Is energy conserved in general relativity ?
̶ The lates collaboration with Tetsuya ̶
Sinya AOKICenter for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University

CGPQI
QI

Center for Gravitational Physics andQuantum Information
Yukawa Institute for Theoretical Physics, Kyoto University

Non-perturbative Analysis of Quantum Field Theory
and its Application
September 22, 2022, Osaka University Hall, Osaka University

Congratulations, Tetsuya, on your 60th year (還暦）！

Tetsuya is 2 years younger than I. I became a graduate student@Hongo, in1982,
while Tetsuya became a graduate student@Komaba, in 1984, and I have known
him since then. Collaborations with Tetsuya started in 1995 on lattice QCD:
“Manifestation of Sea Quark E ects in the Strong Coupling Constant in Lattice QCD”
Phys. Rev. Lett. 74 (1995) 22-25.
We together wrote 40 papers, 3 FLAG reviews, and 57 proceedings so far.

ff

Today I will talk on the latest collaboration with Tetsuya.

References:
S. Aoki, T. Onogi and S. Yokoyama, “Conserved charge in general relativity”,
Int. J. Mod. Phys. A36 (2021) 2150098, arXiv:2005.13233[gr-qc].
S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy Current, andGravitation”, Int. J. Mod. Phys. A36 (2021)2150201, arXiv:2010.07660[gr-qc].
S. Aoki and T. Onogi, “Conserved non-Noether charge in general relativity:
Physical de nition vs. Noetherʼs 2nd theorem”, to appear in IJMPA,
arXiv:2201.09557[hep-th], https://doi.org/10.1142/S0217751X22501299.
S. Aoki, “Noetherʼs 1st theorem with local symmetries”, arXiv:2206.00283[hep-th].

fi

Shuichi YokoyamaA simple question I would like ask you.

Is energy conserved in general relativity (GR) ?
Please consider your answer to this question. Yes or No ? Why ?

In a at spacetime, energy is a conserved charge of time translational symmetry.

Time translational symmetry not only de nes “energy” but also leads to its
conservation. (Noetherʼs 1st theorem)

Does Noetherʼs 1st theorem lead to a conserved energy in GR ?
<latexit sha1_base64="aG4tLmQnH/Qm7EJ+00ivNVdMjfM=">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</latexit>fi

fltime translation 2 general coordinate transformation (gauge symmetry)Another perspective on this problem from the Einstein equation
gravitymatter1
Rµ⌫gµ⌫ R + ⇤gµ⌫ = 2Gd Tµ⌫
2<latexit sha1_base64="5+m9MCg5mjOk98HtzxYRZSyThKg=">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</latexit>

 = 4⇡GN<latexit sha1_base64="wDhrsqj8qf4tXKFOGDj5LP75CWc=">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</latexit>

energy momentum tensor (EMT)
Bianchi identity for Rµ⌫<latexit sha1_base64="mNB+dR6KtZPn60KShA93LgJ7fes=">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</latexit>

covariant conservation<latexit sha1_base64="zkcMwb6WEIS2yDAih5Kt0UT384o=">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</latexit>

rµ (

p

µ

gT ⌫ ) = @µ (pµ

gT ⌫ ) +

µ p
µ↵ (

gT

↵

⌫)

<latexit sha1_base64="nFmDlfSLrJVfD75JJPfWKQy5OCo=">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</latexit>

but what we need for a conserved energy is

2nd and 3rd terms are obstructions.

↵ p
µ⌫ (

@µ (

p

gT µ ↵ ) = 0

gT µ ⌫ ) = 0

I. Noetherʼs Theorem
for “gauge” theory

S. Aoki and T. Onogi, “Conserved non-Noether charge in general relativity:
Physical de nition vs. Noetherʼs 2nd theorem”, to appear in IJMPA,

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arXiv:2201.09557[hep-th], https://doi.org/10.1142/S0217751X22501299.

1. Naive derivation of conserved current<latexit sha1_base64="XqC+7Acri7mcH/1zH60yf+KjcD8=">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</latexit>

S⌦ =

Z

d

<latexit sha1_base64="1sJkLHxYpWI6itaRvCLold49mho=">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</latexit>d x L( ,,µ )⌦:= ✓(x)F0 ( ) + ✓,µ (x)F1µ ( ) + · · ·<latexit sha1_base64="UJUZNWpxMnUQ6qw+P4Va+SC8Cqg=">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</latexit>

In nitesimal local transformation<latexit sha1_base64="QG+V7n7BRs4dy/BEABZITVy48xE=">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</latexit>

Z



@L
S⌦ =
d x
@
⌦
d

,µ := @µ

@µ

✓

@L
@ ,µ

◆

+

d

d x @µ
⌦

✓

@L
@ ,µ

◆

=0
µ

invariance:= E

current := N'0on-shell conserved current<latexit sha1_base64="aOj1Nsgk+kjDwUaN2XgfQbGsPYw=">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</latexit>

EoM (Equation of Motion)Z<latexit sha1_base64="z07TDxKFDdlwK4cGJyaMvHauubA=">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</latexit>[✓]a region ⌦ is arbitrary<latexit sha1_base64="KTnv/2iM//sPvYgNqoQu3ZTsdpk=">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</latexit>@µ N µ [✓] =<latexit sha1_base64="5DtiKtTAkmG1WS0rSzy42gExRV0=">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</latexit>E<latexit sha1_base64="2vWJZwNOd0MBK4wHENuQFajzRf8=">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</latexit>✓(x) = ✓0 : constant<latexit sha1_base64="C4diGt6VTIWRsH2R3TcQ8z1RDfI=">AAACw3ichVHNahRBEP4yURPXn2ziRfDSuEQiyFIbJAmBQEAUDyL5cZPAzmboGTvZZuePmd6FOMwL+AIePCmIiI+hB1/AQx5BPW7AiwdrZieIBrWa7v7qq/qqq7vd2NepITqesCbPnb8wNX2xdunylasz9dm5nTQaJJ5qe5EfJXuuTJWvQ9U22vhqL06UDFxf7br9e0V8d6iSVEfhE3MUq24gD0N9oD1pmHLq+4/37WDgkFhdy0ooOrbpKSMd6toRK8Wpm4s1kdmxTIyWvngkqugpYcc97WR3uEKeiwcOLRTEbafeoCaVJs6CVgUaqGwjqr+FjaeI4GGAAAohDGMfEimPDlogxMx1kTGXMNJlXCFHjbUDzlKcIZnt83rIXqdiQ/aLmmmp9vgUn2fCSoF5+kzvaESf6D19oR9/rZWVNYpejnh3x1oVOzPPr29//68q4N2g90v1z54NDrBS9qq597hkilt4Y/3w2YvR9urWfHaLXtM37v8VHdMHvkE4PPHebKqtl6jxB7T+fO6zYGex2VpqLm7ebayvVF8xjRu4iQV+72Ws4yE20OZzP+IrRjix7lt9K7HMONWaqDTX8JtZ+U+Oh62B</latexit>

N
µ
N0 :=

µ

[✓0 ]
@L
=
F0 ( )✓0@ ,µconserved current for a global symmetry ?

