Geometric conservation in curved spacetime and entropy（曲がった時空の幾何的保存則とエントロピー）
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

小研究会「一般相対論と幾何」(GRGeo)
２０２４年２月８日ー９日
名古屋大学ES総合館6階ES635号室、名古屋大学東山キャンパスReference
SA, Y. Hidaka, K. Kawana and K. Shimada, “Geometric conservation in curved
spacetime and entropy’’, arXiv:2312.09712[hep-th]

関連論文
SA, T. Onogi and S. Yokoyama, “Charge conservation, Entropy Current, and Gravitation”,
Int. J. Mod. Phys. A36 (2021)2150201.
SA and K. Kawana, “Entropy and its conservation in expanding Universe’’,
International Journal of Modern Physics A38 (2023) 2350072 [arXiv:2210.03323 [hep-th]].

SA, T. Onogi and T. Yamaoka, “Energies and a gravitational charge for massive
particles in general relativity’’, [arXiv:2305.09849 [gr-qc]].

I. Introduction（研究の目的）動機
問い：

一般相対性理論における共変的な保存量は存在するか？存在
するとしたら、その物理的意味は？

これまでの研究からの結論
1. 一般相対性理論では、ネーターの第２定理のため、共変的で保存するエネルギーは定義できない。
Einsteinの擬テンソル（共変でない）、準局所エネルギー（エネルギー密度？）2. 物質のエネルギーは共変的に定義されるが、一般には保存しない。

時間の関係で、この結論に関しての議論はしない。興味のある方は以下の論文を参照。
SA and T. Onogi, “Conserved non-Noether charge in general relativity: Physical definition vs. Noether’s
2nd theorem”, Int. J. Mod. Phys. A36 (2022) 2250129,

本講演での結論
1. 曲がった空間上のあるクラスのエネルギー運動量テンソルに対して幾何的・共変的な保存則が存在する。
2. 完全流体の場合、幾何的保存量は「エントロピー」となる。

内容
I. IntroductionII.

Set up （設定）III. Conserved current and conserved charge
IV. Geometric conservation and entropy
V. Conclusion

II. Set up（設定）1. エネルギー運動量テンソルの分解
エネルギー運動量テンソル (EMT)Tµ⌫ = "uµ u⌫ + Pµ⌫ ,<latexit sha1_base64="b9UiFf9kK+1bvU0k5NRJwzkLhT8=">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</latexit>": energy density<latexit sha1_base64="z7I5EeMLvSYhETqWbU533SvVuQU=">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</latexit>

この講演ではHawking-Ellis type Iを考える

Pµ⌫ u⌫ = uµ Pµ⌫ = 0µ

u : a time-like unit vector<latexit sha1_base64="Kl1WD+3jPD1STQQzE46r08ogHH0=">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</latexit>Pµ⌫ : pressure tensor<latexit sha1_base64="Pf4Tx61sVqUpvC3AxPW/4YynqR4=">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</latexit>

Hawking&Ellis 1973, Martin-Moruno&Visser 2018
3+1次元の場合、type I は古典的な物質に対しては標準的な分解

rµ T<latexit sha1_base64="DV5Aj3ElTfLO+swdlCHEspCCHWI=">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</latexit>

保存則

µ⌫

=0
rµ Niµ = 0<latexit sha1_base64="pbkBmkYFW8zsrYRoNZkRjyElu7o=">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</latexit>

他に保存カレントがあってもよい（電荷やバリオン数など）

i = 1, 2 · · · , f<latexit sha1_base64="tOxdhsvIKiSWyz8/K7ii0OoyBfo=">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</latexit>2. 「初期」超平面
空間的な「初期」超平面ℋd−1を選ぶµ<latexit sha1_base64="ac/tp6+1/aUj+JVp33yKbNyocEQ=">AAACa3ichVG7TgJBFD2sL8QHqI1RCyPBYEMG44NYmdhYqgiSAJLddcAN+8o+iEj8AUsbC200Mcb4GTb+gAWfYIwVJjYWXpZNjBL1bmbnzJl77py5I5mqYjuMNQNCT29f/0BwMDQ0PDIajoyNZ23DtWSekQ3VsHKSaHNV0XnGURyV50yLi5qk8j2putHe36txy1YMfdepm7yoiRVdKSuy6BCVc/cLmhs/WihFoizBvJjtBkkfROHHlhG5RQEHMCDDhQYOHQ5hFSJs+vJIgsEkrogGcRYhxdvnOEGItC5lccoQia3Sv0KrvM/qtG7XtD21TKeoNCxSziLGntgda7FHds+e2cevtRpejbaXOs1SR8vNUvh0Mv3+r0qj2cHhl+pPzw7KSHleFfJuekz7FnJHXzs+b6XXdmKNeXbNXsj/FWuyB7qBXnuTb7b5zgVC9ADJn+3uBtnFRHIlsby9FF1P+U8RxDTmEKd+r2Idm9hCxuvzGS5wGXgVJoQpYaaTKgR8zQS+hRD7BMu8jKA=</latexit>uµ (x)<latexit sha1_base64="ac/tp6+1/aUj+JVp33yKbNyocEQ=">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</latexit>u (x)ℋd−1上ではEMTはゼロでない。
uµ (x) 6= 0 at 8 x 2 Hd 1<latexit sha1_base64="8PIK/En1RVUEGN4+Wi3+LjcaXDA=">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</latexit>ℋd−1は連結でなくても良い。

Hd 1<latexit sha1_base64="2hZ5E753/lmm4PeXl2TAxu92WIk=">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</latexit>aℋd−1上の座標y (a = 1,2,⋯, d − 1)uµ (xP (y))<latexit sha1_base64="WWsC17C12Y4ZTn0mJypOes7cM4o=">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</latexit>

yb<latexit sha1_base64="HmrS94BJAVql+juRkn/L92vTIAc=">AAACZnichVHLSsNAFD2Nr1q1rYoouCmWiqsyFV+4Krpx2Yd9QK0lidMamiYhSQu1+AOCW7twpSAifoYbf8BF/0BxWcGNC2/TgGhR7zAzZ87cc+fMjGSoimUz1vEIQ8Mjo2Pecd/E5JQ/EJyeyVp63ZR5RtZV3cxLosVVReMZW7FVnjdMLtYkleek6m5vP9fgpqXo2r7dNHixJlY0pazIok1UunkolYJhFmVOhAZBzAVhuJHQg7c4wBF0yKijBg4NNmEVIixqBcTAYBBXRIs4k5Di7HOcwkfaOmVxyhCJrdJYoVXBZTVa92pajlqmU1TqJilDiLAndse67JHdsxf28WutllOj56VJs9TXcqMUOFtIv/+rqtFs4/hL9adnG2VsOV4V8m44TO8Wcl/fOGl309upSGuZXbNX8n/FOuyBbqA13uSbJE9dwkcfEPv53IMguxqNbUTXk2vh+I77FV4sYgkr9N6biGMPCWTo3ArOcYG251nwC3PCfD9V8LiaWXwLIfQJx7yK1w==</latexit><latexit sha1_base64="/juufPuPaFfXt1v8oGNSmrFUu/I=">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</latexit>xP (y)
Hd 1<latexit sha1_base64="plTEqn3aJpBJRnnlzebJOF8K/HA=">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</latexit>

