Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole

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.

Full text

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
41 + Φ
1−Φ−1−Φ
1 + Φ
X2
1
R2i,(23)
gi0=4vγ 2Φhδi1+Φ
41 +Φ
1−Φδi1−1−Φ
1 + Φ
X1Xi
R2γδi1−1i,(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
+ 4u1v2γ2h1−1
u1v∂Φ
∂t +u· ∇Φie1,(29)
where
∂Φ
∂t =vγX1
R2Φ,(30)
∇Φ = e1
∂
∂x1+e2
∂
∂x2+e3
∂
∂x3Φ
=−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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