This seems good, but there exist infinitely many conserved currents not only fora constant ✓0 but also for an arbitrary function ✓(x).
fi<latexit sha1_base64="YLUKfN8azd9ilm20bO2GDWbIcmY=">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</latexit>2. Maximal use of local symmetry:= ✓(x)F0 ( ) + ✓,µ (x)F1µ ( ) + · · ·<latexit sha1_base64="UJUZNWpxMnUQ6qw+P4Va+SC8Cqg=">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</latexit><latexit sha1_base64="ygX6LM0FlEsxVVZZ9kbr2Mf1vgg=">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</latexit>

S⌦ =

Z

dd x [E F0⌦@µ (E F1µ ) + · · · ] ✓ +=0<latexit sha1_base64="hVA5xEZRTetPhgPeAc2G7Pma394=">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</latexit>Z⌦

dd x @µ (N µ [✓] + E F1µ ✓ + · · · ) = 0=0<latexit sha1_base64="hVA5xEZRTetPhgPeAc2G7Pma394=">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</latexit>

1. Choose ✓(x) = ✓,µ (x) = · · · = 0 at a boundary of an arbitrary ⌦.
<latexit sha1_base64="TynRorYskHWbKCNQHJmX1zbCmg0=">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</latexit><latexit sha1_base64="TPChfvay9usZiPFw1yP9yfPAL2I=">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</latexit>

E F0@µ (E F1µ ) + · · · = 0

o -shell constraint of EOM

2. General ✓ and an arbitrary choice of ⌦.
<latexit sha1_base64="lFK57JHCQa+1RnR1MwyyMoDar0M=">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</latexit>@µ K µ [✓] = 0.

o -shell conservation<latexit sha1_base64="toPIfgKM5A0vmuq3CCAPEVHgrcA=">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</latexit><latexit sha1_base64="Qug9ZShSD6zjtoOmY0o2PW1VqFM=">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</latexit>µ<latexit sha1_base64="M6BVANQc+qngdgC9thJ3N2IrYgA=">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</latexit>µK [✓] := N [✓] + E

F1µ ✓ + · · ·3. ✓ is an arbitrary function.
<latexit sha1_base64="4qagktFjiOZpO8QGo3lbiFqOx7g=">AAACinichVE9TxsxGH5yhQKBllAWpC4nQqpOJ+cqla8FqQwdSdIkSBBFvqsDFpe7k8+JFCL+QPsDGDqBhKqqU9d2ZOEPMOQnVIxBYmHoe5eTKhq1fS3bjx+/z+vHthN6MtKMDTLGo4nJx1PTM9nZuSdP53MLz2pR0FGuqLqBF6hdh0fCk76oaqk9sRsqwduOJ+rO0Zt4v94VKpKB/073QtFo8wNftqTLNVHNXOGVZa7s60Oh+YopI5P7JleO1Iqrntnq+G6cZjVzeWaxJMxxUExBHmnsBLnP2Md7BHDRQRsCPjRhDxwRtT0UwRAS10CfOEVIJvsCJ8iStkNZgjI4sUc0HtBqL2V9Wsc1o0Tt0ikedUVKEwV2zb6wIbtiX9lPdv/XWv2kRuylR7Mz0oqwOf9hqXL3X1WbZo3D36p/etZoYS3xKsl7mDDxLdyRvnt8OqxslAv9F+yc3ZD/MzZgl3QDv3vrXpRE+ROy9AHFP597HNRsq/jaskt2fmst/YppPMcyXtJ7r2ILb7GDKp37Ed/wHT+MOcM21o3NUaqRSTWLeBDG9i8YNZbf</latexit>µ@µ A = 0<latexit sha1_base64="bkX8B7DOkrSP1FWzkfE5pZBnNGU=">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</latexit>

✓

◆

ff

ff

◆

@L
@L ⌫
µ
=
F0 + E F1 + · · · ✓ +F1 + · · · ✓ ⌫ + · · ·
@ ,µ
@ ,µ
|
{z
}
|
{z
}
:=Aµ

:=B µ⌫

Aµ + @⌫ B ⌫µ = 0

B µ⌫ + B ⌫µ = 0<latexit sha1_base64="LfMl2nTToVNx2v6DXLLpDeOHUjc=">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</latexit>

another o -shell conservationff✓<latexit sha1_base64="ROcGMtnZwTm2htpSCpDROpBq4lE=">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</latexit>

<latexit sha1_base64="JlVFHLvKClo40gzyQjP2oN6vSzc=">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</latexit>

Non-dynamical (o -shell) conservationµ@ µ Aµ = 0<latexit sha1_base64="bkX8B7DOkrSP1FWzkfE5pZBnNGU=">AAACenichVHLSgMxFD0d3/VVFUFwUyyKIpTbUlQEQXHj0ldVsFpmxqjBeTGTFrT4A/6AC1cKIupnuPEHXPgJ4lLBjQvvTAdERb0hycnJPTcnieFZMlBEjwmtobGpuaW1Ldne0dnVnerpXQvcim+Koularr9h6IGwpCOKSipLbHi+0G3DEuvGwXy4v14VfiBdZ1UdemLL1vccuStNXTFVTvWXPN1XUrfKJbuSntsOxxkqpzKUpSjSP0EuBhnEseimrlDCDlyYqMCGgAPF2IKOgNsmciB4zG2hxpzPSEb7AsdIsrbCWYIzdGYPeNzj1WbMOrwOawaR2uRTLO4+K9MYpge6phe6p1t6ovdfa9WiGqGXQ56NulZ45e6TgZW3f1U2zwr7n6o/PSvsYiryKtm7FzHhLcy6vnp0+rIyvTxcG6ELemb/5/RId3wDp/pqXi6J5TMk+QNy35/7J1jLZ3MT2fxSITM7FX9FKwYxhFF+70nMYgGLKPK5NZzjBreJd21IG9PG66laItb04UtohQ+sRJHm</latexit>@µ K [✓] = 0

generic consequences of Noetherʼs 2nd Theorem for gauge theory<latexit sha1_base64="7P/AMPHqUmlZ9nzEFZMacGFYrIg=">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</latexit>µµN [✓] = K [✓]E

F1µ ✓ ' K µ [✓]N0µ = Aµ<latexit sha1_base64="iG+VjikpM+/ROoncLclNigU1LXQ=">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</latexit>

E F1µ ' Aµff

ff

Conserved currents from Noetherʼs 1st theorem for gauge theory are equal too -shell conserved currents form Noetherʼs 2nd theorem up to EoM.