Hd 1 = {xµP (y) | y 2 Hd 1 }<latexit sha1_base64="qqNcQ4pheKyk89ryydfJqQIt1TA=">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</latexit>

Hd−1はℝd−1内のd − 1次元の（必ずしも連結ではない）部分空間

ya<latexit sha1_base64="xWNrohq1Zo/ps7OPaYMZT97miFg=">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</latexit>3. 時間的曲線群と超平面の族
超平面ℋd−1から点xP(y)を選び、以下の式で時間的曲線群x μ(τ, y)を定義する。
<latexit sha1_base64="68hSFTaJNlez5UIgtBtu/VaahR8=">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</latexit>dxµ (⌧, y)= uµ (xµ (⌧, y)),d⌧xµ (0, y) = xµP (y)<latexit sha1_base64="97BhMYIedyGKlRawA85qG3ZB0FA=">AAACpHichVFNSxxBEH2OJppNjKteAl4GF2UlstQGvxAkgpccclhdVwXHHWbG1jTOFzM9i+viH/AP5JBTBJGQn+ElkKse/AkhRwNePFgzOxASiammu1+9qldd3W2HrowV0XWP1tv35Gn/wLPC8xeDL4eKwyMbcZBEjmg4gRtEW7YVC1f6oqGkcsVWGAnLs12xaR+spPHNlohiGfjrqh2KHc/a9+WedCzFlFl8e9g0vKRsKCuZbk/pS5lr1sqMXxvSVyY105i+awhlGdN6kqUfllOXBVNmsUQVykx/CKo5KCG3WlA8h4FdBHCQwIOAD8XYhYWYxzaqIITM7aDDXMRIZnGBYxRYm3CW4AyL2QNe99nbzlmf/bRmnKkdPsXlGbFSxwRd0Re6oW/0lX7Q3T9rdbIaaS9t3u2uVoTm0Mmr+u1/VR7vCh9+qx7tWWEPC1mvknsPMya9hdPVt44+3tQX1yY6k3RKP7n/z3RNF3wDv/XLOVsVa59Q4A+o/v3cD8HGm0p1rjK7OlNaXsi/YgBjGEeZ33sey3iHGhp87hm+4xJX2qT2XqtrjW6q1pNrRvGHac17uSWgZQ==</latexit>xµ (⌧, y) = xµP (y) +Z ⌧d⌘ uµ (x(⌘, y))0曲線群を用いて超平面の族を構成。
Hd 1 (⌧ )<latexit sha1_base64="F76W+sOcrfzZiyC4BOPcG/gHlyg=">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</latexit><latexit sha1_base64="eP526HxtwpdSzEqek8KKCcYwnoI=">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</latexit>uµ (⌧, y)

Hd 1 (⌧ ) := xµ (⌧, y) | 9 ⌧,8 y 2 Hd 1<latexit sha1_base64="qBoSEIo0F7LF5qMXYDdfWnW25IE=">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</latexit>

Hd 1 (0) = Hd 1<latexit sha1_base64="npJHC+GKqP2/emEWVWpsRFXQ9ds=">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</latexit>xµ (⌧, y)<latexit sha1_base64="By4hGnjCMiS5rYE3aqgZlTQeEzA=">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</latexit>uµ (0, y)<latexit sha1_base64="HQT1wu9G+fwiCWBLoMgnoBd3Bec=">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</latexit><latexit sha1_base64="t+nNlQcuI9mjNifZiw/1r581OwY=">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</latexit>

Hd 1 (0)xµ (0, y)<latexit sha1_base64="/O+pmcWfF6qvSJ94ajjyTbxn2zc=">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</latexit>

以下では簡単のために

u μ(τ, y) := u μ(x(τ, y))などの記号法を使う。

4. 「3+1」 分解µ@xe⌫a :=@y a<latexit sha1_base64="g8piMX/knkumJDF34q9LJxqBsK4=">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</latexit>

新しい座標をy A = (y 0 = τ, y a)と取る。(3+1分解）A<latexit sha1_base64="IB3CBH/fjgtTwoNLjLxT/BWGJo0=">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</latexit>g̃AB dy dy

B

2

=

2

a

a

b

hab := gµ⌫ eµa e⌫bb<latexit sha1_base64="wMQi6i3oRIbxXI6J6Bo1E7N83VY=">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</latexit>d⌧ + hab (dy + N d⌧ )(dy + N d⌧ )Na := gµ⌫ uµ e⌫a<latexit sha1_base64="+GR2xTW/omwl4Kh43Nf117BmwVE=">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</latexit><latexit sha1_base64="3q03VnwOQQMEaPxS0+WtCnY79zM=">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</latexit>g̃AB =✓<latexit sha1_base64="6kyAjSarjinmk1YRPtqObqwK78M=">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</latexit>

g̃

AB

=

1
2

1,
Na ,
✓

Nb
hab

1,
N a,

◆

<latexit sha1_base64="HX8vU7OQGTr03VD3IZ/SY84me68=">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</latexit>

◆

b

N
2 ab
B

<latexit sha1_base64="J5QrAb3ewvd1H6q7S/+ob/lKng0=">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</latexit><latexit sha1_base64="kgFRyGjJYxh/uht7j4LiePwYGSg=">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</latexit>

unit normal to Hd 1 (⌧ ):

a
b
@y
@y
B ab := g µ⌫ µ ⌫ = hab
@x @x

0
A

2

nµ (⌧, y)( > 0)<latexit sha1_base64="+wwy4LLQD8nW8xQgHk0F/o5wt3U=">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</latexit>uµ (⌧, y)<latexit sha1_base64="qBoSEIo0F7LF5qMXYDdfWnW25IE=">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</latexit><latexit sha1_base64="T0q+MCaA/tmDf2HENkbt2ip+ek0=">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</latexit>

