Waves of space-time from a collapsing compact object
Jaime Mendoza Hernández, Juan Ignacio Musmarra, Mauricio Bellini
aa r X i v : . [ g r- q c ] D ec Waves of space-time from a collapsing compact object Jaime Mendoza Hern´andez ∗ , Juan Ignacio Musmarra † , , Mauricio Bellini. ‡ Departamento de F´ısica, Centro Universitario de Ciencias Exactas e Ingenier´ıas,Universidad de Guadalajara Av. Revoluci´on 1500,Colonia Ol´ımpica C.P. 44430, Guadalajara, Jalisco, M´exico. Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata, Argentina. Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR),Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Mar del Plata, Argentina.
We study the partial time dependent collapse of a spherically symmetric compact object withinitial mass M + M and final mass M and the waves of space-time emitted during the collapsevia back-reaction effects. We obtain exact analytical solutions for the waves of space-time in anexample in which M = M = ( M + M ) /
2. The wavelengths of the space-time emitted wavesduring the collapse have the cut (we use natural units c = ~ = 1): λ < (2 /b ), (1 /b )-being the timescale that describes the decay of the compact object. I. INTRODUCTION AND MOTIVATION
In geometrodynamics[1, 2] particles and fields are not considered as foreign entities that are immersed in geometry,but are regarded as manifestations of geometry, property. Under certain conditions boundary conditions must beconsidered in the variation of the action[3]. In the event that a manifold has a boundary ∂ M , the action should besupplemented by a boundary term, so that the variational of the action to be well-defined[4]. However, this is not theonly manner to study this problem. As was recently demonstrated[5], there is another way to include the flux thatcross a 3D-hypersurface that encloses a physical source without the inclusion of another term in the Einstein-Hilbert(EH) action I EH = Z d x √− g (cid:18) R κ + L m (cid:19) . (1)This is an important fact, because the study of different sources can help to develop a more general and powerfulrelativistic formalism, where dynamical systems can be studied in a theoretical manner, in order to make a betterdescription of the physical reality. In this work we shall deal only with classical systems, in order to describe wavesof space-time. These terms can be seen as a source, described by an extended manifold (with respect to the Riemannbackground), which has geometrodynamical physical consequences on a gravito-electromagnetic physical system[6].On the another hand, the understanding of a collapsing system is a very important issue in theoretical physics,mainly in astrophysics and cosmology physics. Some extensive investigations were made related to a sphericallysymmetric collapse driven by a scalar field [7]. More recently, this issue has been analytically explored in [8, 9]. Thestudy of a collapse for a fluid with an heat flux has been treated in [10]. An interesting issue to study is the evolutionof the global topology of space time during a collapse driven by a scalar field that can avoid the final singularity. Thistopic was explored in a recent work [11] jointly with the geometrical back-reaction of space time produced by thiscollapse. However, in this work we shall consider a semi-Riemannian manifold, so that ∇ ǫ g αβ = 0.In this work we are interested in the study of a partial collapse of some compact object with initial mass M + M ,that is reduced during this collapse to a final state with mass M , such that during the transition it transfers a mass M e − b t to the space-time, and therefore the collapse generates a spherically symmetric wave of space-time. Thepaper is organised as follow: in Sect. (II) we introduce the physical source for a object that transfers mass to a waveof space-time and we