Role of equation of states and thermodynamic potentials in avoidance of trapped surfaces in gravitational collapse
RRole of equation of states and thermodynamic potentials in avoidance of trappedsurfaces in gravitational collapse
Rituparno Goswami ∗ and Terricia Govender † Astrophysics and Cosmology Research Unit, School of Mathematics,Statistics and Computer Science, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, South Africa. National Astrophysics and Space Science Program,School of Physics and Chemistry, University of KwaZulu-Natal,Private Bag X54001, Durban 4000, South Africa
In this paper we consider the novel scenario where a spherically symmetric perfect fluid star is un-dergoing continual gravitational collapse while continuously radiating energy in an exterior radiatingspacetime. There are no trapped surfaces and the collapse ends to a flat spacetime. Also the collaps-ing matter obeys the weak and dominant energy conditions at all epoch. Our analysis transparentlybrings out the role of the equation of state as well as the bounds on the thermodynamic potentialsto realise such a scenario. We argue that, since the system of Einstein field equations allows for sucha scenario for an open set of initial data as well as the equation of state function in their respectivefunctional spaces, these models are generic and devoid of the problems and paradoxes related tohorizons and singularities. The recent high resolution radio telescopes should in principle detect thepresence of these compact objects in the sky.
PACS numbers: 04.20.Cv, 04.20.Dw
I. INTRODUCTION
When a continually collapsing star crosses it’s ownSchwarzschild radius, it gets trapped. For all the col-lapsing shells, ingoing as well as outgoing null wavefrontsnormal to these shells converge and hence the mattermust collapse to a central singularity. Existence ofthese closed trapped 2-surfaces (the collapsing shellsafter crossing the Schwarzschild radius) is key to all thesingularity theorems developed by Hawking, Penroseand Geroch [1]. The process of formation of trappedsurfaces, trapped regions and the boundary of thesetrapped regions, is also central to the black hole physics.In the context of general relativity, these were firsthighlighted by Oppenheimer, Snyder and Datt (OSD)[2], for collapse of a pressureless homogeneous dust.It was shown that the entire star gets trapped muchbefore the formation of the central singularity andhence the central singularity can never be seen by faraway observers. Although the OSD model is extremelyidealised, Penrose argued that any realistic gravitationalcollapse should be qualitatively similar to this modelwhich became his famous cosmic censorship conjecture[3].This censorship conjecture has no formal mathemati-cal proof/disproof till now. However there are numerouscounterexamples for which the idealised picture of theOSD collapse does not hold. There are possibilitiesof naked singularities that can be visible to faraway ∗ Electronic address: [email protected] † Electronic address: [email protected] observers, before being trapped ( see for example [4]and the references therein). In all these examplesthe formation of trapped surfaces are delayed by thepresence of matter shear or Weyl curvature [5], so thatpart of the singularity becomes naked. But still, in allthese counterexamples the trapped surfaces are presentin the spacetime.Another novel picture of gravitational collapse of amassive star was first considered by [6], where a collaps-ing star is continuously radiating and losing it’s mass,and therefore the surface of the star never crosses theSchwarzschild radius. There is no trapped region in thespacetime, and when the shells reach the central point,all the matter is radiated away, leaving a flat spacetime.The exact mechanisms by which the collapsing star cantransfer radiation/matter to the external spacetime wasstudied by numerous authors thereafter. One can matchthe collapsing star with a non-comoving (evaporating)boundary to a purely radiating Vaidya exterior [7–12].The matching can also be done at a comoving stellarboundary to a generalised Vaidya exterior (see [13] andthe references therein). The matter form for all themodels considered in this scenario are very specific,like a specific kind of self interacting scalar field [14],or matter fields that have a specific form of negativepressures in the later stages of collapse.In this paper, we carefully investigate the role of theequation of state for the isentropic perfect fluid stellarmatter, that can give rise to a collapse without anytrapping together with the weak and dominant energyconditions being satisfied. Our analysis brings outtransparently the classes of equation of state functionsas well as the bounds on the thermodynamic potentials a r X i v : . [ g r- q c ] M a r to realise such a scenario. We explicitly relate ourresults to the enthalpy and acceleration potentials ofthe collapsing matter. We show that these classes havenon-zero measure in the function space. Therefore,these models are generic in nature and devoid of theproblems