Energy Contents of Plane Gravitational Waves in Teleparallel Gravity

Transcription

1 Energy Contents of Plane Gravitational Waves in Teleparallel Gravity M. harif and umaira Taj Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. Abstract The gravitational energy-momentum and its related quantities for plane gravitational waves are evaluated in the framework of teleparallel theory using Hamiltonian approach. We find that energy-momentum is non-positive while the angular momentum becomes constant. This shows consistency with the results available in literature. Further, the gravitational pressure is investigated. Keywords: Teleparallel gravity; Plane gravitational waves; Energy. 1 Introduction General Relativity shows consistency with pecial Relativity in different aspects, particularly with the principle that nothing travels faster than light. Thereby the alterations in the gravitational field, i.e., gravitational waves cannot be experienced everywhere instantly as they must propagate at exactly the speed of light. These waves travel outwards from the source, carrying energy and information about the astrophysics of the source. Einstein showed that accelerating masses emit gravitational waves. msharif@math.pu.edu.pk sumairataj@ymail.com 1

2 Gravitational waves, by definition, have zero energy-momentum tensor. Thus their existence was questioned. However, the theory of General Relativity (GR) predicts the existence of gravitational waves as solutions of the Einstein field equations [1]. Indeed this problem arises because energy is not well-defined in GR, as the strong equivalence principle refutes the energy localization of the gravitational field. Ehlers and Kundt [2] resolved this problem for gravitational waves by analyzing a sphere of test particles in the path of plane-fronted gravitational waves. They showed that these particles acquired a constant momentum from the waves. Weber and Wheeler [3] gave the similar discussion for cylindrical gravitational waves. Qadir and harif [4] explored an operational approach, embodying the same principle, to show that gravitational waves impart momentum. Rosen and Virbhadra [5] obtained the energy and momentum densities of cylindrical gravitational waves in Einstein prescription and found them to be finite and reasonable. ome authors [7]-[8] argued that the localization of energy-momentum becomes more transparent in the framework of teleparallel equivalent of General Relativity (TEGR). Møller [9] was the first who observed that the tetrad description of the gravitational field could lead to a better expression for the gravitational energy-momentum than does GR. harif and Nazir [10] investigated the energy of cylindrical gravitational waves in GR and teleparallel gravity. Andrade et al. [11] considered the localization of energy in Lagrangian framework of TEGR. Maluf et al. [12] derived an expression for the gravitational energy, momentum and angular momentum using the Hamiltonian formulation of TEGR. Maluf et al. [13] evaluated the energymomentum flux of linear plane gravitational waves. Maluf and Ulhoa [14] showed that gravitational energy-momentum of plane-fronted gravitational waves is non-positive. This paper is the elaboration of the above procedure by evaluating energy and its content for linear plane and plane-fronted gravitational waves. Next section gives some basic concepts of TEGR and energy-momentum expressions. ection 3 is devoted for the evaluation of energy and its relevant quantities for linear plane gravitational waves. In section 4, we calculate these quantities for plane-fronted gravitational waves. ummary and discussion is presented in section 5. 2

3 2 Hamiltonian Approach: Energy-Momentum in Teleparallel Theory In teleparallel theory, e a µ is a non-trivial tetrad from which the Riemannian metric is obtained as a by product whose inverse satisfies the relation g µν = η ab e a µe b ν (1) e a µe b µ = δ a b, e a µe a ν = δ µ ν. (2) Here, the spacetime indices represented by Greek alphabets (µ, ν, ρ,...) and tangent space indices represented by Latin alphabets (a, b, c,...) run from 0 to 3. Time and space indices are denoted according to µ = 0, i, a = (0), (i). The torsion tensor is defined as which is related to the Weitzenböck connection [11] T a µν = µ e a ν ν e a µ (3) Γ λ µν = e a λ ν e a µ. (4) This is the connection of the Weitzenböck spacetime on which the teleparallel theory of gravity is defined. The tensor Σ abc is defined as Σ abc = 1 4 (T abc + T bac T cab ) (ηac T b η ab T c ) (5) which satisfies the antisymmetric property, i.e., Σ abc = Σ acb. The total energy-momentum is defined as P a = d 3 x i Π ai, (6) V where V is an arbitrary space volume. The angular momentum is defined as M ik = 2 d 3 xπ [ik], V = 2κ d 3 xe[ g im g kj T 0 mj + (g im g 0k g km g 0i )T j mj] (7) V 3

