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Commun. Theor. Phys. 61 (2014) 270–272 Vol. 61, No. 2, February 1, 2014

Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole∗

HE Guan-Sheng ( ) and LIN Wen-Bin ( )†

School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China

(Received August 9, 2013)

Abstract The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of

weak ﬁeld and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an

application, we derive the dynamics of particle and photon in the weak-ﬁeld limit for the moving Schwarzschild black

hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional

centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian

gravitational force.

PACS numbers: 95.30.Sf, 04.20.Cv, 04.20.Jb, 04.25.Nx, 04.70.Bw

Key words: Schwarzschild black hole, Harmonic coordinates, post-Newtonian dynamics, moving source

1 Introduction

The gravitational ﬁeld of moving source and the re-

lated dynamics of particle or photon are of great interest

to the relativity community.[1−5] Aichelburg and Sexl de-

rived the metric and investigated the gravitational ﬁeld

for a massless point particle moving with the velocity of

light, via the combination of the isotropic Schwarzschild

metric and a singular Lorentz transformation.[6] Kopeikin

and Sch¨afer developed the Lorentz covariant theory of

propagation of light in the weak gravitational ﬁelds of N

point masses with arbitrary velocities, in which the au-

thors derived the explicit formulation of light deﬂection

in term of the retarded time and investigated the light

propagation for moving gravitational lenses in detail.[7]

With the combination of a nearly Lorentz transformation

and the isotropic Schwarzschild metric, Adler obtained

the parameterized metric of a point mass for the motion

in any direction.[8] Wucknitz and Sperhake presented the

“2 + 1”D space-time metric of the gravitational lens with

an arbitrary velocity in the weak-ﬁeld limit, and investi-

gated the eﬀects of the lens’ high velocity on the dynamics

of the particle and photon in detail.[9]

In this paper we present the exact metric of a uni-

formly moving Schwarzschild black hole in the harmonic

coordinates. Note the metric outside the horizon of

Schwarzschild black hole is same as that of point mass,

we do not discriminate them through this paper. As an

application, we derive explicit formulations for the dynam-

ics of photon and particle in the weak-ﬁeld limit of the

gravitational source whose velocity is close to the speed of

light.

This paper is organized as follows. Section 2 presents

the derivation of the harmonic metric of a uniformly mov-

ing point mass. Section 3 calculates the dynamics of pho-

ton and particle in the weak ﬁeld. Conclusion is given in

Sec. 4. We use unit with c= 1 throughout. Since the grav-

itational ﬁeld of the static point mass can be described by

Schwarzschild metric, we do not distinguish the metric (or

ﬁeld) of the point mass from that of Schwarzschild black

hole.

2 Metric for Uniformly Moving Point Mass

in Harmonic Coordinates

The energy-momentum tensor of a static point mass

mlocated at the coordinate origin (0,0,0) can be written

as

¯

T00 =mδ3(¯

x),(1)

¯

T0i=¯

Ti0= 0 ,(2)

¯

Tij = 0 ,(3)

where ¯

x≡(¯x1,¯x2,¯x3), and δ3is 3D Dirac delta function.

i, j = 1,2,3.

The gravitational ﬁeld for the point mass is described

by Schwarzschild metric, the harmonic form of which can

be written as[10]

¯g00 =−1 + Φ

1−Φ,(4)

¯g0i= 0 ,(5)

¯gij = (1 −Φ)2δij + Φ2(1 −Φ)

(1 + Φ)

¯xi¯xj

r2,(6)

where δij denotes Kronecker delta. Gis gravitational con-

stant, Φ = −Gm/r is Newtonian potential, and r2=¯

x2.

∗Supported by the Program for New Century Excellent Talents in University under Grant No. NCET-10-0702, the National Basic

Research Program of China (973 Program) under Grant No. 2013CB328904, and the Ph.D. Programs Foundation of Ministry of Education

of China under Grant No. 20110184110016

†Corresponding author, E-mail: wl@swjtu.edu.cn

c

2013 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

No. 2 Communications in Theoretical Physics 271

The energy-momentum tensor of a uniformly moving

point mass can be obtained from the energy-momentum

tensor of the static point mass via Lorentz transformation

with

t=γ(¯

t+v¯x1), x1=γ(¯x1+v¯

t),

x2= ¯x2, x3= ¯x3,(7)

here we assume the velocity vector vof the point mass is

along the positive x1-axis without loss of generality, and

γ= (1 −v2)−1/2is Lorentz factor. We have

T00 =γ2m δ3(X),(8)

T01 =vγ2m δ3(X),(9)

T11 =v2γ2m δ3(X),(10)

where X≡(X1, X2, X3)≡(γx1−γ vt, x2, x3).

The multipole expansion of the energy-momentum ten-

sor of a uniformly moving point mass can be written as

T00 =

0

T00 +

2

T00 +···

= (m+mv2+···)δ3(X),(11)

T01 =

1

T01 +

3

T01 +···

=v(m+mv2+···)δ3(X),(12)

T11 =

2

T11 +

4

T11 +···

=v2(m+mv2+···)δ3(X).(13)

The multipole expansion of the uniformly moving point

mass and momentum can be calculated as follows[10]

0

M≡Z0

T00 d3x=m , (14)

2

M≡Z(

2

T00 +

2

T11) d3x= 2v2m , (15)

1

Pi≡Z1

T0id3x=mvδi1,(16)

where x= (x1, x2, x3), and the multipole ﬁelds for the

uniformly moving point mass can be written as

Φ = −G

0

M

r=−Gm

r,(17)

ζi=−4G

1

Pi

r=−4Gmv

rδi1,(18)

Ψ = −G

2

M

r= 2v2Φ,(19)

where ζis a new vector potential, and Ψ is called as the

second potential. The metric tensor based on these multi-

pole ﬁelds, which is also regarded as the post-Newtonian

approximation, can be written as[10]

g00 ≃ −1−2Φ −2Φ2−2Ψ = −1−2Φ −2Φ2−4v2Φ,(20)

g0i≃ζi= 4vΦδi1,(21)

gij ≃δij −2δijΦ.(22)

The above multipole ﬁelds or the corresponding metric

tensor can only describe the far ﬁeld for the slowly moving

gravitational source. Therefore, we need a more accurate

metric tensor to describe the gravitational ﬁeld in the near

region of one moving black hole, or in both the near and

far regions of the gravitational source with a relativistic

velocity. In this paper the near region is deﬁned as the

one in which Φ ∼1.