3. Consequences in GR<latexit sha1_base64="Hd6A05EmMICj3034B3CBXvaOmno=">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</latexit>

Gravity ActionSG,⌦ =

Z

1 p
LG =
g(R
2
<latexit sha1_base64="3CAQPUh1I1avXOlqypiS0Wup4FU=">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</latexit>dd x LG⌦(in nitesimal) general coordinate transformations<latexit sha1_base64="wpS2JnnG2R86KZvtC3Ue8PTrBmc=">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</latexit>⇠ gµ⌫ (x) =⇠ ↵ ,µ (x)g↵⌫ (x)<latexit sha1_base64="LOQcJZaCuHinu3xta79P5OmjR8U=">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</latexit>⇠ ↵ ,⌫ (x)gµ↵ (x)µ@ µ KG[⇠] = 0<latexit sha1_base64="UcXsFMEO6reAoXB4yvpMXsRloYU=">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</latexit><latexit sha1_base64="obsyVDLmfb+Jf4ZENnFHm/9ksdo=">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</latexit>⇠ SG,⌦ = 0

Noetherʼs 2nd Theorem

fi

µ
µ
x
=
⇠
⇠

2⇤)

o -shell conserved currents
h
i
p
1
µ
KG
[⇠] =
gr⌫ r[µ ⇠ ⌫] = Aµ ⌫ ⇠ ⌫ + B µ ⌫ ↵ ⇠ ⌫ ,↵ + C µ ⌫ ↵ ⇠ ⌫ ,↵2<latexit sha1_base64="h8nhQwfsWMMxb0xS4o0TBlIDVK8=">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</latexit>µ@ µ KG[⇠] = 0<latexit sha1_base64="UcXsFMEO6reAoXB4yvpMXsRloYU=">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</latexit>o -shell conserved current density (covariant)conserved for an arbitrary vector ⇠<latexit sha1_base64="nrd0vFdoXs0xN8clHPulO1SNRQY=">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</latexit>

Komar energy, ADM energy, Wald entropy, asymptotic charges
referred as Quasi-local (energy)@ µ Aµ ⌫ = 0<latexit sha1_base64="JqoDw4iWVzRYSd1EK3COiaLYKzA=">AAACf3ichVHLShxBFD3TGqNj1DFuRDeDgzGr4bZINIHASDYux8eoYJumu1Ojhf2iu3pgHAZc5weycKUQRVzoP2STH3DhJ4hLBTdZ5E5PgxhJcouqOnXqnlunquzQlbEius5pPb0v+l72D+QHXw0NjxRGX6/HQRI5ouYEbhBt2lYsXOmLmpLKFZthJCzPdsWGvfeps7/REFEsA39NNUOx7Vk7vqxLx1JMmYUJI7QiJS3XNLykuPiZx1bbNPzkI5mFEpUpjeJzoGeghCyqQeEUBr4ggIMEHgR8KMYuLMTctqCDEDK3jRZzESOZ7gu0kWdtwlmCMyxm93jc4dVWxvq87tSMU7XDp7jcI1YWMU1XdEZ39JPO6YZ+/bVWK63R8dLk2e5qRWiOfB1fffivyuNZYfdR9U/PCnUspF4lew9TpnMLp6tv7H+7W/2wMt16Q8d0y/6P6Jp+8A38xr3zfVmsHCLPH6D/+dzPwfpsWX9Xnl2eK1UWsq/oxySm8Jbfex4VLKGKGp97gBNc4FLLaTNaWaNuqpbLNGN4Etr731wWk48=</latexit>o -shell conserved current density (non-covariant)Einsteinʼs pseudo-tensorphgAµ ⌫ =
2Rµ ⌫ + g µ↵ ⌫,↵2
<latexit sha1_base64="RoLZYxQje0LVW9qcvoHvuTO9rU0=">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</latexit>

EoM
i gµ⌫ p
µ
µ
µ
⇡
g
(T
+
t
⌫
⌫)
↵ ,⌫
<latexit sha1_base64="b0SYRDDOewTjbaO3Bvuni9SfGWc=">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</latexit>

g↵



µ
1
tµ ⌫ :=
Rµ ⌫ + ⌫ (R
2
2

ff

ff

ff

<latexit sha1_base64="mnEBOzi+zcM3vetEXHpBP1UdNLU=">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</latexit>2⇤) + g µ↵

⌫,↵

g↵

µ
↵ ,⌫

<latexit sha1_base64="L3gINo2po24Y2uj99jeZ+A7ZsYA=">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</latexit>

Matter Action

S ,⌦ =

Z

dd x L (g, ,

,µ )

⌦

@µ K µ [⇠] = 0<latexit sha1_base64="mTXXzpsSHyKup/L0skvIenH0WFY=">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</latexit><latexit sha1_base64="R/twjnrGdjrskZrsSGJ+paaxleo=">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</latexit>⇠ S ,⌦ = 0µ
Kscalar[⇠] = 0<latexit sha1_base64="QOL19wu1zCH8xzNpMQF7OVgaV9A=">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</latexit>pµKvector [⇠] = @⌫ [<latexit sha1_base64="OZv8uq0cUbtKO+qqhYsczzhI3h8=">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</latexit>gF µ⌫ A↵ ⇠ ↵ ]“Energy” de ned from o -shell conserved currents ?
µ@ µ KG[⇠] = 0<latexit sha1_base64="NC0/7Lt/sYOSAbUkQzVOWn9/Wrg=">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</latexit>

identity

Conserved current from Noetherʼs 1st theoremc.f.

