Woosley and Janka (2005), Muno (2006)]. Collapsing gas clouds larger than \(\sim100M_{\rm o}\) are expected to dissipate from radiation pressure, which would prevent more massive stars from forming, and consequently sets an approximate upper limit for stellar mass black holes.

Observational evidence for stellar-mass black holes comes primarily from X-ray astronomy. A black hole in a close binary orbit can form an accretion disk (see Figure 1) of matter pulled off a normal stellar companion. In its inspiral, disk material is heated by shearing, and becomes a strong X-ray emitter. If observations reveal a point-like X-ray emitter with a mass inferred from orbital dynamics to be above that possible for a neutron star, it becomes a black hole candidate. At present there are over a dozen such black hole candidates, including the first object to be identified as a black hole, the X-ray source Cygnus X-1 (estimated mass \(10\pm3M_{\rm o}\)) [Casares (2006)].

The other class of observationally-supported black holes is that of supermassive black holes, ranging from hundreds to billions of solar masses. Evidence for the existence of black holes at the upper end of this range is overwhelming. By contrast, the existence of black holes with mass roughly of order \(10^2\) to \(10^4M_{\rm o}\ ,\) referred to as "intermediate-mass black holes," is speculative. The supermassive holes may have begun as primordial or stellar-mass black holes, but have grown through absorption of stars or gas, or through mergers with other holes [Ferrarese and Ford (2005)].

Evidence for supermassive black holes originally came from quasars, small but intense radio sources seen at cosmological distances. Their huge luminosities implied high mass, while their rapid variability implied extremely small size. More recently, measurements of Doppler shifts of stars and gas in the centers of galaxies have shown that compact objects of mass greater than \(10^{6}M_{\rm o}\) reside in the cores of most galaxies; the very small size of these central objects rules out any plausible alternative to the black hole explanation. The black hole at the center of our own Milky Way galaxy has a measured mass of \(\sim3\times10^6M_{\rm o}\ ,\) based on the orbits of stars near the radio source Sagittarius A* associated with the Galactic core [Ghez et al. (2005); see Fig. 4].

## Gravitational waves from black holes

Black holes are of particular interest for gravitational waves, and vice versa. For gravitationally-bound systems, the typical maximum gravitational wave amplitude (dimensionless strain) is \(h\sim (2GM/c^2)^2/(RD)\sim R_H^2/(RD)\ ,\) where \(D\) is the distance to the system, \(M\) and \(R\) are the mass and size of the system, and \(R_H\) is the horizon radius of a black hole of the same mass. From this it is clear that strongest gravitational waves will involve systems at or near black hole compactness: in particular, systems containing black holes in close proximity to one another [Flanagan and Hughes (2005)].

Conversely, a system containing *only* black holes would not be
expected to radiate *any* sort of radiation other than
gravitational waves, and in any system containing black holes,
gravitational waves provide the most accurate probes of conditions
deep in their gravitational wells. In particular, gravitational wave
observations could definitively test whether a compact mass is truly a
black hole [Cutler and Thorne (2002)].

The strongest intrinsic source of gravitational waves in the present Universe is the collision of two comparable-massed black holes. A sufficiently tight black hole binary system will lose energy through emission of gravitational waves, causing the orbit to circularize and then slowly shrink over time.

When the orbital radius approaches the Schwarzschild radius of the system, complex nonlinear dynamics come into play and the final stages of the inspiral and merger can only be modeled through numerical simulations. These simulations are quite challenging, but recent breakthroughs [Pretorius(2005)(See Fig. 5), Campanelli et al.(2006), Baker et al.(2006)] have led to the first complete inspiral-merger gravitational waveforms. After the black holes have coalesced, they form a single highly-distorted black hole which quickly settles down to a quiescent Kerr state through emission of quasinormal "ringdown" gravitational waves [Pretorius (2005), Campanelli et al. (2006), Baker et al. (2006)].

Another gravitational emission mechanism for supermassive black holes would come from their capture of compact stellar objects, particularly stellar-mass black holes. Unlike comparable-mass mergers, these captures would be on highly eccentric orbits that would not circularize before merger, resulting in intricate rapidly precessing orbits in the final months or years of inspiral. The consequent gravitational-wave signal would allow for exquisitely precise measurements of the parameters of the supermassive black hole, as well as tests of any deviation from the predictions of general relativity [Cutler and Thorne (2002)].

For stellar-mass black holes (tens of \(M_{\rm o}\)), the final stages of inspiral and merger occur at frequencies of hundreds to thousands of Hz, a regime targeted by ground-based gravitational-wave observatories. Systems of supermassive black holes that are millions of \(M_{\rm o}\) emit at frequencies of tens of mHz, which will be detectable by proposed space-based gravitational-wave detectors. Black holes in the billions of \(M_{\rm o}\ ,\) or a stochastic background of signals from more moderate supermassive black holes at earlier stages of their inspiral, might emit waves at nHz frequencies that would be measurable through years-long correlated timing measurements of radio pulsars [Jenet et al. (2005)].

## References

J. G. Baker et al., *Physical Review Letters* **96**, 111102 (2006).

M. Campanelli et al., *Physical Review Letters* **96**, 111101 (2006).

J. Casares, "Observational evidence for stellar-mass black holes", astro-ph/0612312 (2006).

C. Cutler and K. S. Thorne, "An Overview of Gravitational-Wave Sources", gr-qc/0204090 (2002).

L. Ferrarese and H. Ford, *Space Science Reviews* **116**, 523 (2005) astro-ph/0411247.

E. Flanagan and S. A. Hughes, *New Journal of Physics* **7**, 204 (2005).

A. M. Ghez et al., *The Astrophysical Journal* **620**, 744(2005) astro-ph/0306130.

F. A. Jenet at al., *The Astrophysical Journal* **625**, L123 (2005) astro-ph/0504458.

H. MacGibbon and B. J. Carr, *The Astrophysical Journal* **371**, 447 (1991).

C. W. Misner, K. S. Thorne and J. A. Wheeler,
*Gravitation* (Freeman, San Francisco, 1973).

P. Muno, "Which Stars Form Black Holes and Neutron Stars?", astro-ph/0611589 (2006).

F. Pretorius, *Physical Review Letters* **95**, 121101 (2005).

R. M. Wald, *General Relativity,* (University of Chicago Press, Chicago, 1984),

S. Woosley and T. Janka, *Nature Physics* **1**, 147, astro-ph/0601261 (2005).

**Internal references**

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

## Suggested reading

K. S. Thorne, *Black Holes and Time Warps: Einstein's Outrageous Legacy* (W. W. Norton, New York, 1994).

E. F. Taylor and J. A. Wheeler, *Exploring Black Holes: Introduction to General Relativity*
(Benjamin Cummings, 2000).

S. L. Shapiro and S. A. Teukolsky,
*Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects*
(Wiley-Interscience, 1983).

R. H. Price *The Physical Basis of Black Hole Astrophysics*, Chap.5
in *100 Years of Relativity, Spacetime Structure: Einstein and Beyond*
(World Scientific, Singapore, 2005).