Collapsing Spheres Satisfying An "Euclidean Condition"
aa r X i v : . [ g r- q c ] A p r COLLAPSING SPHERES SATISFYING AN“EUCLIDEAN CONDITION”
L. Herrera ∗ and N. O. Santos , , † Escuela de F´ısica, Facultad de Ciencias,Universidad Central de Venezuela, Caracas, Venezuela. School of Mathematical Sciences, Queen Mary,University of London, London E1 4NS, UK. Laborat´orio Nacional de Computa¸c˜ao Cient´ıfica, 25651-070 Petr´opolis RJ, Brazil. Universit´e Pierre et Marie Curie, LERMA(ERGA) CNRS - UMR 8112,94200 Ivry, France.
November 8, 2018
Abstract
We study the general properties of fluid spheres satisfying theheuristic assumption that theirs areal and proper radius are equal(the Euclidean condition). Dissipative and non-dissipative models areconsidered. In the latter case, all models are necessarily geodesic anda subclass of the Lemaˆıtre-Tolman-Bondi solution is obtained. In thedissipative case solutions are non-geodesic and are characterized bythe fact that all non-gravitational forces acting on any fluid elementproduces a radial three-acceleration independent on its inertial mass. ∗ e-mail: [email protected] † e-mail: [email protected] and [email protected] Introduction
Analytical or numerical solutions to Einstein equations describing dissipativegravitational collapse are thought to be useful not only for describing specificastrophysical phenomena, but also as test-bed for probing cosmic censorshipand hoop conjecture among other important issues.In this work we consider a large family of solutions, derived from theheuristic assumption that the areal radius of any shell of fluid, which is theradius obtained from its area, equals the proper radial distance from the cen-tre to the shell. Since these two quantities are always equal in the Euclideangeometry, systems described by solutions satisfying such a condition will becalled “Euclidean stars”.Some of the models are necessarily dissipative. This is appealing froma physical point of view, since gravitational collapse is a highly dissipativeprocess (see [1, 2, 3] and references therein). This dissipation is required toaccount for the very large (negative) binding energy of the resulting compactobject of the order of − erg.The resulting dissipative models have a distinct dynamical property, namelyall non-gravitational forces acting on any fluid element, produce a radialthree-acceleration being independent of the inertial mass density of the fluidelement. This behaviour, which is characteristic of the gravitational force, isnow shared, due to the Euclidean condition, by all forces. The specific caseof the shear-free and conformally flat fluid, are considered in detail.Non-dissipative models are necessarily geodesic, belonging to the Lemaˆıtre-Tolman-Bondi (LTB) solutions (more specifically to the parabolic subclass).They may describe collapsing dust or, more generally, anisotropic fluids [4]. We consider a spherically symmetric distribution of collapsing fluid, boundedby a spherical surface Σ. The fluid is assumed to be locally anisotropic withprincipal stresses unequal and undergoing dissipation in the form of heatflow. Choosing comoving coordinates inside Σ, the general interior metriccan be written ds − = − A dt + B dr + R ( dθ + sin θdφ ) , (1)2here A , B and R are functions of t and r and are assumed positive. Wenumber the coordinates x = t , x = r , x = θ and x = φ .The matter energy-momentum T − αβ inside Σ has the form T − αβ = ( µ + P ⊥ ) V α V β + P ⊥ g αβ + ( P r − P ⊥ ) χ α χ β + q α V β + V α q β , (2)where µ is the energy density, P r the radial pressure, P ⊥ the tangentialpressure, q α the heat flux, V α the four-velocity of the fluid and χ α a unitfour-vector along the radial direction. These quantities satisfy V α V α = − , V α q α = 0 , χ α χ α = 1 , χ α V α = 0 . (3)The four-acceleration a α and the expansion Θ of the fluid are given by a α = V α ; β V β , Θ = V α ; α , (4)and its shear σ αβ by σ αβ = V ( α ; β ) + a ( α V β ) −