N µ (x)<latexit sha1_base64="dRWFr9hCi/1pV2RZ2OHX+yrbR0o=">AAACbHichVG7SgNBFD1ZXzG+4qMQRAiGiDZhVoIGK8HGSowajWiU3XXUwX2xj2AM/oCthYVaKIiIn2HjD1j4CSLYRLCx8GazIBrUO8zMmTP33Dkzo9q6cD3GniJSU3NLa1u0PdbR2dXdE+/tW3Et39F4XrN0yymoist1YfK8JzydF2yHK4aq81V1f7a2v1rijissc9kr27xoKLum2BGa4hG1lpjf3DD8sYPxrXiSpVkQiUYghyCJMBas+A02sA0LGnwY4DDhEdahwKW2DhkMNnFFVIhzCIlgn+MIMdL6lMUpQyF2n8ZdWq2HrEnrWk03UGt0ik7dIWUCKfbIblmVPbA79sw+fq1VCWrUvJRpVutabm/1HA8uvf+rMmj2sPel+tOzhx1kA6+CvNsBU7uFVteXDk+rS9OLqcoou2Iv5P+SPbF7uoFZetOuc3zxDDH6APnnczeClYm0PJnO5DLJmWz4FVEMYQRj9N5TmMEcFpCncw2c4BwXkVdpQBqShuupUiTU9ONbSKOf4seMog==</latexit>

One can freely add a trivially conserved term as
J µ (x) = N µ (x) + @⌫ f µ⌫ (x), f ⌫µ (x) =<latexit sha1_base64="MH/9KR/VTp6aIopWNgSjz2xkJss=">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</latexit>f µ⌫ (x)arbitrary anti-symmetric function

which by substitution of their values for ∆x and δu, provided f is assumed to depend on only first
derivatives of the u’s, goesµ over intoµµ⌫
KG [⇠]orA
corresponds
to
@
f
⌫
⌫
#
!"
#
!"
∂f
∂f !
∂f
∂f
!
!!
u
+
f
p
(x)
+
p
(x) = 0p(x) +−i!
!!
∂x∂ui ∂ui∂ui<latexit sha1_base64="JTB5BHRoiUAeYacOL9x4urXtgsQ=">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</latexit>

Thus quasi-local energy or Einsteinʼs (pseudo-tensor) energy(identically in p(x), p! (x), p!! (x)). This system of equations has solutions for as few as two functions
cannot be a physical energy.
u(x) actually containing the derivatives, namely
$
%
−u1
e
f = (u!1 − u!2 ) Φ u1 − u2 , !
,!
u1 − u2

E. Noether, Gott. Nacho. 1918(1918)235-257 [arXiv:physics/0503066[physics]].

ff

fi

where Φ stands for an arbitrary function of the specified arguments.
Hilbert enunciates his assertion to the effect that the failure of proper laws of conservation ofenergy is a characteristic feature of the “general theory of relativity.” In order for this assertion to
hold good literally, therefore, the term “general relativity” should be taken in a broader sense thanusual, and extended also to the forgoing groups depending on n arbitrary functions.27

II. Matter energy in GR
andits (non) conservation

S. Aoki, “Noetherʼs 1st theorem with local symmetries”, arXiv:2206.00283[hep-th].

Since quasi-local energy or Einsteinʼs energy is not physical,
we should consider an alternative.
While EOM is derived from the total action, a matter action alone is invariant undergeneral coordinate transformations.
Z
S ,⌦ =
dd x L (g, , ,µ )⇠ S ,⌦ = 0<latexit sha1_base64="L3gINo2po24Y2uj99jeZ+A7ZsYA=">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</latexit><latexit sha1_base64="R/twjnrGdjrskZrsSGJ+paaxleo=">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</latexit>⌦↵<latexit sha1_base64="wpS2JnnG2R86KZvtC3Ue8PTrBmc=">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</latexit>⇠ gµ⌫ (x) =⇠ ,µ (x)g↵⌫ (x)↵<latexit sha1_base64="9uBM0T/5Xg2zSjami8KWiMc2IjU=">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</latexit>⇠ ,⌫ (x)gµ↵ (x)⇠:= ⇠ µ (x)Fµ ( ) + ⇠ µ ,⌫ (x)Fµ ⌫ ( ) + · · ·Since transformations do not commute with derivatives, we de ne
¯⇠

:=

⇠

,

<latexit sha1_base64="xyzLyNaGInOmjnxOtpJQaZJF3CQ=">AAACqXichVFNS9xAGH5Mv3Stda0XoZfQRVHEZVZEpSAI9tCjq11dNBImcdwOTiYhmSzVxT/gH/DgyYJI25/RS4+99LD9B8Wjgpce+m42Iq2ob8jMM8/7Pu88M+NFSiaGsXaP9ejxk6fPevsK/c8HXgwWh16uJWEa+6LmhyqM6x5PhJJa1Iw0StSjWPDAU2Ld213q5NebIk5kqN+bvUhsBbyh5Y70uSHKLb51PB4720IZ7jofpd1wW06QOjo9sBfsqXFHc09RJkhtyrrE25PXnE4zKkgn3GKJlVkW9m1QyUEJeSyHxTM42EYIHykCCGgYwgocCX2bqIAhIm4LLeJiQjLLCxygQNqUqgRVcGJ3aWzQajNnNa07PZNM7dMuiv6YlDZG2U/2mV2w7+wr+83+3NmrlfXoeNmj2etqReQOHo6sXj2oCmg2+HCjutezwQ7mM6+SvEcZ0zmF39U3948uVt+sjLbG2Cd2Tv5PWJt9oxPo5qV/WhUrxyjQA1T+v+7bYG26XJktT1dnSovz+VP04hVeY5zuew6LeIdl1GjfL/iBNn5Zk1bVqlsb3VKrJ9cM45+w/L+At6RF</latexit>¯⇠ gµ⌫ =⇠x<latexit sha1_base64="7nl7vPgbtubxaNta3jjmxp6vdnA=">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</latexit>¯⇠¯,↵ = @↵ ( ⇠<latexit sha1_base64="okz7WxAfLWaMbXvbX2OPDpV5luc=">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</latexit>(rµ ⇠⌫ + r⌫ ⇠µ ))commute with derivatives¯⇠ = ⇠ µ [Fµ ( )|{z=F̄µ ( )

µ
⌫
]
+⇠
F
,µ
,⌫ µ ( ) + · · ·}fi<latexit sha1_base64="67/mkEzuRSqWxEplPb1icqqJgHU=">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</latexit>1. Noetherʼs theorem for a matter action in GR<latexit sha1_base64="+3PL0YoDDJC7Gi7Ooo/JVvidqzY=">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</latexit><latexit sha1_base64="+KvtR6xjNqUnn4sIPn1M+PkusZo=">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</latexit>⇠ S ,⌦ =Z<latexit sha1_base64="kyAkKFzGlHnD/rro+ibXvMpkKBg=">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</latexit>

