2007 春季NCTS&CMS專題課程 Topical Program in Analysis and Geometry:Gravitational radiation and the Bondi mass

Gravitational radiation and the Bondi mass

National Center for Theoretical Sciences, Mathematics Division March 16th

, 2007

Wen-ling Huang Department of Mathematics University of Hamburg, Germany

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Gravitational waves (GW)

• predicted by Einstein’s general relativity

• time dependent solutions of Einstein’s field equations which radiate or transport energy

• created by accelerated masses • radiate with the light speed

Examples for GW

• Binary pulsars

• Collapsing black holes • Double star

Detection of GW

Gravitational waves encode the history of the universe.

However, they are very weak and have not been detected yet.

How weak are they ?

For black holes that weigh about 10 times as much as the sun and which are a billion light-years away, the wave strength is about 10−21 when they arrive at the Earth.

Therefore the waves produce tides in the earth’s ocean by 10−21_· 107 = 10−14 meter, or 10 times the diameter of an atom’s nucleus. There exist several GW detectors on the world.

GW Detectors

• LIGO – USA

• VIRGO – France + Italy

• Geo 600 – Germany and GB • TAMAX – Japan

Detection of GW (2)

Although they have not been detected yet, the existence of grav-itational 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.

GW and energy

A fundamental conjecture is that gravitational waves can not carry away more energy than they have initially in an isolated gravitational system.

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Bondi’s radiating space-time (1) Historical notes

• Late 1950s-early 1960s:

Pirani, Bondi, Robinson, Trautman and others • Bondi, Van der Burg, and Metzner:

– vacuum solutions of Einstein’s field equations outside an isolated (i.e. spatially bounded) axisymmetric system

– definition of mass along outgoing null hypersurfaces in the limit r _{→ ∞}

– the total mass, measured at null infinity, is non-increasing with respect to the retarded time

Ref.: H. Bondi, M. van der Burg, A. Metzner, Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London A 269(1962), 21-52.

Bondi’s radiating space-time (2) Historical notes

• Sachs 1962:

Generalization of the work of Bondi, Van der Burg, Metzner to asymptotically flat space-times

Ref.: R. Sachs, Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London, A 270(1962), 103-126.

Bondi’s radiating space-time (3) Historical notes

Interpretation of Bondi mass as the total mass of the isolated physical system measured after the loss due to the gravita-tional radiation up to that time

Bondi’s radiating space-time (4) Retarded time

Minkowski space-time:

• photon emitted when t = t0 takes a certain amount of time to

reach an observer located at distance r _{≥ 0 from the source,} so the observer notices it when t = t₁,

t₁ = t₀ + r

c = t0 + r (where speed of light c := 1) • the time t₀ = t_{1 − r} is defined as retarded time.

• set u = t _{− r}:

the hypersurface defined by u = t _{− r = k}, k a constant, is the future directed null cone with vertex r = 0, t = k.

Bondi’s radiating space-time (5) Bondi’s radiating space-time

Vacuum space-time (T_ij = 0) with Bondi’s radiating metric

gBondi =



+ r2e2γU2cosh 2δ + r2e−2γW2cosh 2δ + 2r2U W sinh 2δ ₋ V r e

2_β

du2 −2e2βdu dr _{− 2r}2e2γU cosh 2δ + W sinh 2δdu dθ

−2r2e−2γW cosh 2δ + U sinh 2δsin θ du dϕ

+r2e2γ cosh 2δ dθ2 + e−2γ cosh 2δ sin2θ dϕ2 + 2 sinh 2δ sin θ dθ dϕ

• β, γ, δ, U, V , W : functions of (u, r, θ, ϕ) which are smooth for 0 < r_{0 ≤ r, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π}

• u: retarded time, r: radius function, θ, ϕ: spherical coordinates.

Bondi’s radiating space-time (6) Examples

(i) Minkowski space-time:

g_Mink = _−dt2 + dr2 + r2 dθ2 + sin2 θdϕ2
.

Using the retarded time u = t _{− r, the metric can be written} as

Bondi’s radiating space-time (7) Examples

(ii) Schwarzschild space-time:

g_Schw = _{− 1 −} 2m r
dt

2 ₊ dr2

1 ₋ 2m_r + r

2 _dθ2 _{+ sin}2 _θdϕ2
_,

where m is the mass.

