THE ENERGY-MOMENTUM IN GRAVITATIONAL
RADIATION
Introduction Gravitational waves are predicted by Einstein’s general
relativity. They are time dependent solutions of the Einstein field equations
which radiate or transport energy. Although they have not been detected
yet, the existence of gravitational waves has been proved indirectly from
observations of the pulsar PSR 1913+16. This rapidly rotating binary system should emit gravitational radiation, hence lose energy and rotate faster.
The observed relative change in period agrees remarkably with the theoretical value.
In [1, 2, 3], Bondi, van der Burg, Metzner and Sachs studied systematically
the theory of gravitational radiation. They assumed the vacuum spacetime
L3,1 , gBondi (possibly with black holes) takes the following Bondi’s radiating metric
V
gBondi = − − e2β + r2 e2γ U 2 cosh 2δ + r2 e−2γ W 2 cosh 2δ
r
+2r2 U W sinh 2δ du2 − 2e2β dudr
−2r2 e2γ U cosh 2δ + W sinh 2δ dudθ
−2r2 e−2γ W cosh 2δ + U sinh 2δ sin θdudψ
+r2 e2γ cosh 2δdθ2 + e−2γ cosh 2δ sin2 θdψ 2
+2 sinh 2δ sin θdθdψ
(1)
where β, γ, δ, U , V and W are functions of x0 = u, x1 = r, x2 = θ, x3 = ψ
which are smooth for r ≥ r0 > 0, u is a retarded coordinate, r is a luminosity
distance, θ and ψ are spherical coordinates, 0 ≤ θ ≤ π, 0 ≤ ψ ≤ 2π. The
outgoing radiation condition implies that the following asymptotic behaviors
hold for r sufficiently large
1
c(u, θ, ψ)
+O 3 ,
r
r
1
d(u, θ, ψ)
+O 3 ,
δ =
r
r
2
2
c +d
1
+
O
,
β = −
4r2
r4
1
c,2 + 2c cot θ + d,3 csc θ
U = −
+
O
,
r2
r3
γ =
1
2
THE ENERGY-MOMENTUM IN GRAVITATIONAL RADIATION
1
d,2 + 2d cot θ − c,3 csc θ
+
O
,
r2
r3
1
V = −r + 2M (u, θ, ψ) + O
r
∂f
(where f,i = ∂xi for i = 0, 1, 2, 3). The following conditions are assumed:
Condition A: Each of the six functions β, γ, δ, U , V , W together
with its derivatives up to the second orders are equal at ψ = 0 and
2π.
Condition B: For all u,
Z 2π
Z 2π
c(u, 0, ψ)dψ = 0,
c(u, π, ψ)dψ = 0.
W
0
= −
0
Furthermore, the physics requires that the retarded time u = constant is
a null hypersurface, and this is the cases in the Minkowski spacetime (where
u = t − r) and in the Schwarzschild spacetime (where u = t − r − 2m ln r −
2m).
The ADM mass at spatial infinity Arnowitt, Deser and Misner defined the total energy-momentum at spatial infinity of an asymptotically flat
spacetime as follows: Let (M 3 , gij , hij ) be an asymptotically flat spacelike
hypersurface such that, outside a compact subset, M is diffeomorphic to R3
minus a ball with the metric g, and the symmetric 2-tensor h satisfies the
following asymptotic conditions:
1
1
1
,
∂k gij = O 2 ,
∂k ∂l gij = O 3 ,
gij = δij + O
r
r
r
1
1
hij = O 2 ,
∂k hij = O 3 .
r
r
The ADM total energy E and the ADM total linear momentum Pk are
Z
1
(∂j gij − ∂i gjj ) ∗ dxi ,
E =
lim
16π r→∞ Sr
Z
1
Pk =
lim
(hki − gki hjj ) ∗ dxi ,
8π r→∞ Sr
where Sr is the sphere with radius r in R3 .