obtain, by varying the action, the new Einstein’s equations with sources and the equation forthe trace of the space-time wave . In Sect. (III) we describe the problem which we are interested in solving, thepartial collapse of a spherically symmetric compact object that transfers a part of its mass to the space-time wave.In Sect. (IV) we propose a particular dynamical evolution for the mass that decreases exponentially with time. Weobtain the wave equation for the space-time and we found a exact analytical solution for this equation in a particularcase. Finally, in Sect. (V) we develop some final comments. ∗ E-mail: [email protected] † [email protected] ‡ Corresponding author : [email protected]
II. PHYSICAL SOURCES FROM BOUNDARY CONDITIONS IN THE VARIATION OF THE ACTION
We consider the variation of the Einstein-Hilbert (EH) action with respect to the metric tensor δI EH = 12 κ Z d x √− g h δg αβ (cid:16) R αβ − g αβ R + κ T αβ (cid:17) + g αβ δR αβ i = 0 , (2)where the stress tensor T αβ is defined in terms of the variation of the Lagrangian T αβ = 2 δ L m δg αβ − g αβ L m , (3)and describes the physical matter fields. By considering a flux δ Φ of δW α through the 3D closed hypersurface ∂M , weobtain that: g αβ δR αβ = ∇ α δW α = δ Φ, with δW α = δ Γ αβγ g βγ − δ Γ ǫβǫ g βα = g βγ ∇ α δ Ψ βγ [14]. To formalise the studyof the system we shall consider boundary conditions that can describe waves of space-time with external sources. Inour case the waves of space-time propagates due to the partial collapse of the spherically compact object, which isthe source that produce the waves. In order for describe a dynamical system with a source that generates waves ofspace-time, we shall consider the case where δI EH δS = 0 implies that δg αβ δS [ G αβ + κ T αβ ] + g αβ δR αβ δS = 0 → δg αβ δS [ G αβ + κ T αβ ] | {z } =Λ g αβ + (cid:3) χ = 0 , (4)that guarantees the preservation of the action, and the recovering of the Einstein’s equations with sources. Further-more, χ ( x ǫ ) ≡ g µν χ µν is a classical scalar field, and χ µν = δ Ψ µν δS describes the waves of space-time produced by thesource through the 3D hypersurface. Therefore, the condition δI EH δS = 0, with boundary conditions included, in thecase described by (4), can be written as G αβ + κ T αβ = Λ ¯ g αβ , (5) (cid:3) χ = δ Φ δS = Λ g αβ δg αβ δS , (6)such that g αβ δg αβ δS = [1 / ( − g )] d ( − g ) dS = ddS [ln( − g )], and ( − g ) is the positive value of the determinant for the metricthat describes the source: g µν . Therefore, the variation of the flux that cross the 3D-gaussian hypersurface withrespect to the line element S , is described by δ Φ δS = Λ g αβ δg αβ δS = Λ ddS [ln( − g )] , (7)that is the source of the wave equation (6). Here, the cosmological constant Λ is related with the flux Φ and thevolume of the manifold: √− g : Λ = 12 d Φ d [ln [ √− g ]] , (8)where we have made use of the fact that g αβ δg αβ δS = ddS [ln( − g )] = 2 ddS [ln( √− g )]. When δ Φ δS is linear with ddS [ln( − g )],hence Λ is a constant. This is the case we shall consider in this work. III. PARTIAL COLLAPSE OF A COMPACT OBJECT
In order to consider the dynamical collapse of a spherically symmetric compact object where the mass evolves inagreement with (12), we must consider the initial Schwarzschild metric of the object with mass M + M dS = ¯ g αβ dx α dx β = f ( r ) dt − f ( r ) dr − r d Ω , (9)where f ( r ) = 1 − G ( M + M ) /r and d Ω = dθ + sin ( θ ) dφ . During the collapse the mass of the compact objectis dynamically reduced from ( M + M ) to M , so that the difference of mass is transferred to the wave of space-time. The transition between both states is described by the function M ( t ), such that M ( t = 0) = M + M andlim t →∞ M ( t ) → M . Furthermore, f ( r ) → f ( r, t ) and the metric during the transition is: dS = g αβ dx α dx β = f ( r, t ) dt − a ( t ) (cid:20) dr f ( r, t ) − r d Ω (cid:21) , (10)with f ( r, t ) = 1 − GM ( t ) /r and a ( t ) ≤ a is the radius of the compact object, which will be considered as proportionalto the time dependent mass: a ( t ) = C G M ( t ). Here, C = a G ( M + M ) > a is the initial radiusof the compact object. Notice that (10) is the metric that describes the physical source of the wave of space-time and(9) is the metric that characterizes the initial state on which will propagates the wave produced by the source. Atthe end of the collapse the remaining mass is M = lim t →∞ M ( t ), and the residual metric will be the same than (9),but with a mass M . It is expected that, during the collapse, where the compact object losses a mass ∆ M = M . Inabsence of dissipative effects, it is expected that the lost energy by the star ∆ E = M c , be transferred to the waveof space-time, that will spread at the speed of light as a spherical wave.The equation of state : ω ( t, r ) = p ( t, r ) /ρ ( t, r ) (here p and ρ are respectively pressure and energy density), for thisspherically symmetric compact object that has a radius a ( t ) = C M ( t ), that decreases with time, is ω ( t, r ) = Λ M (cid:2) (cid:0) ( M G ) − GM r (cid:1) + r (cid:3) − r ¨ M M [ r − GM ] − ˙ M r Λ M [ r − GM ( r − GM )] − ˙ M (4 GM r − r ) , (11)where a ( M + M ) M ( t ) > r ≥ G M ( t ), with a G ( M + M ) >
2. It is expected that during the collapse ω > IV. AN EXAMPLE
To examine the theory, we shall consider the case where the evolution of the mass is M ( t ) = M e − b t + M , (12)where t ≥ b is some positive constant such that 1 /b give us the scale of decaying of the compact object.To evaluate the equation of state inside the star, we shall consider a radius r = α GM ( t ) which is smaller than thestellar radius and bigger than the dynamic Schwarzschild one: 2 < α ≤ a G ( M + M ) For the case (12), the equation ofstate of the system for a ( t ) = a ( M + M ) M ( t ), is ω ( t, r ) = Λ (cid:2) M ( α − (cid:3) − h M ¨ M ( α −
2) + α ˙ M i Λ [ M ( α − ] + α (3 α −
4) ˙ M . (13)Notice that the equation of state in the limit case where t → ∞ islim t →∞ ω ( t, r ) → . (14)Furthermore, in the limit case where α →
2, we obtain the equation of statelim α → ω ( t, r ) → − . (15)In the figure (1) we have plotted ω ( r, t ) as a function of t , for different α -values corresponding to different r -valuesof the interior of the star. We have considered M = M = 0 . M J . Notice that on the Schwarzschild radius ω ( α = 2) = − /
2, but for different (and larger) α -values, the equation of state tends asymptotically to ω → ω ( r, t ) as a function of t for different b -values, using M = M = 0 . M J and α = 2 .
5. Notice that initially ω <
0, but later ω → χ ( t, r, θ, φ ), as χ ( t, r, θ, φ ) = ∞ X n =0 χ n ( t, r, θ, φ ) . (16)For modes that include waves of gravitational origin, must be with l = 2, and − ≤ m ≤
2, so that χ n ( t, r, θ, φ ) = X m = − h A n,l,m χ n,l,m ( t, r, θ, φ ) + A † n,l,m χ ∗ n,l,m ( t, r, θ, φ ) i , (17)such that the modes χ n,l,m ( t, r, θ, φ ), are: χ n,l,m ( t, r, θ, φ ) = (cid:18) E n,l ~ (cid:19) R n,l ( r ) Y l,m ( θ, φ ) τ n ( t ) , (18)We can expand the field χ as a superposition χ n,l,m ( t, r, θ, φ ) ∼ R n,l ( t, r ) Y l,m ( θ, φ ), where the functions Y l,m ( θ, φ )are the usual spherical harmonics Y l,m ( θ, φ ) = s (2 l + 1)( l − m )!4 π ( l + m )! P lm ( θ ) e i mφ , (19)where ! denotes the factorial and P lm ( θ ) are the Legendre polinomials. In this work we shall consider l = 2 becausewe are dealing with waves of space-time and − ≤ m ≤ A. Waves of space-time
