# Presentation: 802495

## Proc. Natl. Acad. Sci., 101,  9545-9550 (2004)

## Endpoints of Gravitational Collapse

-   White Dwarves: Quantum Electron Degeneracy
-   Neutron Stars: Quantum Neutron Degeneracy
-   Black Holes: Ultimate Collapse to Spacetime Singularity
- Black Holes are the expected result of
*classical* general relativity when no known pressure
- or matter eq. of state can counteract gravity
- Seem to be ‘everywhere’ - in galactic nuclei and
- as massive stars which have exhausted their nuclear fuel.

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## Black Holes and Entropy 

- A single classical solution usually has ***no ******	******	******entropy***: What is the “entropy” of the 		Coulomb potential *** = Q/r* ?
- But any object or “information” falling into a 	black hole *disappears *(irreversibly?)
- Black holes are (externally) characterized by 	only a few *conserved* parameters *(M, L, Q)*
*So? Both of these are also true in EM if objects *
*	**	**“**fall into” the Coulomb singularity at r=0*
- Yet if entropy of matter/radiation falling into a 	black hole disappears, so S < 0, doesn’t
- 		this ***violate the second law ??***

## Generalized Second Law ?

- Bekenstein suggested dividing *A* by
- 				***L****Pl****2****= G**~*/*c**3*
*	**	*and *S**BH** = **** k**B **A/*l*Pl**2   *with **
*If* the total entropy* S = S**thermo **+ S**BH*
- he showed ***S > 0* always (***GSL***)
- Then Hawking found black holes could emit 	***thermal*** radiation at
*	**	**	**T**H **= **~ **c**3**/(8 ****G k**B **M)*
- Apparently then *dE = T dS* fixes  = 1/4
*Great! But ...*

## A few questions remain unaswered

*E **** T**-1*  implies _*negative*_ heat capacity
- 			_*dE*_* ** <  0**	**(unstable)*
- 			*dT*
- Can one apply equilibrium thermodynamics to
- 		 an unstable system?
- Classical limit* **~ **** **0, T**H **** **0  **but**  S**BH **** **(?!)*
*S **** A* is *non-extensive **and** **HUGE! *(GSL is *weak*)
- Can pure states evolve into mixed states?
- What about* S = k**B** ln W  ? *(Boltzmann asks)
- What is the *statistical* interpretation of *S**BH* ?

## Statistical Entropy of a Relativistic Star

*S = k**B** ln W(E)* (microcanonical) equivalent
- 			to      	*S = - k**B** Tr (**** ln ****)*
- Maximized by canonical thermal distribution
*Eg.** **Black body radiation** E ~ V T**4  **, S ~ V T**3   **so*
*	**	**	**    **S ~ V**1/4  **E**3/4 **~ R**3/4 **E**3/4*
- For a fully collapsed star* **E ~ M**, and **R **·** (few) M**, so*
*	**	**	**    ****S ******·  ******(few) k******B****** (M/M******Pl******)******3/2***
*S**BH *is a factor* ****(M/M******Pl******)******1/2******  ***_*larger*_* or ****10******19***  for *M = M** ̄*
- There is no way to get *S**BH* by any ``standard” statistical 	thermodynamic counting of states

## Quantum Effects in Gravitational Collapse

**Quantum Effects in Gravitational Collapse**

**Classical**** GR + M + Matter EOS**
**	****	******** ****+ 3p > 0****     ********    black holes**
**Trapped Surface ******** ****Singularity**
**Hawking Effect  ******** ****Negative**** ****Heat Capacity**
**When**** ****~**** ******** 0****, ****Quantum**** ****Information Paradox**
**	****     ****	****	**** ****S****BH ****= 4********k****B ****GM****2****/****}****c****       ****(****HUGE!****)**
**But Quantum Fluctuations**** ****Large**** ****near****  ****R****S****=2GM/c****2**
**A Phase Transition near**** ****R****S ******** Vacuum Energy Changes**
**Dark Energy****  Repulsive Core ******** = - p > 0 **** ********  ******** + 3p < 0****   **
**No trapped surface, **_**No singularity**_
**A GBEC phase transition stabilizes a high density, compact  **
**cold stellar remnant to further gravitational collapse**

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## Gra(vitational) Va(cuum) Condensate StarsGravastars as Astrophysical Objects

- Cold, Dark, Compact, Arbitrary *M*
- Accrete Matter like a Black Hole but Matter
- 		does not disappear down a “Hole”
- May be re-emitted by ultra-relativistic	shell
- More Efficient Central Engine for  Ray Bursters, 		High-Energy Jets, AGN’s, Cosmic Rays?
- Support Angular Momentum, Magnetic Fields
- Gravitational Wave Signatures (Normal Modes)
- 	     Quantum Non-singular Alternative to 			  Classical Black Holes for
- 		Final State of Gravitational Collapse

## Gravitational Vacuum Condensates

**Gravity is a theory of spin-2 ****bosons**
**Interactions are ****attractive**** and become ****strong**** near ****r = R****S**
**Any scalar ****condensate**** order parameter ******** has EOS                                                                                                          ****	****	****      ****p*****v***** = -*********v***** ****= -V(********)**
**Relativistic Entropy Density ****s**** is (for μ = 0)****	****	****	****   ****	****     ****	****      ****	****     ****Ts = p + ******** = 0**** **
**Zero Entropy for a ****single macroscopic quantum state****:  **
**	****	****	****     ****S = k****B ****1n ****W**** = 0**** **
**p =-******** ****violates**** the energy condition, p + 3p ≥ 0 (if ********> 0) needed to ****	****	****prove the classical singularity theorems**
**Dark Energy ******** > 0**** acts as a ****repulsive**** core ****	**

**A GBEC phase transition stabilizes a high density, compact  **
**cold stellar remnant to further gravitational collapse**

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