In 1979, Schoen-Yau [4] proved the the positive mass theorem: If
a spacetime satisfies the dominant energy condition, i.e., for any timelike
vector W , Tuv W u W v ≥ 0, and T uv Wu is a non-spacelike vector, then, for
asymptotically flat initial data set (M 3 , gij , hij ),
sX
P2k .
E≥
k
E = 0 implies that the spacetime is flat over M . This solved the positive
mass conjecture in general relativity. In 1981, Witten [5] found a new proof
THE ENERGY-MOMENTUM IN GRAVITATIONAL RADIATION
3
by using spinors and the Dirac operator.
The Bondi mass at null infinity Now let us go back to the Bondi’s
radiating vacuum spacetime. On the null hypersurface {u = u0 } the Bondi
energy-momentum is defined as [1, 2, 3]:
Z
1
mν (u0 ) =
M (u0 , θ, ψ)nν dS
4π S 2
for ν = 0, 1, 2, 3, where n0 = 1, n1 = sin θ cos ψ, n2 = sin θ sin ψ, n3 = cos θ.
The Bondi energy-momentum is the total energy-momentum measured after
the loss due to the gravitational radiation up to that time. In the paper [1],
Bondi proved that the Bondi mass is non-increasing w.r.t. u, i.e., more and
more energy is radiated away.
It is a trivial consequence of a conserved stress-energy tensor with a positive timelike component that physical systems cannot radiate away more
energy than they have initially. However, the gravitational field does not
have a well-defined stress-energy tensor. Is it possible that a finite gravitational system might be able to radiate arbitrarily large amounts of energy
? The conjecture that this is impossible is known as the positive mass conjecture at null infinity. There is no mathematical setting available of this
conjecture in general spacetimes. In Bondi’s radiating vacuum spacetimes,
the conjecture says that the Bondi mass must be nonnegative.
The outlines of the proof that the Bondi mass is nonnegative were given by
Schoen-Yau[7] by solving the Jang’s equation and physicists (Israel-Nester,
Horowitz-Perry, Ashtekar-Horowitz, Ludvigsen-Vickers, Renla-Tod, etc. [8])
by using Witten’s spinor argument. However, no mathematical detail was
provided in any of those proofs. The idea is to choose certain spacelike
hypersurfaces approaching to null infinity. These spacelike hypersurfaces
are asymptotically hyperbolic with the nontrivial second fundamental forms
in the Bondi’s radiating spacetimes. Therefore, it requires to establish the
positive mass theorem for these spacelike hypersurfaces. In 2002 X. Zhang
[9, 10] was able to find a complete and rigorous proof of this positive mass
theorem near null infinity by using Witten’s method. Recently, together
with Yau and Zhang, we prove :
Theorem
1 (W.-l. Huang, S. T. Yau, X. Zhang, 2005 [11]). Let
L3,1 , g̃ be a vacuum Bondi’s radiating spacetime with metric g̃ given by
(1). Suppose that Condition A and Condition B hold. If M(u0 , θ, ψ) is
constant or c|u=u0 = d|u=u0 = 0 for some u0 , then
s X
m2i (u)
m0 (u) ≥
1≤i≤3
4
THE ENERGY-MOMENTUM IN GRAVITATIONAL RADIATION
for all u ≤ u0 . If the equality holds for some u ∈ (−∞, u0 ], then L3,1 is flat
in the region foliated by all spacelike hypersurfaces which are given by
p
1
u = u0 + 1 + r2 − r + o( 4 )
r
for r sufficiently large. In particular, if the equality holds for all u ≤ u0 ,
then L3,1 is flat in the region {u ≤ u0 }.
The ADM energy-momentum and the Bondi energy-momentum
It is a fundamental problem in gravitational radiation what the relation is
between the ADM energy-momentum and the Bondi energy-momentum.