⇣

⌘

d
(d
x)
⇠

dd x ¯⇠ L + L ,µ ⇠ xµ + @µ ( ⇠ xµ )L = 0|{z}⌦=@µ (L ⇠ µ )¯⇠ L = @L ¯⇠ gµ⌫ + @L ¯⇠ + @L @µ ¯⇠ = @L ¯⇠ gµ⌫ + E ¯⇠ + @µ
@gµ⌫
@
@ ,µ
@gµ⌫

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@L ¯
µ
¯
g
+
E
+
@
N
[⇠] = 0
⇠ µ⌫
⇠
µ
@gµ⌫
p
=
gT µ⌫ EMT

✓

@L ¯
⇠
@ ,µ

@L ¯
N [⇠] := L ⇠ +⇠@ ,µ<latexit sha1_base64="cFiBJ8u4MaZ2dT467HkPK3Zj2Cs=">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</latexit>µµ<latexit sha1_base64="yflmpTZJCkfqU2YdhHCTuT4mEDQ=">AAACe3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuIKdjGFBcWlVTrptcqhMRVx+SWxuSV1sYLKBvoWVqYGZkaKRjoGRiYGxmbgRhG5iZGxgqGQBEQUGaAgoB8geUMMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNZuACRgAslBVwM8KM9AzN9EwCTZQdLKBRwcEgzaDEoAEMb3MGBwYPhgCGUKC9NQyLGFYzrGH8y6TMpMWkA1HKxAjVI8yAAphMASNWkxw=</latexit><latexit sha1_base64="5ZijBmOa1dOlQvjBvehRRmxgcLQ=">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</latexit>µµK [⇠] := N [⇠]2nd theoremp2gT µ⌫ ⇠⌫ + E (⇠ ⌫ F⌫ µ + · · · )rede nition of current<latexit sha1_base64="5jQWypz3fCSObNgM4ibQliXW7E0=">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</latexit>µµ

J [⇠] := N [⇠]

p
p
µ⌫
⌫
µ
K [⇠] = 2
gT ⇠⌫ + E (⇠ F⌫ + · · · ) ⇡ 2
gT µ⌫ ⇠⌫µ

fi

trivial part◆

2. Our proposal for matter energy current
We keep the following property for a matter energy.
A matter energy is associated with global time translation.

µ0<latexit sha1_base64="UJREwKu6xm0/GoQqg6cIZ8EUjEg=">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</latexit><latexit sha1_base64="Abi452uml/UcH5LwZ7OZ4V42C8k=">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</latexit>

J [⇠T ]

matter energy current@µ J µ [⇠T ] = @µ N µ [⇠T ]<latexit sha1_base64="t2C8VsphzBaEYMeh6azntiCmwaM=">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</latexit>

⇠Tµ =

µ

@µ K µ [⇠T ] =| {z }p2

EoMp
gT µ⌫ ¯⇠T gµ⌫ + E ¯⇠T ⇡gT µ⌫ (rµ (⇠T )⌫ + r⌫ (⇠T )µ )

=0

EMT

This current is not conserved in general curved spacetime.

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E :=total energyZ[dx0 :fixd 1µx]µ J [⇠T ] =

Z

[d

d 1

x]µ

p

g T µ0x0 :fix

same as energy in at spacetime

Evacuum = 0 by definition, since vacuum has T µ⌫ = 0.
fl<latexit sha1_base64="vaRNwxf2pZ0kBXaAvLvg5N8G9cE=">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</latexit>matter energy (non) conservationµcondition @µ J [⇠T ] ⇡<latexit sha1_base64="gCSBK1gffR/K/hmLkANJTmidWCI=">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</latexit>

p

gT

µ⌫

<latexit sha1_base64="/Do5Cw07uCTKMeja5Ng/wik8Jsg=">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</latexit>(rµ (⇠T )⌫ + r⌫ (⇠T )µ )µ

J [⇠T ] ⇡p

gT µ 01. If a metric gµ⌫ doesn’t depend on x0 , ⇠Tµ is a Killing vector and satisfies<latexit sha1_base64="INXNKToOw5/VQ1qPQtT7MLBok4w=">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</latexit><latexit sha1_base64="niFpi5Nk19T6tdiNCMJoKiaWaMY=">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</latexit>rµ (⇠T )⌫ + r⌫ (⇠T )µ = 0<latexit sha1_base64="xtXEuJprW7zbWjEbhMsgjNm+iqI=">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</latexit>

E=

Z

[d

d 1

x]µ

p

g T µ0@µ J µ [⇠T ] ⇡ 0conserved energy

x0 :fix

2. Even if ⇠Tµ is NOT a Killing vector but T µ ⌫<latexit sha1_base64="Y9V8aeZyQLJ9+0iJjeoeBAWf7uE=">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</latexit><latexit sha1_base64="1GfetbU1QG2GVuXEtK+wy5fzKmg=">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</latexit>

(T )

µ

conserved matter current<latexit sha1_base64="HBI37k0eWgQ8n3pDvNlMSzIBxmQ=">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</latexit>⌫⌫ rµ ⇠ T =µ(T ) ⌫⌫µ0 = 0⌫µ0 = 0 is satisfied,@µ J µ [⇠T ] ⇡ 0<latexit sha1_base64="HBI37k0eWgQ8n3pDvNlMSzIBxmQ=">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</latexit>

matter energy is conserved

3. Matter energy E is not conserved for a generic metric gµ⌫ .
<latexit sha1_base64="p6NdZGmaLFJBLcOeRnY76ShEAnU=">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</latexit>3. Our answer to the question in the title

1. A matter energy can be de ned in a curved spacetime.

2. A matter energy is not conserved in a general curved space time.

fi

Next question: Does a conserved matter charge other than energy exist ?

III. Conserved matter charge in GR

S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy Current, andGravitation”, Int. J. Mod. Phys. A36 (2021)2150201, arXiv:2010.07660[gr-qc].
S. Aoki, “Noetherʼs 1st theorem with local symmetries”, arXiv:2206.00283[hep-th].

