Chapter 12. Quantum black holes

Transcription

1 Chapter 12 Quantum black holes Classically, the fundamental structure of curved spacetime ensures that nothing can escape from within the Schwarzschild event horizon. That is an emphatically deterministic statement. But what about quantum mechanics, which is fundamentally indeterminate? 12.1 Hawking Black Holes: Black Holes Are Not Really Black! The uncertainty principle and quantum fluctuations of the vacuum that play a central role in quantum mechanics. As we now explain, because of quantum mechanics it is possible for a black hole to emit mass. Therefore, as Stephen Hawking discovered, black holes are not really black! 399

2 400 CHAPTER 12. QUANTUM BLACK HOLES Deterministic Geodesics and Quantum Uncertainty Our discussion to this point has been classical in that it assumes that free particles follow geodesics appropriate for the spacetime. But the uncertainty principle implies that 1. Microscopic particles cannot be completely localized on classical trajectories because they are subject to a spatial coordinate and 3-momentum uncertainty of the form p i x i h; 2. Neither can energy conservation be imposed except with an uncertainty E t h, where E is an energy uncertainty and t is the corresponding time period during which this energy uncertainty This implies an inherent quantum fuzziness in the 4-momenta associated with our description of spacetime at the quantum level.

3 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK!401 For the Killing vector K K t =(1,0,0,0) in the Schwarzschild metric, ( K K = g µν K µ K ν = g 00 K 0 K 0 = 1 2M ), r from which we conclude that K is Timelike outside the horizon, since then K K < 0, Spacelike inside the horizon, since then K K > 0. As we now make plausible, this property of the Killing vector K permits a virtual quantum fluctuation of the vacuum to be converted into real particles and these are detectable at infinity as emission of mass from the black hole.

4 402 CHAPTER 12. QUANTUM BLACK HOLES v Particle-antiparticle fluctuation Singularity (r = 0) Horizon r = 2M, t = Spacelike ξ p _ ξ Horizon ξ Timelike ξ p u Figure 12.1: Hawking radiation in Kruskal Szekeres coordinates Hawking Radiation Assume a particle antiparticle pair created by vacuum fluctuation near the horizon of a Schwarzschild black hole, such that the particle and antiparticle end up on opposite sides of the horizon (Fig. 12.1). If the particle antiparticle pair is created in a small enough region of spacetime, there is nothing special implied by this region lying at the event horizon: the spacetime is indistinguishable from Minkowski space because of the equivalence principle. Therefore, the normal principles of (special) relativistic quantum field theory will be applicable to the pair creation process in a local inertial frame defined at the event horizon (even if the gravitational field is enormous there).

5 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK!403 v Particle-antiparticle fluctuation Singularity (r = 0) Horizon r = 2M, t = Spacelike ξ p _ ξ Horizon ξ Timelike ξ p u In the Schwarzschild geometry, the conserved quantity analogous to the total energy in flat space is the scalar product of the Killing vector K K t =(1,0,0,0) with the 4-momentum p. Therefore, if a particle antiparticle pair is produced near the horizon with 4-momenta p and p, respectively, the condition K p+k p=0, must be satisfied (to preserve vacuum quantum numbers). For the particle outside the horizon, K p>0 (it is proportional to an energy that is measureable externally). If the antiparticle were also outside the horizon, it too must have K p>0, in which case 1. The condition K p + K p = 0, cannot be satisfied and 2. The particle antiparticle pair can have only a fleeting existence of duration t h/ E (Heisenberg). So far, no surprises; just Quantum Mechanics

6 404 CHAPTER 12. QUANTUM BLACK HOLES v Particle-antiparticle fluctuation Singularity (r = 0) Horizon r = 2M, t = Spacelike ξ p _ ξ Horizon ξ Timelike ξ p u HOWEVER... If instead (as in the figure) the antiparticle is inside the horizon, The Killing vector K is spacelike (K K > 0) The scalar product K p is not an energy (for any observer) In fact, the product K p is a 3-momentum component. Therefore K p can be positive or negative (!!). Thus, there is magic afoot: If K p is negative K p + K p = 0 can be satisfied. Then the virtual particle created outside the horizon can propagate to infinity as a real, detectable particle, while the antiparticle remains trapped inside the event horizon. The black hole emits its mass as a steady stream of particles (!).

7 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK!405 Therefore, quantum effects imply that a black hole can emit its mass as a flux of particles and antiparticles created through vacuum fluctuations near its event horizon. The emitted particles are termed Hawking radiation. The loss of mass by Hawking radiation for a black hole is called black hole evaporation.

