Conserved charge in gravity Sinya AOKI Center for Gravitational - - PowerPoint PPT Presentation
Conserved charge in gravity Sinya AOKI Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University YITP Workshop Strings and Fields, November 16-20, 2020 Kyoto, Japan References: S. Aoki, T. Onogi and S.
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University YITP Workshop Strings and Fields, November 16-20, 2020 Kyoto, Japan
References:
“Conserved charge in general relativity”, arXiv:2005.13233[gr-qc].
“Charge conservation, Entropy, and Gravitation”, arXiv:2010.07660[gr-qc]. Sinya Aoki Tetsuya Onogi Shuichi Yokoyama
Einstein equation
Rµν − 1 2gµνR + Λgµν = 8πGdTµν
gravity matter Tµν(x) = δSmatter δgµν(x)
conservation
rµTµν = 0
Bianchi identity but
∂µTµν 6= 0
Einstein’s pseudo-tensor
∂µ hp |g| (Tµν + tµν) i = 0
rewrite gravitational energy ? Problem: tµν is not covariant under general coordinate transformation. violate the fundamental principle of general relativity !
Quasi-local energy Arnowitt-Deser-Misner (ADM) energy Komar energy, Bondi energy Hamiltonian with Gibbons-Hawking term E = Z dV (local energy) E = Z
r→∞
dS (quasi-local energy)
Q = Z
V
dV J0 = Z
∂V
dSµF 0µ However quasi-local energy cannot tell a distribution of energy. Local energy must exist since quasi-local energy is derived from it. The absence of covariant definition for local energy is the most unsatisfactory point in general relativity. local energy (mass) is a source which generates gravitational fields.
Generic conserved charge in GR.
To give a precise and universal definition of energy by the volume integral
Energy. meaning ?
“Conserved charge in general relativity”, arXiv:2005.13233[gr-qc]
conserved charge
Lξgµν = rµξν + rνξµ = 0
Killing vector
rµ(T µ
νξν) = (rµT µ ν)ξν + 1
2T µν(rµξν + rνξµ) = 0 Q(ξ) = Z
Σ(x0)
dΣ0 √−g T 0
νξν
0 = Z
M
ddx p |g| rµJµ = Z
M
ddx ∂µ ⇣p |g|Jµ⌘ = Z
∂M
dΣµ p |g| Jµ
Jµ := T µ
νξν
rµJµ = 1 p |g| ∂µ ⇣p |g|Jµ⌘ Stokes’ theorem ∂M = ∂Ms Σt2 Σt1 scalar
M ∂Ms Σt2 Σt1
assume dΣkJk = 0 on ∂Ms
Q(Σt2) = Q(Σt1)
Symmetry covariantly conserved vector current
a metric gµν does not contain x0 Killing vector
ξµ = −δµ
conserved energy
E = − Z
Σ(x0)
dΣ0 √−g T 0
Covariant and universal definition of total energy works for an arbitrary asymptotic behavior in an any coordinate system.
E(ξ) = c 16πGd Z
Σ(x0)
dΣ0 pg rνr[0ξν] = c 16πGd Z
∂Σ(x0)
dΣ0k pg r[0ξk] Komar, PR127(1962) 1411 Quasi-local energy This is a Noether charge of the 2nd type for a coordinate transformation ξν. For ξµ = −δµ
0 ,
E = − 2c 16πGd Z
Σ(x0)
dΣ0 √−g R0
c: some constant
Our formula has been known, but rarely used.
The quantity I = Z T µ0ϕµ pg dx1dx2dx3 will be constant, · · · , if the vector ϕµ satisfies the equation rνϕµ + rµϕν = 0.
(unpublished), May-June 1958; Gen. Res. Grav. 34 (2002), 721-762, cited Fock.
These were forgotten in major textbooks (e.g. Landau-Lifshitz) except a few.
footnote 3. a Killing vector field ξa is presented, · · · , ra(Tabξb) = (raTab)ξb+Tabraξb = 0, so R
Σ Tabξbna is conserved, i.e., independent of choice of Cauchy surface Σ.
