A conformally flat accelerated charge metric is found in an arbitrary
dimension $D$. It is a solution of the Einstein-Maxwell-null fluid with a
cosmological constant in $D \ge 4$ dimensions. When the acceleration is zero
our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that
the charge loses its energy, for all dimensions, due to the acceleration.Comment: Latex File, 12 page

A. García

A. Trautman

B. Bertotti

D.A. Lowe

F.R. Tangherlini

H. Stephani

J.D. Jackson

M. Gürses

Metin Gürses

R. Penrose

R. Robinson

T. Levi-Civita

V. Cardoso

Ö. Sarıoğlu

Özgür Sarıoğlu

English

arXiv

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Accelerated Levi-Civita-Bertotti-Robinson metric in D dimensions

A conformally flat accelerated charge metric is found in an arbitrary dimension D. It is a solution of the Einstein-Maxwell-null fluid equations with a cosmological constant in D >= 4 dimensions. When the acceleration is zero, our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that the charge loses its energy, for all dimensions, due to the acceleration

Sarioglu, O

Gurses, M

Middle East Technical University Research Information System

A conformally flat accelerated charge metric is found in an arbitrary dimension D. It is a solution of the Einstein-Maxwell-null fluid equations with a cosmological constant in D ≥ 4 dimensions. When the acceleration is zero, our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that the charge loses its energy, for all dimensions, due to the acceleration

Gürses, M.

Sarıoǧlu, Ö.

Bilkent University Institutional Repository

Gen. Relativ. Gravit. (2005) 37(12): 2015–2022DOI 10.1007/s10714-005-0176-yRESEARCH ARTICLEMetin Gürses · Özgür SarıoğluAccelerated Levi-Civita-Bertotti-Robinsonmetric in D dimensionsReceived: 14 February 2005 / Published online: 22 November 2005C© Springer-Verlag 2005Abstract A conformally flat accelerated charge metric is found in an arbitrarydimension D. It is a solution of the Einstein-Maxwell-null fluid equations with acosmological constant in D ≥ 4 dimensions. When the acceleration is zero, oursolution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that thecharge loses its energy, for all dimensions, due to the acceleration.Keywords Accelerated charge · Einstein-Maxwell-null fluid ·Levi-Civita-Bertotti-Robinson metric1 IntroductionThe Levi-Civita-Bertotti-Robinson (LBR) metric is one of the classical metrics ingeneral relativity. It is the only Einstein-Maxwell solution which is homogeneousand has homogeneous non-null Maxwell field [1–4]. LBR space-time is a productof two spaces of constant curvature, namely it is Ad S2 × S2. Due to this propertyLBR metric arises also in supergravity theories [5–8]. LBR type of space-timesshow up also as the space-time regions closer to the horizons of extreme Reissner-Nordstrom black hole geometries [9, 10]. Its curvature invariants are all constantand the electromagnetic field tensor is covariantly constant. This property leads tothe result that LBR metric is an exact solution of any theory of gravitation coupledwith a U (1) gauge field [9, 11].M. Gürses (B)Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara-TurkeyE-mail: gurses@fen.bilkent.edu.trÖ. SarıoğluDepartment of Physics, Faculty of Arts and Sciences, Middle East Technical University,06531 Ankara-TurkeyE-mail: sarioglu@metu.edu.tr2016 M. Gürses, Ö. SarıoğluIn this work we generalize the LBR metric in D-dimensions by introducing ac-celeration. A similar generalization was done long ago for the Reissner-Nordstrommetric in four dimensions [12]. Recently we extended the Bonnor-Vaidya formal-ism to Reissner-Nordstrom metric in D-dimensions [13, 14]. We used an arbitrarycurve C in D-dimensional Minkowski space-time MD and examined in detail theEinstein-Maxwell-null dust [13] and Einstein-Born-Infeld-null dust field equa-tions [14], Yang-Mills equations [15], and Liénard-Wiechert potentials in evendimensions [16]. In the first three works [13–15], we found some new solutionsgeneralizing the Tangherlini [17], Plebański [18], and Trautman [19] solutions,respectively. The last one, [16], indicates that the accelerated scalar or vectorcharged particles in even dimensions lose energy.Our conventions are similar to the conventions of our earlier works [13, 14],[16]. In a D-dimensional Minkowski space-time MD , we use a parametrized curveC = {xµ ∈ MD| xµ = zµ(τ) , µ = 0, 1, 2, . . . , D − 1 , τ ∈ I } such that τ is aparameter of the curve and I is an interval on the real line R. We define the worldfunction  as = ηµν (xµ − zµ(τ)) (xν − zν(τ )), (1)where xµ is a point not on the curve C . There exists a point zµ(τ0) on the non-space-like curve C which is also on the light cone with the vertex located at thepoint xµ, so that (τ0) = 0. Here τ0 is the retarded time. By using this propertywe find thatλµ ≡ ∂µ τ0 = xµ − zµ(τ0)R, (2)where R ≡ żµ(τ0) (xµ − zµ(τ0)) is the retarded distance. Here a dot over a letterdenotes differentiation with respect to τ0. It is easy to show that λµ is null andsatisfiesλµ,ν = 1R[ηµν − żµ λν − żν λµ − (A − ε) λµ λν], (3)R,µ = żµ + (A − ε)λµ, (4)where A ≡ z̈µ (xµ − zµ(τ0)) and żµ żµ = ε = −1, 0. Here ε = −1 and ε = 0correspond to the time-like and null velocity vectors, respectively. In this workwe shall consider only the case where the velocity vector is time-like (ε = −1).One can also show explicitly that λµ żµ = 1 and λµ R, µ = 1. If one definesa ≡ A/R = λµ z̈µ, thenλµ a, µ = 0. (5)Furthermore defining (letting a0 ≡ a)ak ≡ λµ dk+2 