The quantum nature of black holes Lecture III Samir D. Mathur The Ohio State University
The Hawking process entangled Outer particle escapes as Hawking radiation Inner particle has negative energy, reduces mass of hole! 1 p 2 (00 0 + 11 0 )
Fuzzballs
The simplest solution would seem to be that there is no horizon.. Then the black hole would be like a star But this is not easy: matter near the horizon is pulled in very strongly, and falls in: Black holes have no hair
The Black Hole
Buchdahl s theorem: If we have a perfect fluid ball with radius R < 9 4 M, then the pressure at the origin will diverge (so the ball will have to collapse to a black hole)
The no-hair theorem does not just apply to the classical solution, but also to the quantum state around the horizon =0 = f k â k + f k â k The states are given by f =0 f = f(r)y lm (, )e i!t â k 1...â k 1 0i But for the black hole, there are no normalizable solutions to this equation
Fuzzballs: Consider a black hole in string theory (e.g. The D1D5P black hole) List out all the states of this hole in the microscopic picture (excitations on an effective string) Starting with simple states, construct the string theory solution for the corresponding microstate in the gravity description Instead of getting a black hole, we get a solution with no horizon: fuzzball The fuzzball radiates from its surface just like a piece of coal, so there is no information paradox (SDM 97, SDM+Lunin 01, Bena+warner 2004 )
The fuzzball is expected to have a size 2M + How does this solution escape the theorem Black holes have no hair? Why does it not just collapse to a black hole by Buchdahl s theorem? Let us take some simple gravity solutions that extract the essential property of fuzzballs relevant to this issue (SDM 16)
Toy example: Euclidean Schwarzschild plus time ( neutral fuzzball ) ds 2 = dt 2 +(1 r 0 r )d 2 + 1 dr2 r 0 + r 2 (d 2 +sin 2 d 2 ) r 0 apple < 4 r 0 We can reduce on the direction to again get scalar field in 3+1 gravity. Why does this shell of scalar field not collapse inwards?
Dimensional reduction g = e 2 p 3 = p 3 2 ln(1 r 0 r ) The 3+1 metric is defined as g E µ = e 1 p 3 g µ ds 2 E = (1 r 0 r ) 1 2 dt 2 + (1 dr 2 r 0 r ) 1 2 + r 2 (1 r 0 r ) 1 2 (d 2 +sin 2 d 2 ) The action is S = 1 16 G Z d 4 x p g R E 1 2,µ,µ
The stress tensor is the standard one for a scalar field T µ =,µ, 1 2 ge µ,, which turns out to be T µ = diag{,p r,p,p } = diag{ f,f, f, f} f = 8r 4 (1 3r 2 0 r 0 r ) 3 2 We see that the energy density and radial pressure are positive. The tangential pressures are negative All these quantities diverge as we reach the tip of the cigar. g tt never changes sign, so there is no horizon
So what happened to Buchdahl s theorem? Because the radial pressure diverged, Buchdahl would have discarded this solution as unphysical. But we see that the problem is with the dimensional reduction: the full spacetime is completely smooth
Fuzzballs What happened to Black holes have no hair? Gibbons-Warner 2013: Look over the no-hair theorems which were proved The case of extra dimensions with nontrivial topology was not covered
Main point of fuzzball construction: In string theory we have found states that are horizon sized but do not collapse under their selfgravitation. If we just had canonically quantized 3+1 gravity, for example, there would be no such states h0 i h0 i 1
How does the classical expectation get violated so dramatically?
The fuzzball construction says spacetime ends at the horizon But if a star collapses, then the physics looks quite classical, and so one seems to make the usual black hole with a smooth horizon...
Shell collapses to make a black hole...??
Consider the amplitude for the shell to tunnel to a fuzzball state Amplitude to tunnel is very small But the number of states that one can tunnel to is very large!
Path integral Z = D[g]e 1 S[g] Measure has degeneracy of states Action determines classical trajectory For traditional macroscopic objects the measure is order action is order unity while the But for black holes the entropy is so large that the two are comparable Thus the black hole is not a semiclassical object (SDM 08, 09, Bena, Puhm.. 16)
Summary
1. Is there a mechanism to break the no hair theorem? 2. If not, then what happens to the entanglement of the produced pair? h0 i h0 i 1
Fuzzball complementarity (Gravity-Gravity duality)
(b) The conjecture of fuzzball complementarity: When the collapsing shell tunnels into fuzzballs, then its state keeps evolving in this large space of fuzzball states This evolution can be mapped (approximately) to infall in the traditional black hole geometry
Thus when the collapsing shell reaches its horizon radius, then it does not hit a firewall Instead, one must study its evolution in superspace (the space of all fuzzball configurations) This representation gives a dual description of the physics ( gravity-gravity duality (Plumberg+SDM 11)) This dual description mimics (to a good approximation when E>>T) the picture of infall into the traditional black hole