No application. Let us consider some applications. See also lecture notes by Blau; Shiromizu (Japanese); Sekiguchi (Japanese).
measure term is not specified, though.
metric
ds2 = −(1 + u)(dx0)2 − 2udx0dr + (1 − u)dr2 + r2¯ gijdxidxj
(d-2)-sphere Eddington-Finkelstein EMT f(r) := rd−3 rd−3 T 0
0 = −(d − 2)
16πGd ∂r(rd−3f(r)) rd−2 Energy ξµ = −δµ Killing E = − Z dd−1x √−g T 0
0 = (d − 2)Vd−2
16πGd Z ∞ dr ∂r(rd−3f(r)) = (d − 2)Vd−2rd−3 16πGd Vd−2 := Z dd−2x p det ¯ gij volume of (d-2)-sphere This reproduces known results. For example E = r0 2G4 = M at d = 4 Remark x0 = constant hypersurface is space-like even inside the horizon. The BH energy is independent of the cosmological constant. However the Killing vector becomes space-like inside the horizon. u(r) = −rd−3 rd−3 − 2Λr2 (d − 2)(d − 1)
Komar energy for BH diverges for non-zero cosmological constant. EKomar = c 16πGd Z
r→∞
dd−2x pgr[0ξr] = lim
r→∞
cVd−2 16πGd (d 3)rd−3
(d 2)(d 1)
16πGd at Λ = 0 E = (d − 2)Vd−2rd−3 16πGd
c = d − 2 d − 3 This choice reproduces known results. But it is ill-defined at d=3. Our definition is much more robust and universal.
d=3 AdS
f(r) = r2 L2 − 2GNMθ(r) + G2
NJ2
4r2 , ω(r) = GNJ 2r2 ,
Killing vectors ξµ
T := −δµ 0 , ξS := δµ φ
EMT Energy
T 0
φ = −
M 16πGN ∂r(r3ωr) r T 0
0 = −M
8π δ(r) r E = − Z 2π dφ Z dr r T 0
0 = M
4π
Angular momentum
Pφ = Z 2π dφ Z dr r T 0
φ = J
8
Bandaos-Teitelboim-Zanelli, PRL69(1992)1849.
stationary spherically symmetric metric
ds2 = −f(r)(dx0)2 + h(r)dr2 + r2˜ gijdxidxj
with perfect fluid
T 0
0 = −ρ(r),
T r
r = P(r),
T i
j = δi jP(r)
Einstein equation Oppenheimer-Volkoff equation
1 h(r) = k − 2GdM(r) rd−3 − 2Λr2 (d − 2)(d − 1)
with
M(r) = 8π d − 2 Z r dssd−2ρ(s), M(0) = 0
−dP(r) dr = GdM(r) rd−2 (P(r) + ρ(r)) h(r) ⇢ d − 3 + rd−1 (d − 2)M(r) ✓ 8πP(r) − 2Λ (d − 1)Gd ◆
Oppenheimer-Volkoff, PR55(1939)374.
EOS P = P(ρ) solution to OV equation
radius of compact star R from P(r = R) = 0
Schwarzschild metric with m = M(R)
f(r) = 1 h(r) = k − 2GdM(R) rd−3 − 2Λr2 (d − 2)(d − 1) r > R, ρ(r) = P(r) = 0
Schwarzschild metric
Killing vector
ξµ = −δµ
conserved energy
E = − Z dd−2x Z ∞ dr p |g|T 0
0 = Vd−2
Z R p f(r)h(r)rd−2ρ(r) = (d − 2)Vd−2 8π Z R p f(r)h(r)dM(r) dr = (d − 2)Vd−2 8π " M(R) − Z R M(r) 2 d dr log |f(r)h(r)| #
dM(r) dr = 8π d − 2rd−2ρ(r) gravitational mass felt by distant objects (estimation by quasi-local energy) corrections due to a structure inside star
A similar result can be found in their appendix.
Physical meaning leading correction term (Newtonian limit) gravitational interaction energy at d=4 ∆E ' GdVd−2 Z R dr rM(r)ρ(r) + · · · U4 := −Gd 2 Z dd−1x dd−1y ⇢(~ x)⇢(~ y) |~ x − ~ y| = 4⇡G4 Z R dr rM(r)⇢(r) correction term represents the gravitational interaction energy !