Assuming that the spacetime can be conformally compactified and that
it is asymptotically empty and flat at null and spatial infinity in a certain
sense, Ashtekar and Magnon-Ashtekar [12] demonstrated in 1979 that the
mass at spatial infinity is the past limit of the Bondi mass. Here, the “past
limit” means limu→−∞ mν (u). (In 2003, Hayward [13] proved this theorem
in a new framework for spacetime asymptotics, replacing the Penrose conformal factor by a product of advanced and retarded conformal factors.)
In 1993, Christodoulou and Klainerman [14] proved the global existence of
globally hyperbolic, strongly asymptotically flat (Condition C), maximal foliated vacuum solutions of the Einstein field equations. They also proved
rigorously that the ADM mass at spatial infinity is the past limit of the
Bondi mass in these spacetimes.
In 2004, X. Zhang [15] studied this problem in Bondi’s radiating vacuum
spacetime. He defined the spatial infinity as the t-slices where the “real”
time t is defined as t = u + r. Denote E(t0 ) by P0 (t0 ). Then he verified that
Pν (t0 ) = mν (−∞)
for ν = 0, 1, 2, 3 under the asymptotic flatness assumptions at spatial infinity
which ensure the Schoen-Yau’s positive mass theorem. In this case, the
ADM total energy, the ADM total linear momentum of (spatial) t0 -slice
and the Bondi energy-momentum of (null) u0 -slice satisfy
Z u0 Z 1
(c,0 )2 + (d,0 )2 nν dSdu.
Pν (t0 ) = mν (u0 ) +
4π −∞ S 2
In particular, if there is news c,0 , d,0 , then the ADM total energy is always
greater than the Bondi mass.
Unfortunately, the asymptotic flatness conditions (Condition C) at spatial
infinity in all above works preclude gravitational radiation. In the paper[15]
X. Zhang assumed certain weaker asymptotic flatness conditions at spatial
infinity in order to include gravitational radiation:
THE ENERGY-MOMENTUM IN GRAVITATIONAL RADIATION
Condition D:
γ ∈ C{1,1,1} , δ ∈ C{1,1,1} ,
C{2,2,2} , W ∈ C{2,2,2} , V + r ∈ C{0,0,0} .
β ∈ C{2,2,2} ,
5
U ∈
and proved
Theorem 2 (X. Zhang, 2004 [15]). Let E(t0 ) be the ADM total energy of
spacelike hypersurface Nt0 . Under Condition A, Condition B and Condition D, we have
Z
1
lim
c2 + d2 ,0 dS.
E(t0 ) = m0 (−∞) +
u→−∞
4π
S2
Recently, we have found the relations between the ADM linear momentum of any t0 -slice and the past limit of the Bondi momentum:
Theorem 3 (W.-l. Huang, X. Zhang, 2005 [16]). Let Pk (t0 ) be the ADM
total linear momentum of spacelike hypersurface Nt0 .Under Condition A,
Condition B and Condition D, we have
Z π Z 2π
1
Pk (t0 ) = mk (−∞) +
lim
Pk dψ dθ
8π u→−∞ 0 0
for k = 1, 2, 3, where Pk are given in the appendix of [16]. In axi-symmetric
spacetimes where c = c(u, θ), d = 0 [1],
P1 (t0 ) = m1 (−∞),
P2 (t0 ) = m2 (−∞).
However, one cannot expect the “real” time to be t = u + r in general
(e.g., the Schwarzschild spacetime). In [17], the case
t = u + r + f (r, θ, ψ)
(2)
is studied for r sufficiently large. Here, f is a smooth function which has
the following asymptotic behavior:
f = a1 ln r + a2 (θ, ψ) + a3 (r, θ, ψ)
(3)
for r sufficiently large, where a1 is a constant and a2 , a3 are smooth functions
which satisfy
ai = ai , ai,A = ai,A , ai,AB = ai,AB ψ=0
ψ=2π
ψ=0
ψ=2π
ψ=0
for i = 2, 3, A, B = 2, 3. Moreover,
1
˘ k a3 = O 1 , ∇
˘ l∇
˘ k a3 = O 1 .