zµ(τ0)dτ k+20, k = 0, 1, 2, . . . (6)one can show thatλµ ak, µ = 0, ∀k = 0, 1, 2, . . . (7)In the next section, using [9], we present the LBR metric in arbitrary D-dimensions. In Sect. 3 we give the accelerated LBR metrics in D-dimensions.Here we find the rate of energy loss for all D. In Sect. 4 we consider a specialcurve C where the null fluid disappears. In particular, taking a curve C whichAccelerated Levi-Civita-Bertotti-Robinson metric in D dimensions 2017corresponds to a constant acceleration, we find a solution of the Einstein-Maxwellfield equations in D-dimensions. We show that this is also the LBR metric in anaccelerated reference frame.2 Levi-Civita-Bertotti-Robinson metricsWe assume a spherically symmetric, static space-time in D-dimensions. Amongsuch a class of space-times, LBR geometry is a product space-time Ad S2 × SD−2.It has been studied for several purposes [9, 11], (see also [10]). We have the fol-lowing theorem.Theorem 1 Let gµν be the metric and Fµν be the Maxwell field given bygµν = q2r2[ − tµ tν + c20 kµ kν + r2 hµν], (8)Fµν = c0r(tµ kν − tν kµ), (9)where hµν is the metric of the (D − 2)-dimensional sphere SD−2, c0, q are con-stants, tµ = δ0µ, kµ = δrµ. Then they solve the Einstein-Maxwell field equationswith a cosmological constantGµν = Q2c20[Fµα Fνα − 14(Fαβ Fαβ)gµν]+ gµν, (10)whereQ2 = q2[(D − 3) c20 + 1],  = 12q2[1c20− (D − 3)2]. (11)Here Q is the electric charge. This metric describes the near horizon region of thecharged Tangherlini metric with a cosmological constant [9, 10] and moreover isconformally flat if c0 = 1. In this case the cosmological constant vanishes onlywhen D = 4. All the curvature invariants are constants, being functions of theconstants, c0, Q and D. These metrics describe the geometry of black holes inthe neighborhood of their outer horizons [9–11]. Our purpose in this work is togeneralize this solution when the charge moves on a curve C described in theintroduction.3 Accelerated Bertotti-Robinson metric in D-dimensionsIn this section, we shall generalize the LBR metric by introducing accelerationusing the curve kinematics given in the introduction. We shall consider the casewhen the space-time is conformally flat. We assume that the metric and the elec-tromagnetic vector potential are given bygµν = eψ ηµν, Aµ = S żµ, (12)2018 M. Gürses, Ö. Sarıoğluwhere ηµν is the Minkowski metric, ψ and S are functions of R only. Then wehaveTheorem 2 In an arbitrary dimension Dds2 = R20R2ηµν dxµ dxν, (13)Aµ = e żµR, (14)are the solutions of the Einstein-Maxwell-null (pressureless) fluid equations withcosmological constant ,Gµν = κ[Fµα Fνα − 14(Fαβ Fαβ) gµν]+ κ ρλµ λν + gµν. (15)The energy density of the fluid and the cosmological constant are given byκ ρ = (D − 2) (a1 − z̈α z̈α), (16) = − (D − 2)(D − 4)2 R20, e2 = D − 2κR20, (17)where R0 is a constant, R is the retarded distance described in the introduction, λµis the null vector defined in (2) and a1 is defined in (6). Furthermore, the currentvector Jµ = ∇ν Fµν vanishes for all D (except on the curve C, see Remark 1below).First of all, when the curve C is a straight line, the solution given in Theorem 2reduces to the one in Theorem 1 with c0 = 1. Moreover, we observe that the met-ric and the electromagnetic fields for all dimensions have the same form. This isinteresting because in the case of accelerated charges in the Kerr-Schild geometry,the metric and the electromagnetic fields take different forms in different dimen-sions [13, 14]. Here the only D dependent quantity is the cosmological constant.On the other hand, the energy density of the null fluid depends on the accelerationparameter of the curve C . It is worthwhile to look at the energy loss formula inthis case. Energy flux formula, in general, is given by (see [13]),N = − limR→∞∫SD−2żα Tαβ nβ R−1 RD−20 d, (18)where Tµν is the corresponding energy momentum tensor (of the fluid or theMaxwell field) and nµ is a space-like vector defined through λµ = −żµ − nµ/R(see [13] for more details). We find that NF = 0 for the null fluid distribution. Itsenergy is conserved. The rate of change of the energy of the electromagnetic fieldNE is found asNE = e2 RD−40 κ21{−D−3 (D−22) [−√π( D−12) +2D−1 ( D+22)(D)]+D−2},(19)Accelerated Levi-Civita-Bertotti-Robinson metric in D dimensions 2019for D ≥ 4, where D is the solid angle in D dimensions. Here κ1 is the firstcurvature of the curve C . In calculating (19), we used the integration techniquedeveloped in our earlier works [13]. The quantity NE given above is positive for alldimensions D and hence there is energy loss due to acceleration in all dimensions.In particular, when D = 4, NE = 8π3 e2 κ21 is exactly the energy loss formula dueto the acceleration of a charged particle in flat space-time [20].Remark 1 We obtain the covariant derivative of the Maxwell tensor Fαβ as∇µ Fαβ = λµ (λα ζβ − λβ ζα) (20)whereζµ = 1R[d3 zµdτ 30− a1 d zµdτ0].This vector vanishes if C is a straight line or has constant acceleration, in whichcase Fαβ becomes a covariantly constant tensor field. Since λµ ζµ = 0, then thecurrent vector Jµ mentioned in Theorem 2 is zero everywhere except on the curveC . In fact it takes the formJµ(x) = 1D−2∫Cżµ(τ) δ(x − z(τ )) dτ,where ∇µ Jµ = 0 identically.Remark 2 We note that the Ricci scalar, the Ricci invariant Rαβ Rαβ , and theMaxwell invariant Fαβ Fαβ are all constants.gαβ Rαβ = (D − 1)(D − 4)R20, (21)Rαβ Rαβ = D3 − 8D2 + 21D − 16R40, (22)Fαβ Fαβ = 2κ R20(2 − D). (23)Similarly the curvature invariant Rαβγσ Rαβγσ is also constant due to the confor-mal flatness. All of these invariants are equal to the corresponding invariants ofthe LBR metric (static case).4 Charged particle with constant accelerationIn this section we consider only the Einstein-Maxwell field equations with a cos-mological constant. To achieve this, we look for special curves C such that thenull fluid energy density ρ vanishes.Using the Serret-Frenet frame [13] in MD , we obtainκρ = (D − 2) [−κ̇1 cos θ + κ1 κ2 sin θ cos φ],2020 M. Gürses, Ö. Sarıoğluwhere κ1 and κ2 are the first two curvatures of the curve C and (θ, φ) are thefirst two angular coordinates on SD−2. It is clear that the energy density ρ ofthe null fluid does not have a fixed sign. It changes its sign at different points.There is a non-trivial choice where κ1 is a non-zero constant and κ2 = 0 whichleads to the vanishing of the energy density (ρ = 0). In this case C describes aparticle moving with a constant acceleration and the Maxwell field tensor Fαβ iscovariantly constant. The total energy measured by the observer moving along thecurve C isE = Gµν żµ żν = (D − 2)[κ21 sin2 θ + D − 32R2].This vanishes asymptotically (R → ∞) in the case of LBR space-time, but asymp-totically it is proportional to the square of the first curvature in the space-timecorresponding to our solution.As an example of such a curve, let (using the notation xµ = (t, x,x2, . . . , x D−1))zµ = B (sinh(wτ0), cosh(wτ0) − 1, 0, . . . , 0),where B and w are constants with κ1 = w = 1/B, and τ0 is defined throughcosh(wτ0) = (x + B)[t2 − (x + B)2 − r2 − B2] + 2B t R2B[t2 − (x + B)2] ,where r2 = (x2)2 + (x3)2 + · · · + (x D−1)2. Here R is the retarded distance. It isgiven byR = ± 12B√(t2 − (x + B)2 − r2 − B2)2 + 4B2 (t2 − (x + B)2).This curve has non-zero constant first curvature κ1 and all other curvatures (κi =0, i ≥ 2) are zero. The charged particle has constant acceleration √z̈µ z̈µ = κ1along the x-direction.In four dimensions since the cosmological constant is also zero, we have asolution of the Einstein-Maxwell field equations representing an (constant) accel-erated charged particle. The LBR metric and our solution given above are bothconformally flat and solutions of the Einstein-Maxwell field equations. Further-more, as we mentioned in Remark 2, both have the same curvature invariants.On the other hand LBR metric and our metric correspond to two distinct curves,namely, straight line and (non-zero) constant