EKomar = c(d − 3)Vd−2 8π M(R) Komar energy misses the gravitational interaction energy. Our definition is physically more sensible.
constant density d = 4, Λ = 0 correction R ≥ Rmin = 9GMBH 4 ' 68% of MBH at R = Rmin ρ(r) = ρ0 Size of corrections Correction could be large ! Gas of a total mass M (with negligible interaction) can not becomes a compact star of the same mass M due to the energy conservation, unless an extra energy is released. MBH = 4πR3ρ0 3 fixed E = MBH − πρ0 R(3r2
0 − R2) − 3r2
q r2
0 − R2 sin−1
✓ R r0 ◆ r2
0 :=
3 8πG4ρ0
“Charge conservation, Entropy, and Gravitation”, arXiv:2010.07660[gr-qc].
If the Killing vector is absent (no symmetry), Q[v](t) = Z
Σt
dd−1x p |g| T 0
νvν,
Charge
dQ[v] dt =
dd−1⃗ x
ν∇µvν.
6= 0 in general sufficient condition
ν∇µvν = 0,
“conservation condition” for v
Aµν∂νvµ + Bµvµ = 0,
Aµ
ν := Tµ ν, Bµ = T α βΓβ αµ
fix direction 1st order linear PDE A solution exists (at least locally in t).
dxµ dt = Aµ(x), dv(t) dt = −B(x)v(t).
simultaneous linear ODE initial value is given on a hypersurface at fixed t. vµ = vδµ
µ0
we can define a generic conserved charge in general relativity. Hereafter, we consider the case, the vector is proportional to time direction, and write it as vµ = ζµ dQ[v] dt = 0 What is this conserved charge ? Entropy
S := Q[ζ] =
sµ =
νζν.
entropy current density
∂µsµ = 0
conserved conservation of total entropy (1) This is not a Noether charge, since no symmetry exists. (2) This is not an energy, but reduces to the energy for the Killing vector. (3) The conservation of the entropy is natural, since the theory has time reversal symmetry. (4)It satisfies the 1st law of thermodynamics. (next) Our answer:
nµ: unit time evolution vector T µ
ν = ρnµnν + P ¯
gµ
ν
¯ gµ
ν := δµ ν + nµnν
ds2 = −N 2(dx0)2 + ¯ gijdxidxj (rµT µ
ν)nν = 0
conservation of EMT nµ∂µρ + (ρ + P)K = 0 ζµ = −βnµ conservation condition ρnµ∂µβ − βPK = 0 nµ = ∂xµ ∂η d(ρβ) dη + ρβK = 0
β = β0 ρ0 ρ exp
η
η0
dηK
gν
µrνnµ = rµnµ
K = 1 p |g| ∂µ( p |g|nµ) = d(log √¯ g) dη 1st law of thermodynamics u := ρv: local energy v := √¯ g: local volume
β = 1 T T ds0 dη = du dη + P dv dη if we take s0 := ρn0β = uβ entropy density n0 = 1 N ¯ g := det ¯ gij ds0 dη = du dη β + udβ dη = ✓du dη + P dv dη ◆ β dβ dη = βPK ρ = βP u dv dη 1st law of thermodynamics Variations by dynamical process
ds2 = −(dx0)2 + a2(t)¯ gijdxidxj Freedman-Lemaitre-Robertson-Walker metric
β = β0 ρ0 ρ exp
η
η0
dηK
K = d(log ad−1(t)) dt perfect fluid Tds0 = du + Pdv = 0 T decreases as the Universe expands P 6= 0 (T: constant for P=0) For the closed Universe, it expands, stops and contracts. Time evolution of the Universe is a kind of space piston. Entropy is conserved during this process. T µ
ν = ρnµnν + Pgµν
1st law of thermodynamics
ds2 = e2Ω(u)(dx2 − dτ 2) + u2(e2β(u)dy2 + e−2β(u)dz2) u = τ − x T x
x = T τ x = −T τ τ = −T x τ = e2Ω
4πu
EMT conservation vectors vµ
0 = −δµ τ
vµ
x = δµ x
conserved charges E = Px = V2 4π Z dx u (2Ω0 − uβ02) E = Px = 0 2Ω0 − uβ02 = 0 uβ00 + 2β0 u2β03 6= 0
∀Rab = 0,∃ Rabcd 6= 0
The Ricci flat gravitational wave does not carry energy/momentum.
Killing vector Killing vector Classical general relativity (Ricci flat) gravitational waves Quantum gravity necessary ? graviton has energy/entropy ? conserved charge = energy, etc. generic conserved charge = entropy energy/entropy gravitational collapse binary stars ……… ………