We must calculate the equation (6), which in our example takes the form g αβ ∇ α ∇ β χ ( t, r, θ, φ ) = 6 ˙ M ( t ) rM ( t ) ( r − G M ( t )) Λ , (20)where we have made use of the fact that ddS = U α dd x α . Notice that all the energy transferred by the compact objectto the wave is M c , so that the total energy of the system (star + space-time wave), is conserved. For a co-movingobserver U i = 0 and U = p g in the metric (10), because we are describing the source of the wave in the rightside of the equation (20). In order to calculate the covariant derivatives in the D’Alambertian of the left side in (20),we must use the connections of the metric (9), with a mass M + M . This is because when the collapse begins,the mass of the compact object is M + M and the information that changes the space-time travels with the waveof space-time. Hence, in a given point of the space-time, an observer will feel a gravitational field due to the mass M + M before the wave leads to the observer, and a gravitational field due to the residual mass M , after the wavehas passed the observer. The equation (20), written explicitly, is − r ∂χ n,l,m ∂r − ∂ χ n,l,m ∂r + 2 G ¯ Mr ∂χ n,l,m ∂r + 2 G ¯ Mr ∂ χ n,l,m ∂r − r cos ( θ )sin ( θ ) ∂χ n,l,m ∂θ − r sin ( θ ) ∂ χ n,l,m ∂φ − r (cid:2) G ¯ M − r (cid:3) ∂ χ n,l,m ∂t − r sin ( θ ) ∂ χ n,l,m ∂θ + " M ( t ) rM ( t ) [2 GM ( t ) − r ] Λ = 0 , (21)where we have denoted ¯ M = ( M + M ). We can make variables separation: χ n,l,m ( t, r, θ, φ ) = R n,l ( t, r ) Y l,m ( θ, φ ),and after making r = α M ( t ) in the source with the approximation of it, for sufficiently large time, we obtain thedifferential equations: 2 r ∂R n,l ( t, r ) ∂r + r ∂ R n,l ( t, r ) ∂r − G ¯ M ∂R n,l ( t, r ) ∂r − G ¯ M r ∂ R n,l ( t, r ) ∂r + r [2 G ¯ M − r ] ∂ R n,l ( t, r ) ∂t − l ( l + 1) R n,l ( t, r ) = 3 ¯ M e − b t α Λ( α − , (22)cos ( θ )sin ( θ ) ∂Y l,m ( θ, φ ) ∂θ + 1sin ( θ ) ∂ Y l,m ( θ, φ ) ∂φ + 1sin ( θ ) ( θ ) ∂ Y l,m ( θ, φ ) ∂θ + l ( l + 1) Y l,m ( θ, φ ) = 0 . (23)The solution for Eq. (22) R n,l ( t, r ) = (cid:18) α − (cid:19) e b ( r − t ) ( G ¯ M ) bα Λ (cid:0) G ¯ M − r (cid:1) bG ¯ M H C (1) × Z e − br H C (2) (cid:0) G ¯ M − r (cid:1) − bG ¯ M r (cid:16) H C (1) H C (2) b ( G ¯ M ) − H C (1) H CP (2) G ¯ M + H C (1) H CP (2) r + 2 H C (2) H CP (1) G ¯ M − H C (2) H CP (1) r (cid:17) dr − (cid:18) α − (cid:19) e b ( r − t ) ( G ¯ M ) bα Λ (cid:0) G ¯ M − r (cid:1) − bG ¯ M H C (2) × Z (cid:0) G ¯ M − r (cid:1) bG ¯ M H C (1) e − br r (cid:16) H C (1) H C (2) b ( G ¯ M ) − H C (1) H CP (2) G ¯ M + H C (1) H CP (2) r + 2 H C (2) H C P (1) G ¯ M − H C (2) H CP (1) r (cid:17) dr, (24)where we have denoted the confluent Heun functions H C and the prime confluent Heun functions H CP , as H C (1 , = H C (cid:18) − bG ¯ M, ∓ bG ¯ M, , b ( G ¯ M ) , − b ( G ¯ M ) − l ( l + 1) , G ¯ M − r G ¯ M (cid:19) , (25) H CP (1 , = H C P (cid:18) − bG ¯ M , ∓ bG ¯ M, , b ( G ¯ M ) , − b ( G ¯ M ) − l ( l + 1) , G ¯ M − r G ¯ M (cid:19) . (26)Notice that we have made null the constants in the homogenous contributions to the solution for R n,l ( t, r ) because itis expected that in absence of sources, there will no emission of space-time waves.In the figures (3), and (4) we have plotted the norm of R n,l ( t, r ∗ ) as a function of time, for different values of r ∗ ,and M = M = 2 . M J . Here, r ∗ is the distance where is situated an observer outside the star. We have consideredΛ = 3 / ( b ), with b = 0 . a = 2 . G ( M + M ) and l = 2, because we are dealing with waves of space-time. In thefigure (3) the observer is close to the star (at r ∗ = 4 G ( M + M )), but in (4) is more distant (at r ∗ = 5 G ( M + M )).Notice that the signal is more weak in this last. Because the wavelengths must be smaller than the initial size of thestar, the wavelengths of the emitted space-time waves, must be λ ≡ πk < a = (2 /b ) , (27)where (1 /b ) is the time scale that describes the decay rate of the compact object. Finally, notice that the equationof state change its signature during the collapse in the interior of the compact object for different interior radius, butno for a Schwarzschild one. V. FINAL COMMENTS