Using the retarded time u = t_{−r −2m ln} r−2m  , the metric can be written as g_Schw = _{− 1 −} 2m r
du 2 − 2dudr + r2 dθ2 + sin2 θdϕ2
.

Bondi’s radiating space-time (8) Examples

(iii) Kerr space-time: gKerr = −  1 ₋ 2mr Σ  dt2 ₋ 4mar sin 2 _θ Σ dtdφ + Σ 4dr 2 _{+ Σdθ}2 ₊ _r2 _{+ a}2 ₊ 2mra2 sin2 θ Σ2  sin2 θdφ2 where Σ _{≡ r}2 + a2 cos2 θ, _{4 ≡ r}2 _{− 2mr + a}2,

m is the mass, ma is the angular momentum as measured from infinity.

Retarded time u ? It is an interesting problem to obtain the Kerr solution in the form of a Bondi metric.

Bondi’s radiating space-time (9)

The _{outgoing radiation conditions imply, as r → ∞,} γ = c(u, θ, ϕ) r + O(r −3_), δ = d(u, θ, ϕ) r + O(r −3_), β = ₋c 2 _{+ d}2 4r2 + O(r −4_), U = ₋l(u, θ, ϕ) r2 + O(r −3_), W = ₋¯l(u, θ, ϕ) r2 + O(r −3_), V = _{−r + 2M(u, θ, ϕ) + O(r}−1),

Bondi’s radiating space-time (10)

Bondi’s radiating metric under the outgoing radiation conditions

g_Bondi = ₋1₋ 2M r + O(r −2₎_du2 −21 ₋ c 2 + d2 4r2 + O(r −4₎_dudr +2l + 2cl + 2dl r + O(r −2₎_dudθ +2¯l− 2c¯l− 2dl r + O(r −2₎_{sin θ}_dudϕ +r21 + 2c r + O(r −2₎_dθ2 +r21₋ 2c r + O(r −2₎_sin2 θdϕ2 +r24d r + O(r −2₎_{sin θdθdϕ}2 ,

Bondi’s radiating space-time (11) We have two regularity assumptions

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 ϕ = 0 and 2π. Condition B

For all u and ϕ,

Bondi’s radiating space-time (12) At null infinity:

The Bondi energy-momentum of u₀-slice: m_ν(u₀) = 1 4π I S2 M(u₀, θ, ϕ)nνdS for ν = 0, 1, 2, 3, where

n0 = 1, n1 = sin θ cos ϕ, n2 = sin θ sin ϕ, n3 = cos θ. m₀: “Bondi energy” or “Bondi mass”

m_i: “Bondi momentum”

• Minkowski space-time: mν(u0) = 0.

Bondi’s radiating space-time (13) We define M_{(u, θ, ϕ) = M (u, θ, ϕ) −} 1 2 l,2 + l cot θ + ¯l,3 csc θ
= M (u, θ, ϕ) ₋ 1 2 − 2c(u, θ, ϕ) + c,22(u, θ, ϕ) − csc2 θ c_,33(u, θ, ϕ) + 2 csc θ d_,23(u, θ, ϕ)

+3 cot θ c_,2(u, θ, ϕ) + 2 cot θ csc θ d_,3(u, θ, ϕ). Its u-derivative is

Bondi’s radiating space-time (14) Under condition A and B, we have

I S2 l_,2 + l cot θ + ¯l_,3 csc θ
nνdS = 0. This implies m_ν(u₀) = 1 4π I S2 M(u₀, θ, ϕ) nν dS = 1 4π I S2 M_{(u0, θ, ϕ) n}ν _dS. ν = 0, 1, 2, 3.

Bondi’s radiating space-time (15) Bondi mass loss formula

d du mν = − 1 4π I S2 (c_,0)2 + (d_,0)2nνdS, ν = 0, 1, 2, 3. When ν = 0, this is the famous Bondi mass loss formula

d du m0 = − 1 4π I S2 (c_,0)2 + (d_,0)2dS ≤ 0.

The Bondi mass is non-increasing in u, i.e., more and more energy is radiated away.

Bondi’s radiating space-time (16)

Generalized Bondi energy-momentum loss formula Applying H¨older’s inequality and Cauchy-Schwarz’ inequality, we obtain d du  m_{0 −} s X 1_≤i≤3 m2_i _{≤ 0.}

Bondi’s radiating space-time (17) Positive Mass Conjecture at Null Infinity

In Bondi’s vacuum radiating space-times, the Bondi mass must be nonnegative, i.e., a finite gravitational gravitational system cannot radiate away more energy than it has initially.