一方で@y A µ
ũA =
u =
µ
@x

:= hab N a N b + 1

N aN b<latexit sha1_base64="Pk6e9fCuF3UuIbe3kxCyelp5Xls=">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</latexit>

@xµ
ñA =
nµ =
A
@y

2

A
0

eµa (⌧, y)<latexit sha1_base64="RRVN/xq9V1bhKY2MhbhAlHNKjJ4=">AAACdHichVHLSsNAFD2Nr1ofrboRdFEMFQUJU/GFK9GNS22tClZDEqcazItkUqjFH/AHXLhSFBE/w40/4MJPEJeKblx4mwZERb1hMmfO3HPnzFzds8xAMPaQkFpa29o7kp2pru6e3nSmr389cEPf4CXDtVx/U9cCbpkOLwlTWHzT87lm6xbf0A+WGvsbVe4HpuusiZrHt21tzzErpqEJotRMmqvaTtkOx8pCCydq42pGZgqLIvsT5GMgI44VN3OFMnbhwkAIGxwOBGELGgL6tpAHg0fcNurE+YTMaJ/jCCnShpTFKUMj9oD+e7TailmH1o2aQaQ26BSLhk/KLHLsnl2zZ3bHbtgje/+1Vj2q0fBSo1lvarmnpo8Hi2//qmyaBfY/VX96FqhgLvJqkncvYhq3MJr66uHJc3G+kKuPsnP2RP7P2AO7pRs41RfjcpUXTpGiBuS/P/dPsD6p5GeU6dUpeWExbkUSQxjBGL33LBawjBWUop6c4gKXiVdpWJKlXDNVSsSaAXwJSfkAfjqPpA==</latexit>xµ (⌧, y)<latexit sha1_base64="ZYhwk+gYG/5G4ER05jDgh+NlJEg=">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</latexit>u · n = ũ · ñ =<latexit sha1_base64="6/5OrDa3PPDwyrnOP5+q8/S+OLg=">AAACo3ichVHPSxtBFP7cqo2p1VgvBS+LIWIPDS+laBAqQi8FLxqNCm4I+2O0g5PddXc2YIP/gP+AB09KSyn9M7yIV/Xgn1B6tNBLD75sVqSVtm/Zme99874338w4oZKxJrruMx71Dww+zg3lnww/HRktjD1bi4MkckXdDVQQbTh2LJT0RV1LrcRGGAm75Six7uy87a6vt0UUy8Bf1XuhaLTsbV9uSdfWTDUL84nleoE2ffONpaXyhJkRWca8+dJS3NCzTWs3sT1z+i6dpxfNQpHKlIb5EFQyUEQWS0HhMyx4COAiQQsCPjRjBRsxf5uogBAy10CHuYiRTNcF9pFnbcJVgitsZnd43OZsM2N9zrs941Tt8i6K/4iVJkp0RV/ohs7oK32jX3/t1Ul7dL3s8ez0tCJsjh48X/n5X1WLZ43396p/etbYQjX1Ktl7mDLdU7g9ffvD4c3KXK3UmaIT+s7+j+maTvkEfvuH+2lZ1I6Q5weo/HndD8Haq3Jlpjyz/Lq4UM2eIocJTGKa73sWC3iHJdR53484xwUujZKxaNSM1V6p0ZdpxvFbGI1bpAaf6g==</latexit>

したがって( > 0)
Hd 1 (⌧ )<latexit sha1_base64="F76W+sOcrfzZiyC4BOPcG/gHlyg=">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</latexit>5. K := ∇μ u μ の評価後で使うので K = ∇μ u μ を計算する。
<latexit sha1_base64="LKJa/D0jjQPvoOO6foIW65lJzE8=">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</latexit>

K=g

µ⌫

p
1 µ⌫
1 µ⌫
1
rµ u⌫ = g (rµ u⌫ + r⌫ uµ ) = g Lu (gµ⌫ ) = p Lu (g)
22g
Lu : Lie derivative along with u<latexit sha1_base64="WnSaJU0RNgOKiRQdMKcNmYpebMY=">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</latexit>µ

g := det gµ⌫<latexit sha1_base64="p7uSdvoADA15Wu6xXs8JEGy/0EI=">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</latexit>ũ A = δ0A は定数ベクトルなので y A 座標で計算すると<latexit sha1_base64="DHaFSoSAboKYnvORBxWTY9ZTdUI=">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</latexit>

1
K=p

@

p

g̃ @y 0

g̃

n
p o
p o
1
@ n
@
p
=
(u · n) h =log(u · n) h@⌧(u · n) h @⌧ここで、− g̃ := − det g̃AB = λ 2h = (u ⋅ n)2h を使った。
ただし、h := det hab である。

III. Conserved current and conserved charge（保存カレントと保存電荷）

1. 保存カレントの構成
保存カレントEMTからu μ(x)を用いて以下のように構成

Aoki, Onogi & Yokoyama 2021 での提案の精密化

J µ (x) := T µ ⌫ (x)⇣(x)u⌫ (x) =<latexit sha1_base64="7tGeH0wrjJ7Sl0Kv3V3V3+0iyJI=">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</latexit>"(x)⇣(x)uµ (x)この定義は座標の取り方によらない保存則 ∇μ J μ = 0を満たすようにζ(x)を決める。
@
rµ J (x) = u @µ (⇣") ⇣"K =(⇣") ⇣"K = 0
@⌧
p
 Z ⌧
(n · u) h(0, y)p⇣(⌧, y)"(⌧, y) = ⇣(0, y)"(0, y) expd⌘ K(⌘, y) = ⇣(0, y)"(0, y)(n · u) h(⌧, y)0<latexit sha1_base64="l7p/lY5m8tYI07k3HD4HiPNqxSQ=">AAAC/3ichVFLaxRBEP5mfMX1kVUvgpfGJWFz2LVWfAQhEPAiesnDTQKZuPSMndik58FMz2Ky5ODVPyDoSUGCCOLVsxf/gIf8BPEYwYsHa2YniFkf1XR31Vf1fV3d7SdGZ5Zoz3GPHD12/MTYydqp02fOjtfPnV/K4jwNVDeITZyu+DJTRkeqa7U1aiVJlQx9o5b9zdtFfrmv0kzH0X27lai1UG5Eel0H0jLUqz/3Iukb2fPCXNx9wGvz8ZSYES2RF4GXyNRqacp009tWVnp9maok0yaOplriMCTuFeTBAc+L+WxxEAnPynznDzqiNaozQ716g9pUmhh1OpXTQGVzcX0XHh4iRoAcIRQiWPYNJDIeq+iAkDC2hgFjKXu6zCvsoMbcnKsUV0hGN3nd4Gi1QiOOC82sZAd8iuGZMlNggj7TG9qnT/SWvtCPv2oNSo2ily3e/SFXJb3xpxcXv/+XFfJu8egX6589W6xjuuxVc+9JiRS3CIb8/vaz/cVbCxODSXpFX7n/l7RHH/kGUf9b8HpeLbxAjT+gc/i5R52lq+3Ojfb1+WuN2enqK8ZwCZfR5Pe+iVncwRy6CBw4k84Vh9wn7q77zn0/LHWdinMBv5n74SeL5cLk</latexit>µµ<latexit sha1_base64="2BPL1IYRCOASkCmG6aNXC9yNnbQ=">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</latexit>