4 for an arbitrary volume V of three-dimensional space. The a components of the gravitational energy-momentum flux and matter energy-momentum flux [14] are given by Φ a g = d j φ aj, Φ a m = d j (ee a µt jµ ). (8) Here the quantity φ aj = κee aµ (4Σ bcj T bcµ δ j µσ bcd T bcd ) (9) represent the a component of the gravitational energy-momentum flux density in j direction. In terms of the gravitational energy-momentum, we have a = Φ a g Φ a dt m. (10) For the vacuum spacetime, the above equation reduces to a = Φ a g = d j φ aj. (11) dt If we take a = (i) = (1), (2), (3), then (i) = dt d j ( φ (i)j ). (12) The left hand side of the above equation has the character of force because it contains time derivative of the momentum component. Thereby the density ( φ (i)j ) is considered as a force per unit area, or pressure density. Thus Eq.(12) has the nature of the gravitational pressure [15]. 3 Linear Plane Gravitational Waves First we take linear plane wave as a solution of the Einstein field equations. We impose restriction to just one ploarization of the wave given by [13] ds 2 = dt 2 + dx 2 + [1 f + (t z)]dy 2 + [1 + f + (t z)]dz 2, (13) where (f + ) 2 << 1. The tetrad field, satisfying Eqs.(1) and (2) is e a µ(t, z) = f f+ 4. (14)

5 The non-zero components of the torsion tensor are which give rise to f + T (2)02 = 2 f +, T (2)12 = 1 f + 2, 1 f + T (3)03 = f f +, T (3)13 = f f + (15) T 202 = 1 2 f +, T 212 = 1 2 f +, T 303 = 1 2 f +, T 313 = 1 2 f +. (16) Here dot and prime denote differentiation with respect to t and z respectively. 3.1 Energy, Momentum and Angular Momentum The components of energy-momentum density take the form ] ( ) i Π (0)i f + f + = 1 [ 2κ f + f + = 2κ 1, 1 (f+ ) 2 1 (f+ ) 2 ] ( ) i Π (1)i f + f + f = 1 [2κ + f = 2κ 1 +, 1 (f+ ) 2 1 (f+ ) 2 i Π (2)i = 0 = i Π (3)i. (17) Consequently, we have P (0) = P (1) f + f + = 2κ 1 (f+ ) 2 dydz (18) and P (2) = constant = P (3). We have used f + = f + to evaluate energy. As (f + ) 2 << 1, thus the above components take the form P (0) = P (1) 2κf + f + dydz 0. (19) If we set the constant to be zero then we can say that energy-momentum enclosed by an arbitrary volume of linear plane gravitational wave is nonpositive. The angular momentum turns out to be constant. 5

6 3.2 Gravitational Pressure For linear plane gravitational waves, all the components of gravitational momentum flux density vanish except φ (1)1. Using these values φ (1)1 f + f + = κ 1 (f+ ), 0 = 2 φ(2)1 = 0 = φ (3)1 (20) and the unit vector ˆr = (1, 0, 0) in (i) = d 1 ( φ (i)1 ), (21) dt we obtain dt = κ f + f + dydzˆr. (22) 1 (f+ ) 2 By conversion of surface element dydz into spherical polar coordinates, we have dt = κ f + f + ρ 2 sin θdθdφˆr. (23) 1 (f+ ) 2 Integration over a small solid angle dω = sin θdθdφ of constant radius ρ gives dt = f + f + 1 (f+ ) 2 κρ2 Ωˆr. (24) Replacing dt d(ct), κ = 1 dt = 16π c4 16πG ( c3 16πG in the above equation, we obtain f + f + 1 (f+ ) 2 ) (ρ 2 Ω)ˆr. (25) ince (f + ) 2 << 1, therefore the above equation takes the form dt c4 16πG (f +) 2 (ρ 2 Ω)ˆr. (26) The quantity (f +) 2 c 4 /(16πG) on the right hand side of this equation gives the gravitational pressure exerted on the area element (ρ 2 Ω). By reconsidering Eq.(26), we have ( ) d P c4 dt M 16πGM (f +) 2 ρ 2 Ωˆr. (27) The left hand side interprets the definition of acceleration. Thus we can consider it as the gravitational acceleration field, which acts on the solid angle Ω at a radial distance ρ. 6

7 4 Plane-fronted Gravitational Waves A plane-fronted gravitational wave is given by the line elemenet [1] ds 2 = L 2 (u)[e 2β(u) dy 2 + e 2β(u) dz 2 ] dudv, (28) where the functions L and β satisfy L,uu + Lβ 2,u = 0. Transforming the coordinates from (u, v) (t, x), where u = t x and v = t + x, the metric (28) takes the form ds 2 = dt 2 + dx 2 + L 2 (t, x)[e 2β(t,x) dy 2 + e 2β(t,x) dz 2 ] (29) and the conditions become L,αα + Lβ 2,α = 0, α = 0, 1. The corresponding tetrad field satisfying Eqs.(1) and (2) is e a µ(t, x) = Le β 0. (30) Le β The non-vanishing components of the torsion tensor are T (2)02 = e β (L β + L), T (2)12 = e β (Lβ + L ), T (3)03 = e β ( L L β), T (3)13 = e β (L Lβ ), (31) where dot and prime denote differentiation with respect to t and x respectively. The corresponding non-zero components of T λµν = e a λt aµν are T 202 = e 2β L(L β + L), T 212 = e 2β L(Lβ + L ), T 303 = e 2β L( L L β), T 313 = e 2β L(L Lβ ). (32) 7