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Positive mass theorem at null infinity (1) The Positive mass theorem near infinity

1982, Schoen-Yau:

Solving the Jang’s equation–prescribing the mean curvatures.

1982-, Israel-Nester, Horowitz-Perry, Ashtekar-Horowitz, Renla-Tod, Ludvigsen-Vickers, etc.:

Witten’s method–the Dirac operator.

2006, W.-l. Huang, S.T. Yau, and X. Zhang: Detailed proof.

Ref.: W.-l. Huang, S.T. Yau, and X. Zhang, Positivity of the Bondi mass in Bondi’s radiating spacetimes, Rend. Lincei. Mat. Appl. 17 (2006), 335-349.

Positive mass theorem at null infinity (2) Asymptotically null initial data set

In Minkowski space-time, the spacelike hypersurface t = p1 + r2

has the hyperbolic metric ˘g and the nontrivial second form ˘h, ˘ g = dr 2 1 + r2 + r 2 _dθ2 _{+ sin}2 _θdϕ2
_, ˘h = dr2 1 + r2 + r 2 _dθ2 _{+ sin}2 _θdϕ2

in polar coordinates (r, θ, ϕ) where 0 < r < _{∞, 0 ≤ θ < π,} 0 _{≤ ϕ < 2π.}

Positive mass theorem at null infinity (3) Asymptotically null initial data set

Denote the associated orthonormal frame _{{˘ei} and coframe {˘e}i_} by ˘ e₁ = p1 + r2 ∂ ∂r, e˘ 1 _{= √} dr 1 + r2, ˘ e₂ = 1 r ∂ ∂θ, e˘ 2 _{= rdθ,} ˘ e₃ = 1 r sin θ ∂ ∂ϕ, e˘ 3 _{= r sin θdϕ.} ˘

Positive mass theorem at null infinity (4) Asymptotically null initial data set

An initial data set (M3, g, p) (p is not necessarily symmetric) is asymptotically null of order τ if, outside a compact subset,

• M is diffeomorphic to R3\BR,

• the metric g and the 2-tensor p are

g(˘e_i, e˘_j) = ˘g(˘e_i, e˘_j) + a_ij, p(˘e_i, e˘_j) = ˘p(˘e_i, e˘_j) + b_ij where a_ij and b_ij satisfy

a_ij = O(r−τ), b_ij = O(r−τ), ˘

∇kaij = O(r−τ), ∇˘ kbij = O(r−τ),

˘

Positive mass theorem at null infinity (5)

In Bondi’s radiating vacuum space-time, the spacelike hypersur-face given by u = p1 + r2 _{− r +} (c 2 _{+ d}2₎ u=0 12r3 + a₃(θ, ϕ) r4 + o(r −4₎

is asymptotically null of order 1.

(Since Bondi’s radiating metric is complicated, we have calculated the induced metric g and the second fundamental form h using Mathematica 5.0.)

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Schoen-Yau’s method (1) Jang’s equation:  gij ₋ f i_fj 1 + _|∇f|2  f_,ij p1 + |∇f|2 − hij  = 0.

If Jang’s equation has a solution f which has the asymptotic expansion

f = p1 + r2 + p(θ, ϕ) ln r + o(1)

for r sufficiently large, then p(θ, ϕ) and M(0, θ, ϕ) must be constant.

Schoen-Yau’s method (2) Proof. For r sufficiently large,

J(f ) _≈ ln r r3 4S2 p(θ, ϕ) + p(θ, ϕ) _{− 2M(0, θ, ϕ)} r3 , where 4_S2 = ∂2 ∂θ2 + cot θ ∂ ∂θ + csc 2 _θ ∂2 ∂ϕ2. J(f ) = 0 implies 4_S2 p(θ, ϕ) = 0, p(θ, ϕ) − 2M(0, θ, ϕ) = 0.

There is no nonconstant harmonic function on S2 ⇒ p(θ, ϕ) and M(0, θ, ϕ) = p(θ,ϕ)₂ are constant.