n
p o
@
K=
log
(u · n) h@⌧<latexit sha1_base64="xvxHMxsvZuQhRVCKJoJVg/Gr1Qc=">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</latexit>

保存カレントが決まる。
<latexit sha1_base64="I9YzORCsLECZmKy+pE3mfA0UKmw=">AAAC73ichVHNSuRAEK5k/3Rc19EFWdiDwUEZQYeaRXdFEAQvy578GxWMO3Riq41JOiadgTHMC/gCHgRZBVlkH8OLL+DBRxAvgoIXD9YkkV0V3Q7p+uqr+qqquy3fEaFCPNP0V6/fvH3X0ppre9/+oSPf2TUfyiiwecWWjgwWLRZyR3i8ooRy+KIfcOZaDl+wNiab8YUaD0IhvTlV9/myy9Y8sSpspoiq5rd//DTdqGgqFg3WB4xxY8jc4ooVcdCoD5g1FnA/FI70iKBwakx7RSojSnG4Gah4PcGNOPq3mCmpsUGaez+T3buZMnMbjWq+gCVMlvEUlDNQgGxNyfxvMGEFJNgQgQscPFCEHWAQ0rcEZUDwiVuGmLiAkEjiHBqQI21EWZwyGLEbtK+Rt5SxHvnNmmGitqmLQ39ASgP68BSP8ApP8A+e4+2zteKkRnOWOlkr1XK/2rH9afbmvyqXrIL1v6oXZ1awCqPJrIJm9xOmeQo71de2dq5mx2b64n48wAuafx/P8JhO4NWu7cNpPrMLOXqA8uPrfgrmv5TKX0sj08OFidHsKVrgM/RCke77G0zAd5iCCvW91Lq1Hs3QN/UdfU//labqWqb5CA+WfnQH9qy51w==</latexit>

J µ (⌧, y) =p⇣(0, y)"(0, y)n(0, y) · u(0, y) h(0, y)uµ (⌧, y)pn(⌧, y) · u(⌧, y) h(⌧, y)

2. 保存則と保存電荷
a foliation of ℋd−1(τ)

Md (⌧1 , ⌧2 ) := {Hd 1 (⌧ ) | ⌧1  ⌧  ⌧2 }<latexit sha1_base64="LeD2VIm9tVfcvyXcvq2yK1kWiQU=">AAACyXichVFNa9RAGH4aq9b1o1u9CF5Cl0oLukwWq6UgFLwUSqFf2xY6JSTZ2e3Q2SQms4ttyMmbf8BDTxWKiD/Di3jvoT9BvAgtiOLBN7MR0aK+IfM+88zzvPPOjB8rmWrGToasC8MXL10euVK5eu36jdHq2M31NOolgWgGkYqSTd9LhZKhaGqpldiME+F1fSU2/N0nxfpGXySpjMI1vReL7a7XCWVbBp4myq22Mh54yl7M3dYk117Pde6Z1JiyZx9zJdqaZ/ZAM5+7Weu+kxvdlM1tToV1kY2PxAbZP4Hb4Ins7Gieu9UaqzMT9nnglKCGMpai6mtwtBAhQA9dCITQhBU8pPRtwQFDTNw2MuISQtKsC+SokLdHKkEKj9hdGjs02yrZkOZFzdS4A9pF0Z+Q08YEO2Zv2Cl7z96yj+z7X2tlpkbRyx5lf+AVsTv64vbql/+6upQ1dn65/tmzRhszpldJvceGKU4RDPz9/Zenq7MrE9ld9op9ov4P2Ql7RycI+2fB0bJYOUCFHsD587rPg/VG3XlYn15+UJubKZ9iBHcwjkm670eYwzyW0KR9P+AzvuKbtWA9tZ5Z+wOpNVR6buG3sJ7/APPRrzQ=</latexit>

Z<latexit sha1_base64="vW5Zqvr1rh19NBLfqxaAwChLnAc=">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</latexit>0=
Hd 1 (⌧2 )<latexit sha1_base64="x4fxsXr45qbPL76BaiEdRJoaWSQ=">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</latexit>u<latexit sha1_base64="fvgUxby16o86sfCppIqG6z8plps=">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</latexit>

Md (⌧1 , ⌧2 )

µ

@s Md

d x

p

grµ J µMd (⌧1 ,⌧2 )<latexit sha1_base64="jFJWzUp7sMTjfXA0pMo8VygTKoI=">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</latexit>= Q(Hd 1 (⌧2 ))n<latexit sha1_base64="DUKrkH9gKoE6IPvEGtzNL9RSZQM=">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</latexit>

Q(Hd 1 (⌧1 )) +ここで<latexit sha1_base64="Wvfj4jzYb+Lfp/gQmH+GJeifNRU=">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</latexit><latexit sha1_base64="FpcRc1ilK6MWU3GLF/i33+uvj8k=">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</latexit>dµ<latexit sha1_base64="gtX0sVUqD7/KJEjymOk2bRlY/l4=">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</latexit>