<latexit sha1_base64="rY+4F4xYYyCJEv+D7dcBnI1ud1o=">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</latexit>µ@µ J [⇠] ⇡<latexit sha1_base64="DzmhpEd1NKS6NL90nkHeCclkfxU=">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</latexit>µ

If a special vector ⇣ satisfies<latexit sha1_base64="tRU+OvFlfxDNalrFP2bWkkjQfQU=">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</latexit>pgT µ⌫ (rµ ⇠⌫ + r⌫ ⇠µ )T µ⌫ (rµ ⇣⌫ + r⌫ ⇣µ ) = 0@µ J µ [⇣] ⇡ 0<latexit sha1_base64="LCNfl3d+wY4rssPuF8i4mAHoNIc=">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</latexit>

conserved matter current

This is a conserved Noether current for a global symmetry generated by✓0 ⇥ ⇣ µ (x) for a given gµ⌫ with a constant ✓0 .
<latexit sha1_base64="C1kSYFdfKOd5vy7xk1Tsk2gktiM=">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</latexit>gµ⌫ is a background field for a matter.
<latexit sha1_base64="u0miGdIKTqpitw8ICBB4gxyukxU=">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</latexit>

EOM for gµ⌫ is not assumed.
<latexit sha1_base64="ki7oK1aLrv2DclwYCEY6n0Qmi2s=">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</latexit>

<latexit sha1_base64="DzmhpEd1NKS6NL90nkHeCclkfxU=">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</latexit>

T µ⌫ (rµ ⇣⌫ + r⌫ ⇣µ ) = 0

We take ⇣ µ (x) = (x)⇠Tµ , then the condition becomes<latexit sha1_base64="Jnc+3tIJDbhzgvZ5IcNN1UoaOUE=">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</latexit><latexit sha1_base64="dZO8jfltuSi7kxJNqOofNuUNwPo=">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</latexit>

T 0 0 (x)@0 (x) + T k 0 (x)@k (x) + T µ ⌫ (x) ⌫µ0 (x) (x) = 0

1st order linear PDE

If an initial value (x0 ,8 ~x) is given at x0 , (x) for other x0 is easily obtained.
<latexit sha1_base64="0vSDUoJzbqGQNHXxeBj/S8O4F+A=">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</latexit><latexit sha1_base64="fUUABtpaYwKoe3B3vXEbvRFL8Gc=">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</latexit>

If (x) is constant, a conserved charge is a matter energy.

There exists a conserved matter charge more general than energy in GR.

A solution is known as a Kodama vector for a spherically symmetric system.

Kodamaʼ80

<latexit sha1_base64="DzmhpEd1NKS6NL90nkHeCclkfxU=">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</latexit>

condition

T µ⌫ (rµ ⇣⌫ + r⌫ ⇣µ ) = 0↵<latexit sha1_base64="ofAMgZiWg/iEp0DD/1rTqhnciCM=">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</latexit>

conserved current density

J [⇣] ⇡

pgT µ⌫ ⇣⌫

Condition and conserved current have been proposed to de ne a conserved
charge in general relativity from a di erent point of view in

S. Aoki, T. Onogi and S. Yokoyama, “Conserved charge in general relativity”,
Int. J. Mod. Phys. A36 (2021) 2150098, arXiv:2005.13233[gr-qc].
S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy Current, andGravitation”, Int. J. Mod. Phys. A36 (2021)2150201, arXiv:2010.07660[gr-qc].

fi

ff

What is a physical meaning of this matter conserved charge ?

Homogeneous and isotropic expansing Universeds2 =(dx0 )2 + a2 (x0 )g̃ij dxi dxj
T 00 =⇢(x0 ), T i j = P (x0 ) ji , T 0 j = T i 0 = 0<latexit sha1_base64="5OilFTJzdRbHocLwPMc0SHZnOSM=">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</latexit>

EMT (perfect uid)

Freedman-Lemaitre-Robertson-Walker metric<latexit sha1_base64="h6ChvQn3Yg5L8aIPXzBsKjcBdog=">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</latexit>µ<latexit sha1_base64="WxMhp2q2cs62374WyN0o+7fu0sY=">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</latexit>E(x ) :=

Z

d

d 1

x

p

0

g T 0 = Vd 1 a

d 1

⇢,

Vd 1 :=

Z<latexit sha1_base64="vxYi4+SxwHd41XKlqw4sMiksrv0=">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</latexit>

Ė
=
E

fl

energy0⇢˙ + (drµ T ⌫ = 0

covariant conservationȧ1)(⇢ + P ) = 0
a

(d

ȧ P
1)
6= 0
a⇢

not conserved if P 6= 0
<latexit sha1_base64="vqsTAF3W6Q3e2VatCXbFPHqCKuo=">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</latexit>

d

d 1

p
x g̃.

T µ ⌫ rµ ⇣ ⌫ = 0<latexit sha1_base64="z7UXU1yRauOh6EveY0mzPMKnWaY=">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</latexit>

condition<latexit sha1_base64="tGQ3ZekZcpv3ZCou3dPyknLa27o=">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</latexit><latexit sha1_base64="qPHJiJXZinwfJOI5qWMY15E1v8o=">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</latexit>µ(x ) 0µ0⇣ (x ) =<latexit sha1_base64="0b80gGeYCDveG0ruXzxXxhBywok=">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</latexit>

charge0S(x ) :=

<latexit sha1_base64="MMJkysIdmlGEzK1Eac4eetHirqY=">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</latexit>

⇢˙

0

Z

d

˙
Ṡ
Ė
=
+ =
S
E

d 1

x

(d

p

(d

ȧ
1) P = 0
a

0

d 1

ȧ P
1)
+ (d
a⇢

ȧ P
1)
=0
a⇢

g T 0 = Vd 1 a= E(x0 ) (x0 )<latexit sha1_base64="WZDgd6i3cQcMobQXNKzH3HZOiT4=">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</latexit>⇢

conserved !

Physical interpretation of the matter charge
volume density

energy density

charge densitye(x 0) = ρ(x 0)v(x 0)v(x 0) = a(x 0)d−1

ds
de
dβ
de
dv
= 0β+e 0 =+P 0 β
0
0
( dx
dx
dx
dx
dx )

s(x 0) = e(x 0)β(x 0)

Tds = de + Pdv
1st law of thermodynamics

dβ
a· Pβ
dv β
= (d − 1)
=P 0
0
dx
a ρ
dx e

S

<latexit sha1_base64="mRTF9N9LZV1g10WPHYOtrfHPsGw=">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</latexit>

entropy1/T

inverse temperature

Entropy of the Universe is conserved during its expansion.