, ∇
a3 = O
r
r2
r3
for r sufficiently large.
ψ=2π
6
THE ENERGY-MOMENTUM IN GRAVITATIONAL RADIATION
Theorem 4 (W.-l. Huang, X. Zhang, 2006
[17]). Let E(t0 ) be the ADM
total energy of the initial data set Nt0 , g, h where t is given by (2) with
condition (3). Under Condition A, Condition B and Condition D, we
have
Z π Z 2π
1
c2 + d2 ,0 sin θ dψ dθ
E(t0 ) = m0 (−∞) +
lim
4π u→−∞ 0 0
Z π Z 2π
1
lim
a2,2 sin θ l,0 + a2,3 ¯l,0 dψ dθ.
+
16π u→−∞ 0 0
Theorem 5 (W.-l. Huang, X. Zhang, 2006 [17]). Let
Pk (t0 ) be the ADM
total linear momentum of the initial data set Nt0 , g, h where t is given by
(2) with condition (3). Under Condition A, Condition B and Condition
D, we have
Z π Z 2π 1
Pk (t0 ) = mk (−∞) +
lim
Pk + DP k dψ dθ
8π u→−∞ 0 0
for k = 1, 2, 3, where Pk , DP k are given in the appendix in [17].
References
[1] H. Bondi, M. van der Burg, A. Metzner, Proc. Roy. Soc. London A 269(1962)21.
[2] R. Sachs, Proc. Roy. Soc. London A 270(1962)103.
[3] M. van der Burg, Proc. Roy. Soc. London A 294(1966)112. IV(1982)175.
17(2)(1976)174.
[4] R. Schoen, S.T. Yau, Commun. Math. Phys., 65(1979)45, 79(1981)47, 79(1981)231.
[5] E. Witten, Commun. Math. Phys. 80(1981)381.
[6] X. Zhang, Commun. Math. Phys., 206(1999)137.
[7] R. Schoen, S.T. Yau, Phys. Rev. Lett. 48(1982)369.
[8] W. Israel, J. Nester, Phys. Lett. 85A(1981)259; G. Horowitz, M. Perry, Phys.
Rev. Lett. 48(1982)371; A. Ashtekar, G. Horowitz, Phys. Lett. 89A(1982)181; G.
Horowitz, P. Tod, Commun. Math. Phys. 85(1982)429; M. Ludvigsen, J. Vickers, J.
Phys. A: Math. Gen.15(1982)L67; O. Reula, K. Tod, J. Math. Phys. 25(1984)1004.
[9] X. Zhang, Commun. Math. Phys., 249(2004)529.
[10] X. Zhang, The positive mass theorem near null infinity, Proceedings of ICCM
2004, December 17-22, Hong Kong (eds. S.T. Yau, etc.), AMS/International Press,
Boston, to appear.
[11] W.-l. Huang, S.T. Yau, X. Zhang, Positivity of the Bondi mass in Bondi’s radiating
spacetimes, to appear in Rendiconti Lincei.
[12] A. Ashtekar, A. Magnon-Ashtekar, Phys. Rev. Lett. 43(1979)181
[13] S. Hayward, Phys. Rev. D68(2003)104015.
[14] D. Christodoulou, S. Klainerman, The global nonlinear stablity of Minkowski space,
Princeton Math. Series 41(1993), Princeton Univ. Press.
[15] X. Zhang, On the relation between ADM and Bondi energy-momenta, to appear
in ATMP.
[16] W.-l. Huang, X. Zhang, On the relation between ADM and Bondi energy-momenta
II - radiative spatial infinity, preprint, 2005, gr-qc/0511036.
[17] W-l. Huang, X. Zhang, On the relation between ADM and Bondi energy-momenta
III - perturbed radiative spatial infinity, preprint, 2006, gr-qc/0601022v2.