Q(Hd 1 (⌧ )) :=u µ nµ = 0
on @s Md<latexit sha1_base64="uolEQd3Z0225trrw0dPfZUyzo6k=">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</latexit><latexit sha1_base64="nYEAXlTjUO4ulJqS6EGAoh2/5Jg=">AAACfnichVHLSgMxFD2Or1pfVTeKm2JV3FhTwQeuRDduBF9VwcqQGVMNpjPDTFqoRdz7Ay5cKYiIoB/hxh9w4SeISwU3LrydDoiKekOSk5N7bk4Sy1My0Iw91hn1DY1NzbGWeGtbe0dnoqt7PXCLvi2ytqtcf9PigVDSEVkttRKbni94wVJiw9qfr+5vlIQfSNdZ02VPbBf4riPz0uaaKDPR5zrJwZzHfS25MoNKzuYquXho7gyaiRRLszCSP0EmAilEseQmLpHDDlzYKKIAAQeasAJHQG0LGTB4xG2jQpxPSIb7AoeIk7ZIWYIyOLH7NO7SaitiHVpXawah2qZTFHWflEkMsQd2xV7YPbtmT+z911qVsEbVS5lmq6YVntl53Lv69q+qQLPG3qfqT88aeUyHXiV590Kmegu7pi8dnLyszqwMVYbZOXsm/2fskd3RDZzSq32xLFZOEacPyHx/7p9gfTydmUxPLI+nZueir4ihHwMYofeewiwWsIQsnXuEC9zg1oAxbIwaY7VUoy7S9OBLGNMfhKuSvg==</latexit>

Hd 1 (⌧1 )<latexit sha1_base64="vekqnJdkzZDW4QFvIiQ9ap46r7U=">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</latexit>ZZd⌃µ J µ@s M dd⌃µ J µHd

1 (⌧ )

また、構成法から空間の境界上∂sℳd では<latexit sha1_base64="+JSXgVsl7PEeIAu9adWn+wPvFrc=">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</latexit>d⌃µ J µ / d⌃µ uµ / nµ uµ = 0

したがって、Q(Hd 1 (8 ⌧2 )) = Q(Hd 1 (8 ⌧1 )) := Q<latexit sha1_base64="mHnmaUGZ4QoPmPK8BB5U+74l9LI=">AAACrXichVFBSxtBGH1ua42xbdJ6EbwsDZbk0HRWrIogCF5yNGpUcHWZXSfp4mR32Z0E4pI/0D/QgycFERH8E178Ae3BQ3sXjwpePPhlsyCt1H7DzLx5871v3szYgXQjxdjlgPbi5eCrocxwduT1m7e5/Lv3a5HfCh1Rc3zphxs2j4R0PVFTrpJiIwgFb9pSrNu7i7399bYII9f3VlUnEFtN3vDcuutwRZSVr1SLselwqVe6VrzzyegWt826H3IpTcVb1mSppM/rz+cYpdLcfNXKF1iZJaE/BUYKCkhjyc8fw8QOfDhooQkBD4qwBEdEbRMGGALithATFxJyk32BLrKkbVGWoAxO7C6NDVptpqxH617NKFE7dIqkHpJSxwT7yU7YDbtgp+yK3f+zVpzU6Hnp0Gz3tSKwct/GVu7+q2rSrPD1UfWsZ4U6ZhOvLnkPEqZ3C6evb+99v1mZW56IP7JDdk3+D9glO6cbeO1b56gqlveRpQ8w/n7up2BtsmxMl79UpwoLs+lXZDCODyjSe89gARUsoUbnnuEHfuG39lmraaa23U/VBlLNKP4IrfEAb/Gjjg==</latexit>

保存電荷3. 保存電荷の具体形<latexit sha1_base64="I9YzORCsLECZmKy+pE3mfA0UKmw=">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</latexit>

J µ (⌧, y) =<latexit sha1_base64="gtX0sVUqD7/KJEjymOk2bRlY/l4=">AAACrnichVFNaxRBEH2Z+BE3ajbxIngZXCIb0KVGNAkBIeAl5pTNuptAZh16ZjubJvPFTM9CMuwf8A948KQgIh7yI7z4B3LIRc/iMQEvHqydHRCNH9V01+tX9aqru93YV6kmOpkwJi9cvHR56kpl+uq16zPV2blOGmWJJ9te5EfJtitS6atQtrXSvtyOEykC15db7v7jUXxrIJNUReFTfRDLbiD6odpVntBMOdUnzXpue8I314ZO3rtnDeu2FtnCgrnyyFahdvI/RYdmz26pfiAcO8jsu+b6M/ZOtUYNKsw8D6wS1FDaRlR9Cxs9RPCQIYBECM3Yh0DKYwcWCDFzXeTMJYxUEZcYosLajLMkZwhm93nt826nZEPej2qmhdrjU3yeCStNzNMxvaNT+kjv6Qt9/2utvKgx6uWAvTvWytiZeX6z9e2/qoC9xt5P1T971tjFctGr4t7jghndwhvrB4cvTlsrm/P5HXpNX7n/V3RCH/gG4eDMe9OUmy9R4Q+wfn/u86Bzv2EtNh42H9RWl8uvmMIt3Ead33sJq1jDBtp87hGO8QmfDTI6RtdwxqnGRKm5gV/M2PsB3EGkrQ==</latexit>p⇣(0, y)"(0, y)n(0, y) · u(0, y) h(0, y)

Q(Hd 1 (⌧ )) :=<latexit sha1_base64="tlcYLlty9AwzqkCspu1dGP5qAh8=">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</latexit>

Q(Hd 1 (⌧ )) =

Z
Z

d⌃µ J
Hd

Hd

µ

<latexit sha1_base64="ys2eKCOjbZc387nbE9sSna8kOjA=">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</latexit>d⌃µ =1 (⌧ )

1uµ (⌧, y)pn(⌧, y) · u(⌧, y) h(⌧, y)

d

d 1

Z
p
dd 1 y h(⌧, y) nµ (⌧, y)J µ (⌧, y) =

p

y hnµ
d

Hd

1

d 1

p
y ⇣(0, y)"(0, y) nµ (0, y)u (0, y) h(0, y)µ

確かに τ によらず保存している。

Hd 1 (⌧2 )<latexit sha1_base64="x4fxsXr45qbPL76BaiEdRJoaWSQ=">AAACe3ichVHLSsNAFD2N7/qqCiK4KRZFRcuk+MKV6MalVauClZCkUw1Nk5BMCxr7A/6AC1cWRKT+hRt/wEU/QVwquFHwNg2IinqHmTlz5p47Z2Y0xzQ8wVg9IrW0trV3dHZFu3t6+/pjA4M7nl1ydZ7RbdN29zTV46Zh8YwwhMn3HJerRc3ku1phrbG/W+auZ9jWtjh2+EFRPbSMvKGrgiglNuxnddWMr1cUPzcrVyazQi0pqSkllmBJFkT8J5BDkEAYG3bsGlnkYENHCUVwWBCETajwqO1DBoND3AF84lxCRrDPUUGUtCXK4pShElug8ZBW+yFr0bpR0wvUOp1iUndJGcc4e2A37Jndsxp7ZG+/1vKDGg0vxzRrTS13lP6zka3Xf1VFmgWOPlV/ehbIYynwapB3J2Aat9Cb+vLJ+fPW8ua4P8Gq7In8X7I6u6MbWOUX/SrNNy8QpQ+Qvz/3T7CTSsoLyfn0XGJlNfyKToxiDJP03otYwTo2kKFzT1FFDbeRdykhTUszzVQpEmqG8CWk+Q8SCJIa</latexit>ℋd−1(τ) を左図のような勝手な超平面 Σ