Schoen-Yau’s method (3)

Theorem 1. Let L3,1, g˜
be a vacuum Bondi’s radiating space-time with Bondi-metric g. Suppose that Condition A˜ and Condition B hold.

If there exists u₀ such that M(u₀, θ, ϕ) is constant, then

m₀(u) _≥ s X 1_≤i≤3 m2_i(u) for all u _{≤ u0}.

Schoen-Yau’s method (4) Idea of the proof:

• Let M(u0, θ, ϕ) = p₂ be constant for some u0.

• Without loss of generality, u0 = 0.

• On the hypersurface u = p1 + r2 _{− r +} (c 2 _{+ d}2₎ u=0 12r3 + a₃(θ, ϕ) r4 + o(r −4_),

Jang’s equation has a solution

f = p1 + r2 + p ln r + q,

Schoen-Yau’s method (5) Idea of the proof:

• ADM total energy E(¯g) = p.

• Scalar curvature ¯R _{≥ 2 |Y |}2_{g¯ − 2 divg¯}Y .

• ¯g can be transformed conformally to a metric ˆg with ˆ

R = 0 and E(¯g) _{≥ E(ˆg).}

• The positive mass theorem at spatial infinity implies p = E(¯g) _{≥ E(ˆg) ≥ 0.}

• Bondi mass on the u = 0 slice: m₀(0) = p

Schoen-Yau’s method (6) Idea of the proof:

• From the Bondi mass-loss formula,

m₀(u) _{≥ m0}(u₀) _{≥ 0 ∀u ≤ u0}.

• From the generalized Bondi energy-momentum-loss formula, m₀(u) _≥

s

X

1_≤i≤3

Structuring

1. Gravitational waves

2. Bondi’s radiating space-time

3. Positive mass theorem at null infinity (a) Schoen-Yau’s method

Witten’s method (1)

Let (X, g, h) be an asymptotically null spacelike hypersurface. • total energy: E _{= ˘∇}j_{a1j − ˘∇1trg˘(a) −} a_{11 − δ11trg˘}(a), E_{ν(X) =} 1 16π rlim_→∞ I S_r E nν r dS • total linear momentum:

P _{= b11 − δ11trg˘(b),} P_{ν(X) =} 1 8π rlim_→∞ I S_r P nν r dS S_{r: sphere of radius r in R}3, ν = 0, 1, 2, 3.

Witten’s method (2)

Positive mass theorem (X. Zhang, 2002):

Let (X, gij, pij) be a 3-dimensional asymptotically null initial

data set of order τ = 3. Denote

µ := 1 2 R + (p i i) 2 − pijpij
, ϕj := ∇ipji − ∇jp_ii, σj := 2∇i pij − pji
.

If the initial data set satisfies the dominant energy condition

µ _{≥ max}nqXϕ2_j, q_X (ϕj + σj)2 o , then E_{0(X) − P0(X) ≥} s X i=1,2,3 E_{i(X) − Pi(X)}2 . If equality holds, then

Witten’s method (3)

Remark: The proof of the theorem can still go through if the order τ > 3₂ and the E_{ν − Pν} are finite for ν = 0, 1, 2, 3.

Witten’s method (4)

Theorem 2. Let L3,1, g˜
be a vacuum Bondi’s radiating space-time with Bondi-metric g. Suppose that Condition A˜ and Condition B hold.

If there exists u₀ such that c u=u₀ = d   u=u₀ = 0, then m₀(u) _≥ s X 1_≤i≤3 m2_i(u) for all u _{≤ u0}.

Witten’s method (5) Proof.

• Without loss of generality, assume u0 = 0.

• Choose an asymptotically null spacelike hypersurface X with u = p1 + r2 _{− r +} (c

2 _{+ d}2₎

u=0

12r3 + O(r

−4_).

• E−2P = −_2r33(l,2+l cot θ+¯l,3 csc θ)u=0+

4M (0,θ,ϕ) r3 +O(r−4). • Eν(X) − Pν(X) = mν(0). • E0(X) − P0(X) ≥ q P3 i=1 Ei(X) − Pi(X) 2 . • m0(u) ≥ q P3 i=1 m2i(u)

Open questions

1. What does it mean in physics that M (u₀, θ, ϕ) = constant ? 2. What is the physical meaning of the condition

c(u₀, θ, ϕ) = 0 = d(u₀, θ, ϕ) ? Does it preclude gravitational waves ?