13 Θ( g αβ + V α V β ) . (5)Since we assumed the metric (1) comoving then V α = A − δ α , q α = qB − δ α , χ α = B − δ α , (6)where q is a function of t and r . From (4) with (6) we have for the four-acceleration and its scalar a , a = A ′ A , a = a α a α = A ′ AB ! , (7)and for the expansion Θ = 1 A ˙ BB + 2 ˙ RR ! , (8)where the prime stands for r differentiation and the dot stands for differen-tiation with respect to t . With (6) we obtain for the shear (5) its non zerocomponents σ = 23 B σ, σ = σ sin θ = − R σ, (9)3nd its scalar σ αβ σ αβ = 23 σ , (10)where σ = 1 A ˙ BB − ˙ RR ! . (11)The mass function m ( t, r ) introduced by Misner and Sharp [5] (see also [6])reads m = R R = R ˙ RA ! − R ′ B ! + 1 , (12)We can define the velocity U of the collapsing fluid as the variation of theareal radius with respect to proper time, i.e. U = D T R < , (13)where D T = (1 /A )( ∂/∂t ) defines the derivative with respect to proper time.Then (12) can be rewritten as E ≡ R ′ B = (cid:18) U − mR (cid:19) / . (14)The proper radial three-acceleration D T U of an infalling particle insideΣ can be calculated to obtain D T U = − (cid:18) mR + 4 πP r R (cid:19) + Ea, (15)feeding back this expression into the radial component of the Bianchi iden-tities produces (see [7] for details)( µ + P r ) D T U = − ( µ + P r ) (cid:18) mR + 4 πP r R (cid:19) − E (cid:20) D R P r + 2( P r − P ⊥ ) 1 R (cid:21) − E (cid:20) D T q + 2 q (cid:18) UR + σ (cid:19)(cid:21) . (16)The physical meaning of different terms in (16) is discussed in detail in [1,7, 8]. We would like just to recall that the first term on the right hand sidedescribes the gravitational force term. As expected from the equivalenceprinciple, its contribution to D T U is independent on the inertial mass density4 + P r . The two last terms describe non-gravitational force terms (i.e. theircombination vanishes in a geodesic motion).Two radii are determined for a collapsing spherical fluid distribution bythe metric (1). The first is determined by R ( t, r ) representing the radius asmeasured by its spherical surface, hence called its areal radius . The secondis obtained out its radial integration R B ( t, r ) dr , hence called proper radius .These two radii in general, in Einstein’s theory, need not to be equal, unlikein Newton’s theory. Here we assume those two radii to be equal. Hence withthis condition we can write, B = R ′ , (17)implying from (14) E = 1 . (18)The field equations with this condition become κµ = 1 A ˙ RR + 2 ˙ R ′ R ′ ! ˙ RR , (19) κqAR ′ = − RR A ′ A , (20) κP r = − A " RR − AA − ˙ RR ! ˙ RR + 2 A ′ A RR ′ , (21) κP ⊥ = − A " ¨ RR + ¨ R ′ R ′ − ˙ AA ˙ RR − ˙ AA − ˙ RR ! ˙ R ′ R ′ + " A ′′ A − R ′′ R ′ − R ′ R ! A ′ A R ′ ; (22)while the mass function (12) now reads, m = R ˙ RA ! . (23)It is clear from (23) that if ˙ R = 0 then m = 0 and spacetime becomesMinkowskian. Therefore all Euclidean stars are necessarily non-static . Fur-thermore, using (13), (23) can be rewritten as mR = U . (24)5ence, (24) can be interpreted as the Newtonian kinetic energy (per unitmass) of the collapsing particles being equal to their Newtonian potentialenergy.From (20), we observe that if the system is dissipating in the form of heatflow, the collapsing source needs A ′ = 0, implying because of (7) a α = 0. Thismeans that dissipation does not allow collapsing particles to follow geodesics.Inversely, of course, non-dissipative Euclidean models are necessarily geodesic ,since q = 0 implies because of (7) and (20) that a α = 0.It is interesting to observe that due to the Euclidean condition, the dy-namical equation (15) or (16) becomes, D T U = − (cid:18) mR + 4 πP r R (cid:19) − κqR U , (25)implying that the non-gravitational force term (the last on the right handside) contributes to D t U , for any fluid element, independently on its in-ertial mass density. In other words, the Euclidean condition produces a“gravitational-like” behaviour in non-gravitational forces (which are con-trolled by q ). Thus, the effect of non-gravitational forces amounts to modifythe gravitational force term, leaving a “gravitational-like” force term pro-ducing a radial three-acceleration independent on the inertial mass densityof the fluid element.The Weyl tensor C αβγδ for metric (1) with (17) has the following non zerocomponents, C = A " ¨ RR − ¨ R ′ R ′ + ˙ AA + ˙ RR ! ˙ R ′ R ′ − ˙ RR ! R ′ A ! + A ′′ A − R ′′ R ′ + R ′ R ! A ′ A ) , (26)and all the other non zero components are proportional to (26), R C = − B C = − (cid:18) B sin θ (cid:19) C = A C = (cid:18) A sin θ (cid:19) C = − (cid:18) ABR sin θ (cid:19) C . (27)6ith (11), (21) and (22) we can rewrite (26) like C = AR ′ " κ ( P ⊥ − P r ) A + 2 ˙ Rr σ , (28)showing that for isotropic systems the shear-free conditions implies a confor-mally flat source.We consider next the non-dissipative case. q = 0 As mentioned before, for this case we have from (20) that A ′ = 0 whichmeans