xµ (⌧, y)<latexit sha1_base64="ZYhwk+gYG/5G4ER05jDgh+NlJEg=">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</latexit>に変えても Q の値は変わらない。
@s Md<latexit sha1_base64="FpcRc1ilK6MWU3GLF/i33+uvj8k=">AAACeXichVHLSgMxFD0d3/VVHwvBTbUo4qKkvnElunEj+KoKVobMNGownRlm0oIWf8AfcOFGBRH1M9z4Ay78BHGp4EIX3k4HREW9IcnJyT03J4nlKRloxh5iRk1tXX1DY1O8uaW1rT3R0bkWuEXfFlnbVa6/YfFAKOmIrJZaiQ3PF7xgKbFu7c1V9tdLwg+k66zqfU9sFfiOI7elzTVRZqI753FfS67MoJyzuUouHJp5M5FiaRZG8ifIRCCFKBbdxCVyyMOFjSIKEHCgCStwBNQ2kQGDR9wWysT5hGS4L3CIOGmLlCUogxO7R+MOrTYj1qF1pWYQqm06RVH3SZnEALtnV+yZ3bEb9sjef61VDmtUvOzTbFW1wjPbj3pWXv9VFWjW2P1U/elZYxtToVdJ3r2QqdzCrupLB8fPK9PLA+VBds6eyP8Ze2C3dAOn9GJfLInlE8TpAzLfn/snWBtJZybS40tjqZnZ6Csa0Yt+DNF7T2IG81hEls49wCmucB17M/qMIWO4mmrEIk0XvoQx+gFD3JJI</latexit>@s Md<latexit sha1_base64="FpcRc1ilK6MWU3GLF/i33+uvj8k=">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</latexit>uµ<latexit sha1_base64="fvgUxby16o86sfCppIqG6z8plps=">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</latexit>d⌃µ<latexit sha1_base64="ytqffd3UWoEkv/lxcXxMdbgeJh0=">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</latexit>⌃<latexit sha1_base64="KZvhVc7AzzfbCjtLyeoWV0rUzhE=">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</latexit><latexit sha1_base64="eUYllazU4rmBI1PkjLtxJPyL03g=">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</latexit>

Q(Hd 1 (⌧ )) = Q(⌃) = Q
Hd 1 (⌧1 )<latexit sha1_base64="vekqnJdkzZDW4QFvIiQ9ap46r7U=">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</latexit>

IV. Geometric conservation and entropy（幾何的保存量とエントロピー）

1. ζ の初期条件<latexit sha1_base64="I9YzORCsLECZmKy+pE3mfA0UKmw=">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</latexit>µJ (⌧, y) =p⇣(0, y)"(0, y)n(0, y) · u(0, y) h(0, y)uµ (⌧, y)pn(⌧, y) · u(⌧, y) h(⌧, y)

保存カレントは ζ の初期条件 ζ(0,y) に依存するが、以下の初期条件を考える。
p⇣(0, y)"(0, y)n(0, y) · u(0, y) h(0, y) = 1<latexit sha1_base64="1l82AB8LaRpwddqAoqPg+ZeTFeM=">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</latexit>uµ (⌧, y)pn(⌧, y) · u(⌧, y) h(⌧, y)<latexit sha1_base64="B7g1jhts9l6NyKbLzcwNG8miIQU=">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</latexit>

J µ (⌧, y) =<latexit sha1_base64="IKUJmuMMVwX3LfmrWoiAM02HDMg=">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</latexit>

Q=Z

dd 1 yHd

幾何的保存カレント

幾何的保存量（”重力荷”)

1

これらの定義は座標系に依存しない。初期平面の取り方には依存する。

Q は y a のvolume preserving diffeo では不変。
保存カレントは自明に見えるが、後述のように完全流体の場合はJ μ = エントロピーカレント になる。

2. 幾何的保存カレント
保存則には時間的な u μ(τ, y) が存在し、以下を満たすことが重要。
<latexit sha1_base64="4JgYPFDaKN3towJX5ADqJZbE0u4=">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</latexit>8µuµ (⌧, y) = 0 at 8 ⌧ for y 2/ Hd 1<latexit sha1_base64="PGhkrbhK+RNdlat1HrYFDeYfYvg=">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</latexit>u (⌧, y) 6= 0 at ⌧ for y 2 Hd 1<latexit sha1_base64="NCJ/yLXsq0bU+/fRue+E4glbSEw=">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</latexit>µu (⌧, y)<latexit sha1_base64="6kyAjSarjinmk1YRPtqObqwK78M=">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</latexit>g̃<latexit sha1_base64="DSSuj3LtMDZ/y1Lb4STjdNtfmWE=">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</latexit>

AB

=

2

b

1,
N a,

N
2 ab
B

det B ab= g̃⌧ ⌧ =AB
det g̃

<latexit sha1_base64="fnyu9ZH4oH1Fb/UurijarlvOkGg=">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</latexit>

b :=
y

1

✓

p

a
b
@y
@y
B ab = g µ⌫ µ ⌫@x @x<latexit sha1_base64="kds99dFXkXBGZQC77wbR7QDV9h0=">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</latexit>

cofactor を使うと

1
1

det B ab = p

◆

g̃

1
p
(u · n) h=なので<latexit sha1_base64="RRKaK88IiQ3fGKfcZxTN9Y85uo4=">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</latexit>

幾何的カレントJ µ = buµ<latexit sha1_base64="oO72EBwj6h0LhtND48IVUFONwlE=">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</latexit>