curvature cases, respectively. In spiteof this difference, these two metrics are transformable to each other. Letx ′µ = xµ + skµ1 + 2u + k2 s , (24)where s ≡ ηµν xµxν , u ≡ kµ xµ and kµ is a constant vector with k2 ≡ ηµνkνkµ.Here we choose k0 = 0, Bk1 = −2 and kµ = 0 for all µ > 1. Then it isstraightforward to show thatds2 = R20R′2ηµν dx′µdx ′ν = R20x2 + r2 ηµν dxµdxν, (25)Accelerated Levi-Civita-Bertotti-Robinson metric in D dimensions 2021whereR′ = ± 12B√[(t ′)2 − (x ′ + B)2 − (r ′)2 − B2]2 + 4B2 ((t ′)2 − (x ′ + B)2).This means that our solution is expressed in an (constant) accelerated frame. Thisis the reason why we observe radiation.Remark 3 The infinitesimal version of the conformal transformation (24) (con-formal Killing vector in flat Minkowski space-time) is given in Penrose andRindler [21] asξµ = −2uxµ + skµ. (26)They remark that “· · · the special conformal transformations (four parameterssometimes misleadingly called uniform acceleration transformation · · · ).” Herewe observe that these special transformations really correspond to a constantacceleration.5 ConclusionWe found accelerated LBR metrics which solve Einstein-Maxwell-null fluid fieldequations with a dimension dependent cosmological constant. We have obtainedthe energy loss formula due to the acceleration. In four dimensions it coincideswith the standard energy loss formula in flat space-time. We obtain the LBRmetric when the curve C is a straight line in MD . We also showed that there isanother LBR limit. When the curve C has constant acceleration our solution istransformable (by a special conformal mapping) to the LBR metric.Acknowledgements This work is partially supported by the Scientific and Technical ResearchCouncil of Turkey and by the Turkish Academy of Sciences.References1. Levi-Civita, T.: R. C. Acad. Lincei 26, 519 (1917)2. Bertotti, B.: Phys. Rev. 116, 1331 (1959)3. Robinson, R.: Bull. Acad. Polon. 7, 351 (1959)4. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions ofEinstein’s Field Equations, Cambridge Monographs on Mathematical Physics. (Cambridge:Cambridge University Press), Second Edition (2003)5. Gibbons, G.W., Hull, C.M.: Phys. Lett B 109, 190 (1982); Gibbons, G.W.: in Supersymme-try, Supergravity and Related Topics, eds. del Auglia, F., de Azcaraga, J., Ibanez, L.: WorldScientific, Singapore, p 147, (1985)6. Ferrara, S.: Bertotti-Robinson Geometry and Supersymmetry, Talk given at 12th ItalianConference on General Relativity and Gravitational Physics, Rome, September, (1996)(hep-th/9701163)7. Lowe, D.A., Strominger, A.: Phys. Rev. Lett. 73, 1468 (1994)8. Kallosh, R., Peet, A.: Phys. Rev D 46, 5223 (1992); Kallosh, R.: Phys. Lett B 282, 80 (1992)9. Gürses, M.: Phys. Rev. D 46, 2522 (1992)10. Cardoso, V., Dias, O.J.C., Lemos, J.P.S.: Phys. Rev. D 70, 024002 (2004)11. Gürses, M., Sermutlu, E.: Class. Quantum Grav. 12, 2799 (1995)12. Bonnor, W.B., Vaidya, P.C.: General Relativity papers in honor of Synge, J.L., edited byL. O’ Raifeartaigh (Dublin: Dublin Institute for Advanced Studies) pp 119 (1972)13. Gürses, M., Sarıoğlu, Ö.: Class. Quant Grav. 19, 4249 (2002); Erratum, ibid 20, 1413 (2003)2022 M. Gürses, Ö. Sarıoğlu14. Gürses, M., Sarıoğlu, Ö.: Class. Quant Grav. 20, 351 (2003)15. Sarıoğlu, Ö.: Phys. Rev. D 66, 085005 (2002)16. Gürses, M., Sarıoğlu, Ö.: J. Math. Phys. 44, 4672 (2003)17. Tangherlini, F.R.: Nuovo Cimento 77, 636 (1963)18. Garcı́a, A., Salazar, I.H., Plebański, J.F.: Nuovo Cimento B 84, 65 (1984)19. Trautman, A.: Phys. Rev. Lett. 46, 875 (1981)20. Jackson, J.D.: Classical Electrodynamics, New York: Wiley (1975)21. Penrose, R., Rindler, W.: Spinors and Space-Time: Spinor and twistor methods in space-time geometry Vol. 2 Cambridge University Press, Cambridge, pp 83–84, (1986)

General Relativity and Gravitation

Accelerated Levi-Civita-Bertotti-Robinson Metric in D-Dimensions

Accelerated Levi-Civita-Bertotti-Robinson Metric in D-Dimensions

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