A = A ( t ) and by rescaling t we can have A = 1 . (29)Of course such models are members of the Lemaˆıtre-Tolman-Bondi (LTB)spacetimes [9, 10, 11], furthermore they correspond to the parabolic case.Indeed, the general metric for LTB spacetimes read, ds = − dt + R ′ − K ( r ) dr + R ( dθ + sin θdφ ) , (30)where K ( r ) is an arbitrary function of r .Imposing the Euclidean condition (17) in (30), one obtains K = 0, whichdefines parabolic LTB spacetimes. Further, assuming that the source consistsof pure dust ( P r = P ⊥ = 0) then it follows from the field equations that R ( t, r ) = [ c ( r ) t + c ( r )] / , (31)and κµ = 4 c c ′ c t + c )( c ′ t + c ′ ) , (32)where c ( r ) and c ( r ) are integration functions. Hence the solution reducesto parabolic LTB collapsing dust [9, 10, 11].From (29) and (31) we have for (26), C = (cid:18) (cid:19) c ( c ′ t + c ′ ) ( c t + c ) / σ, (33)7here σ , from (11), is σ = c c ′ − c c ′ ( c t + c )( c ′ t + c ′ ) . (34)In the shear-free case, c = c , the system becomes conformally flat too,and with the freedom for choosing the r coordinate we can assume c = r / recovering the Friedmann critical dust sphere.Of course more general models can be obtained by relaxing the conditionof vanishing pressure, we recall that LTB spacetime is compatible with ananisotropic fluid [12, 13]. q = 0 We consider now the dissipative case. For simplicity we assume the fluid tobe shear-free. In this latter case the line element can be written as [14] ds − = − A dt + B [ dr + r ( dθ + sin θdφ )] , (35)then the Euclidean condition becomes B = ( Br ) ′ → B = f ( t ) (36)implying R = f ( t ) r, (37)where f is an arbitrary function of t .The field equations (19-22) now read, κµ = 3 A ˙ ff ! , (38) κq = − ff A ′ A , (39) κP r = − A " ff − AA − ˙ ff ! ˙ ff + 2 f r A ′ A , (40) κP ⊥ = − A " ff − AA − ˙ ff ! ˙ ff + 1 f A ′′ A + 1 r A ′ A ! . (41)8rom (40) and (41) we have κ ( P ⊥ − P r ) = 1 f A A ′′ − A ′ r ! . (42)From (28) it follows that (in the shear–free case) if the collapsing source isconformally flat, it must be isotropic in its pressures, and vice-versa.The general form of all conformally flat and shear-free metrics is known[15], it reads A = h e ( t ) r + 1 i B, (43)where e is an arbitrary function of t , and B = 1 e ( t ) r + e ( t ) , (44)where e and e are arbitrary functions of t .The Euclidean condition then implies e = 0 , e = 1 f . (45)An approximate solution of this kind has been presented and discussedin [15]. Furthermore, an exact solution is also known [16], which in turn is aparticular case of a family of solutions found in [17]. It reads (see Case IIIin [16]) f ( t ) = ( β + β ) e − αr Σ t (46)and A = ( αr + 1) f, (47)where α , β and β are constants. The above solution satisfies junctionconditions and its physical properties have been discussed in [16]. Thus weshall not elaborate any further on it. Suffice to say at this point that itsphysical properties are reasonable and a thermodynamic analysis brings outthe relevance of relaxational effects on the evolution of the system. Acknowledgments.
L.H. wishes to thank financial support from the FUNDACION EMPRESASPOLAR, the CDCH at Universidad Central de Venezuela under grants PG03-00-6497-2007 and PI 03-00-7096-2008, the Universit´e Pierre et Marie Curie(Paris) and Universitat Illes Balears (Palma de Mallorca).9 eferences [1] L. Herrera and N. O. Santos
Phys. Rev. D , 084004 (2004).[2] L. Herrera, A. Di Prisco, J. Martin, J. Ospino, N. O. Santos and O.Troconis Phys. Rev. D , 084026 (2004).[3] A. Mitra Phys. Rev. D , 024010 (2006).[4] L. Herrera and N. O. Santos Phys. Rep. , 53 (1997).[5] C. Misner and D. Sharp
Phys. Rev. , B571 (1964).[6] Cahill M. and McVittie G.
J. Math. Phys. , 1382 (1970).[7] A. Di Prisco, L. Herrera, G. Le Denmat, M. MacCallum and N.O. San-tos. Phys. Rev. D , 064017 (2007).[8] L. Herrera, A. Di Prisco, E. Fuenmayor and O. Troconis Int. J. Mod.Phys. D , 129 (2009).[9] G. Lemaˆıtre Ann. Soc. Sci. Bruxelles
A 53 , 51 (1933).[10] R. C. Tolman
Proc. Nat. Acad. Sci. , 169 (1934).[11] H. Bondi Mon. Not. R. Astron. Soc. , 410 (1947).[12] J. Gair
Class. Quantum Grav. , 4897 (2001).[13] R. A. Sussman Phys. Rev. D , 025009 (2009).[14] E. N. Glass J. Math. Phys. , 1508 (1979).[15] L. Herrera, G. Le Denmat, N. O. Santos and A. Wang Int. J. Mod. Phys.D , 583 (2004).[16] S. D. Maharaj and M. Govender Int. J. Mod. Phys. D , 667 (2005).[17] L. Herrera, A. Di Prisco and J. Ospino Phys. Rev. D74