Exact Harmonic Metric for a Moving Reissner-Nordström Black Hole
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Exact Harmonic Metric for a Moving Reissner-Nordstr ¨om Black Hole
G. He and W. Lin ∗ School of Physical Science and Technology,Southwest Jiaotong University, Chengdu 610031, China
Abstract
The exact harmonic metric for a moving Reissner-Nordstr¨om black hole with an arbitrary constant speedis presented. As an application, the post-Newtonian dynamics of a non-relativistic particle in this field iscalculated.
PACS numbers: 04.20-q, 04.20.Jb, 04.70.Bw, 95.30.Sf ∗ To whom correspondence should be addressed. Email: [email protected] . INTRODUCTION The motion of a gravitational source can affect the dynamics of particle passing by it, and thiseffect has attracted considerable attention over the last two decades [1–12]. There are severalmethods to calculate this effect. One is to directly solve Li´enard-Wiechert gravitational potentialfrom the field equations, which has been used to study light propagation in the gravitational fieldof an arbitrarily moving N-body system, as well as that with angular momentum [3, 5]. Anothermethod takes advantage of the general covariance of field equations to obtain the metric of themoving source from the known static source’s metric via Lorentz transformation [13]. Recently,this method was employed to derive the time-dependent harmonic metrics of arbitrary-constantmoving Schwarzschild and Kerr black holes [14–16].In this work, we apply a Lorentz transformation to derive the exact harmonic metric for amoving Reissner-Nordstr¨om black hole with an arbitrary constant speed. Furthermore, based onthe metric, we calculate the dynamics of a photon and a particle in the weak-field limit. In whatfollows we use geometrized units ( G = c = 1) . II. EXACT HARMONIC METRIC FOR AN ARBITRARILY CONSTANTLY MOVING REISSNER-NORDSTR ¨OM BLACK HOLE
We start with the harmonic metric of Reissner-Nordstr¨om black hole, which can be writtenas [17] ds = − R − m + Q ( R + m ) dX + (cid:16) mR (cid:17) (cid:20) δ ij + m − Q R − m + Q X i X j R (cid:21) dX i dX j , (1)where m and Q are the rest mass and electric charge of the black hole, respectively. i, j =1 , , , and δ ij denotes Kronecker delta. Notice that here X µ denotes the contravariant vector x ′ µ = ( t ′ , x ′ , y ′ , z ′ ) for display convenience, and R = X + X + X .Since Einstein field equations have the property of general covariance, the harmonic metricof a constantly moving R-N black hole can be obtained via applying a Lorentz boost to Eq. (1).We denote the coordinate frame of the background as ( t, x, y, z ) , and assume the velocity ofthe black hole to be v = v e + v e + v e , with e i ( i = 1 , , denoting the unit vector of3-dimensional Cartesian coordinates. The Lorentz transformation between ( t, x, y, z ) and thecomoving frame ( t ′ , x ′ , y ′ , z ′ ) of the moving hole can be written as x ′ α = Λ αβ x β , (2)2ith Λ = γ , (3) Λ i = Λ i = − v i γ , (4) Λ ij = δ ij + v i v j γ − v , (5)where γ = (1 − v ) − is the Lorentz factor and v = v + v + v . Therefore, the exact harmonicmetric of the moving Reissner-Nordstr¨om black hole can be obtained as follows g = − γ ( R − m + Q )( R + m ) + γ (cid:16) mR (cid:17) (cid:20) v + ( v · X ) ( m − Q ) R ( R − m + Q ) (cid:21) , (6) g i = v i γ (cid:20) R − m + Q ( R + m ) − (cid:16) mR (cid:17) (cid:21) − γ (cid:16) mR (cid:17) m − Q R ( R − m + Q ) × (cid:20) X i ( v · X )+ v i ( γ − v · X ) v (cid:21) , (7) g ij = (cid:16) mR (cid:17) (cid:26) δ ij + m − Q R ( R − m + Q ) (cid:20) X i + v i ( γ − v · X ) v (cid:21) × (cid:20) X j + v j ( γ − v · X ) v (cid:21)(cid:27) + v i v j γ (cid:20)(cid:16) mR (cid:17) − R − m + Q ( R + m ) (cid:21) . (8)If we set Q = 0 , Eqs. (6) - (8) reduce to the harmonic metric of a moving Schwarzschild blackhole with velocity v g = − γ (1 + Φ)1 − Φ + v γ (1 − Φ) + γ Φ (1 − Φ)1 + Φ ( v · X ) R , (9) g i = v i γ (cid:20) − Φ − (1 − Φ) (cid:21) − γ Φ (1 − Φ)1 + Φ (cid:20) X i ( v · X ) R + v i ( γ − v · X ) v R (cid:21) , (10) g ij = (1 − Φ) δ ij + Φ (1 − Φ) R (1 + Φ) (cid:20) X i + v i ( γ − v · X ) v (cid:21)(cid:20) X j + v j ( γ − v · X ) v (cid:21) + v i v j γ (cid:20) (1 − Φ) − − Φ (cid:21) , (11)which are the extension of the exact metric [14] for a Schwarzschild black hole with v = v e .Here R is also equal to p X + X + X . It is worth pointing out that Eqs. (9) - (11), to thefirst post-Minkowskian approximation, are in agreement with the gravitational Li´enard-Wiechertretarded solution [3]. III. DYNAMICS OF PARTICLE IN THE WEAK-FIELD LIMIT
As an application, we apply the harmonic metric to derive the post-Newtonian dynamics of aneutral and non-relativistic particle in the far field of the moving Reissner-Nordstr¨om black hole.3irst, we expand Eqs. (6) - (8) up to an order of /R g = − − v )Φ − − Q R , (12) g i = 4 v i Φ , (13) g ij = (1 − δ ij , (14)where the velocity of the black hole has also been assumed to be non-relativistic, i.e., γ ≃ . Aftertedious but straightforward calculations, up to the order of v /r ( v and r denote typical valuesof velocity and separation of a system of particles, respectively), we can obtain the equation ofmotion of a massive particle as follow d u dt = −∇ (cid:18) Φ+2 v Φ+2Φ + Q R (cid:19) − ∂ ζ ∂t + u × ( ∇× ζ )+ 3 u ∂ Φ ∂t + 4 u ( u · ∇ ) Φ − u ∇ Φ , (15)where u denotes the velocity of the particle, and ζ = 4 v Φ . When the charge of the black holevanishes, this equation reduces to the post-Newtonian dynamics of a non-relativistic particle in thefield of a moving Schwarzschild black hole [14, 18]. IV. CONCLUSION
The metric in harmonic coordinates plays an important role in the post-Newtonian dynamicsand gravitational wave radiation. In this work we obtain the exact metric for a moving Reissner-Nordstr¨om black hole via applying a Lorentz boost to the Reissner-Nordstr¨om metric in the har-monic coordinates. This method can avoid directly solving the Einstein field equations for amoving gravitational source. Based on this metric, we derive the post-Newtonian dynamics ofa non-relativistic particle. This metric can also be used to calculate the deflection and time delayof light passing by a non-static Reissner-Nordstr¨om black hole, as well as Hawking radiation ofthe black hole.
ACKNOWLEDGMENTS
This work was supported in part by the Program for New Century Excellent Talents in Univer-sity (Grant No. NCET-10-0702), the National Basic Research Program of China (973 Program)(Grant No. 2013CB328904), and the Ph.D. Foundation of the Ministry of Education of China4Grant No. 20110184110016). [1] K. H. Look, C. L. Tsou and H. Y. Kuo,
Acta Phys. Sin. , 225 (1974).[2] T. Pyne and M. Birkinshaw, Astrophys. J. , 459 (1993).[3] S. M. Kopeikin and G. Sch¨afer,
Phys. Rev. D , 124002 (1999).[4] M. Sereno, Phys. Lett. A , 7 (2002).[5] S. M. Kopeikin and B. Mashhoon,
Phys. Rev. D , 064025 (2002).[6] M. Q. Miao, S. J. Qing and L. C. An, Acta Phys. Sin. , 049 (2003).[7] M. Sereno, Mon. Not. R. Astron. Soc. , L19 (2005).[8] M. Sereno,
Mon. Not. R. Astron. Soc. , 1023 (2007).[9] S. M. Kopeikin and V. V. Makarov,
Phys. Rev. D , 062002 (2007).[10] C. Bonvin, Phys. Rev. D , 123530 (2008).[11] S. C. Novati, M. Dall’Ora et al., Astrophys. J. , 987 (2010).[12] S. Zschocke and S. A. Klioner,
Class. Quantum Grav. , 015009 (2011).[13] O. Wucknitz and U. Sperhake, Phys. Rev. D , 063001 (2004).[14] G. He and W. Lin, Commun. Theor. Phys. , 270 (2014).[15] G. He and W. Lin, Int. J. Mod. Phys. D , 1450031 (2014).[16] G. He, C. Jiang and W. Lin, under review (2014).[17] W. Lin and C. Jiang, Phys. Rev. D , 087502 (2014).[18] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Rel-ativity , Wiley, New York (1972)., Wiley, New York (1972).