と書ける。

Hd 1
<latexit sha1_base64="df5e5pATk6pLjYbBFtIRrww2F7o=">AAACbHichVG7SgNBFD1Z3/GR+CgEEcREsTHMBnxgJdikjI+YSJSwu07ikn2xOwnExR+wtbBQCwUR8TNs/AELP0EEGwUbC282C6Ki3mFmzpy5586ZGdUxdE8w9hCR2to7Oru6e6K9ff0Dsfjg0KZn11yN5zTbsN2Cqnjc0C2eE7oweMFxuWKqBs+r1ZXmfr7OXU+3rQ3RcPiOqVQsvaxriiBqK5kp+buz8kGyFE+wFAti4ieQQ5BAGFk7foVt7MKGhhpMcFgQhA0o8KgVIYPBIW4HPnEuIT3Y5zhAlLQ1yuKUoRBbpbFCq2LIWrRu1vQCtUanGNRdUk5git2za/bC7tgNe2Tvv9bygxpNLw2a1ZaWO6XY4ej6278qk2aBvU/Vn54FylgMvOrk3QmY5i20lr6+f/yyvrQ25U+zC/ZE/s/ZA7ulG1j1V+1yla+dIEofIH9/7p9gM52S51Nzq+nE8mL4Fd0YwyRm6L0XsIwMssjRuSaOcIqzyLM0Io1J461UKRJqhvElpOkPjiuMdw==</latexit>

1<latexit sha1_base64="khiuWoFOox4atCQKP00BFWLMqJU=">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</latexit>µ

別の表式（計算略） J =μ(dμ1 µ↵1 ···↵d 1
p e
ẽ0a1 ···ad 1 @↵1 y a1 · · · @↵d 1 y ad 1
1)!
g̃
a

したがって、 ∇μ J = 0 と J ∂μ y = 0 が成り立つ。また1
J /p<latexit sha1_base64="HvUhSbL6p69OehRH1J6GoP8qb3I=">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</latexit>µg

3. 完全流体の有効理論
Dubovsky, Hui, Nicolis & Son 2012 を曲がった時空に拡張
力学変数

y a(x)：流体の共動座標、ψ(x)：保存量(一つを仮定）に対応する位相

不変性

ポアンカレ対称性、y a → f a(y) with det(∂b f a) = 1 (volume preserving diffeo)
位相変換：ψ(x) → ψ(x) + c

低エネルギー有効理論<latexit sha1_base64="pb46jFHDJ6tRAepSlM8MlkjtmrU=">AAAChnichVFNLwNBGH6sr6qPFheJy9IQEpqp+IpE0kQijqgiUZrd7aiJ7e7anTZonCX+gIMTiYiIKz/AxR9w8BPEkcTFwdvtJoLgnczMM8+8zzvPzOiOKTzJ2GONUltX39AYago3t7S2RaLtHcueXXQNnjZs03ZXdc3jprB4Wgpp8lXH5VpBN/mKvj1T2V8pcdcTtrUk9xy+XtDyltgUhiaJykZ7Uuq0mhGWVHMbOXU3M6RmvB1XlofzB+rsgD60P5iNxlic+aH+BIkAxBDEvB29QAY52DBQRAEcFiRhExo8amtIgMEhbh1l4lxCwt/nOECYtEXK4pShEbtNY55WawFr0bpS0/PVBp1iUndJqaKPPbBL9sLu2RV7Yu+/1ir7NSpe9mjWq1ruZCNHXam3f1UFmiW2PlV/epbYxKTvVZB3x2cqtzCq+tL+8UtqarGv3M/O2DP5P2WP7I5uYJVejfMFvniCMH1A4vtz/wTLI/HEeHxsYTSWnAy+IoRu9GKA3nsCScxhHmk69xDXuMGtElLiypgyUU1VagJNJ76EkvwAmzOUYA==</latexit>

S=

Z

d

d x

p

<latexit sha1_base64="EJ6FlXw+b9FMNIb/dPiUYJ89M2U=">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</latexit>

gF (b, z)b=<latexit sha1_base64="HcVluisaOuCIqwhNH/gZcz6tIZQ=">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</latexit>

ψ(x) → ψ(x) + c の保存カレント

N1µ (x) :=

p

微分展開

Jµ
=
@µ
b

det B ab , z := uµ @µS@µ= n1 (x)uµ (x)

保存量(”粒子数”）密度n1 = Fz := @z F<latexit sha1_base64="n3Fm2ka1atRadKF0No9dLsJDASQ=">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</latexit>4. 完全流体のエントロピー<latexit sha1_base64="AXUrtnyXOnaKSM30iQwHQmT9edo=">AAADSnichVHNahRBEK6ZRI3rT1a9CF4al8guskvv4k8IBAKBRfSSZLNJIJMMPZPOpEnPT2Z6FrPDvEBeIIecFETEx/DiwauHPIAH8WYEET1YOzNm1ZhYTXdXfV3fV9XdViBFpCg91PSR0XPnL4xdLF26fOXqePna9aXIj0Obd21f+uGKxSIuhce7SijJV4KQM9eSfNnanh2cL/d4GAnfW1S7AV9zmeOJTWEzhZBZ/rC4nhhubHhxWn1am5pOWoaP+cSIdkKV1A0l5AYnTpomRsBCJZgknSLjV+yYvymk086xIGmTu0mrbfZzgpWSoUiVPF43mAy22LGQmcdGEInaqSXSequKiqRfb5sWsWpDRauo8i+SWa7QBs2MnHSahVOBwub88kswYAN8sCEGFzh4oNCXwCDCsQpNoBAgtgYJYiF6IjvnkEIJuTFmccxgiG7j6mC0WqAexgPNKGPbWEXiDJFJYIK+p6/oEX1LX9OP9MepWkmmMehlF3cr5/LAHN+72fn6X5aLu4KtIevMnhVswmTWq8DegwwZ3MLO+b3+/lFnamEiuUOf00/Y/zN6SN/gDbzeF/vFPF84gBJ+QPPv5z7pLLUazQeN+/P3KjOTxVeMwS24DVV874cwA49gDrpga0+0Ha2vJfo7/bP+Tf+ep+pawbkBf9jI6E9XIeHW</latexit>2µ⌫
T (x) := p

EMT↵<latexit sha1_base64="92QdJfLPb5YCBzl0MUZHWaF7ogg=">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</latexit>

↵
@S
2F
@(J
@↵ )
z
µ⌫
=g F+
b
@gµ⌫
g̃ @gµ⌫ (x)<latexit sha1_base64="lyeMHGefAUzPyi6qOpso9jrnsms=">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</latexit>@(J @↵ )
=
@gµ⌫

bz µ⌫
g
2

@b
=
@gµ⌫

@b
Fb b)
b@gµ⌫

2(Fz zb µ↵ ⌫
g g Bab @↵ y a @ y b =2= g ↵ + u↵ ub µ⌫(g + uµ u⌫ )2<latexit sha1_base64="4WOtlcKcr8/HeRR11y4+xfwERUY=">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</latexit>" = Fz z<latexit sha1_base64="qerds3SH3huV6aJEA2j4g2gjVEo=">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</latexit>