We have studied the dynamics of a partial collapse for a spherically symmetric compact object and the waves ofspace-time emitted during the collapse when the collapsing body delivers a part of its mass during the collapse. Here,the waves of space-time are described by back-reaction effects. To study these effects we have modified the boundaryconditions in the minimum action principle by considering a dynamical source through a 3D gaussian wave-like fluxthat cross such closed hypersurface. The effective dynamics is described by the system of equation (5) and (6), where χ is the trace of the wave components: χ αβ = δ Ψ αβ δS on the background curved space-time. Notice that we are dealingwith waves that do not comes from the variation of a quadrupole momentum, but distortions of space-time whichpropagates as waves due to the collapse of a spherically symmetric compact object. The description of the dynamicsis extremely complicated, but, fortunately we have founded exact analytical solutions in an example for which thefinal mass is M = ( M + M ) /
2. The wavelengths of the space-time emitted waves during the collapse have thecut: λ < (2 /b ) = a , where (1 /b ) is the time scale that describes the decay rate of the compact object. Notice thatthe approach here used is different to other used in some previous works[11, 15] in which boundary conditions arerelated to a displacement from a semi-Riemannian manifold to another extended manifold in which ∇ ǫ g αβ = 0, todescribe back-reaction effects. However, in this work we have used a semi-Riemannian manifold to describe boundaryconditions, so that ∇ ǫ g αβ = 0. This is due to the fact, in this work we have considered that the energy lost throughthe compact object is transferred to the wave of space-time. Acknowledgements
J. M. H. acknowledges CONACYT (M´exico) for fellowship support (”Beca de movilidad al extranjero”). J. I. M.and M. B. acknowledge CONICET, Argentina (PIP 11220150100072CO) and UNMdP (EXA852/18) for financialsupport. This research was supported by the CONACyT-UDG Network Project No. 294625 ”Agujeros Negros yOndas Gravitatorias”. [1] R. F. Baierlein, D. H. Sharp, J. A. Wheeler, Phys. Rev. : 1864 (1962).[2] J. A. Wheeler, Adv. Ser. Astrophys. Cosmol. : 27 (1987).[3] J. W. York, Phys. Rev. Lett. : 1082 (1972).[4] G. W. Gibbons, S. W. Hawking, Phys. Rev. D10 : 2752 (1977).[5] L. S. Ridao, M. Bellini, Astrophys. Space Sci. : 94 (2015).[6] J. M. Romero, M. Bellini, Eur. Phys. J.
C79 : 651 (2019).[7] C. Gundlach, Living Rev. Rel. : 4 (1999).[8] R. Goswami, P. S. Joshi, Phys. Rev. D65 : 027502 (2004).[9] R. Giambo, Class. Quant. Grav. : 2295 (2005).[10] R. Sharma, S. Das, R. Tikekar, Gen. Relat. Grav. : 25 (2015).[11] J. Mendoza Hern´andez, M. Bellini, C. Moreno, Phys. Dark Univ. : 100251 (2019).[12] S. Chakrabarti and N. Banerjee, Eur. Phys. J. C77 : 166 (2017).[13] N. Banerjee and S. Chakrabarti , Phys. Rev.
D95 : 024015 (2017).[14] S. W. Hawking, G. F. R. Ellis.
The large scale structure of space-time . Cambridge Monographs on Mathematical Physics.Cambridge University Press. Cambridge, UK (1973).[15] J. Mendoza Hern´andez, M. Bellini, C. Moreno, Phys. Dark Univ. : 100395 (2019). FIG. 1: Plotting of the ω ( r, t ) as a function of t (seconds), for different α -values corresponding to different r -values, with M = M = 0 . M J . For simplicity, we have done M J = 1. Notice that on the Schwarzschild radius ω ( α = 2) = − /
2, butfor different (and larger) α -values, the equation of state tends asymptotically to ω → FIG. 2: Plotting of the ω ( r, t ) as a function of t (seconds), for different b -values and M = M = 0 . M J and α = 2 .
5. Forsimplicity, we have done M J = 1. Notice that initially ω <