or paradoxes related to horizons and singular-ities. The recent high resolution radio telescopes (likeEvent Horizon Telescope) should in principle detect thepresence of these compact objects in the sky as theobservational signatures of these continually collapsingbut non-trapped compact objects will definitely bedifferent from those of a black hole.The paper is organised as follows: In the next sectionwe describe the field equations along with the regularityand energy conditions for the collapsing matter. Section3 presents the general solution of the collapsing perfectfluid in terms of the thermodynamic quantities. Section4 establishes the conditions for no-trapping, with somespecific examples. In section 5 we transparently relateour results to the thermodynamic potentials of thecollapsing matter. Finally in the last section we givesome concluding remarks about the final fate of themodels we have considered here. II. THE SPHERICAL COLLAPSING STAR:FIELD EQUATIONS AND REGULARITYCONDITIONS
We present a general line element for a spherically sym-metric distribution of a collapsing perfect fluid in comov-ing spherical coordinates ( x i = t, r, θ, φ ) as ds = − e ν ( t,r ) dt + e ψ ( t,r ) dr + R ( t, r ) d Ω . (1)For a perfect fluid, the energy-momentum tensor is givenby T tt = − ρ ( t, r ) , T rr = T θθ = T φφ = p ( t, r ) . (2)The quantities ρ and p represents the energy densityand pressure respectively for the matter cloud. The weakenergy conditions are then considered to have been metby the matter fields. Thus, implicating that the en-ergy density (as measured by any local observer) is non-negative and hence for any time-like vector V i we musthave T ik V i V k ≥ , (3)which simplifies to ρ ≥ ρ + p ≥ r ,at a given time t ) as, F = 1 − R ,a R ,a = 1 − G + H, (4) where G ( t, r ) = e − ψ ( R (cid:48) ) , (5) H ( t, r ) = e − ν (cid:16) ˙ R (cid:17) . (6)In terms of this mass function the Einstein field equationsare given as, F (cid:48) = ρR R (cid:48) , (7)˙ F = − pR ˙ R, (8) p (cid:48) = − ν (cid:48) ( ρ + p ) , (9) R (cid:48) ˙ G = 2 ˙ Rν (cid:48) G, (10)Note that (˙) and ( (cid:48) ) depicts partial derivatives withrespect to t and r respectively. We impose the condition F ( t i ,
0) = 0 to preserve regularity at the initial epoch.Since we want to investigate the collapsing class of so-lutions to the Einstein equations, we must impose thecondition ˙
R < R . This impositionresults in the area radius of all shells of the continual col-lapsing cloud to monotonically diminish to zero (form-ing the spacetime singularity), R ( t s ( r ) , r ) = 0 and wheretime t = t s ( r ) is the time taken for a shell labelled r to reach the singularity. At the time t = t i , the radius R = r , due to the use of the scaling freedom for the ra-dial coordinate r . We proceed with an arbitrary scalingfunction, a ( t, r ) ≡ Rr , (11)resulting in, R ( t, r ) = ra ( t, r ) , (12) a ( t i , r ) = 1 , (13) a ( t s ( r ) , r ) = 0 , (14)where ˙ a <
0. The regularity conditions suggests that F ≈ r close to the center. The normal structure of F isthus, F ( t, r ) = r M ( r, a ) , (15)where we have M ( r, a ) to be some general function asrestricted by the regularity conditions and energy condi-tions. III. THE METRIC AS A FUNCTION OFTHERMODYNAMIC QUANTITIES
To obtain the general solution of the metric functionsin terms of the thermodynamic quantities at any epoch,we perform a change of variables from ( t, r ) to ( a, r ). Inthat case for any function Φ( t, r ), we must have˙Φ = Φ ,a ˙ a, (16)Φ (cid:48) = Φ ,r + Φ ,a a ,r . (17)Also for integrating the G = T equation (10), we define, ν (cid:48) R (cid:48) ≡ A ( r, a ) ,a , (18)where A ( r, a ) is an arbitrary function of the coordinates r and a . Now we can directly integrate equation (10) toobtain G ( r, a ) = [1 + r b ( r )] e rA , (19)where b ( r ) is a free function of integration. Thereforefrom (6) we have, e ψ = ( R (cid:48) ) [1 + r b ( r )] e rA . (20)Now we can rewrite the definition of the Misner-Sharpmass in terms of these new variables in the following way: √ a ˙ a = − e ν (cid:113) e rA ab ( r ) + ah ( r, a ) + M ( r, a ) , (21)where we have defined h ( r, a ) = e rA − r . (22)Integrating (21) we obtain the equation for the time takenfor a shell labelled ‘ r ’, to reach the epoch ‘ a ’ as t ( a, r ) = (cid:90) a √ adae ν (cid:112) e rA ab + ah + M . (23)The above is the solution for the scaling function a ( t, r ) inthe integral form. This immediately gives the singularitycurve t s ( r ), which is the collapse end time where the shelllabelled ‘ r ’ diminishes to zero area radius ( R = 0), t s ( r ) = (cid:90) √ adae ν (cid:112) e rA ab + ah + M . (24)We now have to relate the function A ( r, a ) to the ther-modynamic variables of the collapsing matter. To do so,we first write the G = T and G = T equations interms of the new variables ρ = 3 M + r [ M ,r + M ,a a (cid:48) ] a ( a + ra (cid:48) ) , (25) p = − M ,a a . (26)Substituting M ,a into the density equation (25) and re-arranging, we find, a (cid:48) = 3 M + rM ,r − ρa ra ( ρ + p ) . (27)To solve A we first find a (cid:48) from the definition (18) and(10) to obtain, A ,a R (cid:48) = − ( p ,r + p ,a a (cid:48) ) p + ρ , (28) a (cid:48) = − p ,r − ( p + ρ ) A ,a ar ( p + ρ ) A ,a + p ,a . (29) We equate (27) and (29) to solve for A ,a , A ,a = − p ,r [ ra ( ρ + p )] − [3 M + rM ,r − ρa ] p ,a [3 M + rM ,r − ρa ][ r ( ρ + p )] + [ ra ( ρ + p ) ] , (30)where p ,r = c s ρ ,r and p ,a = c s ρ ,a , where c s is the speedof sound. Therefore we can write the function A ( r, a ) as A = (cid:90) a − p ,r [ ra ( ρ + p )] − [3 M + rM ,r − ρa ] p ,a [3 M + rM ,r − ρa ][ r ( ρ + p )] + [ ra ( ρ + p ) ] da. (31)Now we can immediately write the metric function e ν as e ν = exp (cid:26)(cid:90) r XYZ (cid:27) dr (32)where X = − p ,r [ ra ( ρ + p )] − [3 M + rM ,r − ρa ] p ,a , Y = 3 M + rM ,r + a p and finally the last variable is Z = [ ra ( ρ + p ) ][3 M + rM ,r − ρa + a ( ρ + p )]. We notethat the solution of M depends on the equation of state p = p ( ρ ). Once the equation of state is supplied, then allthe metric functions can be written explicitly in terms ofthe thermodynamic quantities of the collapsing matterfield. IV. CONDITIONS FOR NO-TRAPPING
It is well known that for any spherically symmetricspacetimes, a shell labelled ‘ r ’ is trapped, if the MisnerSharp mass enclosed by the shell is greater than the arearadius of the shell. Therefore the spherical 2-surface la-belled by the co-ordinate ‘ r ’ is trapped if F > R whereaswhen
F < R the surface is not trapped. It is obviousthen, that the boundary of the trapped region or the ap-parent horizon is described by the equation F = R . (33)For a continual collapsing matter cloud, regularity con-ditions imply the avoidance of trapped surfaces at theinitial epoch. When the boundary of this cloud is r = r b and we enforce the condition M ( r b ) <
1. Then theavoidance of trapped surfaces for any shell r ≤ r b occursat t = t i . If we want to avoid trapping in the completespacetime we must ensure that througout the spacetime F < R. (34)This obviously implies that G − H > . (35)Using the definition for the function H , we then obtain G − e ( − ν ) ( ˙ R ) > , (36) (cid:16) √ G − e ( − ν ) ˙ R (cid:17) (cid:16) √ G + e ( − ν ) ˙ R (cid:17) > . (37)Since ˙ R < − e ( − ν ) ˙ R < √ G. (38)We can now summarise the conditions for no-trapping inthe following way: Proposition 1.
For continually collapsing sphericallysymmetric perfect fluid with energy density ρ ( r, a ) andpressure p ( r, a ) with r ∈ [0 , r b ] and a ∈ [0 , , if the fol-lowing conditions are satisfied:1. ρ > and ρ + p ≥ ∀ r ∈ [0 , r b ] and ∀ a ∈ [0 , ,2. p ( r, > for all r ∈ [0 , r b ] ,3. r (cid:112) e rA b + ah + M < (cid:112) a (1 + r b ) e rA , ∀ r ∈ [0 , r b ] and ∀ a ∈ [0 , , where the function A ( r, a ) isgiven by equation (31),then the collapsing spacetime will be devoid of any trap-pings in spite of the weak energy condition being satisfiedby the collapsing matter. The second condition above, is to ensure that the col-lapse commences with positive pressure. However in theprocess of collapse the pressure can be negative withoutviolating the energy conditions. The above conditions getremarkably simple in the case of homogeneous collapse,given by the FLRW metric ds = − dt + a ( t ) (cid:0) dr + r d Ω (cid:1) . (39)In this case the equation G = T can be directly inte-grated to give M = 13 ρa . (40)Integrating the energy conservation equation˙ ρ = − aa ( ρ + p ( ρ )) , (41)gives, a = exp (cid:20) − (cid:90) ρρ dρρ + p (cid:21) . (42)The condition for no trapping is now given by (as F is amonotonic function of r at any given epoch) r b Mr b a < . (43)Using equations (40, 42) the condition becomes ρ < r b exp (cid:20) (cid:90) ρρ dρρ + p (cid:21) . (44)To show that the set of equations of state which ensuressuch behaviour is indeed non-empty and open set, weexplore the equation of state p ( ρ ) = p + p ρ, (45) where p and p are constants. Then we can directlyintegrate equation (44) to get the condition1 < r b ( p + ρ + p ρ ) (cid:16) p (cid:17) ρ . (46)We immediately see that for the above condition to besatisfied we must have p <
0, and we can always choose p and p in such a way that at the initial epoch pres-sure is positive but at later stages it becomes negative,although the energy conditions are satisfied. V. RELATING TO THE THERMODYNAMICPOTENTIALS
As defined in [15], for reversible flows of isentropic per-fect fluid with the barotropic equation of state p = p ( ρ ),the enthalpy W and acceleration potential A , can be de-fined in the following way: W = exp (cid:26)(cid:90) ρρ dρ ρ + p ) (cid:27) , (47) A = exp (cid:26)(cid:90) pp dp ( ρ + p ) (cid:27) . (48)These potentials relate directly to the energy and mo-mentum conservation equations respectively. In fact,from equation (10) we can immediately see that at a givenepoch t = t , we can write A = e − ν . (49)This proves that for homogeneous spacetimes A = 1 andthe dynamics of the spacetime is governed only by thematter enthalpy. In terms of the matter enthalpy thecondition for no-trapping for homogeneous matter canbe stated in a very compact way: Proposition 2.