Einstein’s equation is generally covariant, therefore,

the metric of the uniformly moving point mass can be

obtained via applying the Lorentz transformation Eq. (7)

to the harmonic Schwarzschild metric Eqs. (4)–(6). After

performing straightforward but tedious calculations, we

ﬁnally obtain

g00 =−1 + Φ

1−Φ−4v2γ2Φh1+ Φ

41 + Φ

1−Φ−1−Φ

1 + Φ

X2

1

R2i,(23)

gi0=4vγ 2Φhδi1+Φ

41 +Φ

1−Φδi1−1−Φ

1 + Φ

X1Xi

R2γδi1−1i,(24)

gij = (1 −Φ)2δij + Φ21−Φ

1 + Φ

XiXj

R2γδi1+δj1

−4v2γ2Φh1 + Φ

4

1 + Φ

1−Φiδi1δj1,(25)

where R=pX2

1+X2

2+X2

3=rdenotes the distance

between the ﬁeld point and the source point, i.e., the

distance without Lorentz contraction. Notice that Φ =

−Gm/r =−Gm/R.

Since the harmonic coordinate condition is Lorentz co-

variant and Eqs. (4)–(6) are exact, Eqs. (23)–(25) are the

exact harmonic metric for the uniformly moving point

mass or Schwarzschild black hole with an arbitrary ve-

locity. To our knowledge, this formulation has not been

reported before.

It can be seen that this harmonic metric reduce to

the harmonic Schwarzschild metric if one sets v= 0, and

from these equations we can also recover the multipole

ﬁelds of the moving point mass (see Eqs. (20)–(22)) when

v2∼Φ≪1.

3 Dynamics of Particle and Photon in Weak-

Field Limit

As an application, we consider the dynamics of photon

and particle in the far region with Φ ≪min{1,1/vγ2}.

In this region, the metric Eqs. (23)–(25) can be approxi-

mated as follows:

g00 ≃ −1−2Φ −4v2γ2Φ,(26)

gi0≃4vγ2Φδi1,(27)

gij ≃(1 −2Φ)δij −4v2γ2Φδi1δj1.(28)

It can be seen that Eqs. (26)–(28) are consistent with the

“2 + 1”D metric given by Wucknitz and Sperhake.[9]

The dynamics of photon and particle can be obtained

via the geodesic equation. Let u= dx/dtdenote the

contravariant velocity vector of photon. After performing

272 Communications in Theoretical Physics Vol. 61

tedious but straightforward calculations of the Christoﬀel

symbols, we have

du

dt=−[1 + u2+ 2v2γ2(1 + u2

1)]∇Φ + 4vγ2u

×[∇ × (Φe1)] + h4γ2(1 −vu1)(u· ∇Φ)

+ (1 −u2+ 2γ2(1 −v2u2

1))∂Φ

∂t iu

+ 4u1v2γ2h1−1

u1v∂Φ

∂t +u· ∇Φie1,(29)

where

∂Φ

∂t =vγX1

R2Φ,(30)

∇Φ = e1

∂

∂x1+e2

∂

∂x2+e3

∂

∂x3Φ

=−X+ (γ−1)X1e1

R2Φ,(31)

∇ × (Φe1) = X2e3−X3e2

R2Φ.(32)

Here eidenotes the unit vector of xi-axis, and u1denotes

the x1-axis component of the contravariant velocity vec-

tor ufor the convenience of display. Equation (29) is also

valid for the particle dynamics. Let wdenote the parti-

cle velocity, whose amplitude is usually very small. For

w2∼Φ≪1, the particle dynamics can be written as

dw

dt≃ − (1 + 2v2γ2)∇Φ

= (1 + 2v2γ2)X+ (γ−1)X1e1

R2Φ.(33)

Notice that X1,X,R, and Φ are functions of t. This equa-

tion shows that the test particle undergoes an additional

retarded centripetal force due to the relativistic motion of

source, since 2v2γ2is always positive.

On the other hand, it can also be drawn from Eqs. (29)

and (33) that the force induced by the source motion may

be comparable to or even larger than the conventional

Newtonian gravitational force when the velocity of source

is high enough.

4 Conclusion

Basing on the general covariance of Einstein’s equa-

tion, we apply Lorentz transformation to the harmonic

form of Schwarzschild metric, and obtain the metric for

the uniformly moving Schwarzschild black hole in the har-

monic coordinates. To our knowledge, this harmonic met-

ric has not been reported before. When the velocity of

gravitational source is low, the harmonic metric reduces to

the post-Newtonian approximation for the moving point

mass in the weak-ﬁeld limit. As an application, we derive

the dynamics of photon and particle in the far ﬁeld of the

moving Schwarzschild black hole with an arbitrary veloc-

ity. It is found that the relativistic motion of source can

introduce an additional retarded centripetal force on the

test particle, which may be comparable to or even larger

than Newtonian gravitational force. The harmonic metric

obtained in this work might also be helpful in studying the

dynamics of particle or photon in the strong ﬁeld domi-

nated by one fast moving massive black hole.

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