8 4.1 Energy, Momentum and Angular Momentum The components of energy-momentum density are i Π (0)i = 1 [ 4κ( L 1 L)] = 2κ( 1 2 L 2 ), i Π (1)i = 1 [ 4κ(L 0 L)] = 2κ( 1 2 L 2 ), i Π (2)i = 2 [ 2κe β ( 0 L L 0 β)] = 0, i Π (3)i = 3 [ 2κe β ( 0 L + L 0 β)] = 0, (33) where 0 L = 1 L is used to evaluate i Π (1)i. Inserting these values in Eq.(6), we obtain P (0) = P (1) = 2κ d 3 x( 2 1 L 2 ) 0 (34) V while P (2) = constant = P (3) which can be made zero if we choose constant to be zero. Then it follows that P a P b η ab = 0 which is the characteristic of plane electromagnetic wave. The angular momentum turns out to be constant as expected. 4.2 Energy-Momentum Flux The components of gravitational energy flux density φ (0)j for plane-fronted gravitational waves are φ (0)1 = 4κ[L 2 ( 1 β) 2 ( 1 L) 2 ], φ (0)2 = 0 = φ (0)3 (35) which consequently gives energy flux Φ (0) g = 4κ[L 2 ( 1 β) 2 ( 1 L) 2 ] dydz + constant. (36) The components of the gravitational momentum flux Φ (i) g Φ (1) g = 4κ[L 2 ( 1 β) 2 ( 1 L) 2 ] dydz + constant, Φ (2) g = constant = Φ (3) g (37) 8

9 are obtained by using the components of gravitational momentum flux densities φ (1)1 = 4κ[L 2 ( 1 β) 2 ( 1 L) 2 ], φ (1)2 = 0 = φ (1)3, φ (2)1 = 0 = φ (2)2 = φ (2)3, φ (3)1 = 0 = φ (3)2 = φ (3)3. (38) 4.3 Gravitational Pressure Proceeding in a similar way as for linear plane gravitational waves, it follows that dt = [L2 ( 1 β) 2 ( 1 L) 2 c 4 ] 4πGˆr(ρ2 Ω). (39) The term [L 2 ( 1 β) 2 ( 1 L) 2 ]c 4 /(4πG) on the right hand side of the above equation is interpreted as the gravitational pressure exerted on the area element (ρ 2 Ω). Equation (39) can be re-written as ( ) d P = [L 2 ( 1 β) 2 ( 1 L) 2 c 4 ρ 2 ] Ωˆr. (40) dt M 4πGM The left hand side of this equation can be recognized as the gravitational acceleration. We can consider it as the gravitational acceleration field that acts on the solid angle Ω. 5 ummary and Discussion This paper is devoted to discuss energy and the related quantities of plane gravitational waves using the Hamiltonian approach in the teleparallel gravity. We have evaluated energy, momentum, angular momentum and gravitational pressure of linear and plane-fronted gravitational waves. In both linear plane and plane-fronted gravitational waves, it is found that P a P b η ab = 0 which coincides with earlier result [14]. The constant momentum shows correspondence with the result of GR [4]. The angular momentum also turns out to be constant in both cases. Further, we have also evaluated gravitational pressure exerted by gravitational waves. This may be helpful to investigate the thermodynamics of the gravitational field. It is interesting to mention here that results for both linear and plane gravitational waves are exactly the same. This is what one can expect from the analysis. 9

10 We would like to mention here that energy-momentum obtained by using the prescription [12] shows consistency with the results of different energymomentum complexes both in GR and teleparallel gravity. References [1] Misner, C.W., Thorne, K.. and Wheeler, J.A.: Gravitation (Freeman, New York, 1973). [2] Ehlers, J. and Kundt, W.: Gravitation: An Introduction to Current Research, ed. Witten, L. (Wiley, New York, 1962). [3] Weber, J. and Wheeler, J.A.: Rev. Mod. Phys. 29(1957)509. [4] Qadir, A. and harif, M.: Phys. Lett. A167(1992)331. [5] Rosen, N. and Virbhadra, K..: Gen. Rel. Grav. 26(1993)429. [6] Virbhadra, K..: Pramana J. Phys. 45(1995)215. [7] Nashed, G.G.L.: Phys. Rev. D66(2002) [8] Xu,. and Jing, J.: Class. Quantum Grav. 23(2006)4659. [9] Møller, C.: Tetrad Fields and Conservation Laws in General Relativity (Academic Press, London, 1962). [10] harif, M. and Nazir, K.: Commun. Theor. Phys. 50(2008)664. [11] de Andrade, V.C., Guillen, L.C.T. and Pereira, J.G.: Phys. Rev. Lett. 84(2000)4533. [12] Maluf, J.W., da Rocha-Neto, J.F., Toribio, T.M.L. and Castello-Branco, K.H.: Phys. Rev. D65(2002)124001; J. Math. Phys. 35(1994)335. [13] Maluf, J.W., Faria, F.F. and Castello-Branco, K.H.: Class. Quantum Grav. 20(2003)4683. [14] Maluf, J.W. and Ulhoa,. C.: Phys. Rev. D78(2008) [15] Maluf, J.W.: Annalen Phys. 14(2005)