S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy Current, andGravitation”, Int. J. Mod. Phys. A36 (2021)2150201, arXiv:2010.07660[gr-qc].

“In ation”<latexit sha1_base64="Me5VDNZljymxGaRWLqXGnGEiLqw=">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</latexit>

0<latexit sha1_base64="kMcRH5EgR/f9y3bCgrGAoqXKLPM=">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</latexit>

P (x ) =00⇢(x ) = ⇢I ,⇢(x )S. Aoki and K. Kawana (in preparation)

0
ȧ(x
)
0
H(x ) :=
= HI
0
a(x )

de Sitter-like0a(x ) = a0 e

HI x 0exponential growth<latexit sha1_base64="WsSbsHLHb9qwhT9ubSSlM+AnYOA=">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</latexit>(x0 ) =

AH
S=
4GN

0e

(d 1)HI x

0<latexit sha1_base64="OM2czZaAp1/h6dt8Brr4z/IY/Dc=">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</latexit>

AH<latexit sha1_base64="2YXL45MQD2M3shPpUAq/fu7p5ak=">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</latexit><latexit sha1_base64="ghr5vgjU1Pc90xhkM8sndFQvsvc=">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</latexit>

d

2
2
TH
<latexit sha1_base64="uD8wPCqZVSfVxO/Zu4NU0aCRIdY=">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</latexit>

Bekenstein-Hawking entropy !
area of de Sitter horizon
2⇡
1
:=
0 =
HI
TH

initial conditionde Sitter temperature

our interpretation seems reasonablefl<latexit sha1_base64="U0lYBPhCWDdr8Pstgx0OxKfwKzk=">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</latexit>

IV. Conclusion and discussion

Is energy conserved in GR ?
1. A matter energy can be de ned in a curved spacetime as a generalization ofenergy in a at spacetime.
2. A matter energy is not conserved in a general curved spacetime.
3. GR does not provide a conserved total energy in a general curved spacetime.
4. Instead, there always exists a conserved matter charge more general thanenergy in a curved spacetime, which is regarded as entropy for special cases.

Our method can be applied to gauge theories.
A. Electric charge is a matter U(1) Noether charge w/o gauge xing.
B. Non-abelian matter gauge charge can be de ned.

fi

fi

fi

fl

Ref. Sinya Aoki, “Noetherʼs 1st theorem with local symmetries”, arXiv:2206.00283[hep-th].

Further con rmations
1. Colliding gravitational waves and singularities

2. Binary star and energy

S. Aoki, T. Onogi and T. Yamaoka (work in progress)

3. Expanding universe and entropy conservationfi

and more.

S. Aoki (to appear)

S. Aoki and K. Kawana (in preparation)

Congratulations again ！

Tetsuya, I wish you a happy, healthy and fruitful life even after 60th.

Let us start a new collaboration, the Kanreki (還暦) collaboration, from now on !

Thank you !

BackupExamples of conserved energy
S. Aoki, T. Onogi and S. Yokoyama, Int. J. Mod. Phys. A36 (2021) 2150098.

Schwarzschild black hole energy

ds 2 = − (1 + u)dτ 2 − 2udτdr + (1 − u)dr 2 + r 2dΩ2d−22Λr 2
u(r) := δu(r) −(d − 2)(d − 1)<latexit sha1_base64="9919Mqcc5Yd3vW8ws8kDjMyudVc=">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</latexit>

2GMθ(r)δu(r) := −
r d−3

T
gu

sin

y

it
r
la

⌧>

⇠µ =

µ
⌧

0

ho

riz

1

r=0

on

sha1_base64="auLX+sboOLNJcaMeAeFfbFvEUAw=">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</latexit>

singularity at r = 0<latexit sha1_base64="WTPlmFN4zDkB58hVj1nhxoosyjw=">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</latexit>

sha1_base64="J/z2EsDORPQ9n9QTR/aVu60EPMg=">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</latexit><latexit

θ(r) with θ(0) = 0 handles

stationary Killing vector<latexit<latexit sha1_base64="vZreT8mxXMGNcDn+A15dHtCksY4=">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</latexit>

Eddington-Finkelstein coordinatesr<latexit

sha1_base64="UPsOu0OBs5zoFu1BzCroEsF2GAA=">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</latexit>

=

rg

⌧

⌧

<latexit
sha1_base64="a0BCgk2O9XAhjA1UXT8Vg6DV/Ag=">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</latexit>=

τ constant surface

0<latexit

sha1_base64="8CqfWT1HQcgBB60w1F0VRzlftb8=">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</latexit><

0X<latexit sha1_base64="4epq8t9frELOcYxaC8xj8/dG4lg=">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</latexit>

0<latexit sha1_base64="g1VRuu+q3phcAjkYwC9klX8WDq4=">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</latexit>

⌧
!
1

1
gµ⌫ R + ⇤gµ⌫ = 2Tµ⌫2<latexit sha1_base64="gtZ+52CjuVp6n9ho7gUw66upt8k=">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</latexit>

Einstein equation

Rµ⌫Tµ⌫ = 0 at r 6= 0
<latexit sha1_base64="64mCLgZc6T6ANRpeOG6PfWorWQg=">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</latexit>

If we calculate carefully, we obtain⌧d 2 @r (rd 3 u)
T ⌧ =
=
d
2
16⇡G
r

(d

2 d 3
1
@
u)
r (r
i
T i=
=
d
3
16⇡G
r

1 @r (r)8⇡ rd 3<latexit sha1_base64="+jfwlwyh9QceiNdpH6PvdlRq8hI=">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</latexit>2)M (r)
r
=
T
r
8⇡
rd 2

rd 3 u(r) =<latexit sha1_base64="wmZh9JcEzbbhT9EeITotqlxof2I=">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</latexit>

2GM ✓(r)<latexit sha1_base64="q3kl2c1ZDTwt4+I9U0XLEbEOrYw=">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</latexit>

black hole is not a vacuum solution to Einstein equation.
cf. Coulomb potential by a point charge is NOT a vacuum solution to Maxwell eq.