8 406 CHAPTER 12. QUANTUM BLACK HOLES The Hawking mechanism has been described by loose analogy with a rather nefarious financial transaction Suppose that I am broke (a money vacuum), but I wish to give to you a large sum of money (in Euros, E). Next door is a bank Uncertainty Bank and Trust that has lots of money in its secure vault but has shoddy lending practices. Then 1. I borrow a large sum of money E from the Uncertainty Bank, which will take a finite time t to find that I have no means of repayment. We hypothesize E t h, where h is constant, since the bank will be more diligent if the amount of money is larger. 2. I transfer the money E to your account. 3. I declare bankrupcy within the time t, leaving the bank on the hook for the loan. A virtual fluctuation of the money vacuum has caused real money to be emitted (and detected in your distant account!) from behind the seemingly impregnable event horizon of the bank vault.

9 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK! Mass Emission Rates and Black Hole Temperature The methods for obtaining quantitative results from the Hawking theory are beyond our scope, but the results can be stated simply: The rate of mass emission can be calculated using relativistic quantum field theory: dm dt = λ h M 2, where λ is a dimensionless constant. The distribution of energies emitted in the form of Hawking radiation is equivalent to a blackbody with temperature T = h ( ) 8πk B M = hc3 8πk B GM = M K. M Advanced methods of quantum field theory are required to prove this, but there are suggestive hints from the observations that 1. A black hole acts as a perfect absorber of radiation (as would a blackbody). 2. Hawking radiation originates in random fluctuations, as we would expect for a thermal emission process. Integrating dm/dt = λ h/m 2 for a black hole assumed to emit all of its mass by Hawking radiation in a time t H, we obtain M(t)=(3λ h(t H t)) 1/3. for its mass as a function of time.

10 408 CHAPTER 12. QUANTUM BLACK HOLES Mass, Temperature Mass Temperature Time t H Figure 12.2: Evaporation of a Hawking black hole. From the results T = h 8πk B M dm dt = λ h M 2 M(t)=(3λ h(t H t)) 1/3 the mass and temperature of the black hole behave as in Fig Therefore, The black hole evaporates at an accelerating rate as it loses mass. Both the temperature and emission rate of the black hole tend to infinity near the end. Observationally we may expect a final burst of very high energy (gamma-ray) radiation that would be characteristic of Hawking evaporation for a black hole. Note that temperature increases as mass is emitted: The black hole has a negative heat capacity (it heats up as it cools down ).

11 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK! Miniature Black Holes How long does it take a black hole to evaporate by the quantum Hawking process? From the mass emission rate formula we may estimate a lifetime for complete evaporation as t H M3 3λ h ( ) M 3 s. 1g A one solar mass black hole would then take approximately times the present age of the Universe to evaporate (with a corresponding blackbody temperature of order 10 7 K). It s pretty black! However, black holes of initial mass g or less would have evaporation lifetimes less than or equal to the present age of the Universe, and their demise could be detectable through a characteristic burst of high-energy radiation. The Schwarzschild radius for a g black hole is approximately cm, which is about 1 5 the size of a proton. To form such a black hole one must compress grams (mass of a large mountain) into a volume less than the size of a proton! Early in the big bang there would have been such densities. Therefore, a population of miniature black holes could have formed in the big bang and could be decaying in the present Universe with a detectable signature. No experimental evidence has yet been found for such miniature black holes and their associated Hawking radiation. (They would have to be relatively nearby to be seen easily.)

12 410 CHAPTER 12. QUANTUM BLACK HOLES Black Hole Thermodynamics The preceding results suggest that the gravitational physics of black holes and classical thermodynamics are closely related. Remarkably, this has turned out to be correct. It had been noted prior to Hawking s discovery by Jacob Bekenstein that there were similarities between black holes and black body radiators. The difficulty with describing a black hole in thermodyanamical terms was that a classical black hole permits no equilibrium with the surroundings (absorbs but cannot emit radiation). Hawking radiation supplies the necessary equilibrium that ultimately allows thermodynamics and a temperature to be ascribed to a black hole.

13 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK!411 Hawking Area Theorem: If A is the surface area (area of the event horizon) for a black hole, Hawking has proven a theorem that A cannot decrease in any physical process involving a black hole horizon, da dt 0. For a Schwarzschild black hole, the horizon area is A=16πM 2 da dm = 32πM, which can be written (h = Planck, k B = Boltzmann). dm = h ( ) 8πk B M d kb A 4h But de = dm is the change in total energy and the temperature of the black hole is T = h/8πk B M. Therefore, we may write the preceding as de = TdS }{{} 1st Law ds 0 }{{} 2nd Law S k B 4h A }{{} entropy which are just the 1st and 2nd laws of thermodynamics, provided that is entropy! S (surface area of black hole)

14 412 CHAPTER 12. QUANTUM BLACK HOLES Evaporation of a black hole through Hawking radiation appears to violate the Hawking area theorem in that the black hole eventually disappears. However, The area theorem assumes that a local observer always measures positive energy densities and that there are no spacelike energy fluxes. As far as is known these are correct assumptions at the classical level, but they may break down in quantum processes. The correct quantum interpretation of Hawking radiation and the area theorem is that 1. The entropy of the evaporating, isolated black hole decreases with time (because it is proportional to the area of the horizon). 2. The total entropy of the Universe increases because of the entropy associated with the Hawking radiation itself. 3. That is, the area theorem is replaced by Generalized 2nd Law: The total entropy of the black hole plus exterior Universe may never decrease in any process.