For a collapsing spherically symmetrichomogeneous perfect fluid cloud, if the matter energy den-sity is strictly bounded by the matter enthalpy by the fol-lowing relation ρ < r b W , (50) then there will not be any trapped surfaces in the space-time. For inhomogeneous spacetimes the relation is morecomplicated. However, at any given epoch, we can re-late the function A to the acceleration potential as[ln A ] (cid:48) = − A a (cid:18) a + 3 M + rM ,r − ρa a ( ρ + p ) (cid:19) . (51)Also, for inhomogeneous spacetimes the matter enthalpyis written in terms of the metric functions as W = exp (cid:26) − (cid:90) tt e − ν (cid:18) ˙ ψ + 2 ˙ aa (cid:19) dt (cid:27) . (52)Using the above two equations we can implicitly relatethese potentials to Proposition 1, to see the role of thesepotentials in avoidance of trapped surfaces. VI. DISCUSSIONS: THE FINAL FATE OF THECOLLAPSE
In the previous two sections we gave the conditions onthe equations of state, thermodynamic quantities and po-tentials that ensures no trapped surfaces in the collapsingperfect fluid spacetime. The next obvious question wouldbe, what will be the final outcome of such a collapse?This question is important, because if the final fate is astrong curvature naked singularity (the singularity hasto be naked in the absence of trapping), then these mod-els would definitely be unphysical. The key property ofthese models are: the collapsing matter cloud continu-ously throws out radiations and matter in an externalradiating spacetime, such that the cloud never crossesit’s own Schwarzschild radius. Thus, to determine thefinal outcome we must take into account the external ra-diating spacetime. One of the most common spacetimesthat can be matched with the interior across a comovingboundary r = r b , is the generalised Vaidya spacetime, ds = − (cid:18) − M ( v, r v ) r v (cid:19) dv − dvdr v + r v d Ω . (53)This spacetime describes a combination of Type I and
Type II matter fields and therefore is ideally suited forour model. Matching the first and second fundamentalforms across a co-moving matching surface gives (cid:20) FR (cid:21) int = (cid:20) M ( v, r v ) r v (cid:21) ext . (54)Now as by construction the LHS of the above equationis restricted to be less than unity, so will the RHS. There-fore for any observer at the exterior spacetime the limit ofthe generalised Vaidya mass M ( v, r v ), to the generalisedVaidya radius r v (at the central singularity r v = 0), mustbe a non-negative number less than unity. This will thengive rise to two possible end states: 1. If the limit is non-zero, there will be a naked conicalsingularity at the centre. These are weak curvaturesingularities and can be resolved by extending thespacetime through them.2. If the limit goes to zero, then there is no singularityand the collapse ends to a flat spacetime.Thus in both cases the collapse ends in a flat spacetime.Fig. 1 describes the schematic diagram of the completespacetime considered here. An important point to notehere is that there exist open sets of equation of statefunctions in the functional space and also open sets ofinitial data, for which these models are possible. There-fore the classes of these models have non zero measure FIG. 1: A schematic diagram of the complete spacetime. in the respective function spaces. Therefore, these mod-els are generic in nature and devoid of the problems andparadoxes related to horizons and singularities. Hence, inprinciple, these can describe certain kinds of astrophys-ical collapsing objects, which can be detected by recenthigh resolution telescopes. [1] S. W. Hawking and G. F. R. Ellis,