1
∇
=0
(r)
2

1
∇
∝ δ(x)(r)2

r 6= 0Einstein/Maxwell equations are distributional equations.
energy of black hole<latexit sha1_base64="30MwZHRx5RCQdtfpbIXxqxqduEI=">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</latexit>

EBH =

Z

d

d 1

x

p

⌧

gT ⌧ =

EBH = M(d

2)⌦d 28⇡<latexit sha1_base64="urIpaG7ISuSMBPuP+gldDQXou6w=">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</latexit>

at d = 4<latexit sha1_base64="7jqB/UCytThA8K8toSJdDpyXGkU=">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</latexit>

Z 1
0

dr@r (M ✓(r)) =(d

2)⌦d 2M
8⇡

4-1. Oppenheimer-Volkoff equation

A µ

s ∇µ (T ) 0 =Energy of compact starovariantly conLetus
considerfor
2 the energy 0momentum
2
2 tensor2i thejenergyasds=f(r)(dx)+h(r)dr+rg̃
dx
dx
stationary spherically symmetricij
fluid, given by"
F (r) ,with perfect(17) uid EMT
T 0 0 = −ρ(r), T r r = P (r), Tji = δji P (r), (23)where ρ(r) is the energy density and P (r) is the pressure. The Oppenheimer-Volkoff
1
2GM (R) equation[19, 20] for the
Schwarzschild metricf (r)(5)= with= 11/f (r) in the second= Vd−2 ρ, outside(18) star
metric eq.
term replacedh(r)rby another function h(r) becomes()
Z R
caserof1 [17].
(r)
MdP(R)(r)= 4⇡ GNdrρ(r)M
r2 ⇢(r) (r) ADMPmass
uantity, thanks
−
= 0
1+
h(r)
d−2
dr
r
ρ(r)
nsformation by
*
(
)+
d−1
r
2Λ
&
× d−3+
8πP (r) −, (24)µν(19)(d − 2)M (r)(d − 1)GN<latexit sha1_base64="lmwk/JQSZVmmZf5H4fC7XKGo+ew=">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</latexit><latexit sha1_base64="+b3CImB45OF74BuH5qLGXJei3Gk=">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</latexit>

R

fl

⇢(r)

Z

<latexit sha1_base64="KbKXmJnZhcWCJ2s3lPBl3HuXNVE=">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</latexit>

our energy

E=

2

Z 1

dr

4⇡G4

Z R

d x

E = M (R)observed
gravitational mass
Newtonian limit

g T 0 = 4⇡

G
E'
2

Z

Z R
0

0

<latexit sha1_base64="hfTjgXk4RE4Ndg4yPQBokc6BFww=">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</latexit>

00<latexit sha1_base64="HTTXGVIYPVXHu6n6EjaH7MtjH0c=">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</latexit>

di erence

p

dr

p

p
dr f (r)h(r)r2 ⇢(r)f (r)h3 (r) rM (r) (⇢(r) + P (r))

correction due to theinternal structure of the star:=<latexit sha1_base64="HILrV+OtZxaDUUolH73W15XamS8=">AAACbnichVHLSsNAFD2N7/qqCiKIWCwVV+VWFEUQCiq4rI9WoRZJ4rQG0yQk04IWf8C9uBAUBRHxM9z4Ay78BHEjVHDjwts0ICrqDZM5c+aeO2fuaI5peJLoMaQ0Nbe0trV3hDu7unt6I339Wc8uu7rI6LZpu5ua6gnTsERGGtIUm44r1JJmig1tb6G+v1ERrmfY1rrcd0S+pBYto2DoqmQqNzcf3VoUplSjS9uRGCXIj+hPkAxADEGk7cg1trADGzrKKEHAgmRsQoXHXw5JEBzm8qgy5zIy/H2BQ4RZW+YswRkqs3v8L/IqF7AWr+s1PV+t8ykmD5eVUcTpgW6oRvd0S0/0/mutql+j7mWfZ62hFc5279HQ2tu/qhLPErufqj89SxQw63s12LvjM/Vb6A195eCktja3Gq+O0yU9s/8LeqQ7voFVedWvVsTqKcL8AMnv7f4JspOJ5HSCVqZiqdngKdoxjDFMcL9nkMIy0sj4HTvGGc5DL8qgMqKMNlKVUKAZwJdQJj4ADpqNIA==</latexit>E<latexit sha1_base64="4rcG8Rd1+CUDNnn9hx1fQpUMr8o=">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</latexit>3d x ⇢(x) (x) < 0(x) =Newton potential

Z

Z

⇢(y)
d y
|x y|
3

G
ADM mass M (R) ' E0 +E0 : energy without gravity
d3 x ⇢(x) (x)2Z
our energy E ' E0 + G d3 x ⇢(x) (x)matter energy in Newton potential<latexit sha1_base64="QhQlAxGlr9theUI57ecSShh9b/M=">AAACp3ichVFNTxNRFD2MH2AVKbIxYTOhwZRgmttClLgiGqIbAxRbmjDYzAyv9IX5cua1ASb9A/wBFqwwMUr4GWxYsSPCTyAuMXHjgtvpJEQJeifz3rnn3XPfee9ZgSMjRXTep925e+9+/8CDzMNHg4+HssNPqpHfCm1RsX3HD2uWGQlHeqKipHJELQiF6VqOWLY23nTXl9sijKTvfVBbgVh1zXVPNqRtKqbq2dfv8+UJ3YikKz7pc3WajN8aPgv0UseQntLXPk5tGs91I2z6+diwGvpmh8uDprzO6tkcFSgJ/SYopiCHNBb87FcYWIMPGy24EPCgGDswEfG3giIIAXOriJkLGclkXaCDDGtbXCW4wmR2g8d1zlZS1uO82zNK1Dbv4vAfslLHOJ3SAV3SMR3SBf2+tVec9Oh62eLZ6mlFUB/aebr0678ql2eF5rXqn54VGphJvEr2HiRM9xR2T9/e3r1celUej5/RZ/rB/vfpnI74BF77p/1lUZT3kOEHKP593TdBtVQoviiUFqdzszPpUwxgFGPI832/xCzeYQEV3vcbTvAdZ9qENq9VtVqvVOtLNSP4IzTzCiyooPQ=</latexit><latexit sha1_base64="Jyrsd7skKOjrjfAt1kzT0EH8FlA=">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</latexit><latexit sha1_base64="9TyYIgn+fGvw2dR1/7FaP1mrx5k=">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</latexit>

consistent with our “derivation”<latexit sha1_base64="xbx/XnSIS5o15KwSG8IVVN4ZOt8=">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</latexit>

ADM massG
M (R) ' E +2Z3d xr (x) · r (x) = E0fi

ff

energy of gravitational eld ?

G
2Z⇢(x)⇢(y)d xd y|x y|3

3a factor “1/2”