15 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK! The four laws of black hole dynamics The considerations of this section permit the formulation of four laws of black hole dynamics that are analogous to the four laws of classical thermodynamics: 1. Zeroth Law: The surface gravity κ of a stationary black hole is constant over its event horizon. This is analogous to the zeroth law of thermodynamics that the temperature T is constant for a system in thermal equilibrium. 2. First Law: Energy is conserved because of a relation δm = 1 κδa+ωδj+ ΦδQ, 8π where M is the mass, κ is the surface gravity, A is the area of the horizon, Ω is the angular velocity, J is the angular momentum, and Φ is the electrostatic potential for the black hole. This is the analog of the first law of thermodynamics. 3. Second Law: The area theorem or its quantum generalization da dt 0 ds 0 S k B 4h A This is the analog of the second law of thermodynamics. 4. Third Law: The surface gravity κ of a black hole cannot be reduced to zero by a finite series of operations. Analogous to the Nernst form of the third law of thermodynamics that the temperature T cannot be reduced to zero in a finite series of operations.

16 414 CHAPTER 12. QUANTUM BLACK HOLES Thus the four laws of black hole dynamics are analogous to the four laws of thermodynamics if an identification is made between Temperature T and some multiple of the surface gravity κ. Entropy S and some multiple of the event horizon area A. The four laws of black hole dynamics are more well-grounded theoretical conjecture than established law at this point.

17 12.1. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK! Gravity and Quantum Mechanics: the Planck Scale The preceding results for Hawking radiation are derived assuming that the spacetime in which the quantum calculations are done is a fixed background that is not influenced by the propagation of the Hawking radiation. This approximation is expected to be valid as long as E << M, where E is the average energy of the Hawking radiation and M is the mass of the black hole. This approximation breaks down on a scale given by the Planck mass ( ) 1/2 hc M P = = GeV c 2 = kg. G For a black hole of this mass, the effects of gravity become important even on a quantum ( h) scale, requiring a theory of quantum gravity. We don t yet have an adequate theory of quantum gravity. Note that one approaches the Planck scale near the endpoint of Hawking black hole evaporation. Therefore, we do not actually know yet what happens at the conclusion of Hawking evaporation for a black hole.

18 416 CHAPTER 12. QUANTUM BLACK HOLES 12.2 Black holes and information Entropy is related to information content because it is equal to the logarithm of the number of microscopic configurations that leave the macroscopic description of an object unchanged. Black hole evaporation by the Hawking mechanism leads to apparent paradoxes associated with this relationship. This may be illustrated by noting that Hawking radiation is produced randomly by vacuum fluctuations, so in the simplest picture it contains no information. Thus, the information content of the matter from which the black hole formed appears to be lost to the Universe if the black hole then decays completely away by Hawking radiation. This is a complex issue that is not fully resolved. Some contend that perhaps this means that the Universe does not conserve information. Others have conjectured that a (future) quantum gravity treatment of black hole evaporation may resolve the issue.

19 12.2. BLACK HOLES AND INFORMATION The holographic principle The entropy relation implies that S k B 4h A, The entropy of a black hole is proportional to the surface area of its horizon. But the information content of the black hole is associated with its entropy and The information content of a region of space is usually thought of as being proportional to the volume of that region, not to the area of a bounding surface. This has led to a proposed solution of the black hole information paradox called the holographic principle. The holographic principle asserts that the description of a volume of space can be thought of as being encoded on a 2-dimensional boundary of that region. For a black hole, this implies that Surface fluctuations of the event horizon must in some way contain a complete description of all the objects that have ever fallen into the black hole.

20 418 CHAPTER 12. QUANTUM BLACK HOLES The holographic universe Even more speculatively, the holographic principle has been extended to a cosmological statement that Perhaps the entire universe should be thought of as a two-dimensional information structure painted on the cosmological horizon. Extension of the holographic principle to a cosmological statement may be motivated by noting that A universe with a cosmological horizon resembles in some ways the interior of a black hole. In this view the entire Universe is a kind of gigantic hologram and Our perception that it has three (rather than two) spatial dimensions is an illusion rooted in an effective description of the actual Universe that is valid only at low energies. The AdS/CFT correspondence (more generally gauge/gravity duality) described in later chapters is a specific implementation of such ideas.