T µ ⌫ = (FFb b) ⌫µ + (Fz z<latexit sha1_base64="uO4gWnrlMf3voIeIPmXI41jCOkU=">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</latexit>

Pd" = zdn1<latexit sha1_base64="WibnXAkDTZ4UmiCZs6HSwLskX/0=">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</latexit>

ε+P

Fb db

s = b, T =<latexit sha1_base64="qplduvpnZETG85M/HjunF+hPdaY=">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</latexit>

Fb b)uµ u⌫

F

P = F Fb b
n 1 = Fz<latexit sha1_base64="5yb6Xqkqc0Lgo4qfsqN8M216+GA=">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</latexit><latexit sha1_base64="Sgq+J3lj2KPoUbzZDPC7ah3pkkA=">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</latexit>

熱力学関係式 dε = Tds + μ1dn1 と比較
Fb , µ 1 = z

μ1 : n1 に対する化学ポテンシャル

b はエントロピー密度！
ε + P − μ1n1 = − Fbb = Ts もう１つの熱力学関係式も成り立つ

V. Conclusion（結論）結論
1. 重力系では幾何的保存則が常に成り立つ。
µ
u(⌧, y)µ
p
J (⌧, y) =n(⌧, y) · u(⌧, y) h(⌧, y)<latexit sha1_base64="QNirLeZEZJISQv9jYu2Q5JoicGs=">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</latexit>

保存カレント<latexit sha1_base64="IKUJmuMMVwX3LfmrWoiAM02HDMg=">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</latexit>

重力荷

Q=

Z

dd 1 yHd

1

2. 完全流体の場合、幾何的保存量はエントロピー。

解釈
1. 電磁気力の素が電荷であるように、重力の素（重力荷）は”エントロピー”。

c.f. T. Jacobson 1995, E.P. Verlinde 2011.
2. アインシュタイン方程式 Gμν + Λgμν = 2κTμν を通して、重力側にも幾何的保存則は存在。その意味は？

今後
1. 散逸のある流体の場合の幾何的保存則は物理的に何を意味するか？
重力系ではエネルギーではなくエントロピーが保存する？

2. 幾何的保存則の（面白い）応用例。求む専門家の協力／助言。

ご清聴ありがとうございます。

BackupB1. より一般的な 「3+1」 分解
ADM分解

y A = (y 0, y a), τ = f(y 0), f′(y 0) > 0と取る。

ds2 =(f 0 )2 (dy 0 )2 + hab (dy a + f 0 N a dy 0 )(dy b + f 0 N b dy 0 )<latexit sha1_base64="a86lzTZ5C05ZtZZhCmw1Fr+bYqw=">AAACyXichVFNSxxBEH1OjDGbqJt4CXgZXAwrkqVWgpFAQMhFEMSPrAquu/TM9GrjfGVmdsm6ePKWP+DBk4Eg4s/wEnL34E8IuQgKISGH1PROCEZiqunuV6/qVVd3W6Gr4oTovMe403u3717//dyDhwODQ/lHj1fioBnZsmIHbhCtWSKWrvJlJVGJK9fCSArPcuWqtf06ja+2ZBSrwH+TtEO54YlNXzWULRKm6nnHiWuT5ivzWbFRq4aR8mTVZbUjxpkuOu0apWDC3Kp3hLWrGcHu72Rznl2dZeqYdT1mdWP1fIFKpM28CcoZKCCzhSB/hCocBLDRhAcJHwljFwIxj3WUQQiZ20CHuYiR0nGJXeRY2+QsyRmC2W1eN9lbz1if/bRmrNU2n+LyjFhpYozO6Jgu6ROd0Bf6+c9aHV0j7aXNu9XVyrA+9P7J8rf/qjzeE2z9Ud3ac4IGpnWvinsPNZPewu7qWzv7l8svl8Y6T+kDfeX+D+mcTvkGfuvK/rgolw6Q4w8o//3cN8HKZKk8VZpafF6Ymc6+oh8jGEWR3/sFZjCLBVT43M+4wHf8MOaMt8Y7Y6ebavRkmmFcM2PvF6Juq9I=</latexit>✓<latexit sha1_base64="W1YmS23S4Oq0TkUHGgWcYAZS+Dk=">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</latexit>g̃AB =(f 0 )2 ,f 0 Na ,f 0 Nbhab◆<latexit sha1_base64="F5u+YdlLOl0x9Pk4tR4J2XMUYj0=">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</latexit>

g̃

AB

=

1
(f 0 )2✓

1,
f 0N a,0

b

fN(f 0 )2 B ab◆<latexit sha1_base64="WaDMPo9mFr0BMwfFsxCxcMZpj4Q=">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</latexit>

0<latexit sha1_base64="g5J48d+og3YAJA/pYhg6Bs2dJhs=">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</latexit>

2<latexit sha1_base64="vXL6cffusbjkdr5myn/vjN48C9Y=">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</latexit>g̃ = (f ) h

ñA =

f

0

0
A

ũA =

A
0
f0

nµ (⌧, y)<latexit sha1_base64="+wwy4LLQD8nW8xQgHk0F/o5wt3U=">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</latexit>uµ (⌧, y)<latexit sha1_base64="qBoSEIo0F7LF5qMXYDdfWnW25IE=">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</latexit>u · n = ũ · ñ =<latexit sha1_base64="p4N1loUw1JLxLn36Pq1JlksUYNE=">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</latexit>eµa (⌧, y)<latexit sha1_base64="RRVN/xq9V1bhKY2MhbhAlHNKjJ4=">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</latexit>xµ (⌧, y)<latexit sha1_base64="ZYhwk+gYG/5G4ER05jDgh+NlJEg=">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</latexit>p
1
K = p Lug̃g̃<latexit sha1_base64="jifVdvyeF4UBhiHEF9u94+Orht0=">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</latexit>μK = ∇μ u の計算
Lu

p

p

A

p

00

p

p

d
@y
1
f
g̃
Hd 1 (⌧ )g̃ = ũ @Ag̃ +det= 0 @0gg̃ = @0 00B02
dt ✓ @(y ) ◆t=1
f
(f )
f
A
00
d
@y
f
det
=1
y 0 = y tu
0
B
dt
@(y )
(f 0 )2pp
1@(u · n) hpK== @⌧ log[ (u · n) h]
前と同じ結果00@y(u · n) hf
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