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. Yokoyama, “Conserved charge in general relativity”, arXiv:2005.13233[gr-qc]. S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy, and Gravitation”, arXiv:2010.07660[gr-qc]. Sinya Aoki Tetsuya Onogi Shuichi Yokoyama

O. Motivation Energy in general relativity

Einstein equation conservation Problem: gravitational energy ? rewrite Einstein’s pseudo-tensor Bianchi identity but matter (Conserved) energy in general relativity gravity violate the fundamental principle of general relativity ! R µ ν − 1 T µ ν ( x ) = δ S matter 2 g µ ν R + Λ g µ ν = 8 π G d T µ ν δ g µ ν ( x ) r µ T µ ν = 0 ∂ µ T µ ν 6 = 0 ∂ µ hp i | g | ( T µ ν + t µ ν ) = 0 t µ ν is not covariant under general coordinate transformation.

Quasi-local 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. Arnowitt-Deser-Misner (ADM) energy cf. Gauss’s law in electromagnetism However quasi-local energy cannot tell a distribution of energy. Hamiltonian with Gibbons-Hawking term Komar energy, Bondi energy Z Z E = dV (local energy) E = dS (quasi-local energy) r →∞ Z Z dS µ F 0 µ Q = dV J 0 = V ∂ V *

Our aim I. Invariant under time translation (Killing vector exists) Generic conserved charge in GR. II. Without time translation symmetry To give a precise and universal definition of energy by the volume integral of local energy if exists and extend it to more general cases. Energy. meaning ?

I. Conserved charge with symmetry S. Aoki, T. Onogi and S. Yokoyama, “Conserved charge in general relativity”, arXiv:2005.13233[gr-qc]

conserved charge Stokes’ theorem Symmetry assume scalar covariantly conserved vector current Killing vector L ξ g µ ν = r µ ξ ν + r ν ξ µ = 0 ν ) ξ ν + 1 r µ ( T µ ν ξ ν ) = ( r µ T µ 2 T µ ν ( r µ ξ ν + r ν ξ µ ) = 0 Z √− g T 0 Q ( ξ ) = d Σ 0 ν ξ ν J µ := T µ ν ξ ν Σ ( x 0 ) Z Z Z ⇣p | g | J µ ⌘ | g | r µ J µ = * p p d d x d d x ∂ µ | g | J µ d Σ µ 0 = = ∂ M M M 1 ⇣p | g | J µ ⌘ r µ J µ = ∂ µ p | g | ∂ M = ∂ M s � Σ t 2 Σ t 1 Σ t 2 ∂ M s M d Σ k J k = 0 on ∂ M s Q ( Σ t 2 ) = Q ( Σ t 1 ) Σ t 1

ex. stationary space time Quasi-local energy Killing vector conserved energy Covariant and universal definition of total energy works for an arbitrary asymptotic behavior in an any coordinate system. cf. Komar energy ξ µ = − δ µ 0 a metric g µ ν does not contain x 0 Z √− g T 0 E = − d Σ 0 0 Σ ( x 0 ) Komar, PR127(1962) 1411 p� g r ν r [0 ξ ν ] = p� g r [0 ξ k ] c Z c Z E ( ξ ) = d Σ 0 d Σ 0 k 16 π G d 16 π G d Σ ( x 0 ) ∂ Σ ( x 0 ) c : some constant This is a Noether charge of the 2nd type for a coordinate transformation ξ ν . 2 c Z √− g R 0 For ξ µ = − δ µ E = − d Σ 0 0 , 0 16 π G d Σ ( x 0 )

Our formula has been known, but rarely used. No application. See also lecture notes by Blau; Shiromizu (Japanese); Sekiguchi (Japanese). Let us consider some applications. These were forgotten in major textbooks (e.g. Landau-Lifshitz) except a few. 1. V. Fock, TheTheory of Space, Time and Gravitation (Pergamon Press, New York 1959) p� g dx 1 dx 2 dx 3 will be constant, · · · , if the vector Z T µ 0 ϕ µ The quantity I = ϕ µ satisfies the equation r ν ϕ µ + r µ ϕ ν = 0. 2. A. Tautman, Kings Collage lecture notes on general relativity, mimeographed note (unpublished), May-June 1958; Gen. Res. Grav. 34 (2002), 721-762, cited Fock. 3. A. Tautman’s lecture notes was cited by Komar in PRD127(1962)1411. 4. R. Wald, General Relativity (The University of Chicago Press, Chicago, 1984), p.286, footnote 3. a Killing vector field ξ a is presented, · · · , r a ( T ab ξ b ) = ( r a T ab ) ξ b + T ab r a ξ b = 0, Σ T ab ξ b n a is conserved, i.e., independent of choice of Cauchy surface Σ . R so measure term is not specified, though.

I-1. Schwarzschild black hole Killing volume of (d-2)-sphere Energy This reproduces known results. For example Remark The BH energy is independent of the cosmological constant. However the Killing vector becomes space-like inside the horizon. metric EMT Eddington-Finkelstein (d-2)-sphere ds 2 = − (1 + u )( dx 0 ) 2 − 2 udx 0 dr + (1 − u ) dr 2 + r 2 ¯ g ij dx i dx j u ( r ) = − r d − 3 2 Λ r 2 0 r d − 3 − ( d − 2)( d − 1) ∂ r ( r d − 3 f ( r )) f ( r ) := r d − 3 0 = − ( d − 2) ξ µ = − δ µ T 0 0 0 16 π G d r d − 2 r d − 3 Z ∞ dr ∂ r ( r d − 3 f ( r )) = ( d − 2) V d − 2 r d − 3 0 = ( d − 2) V d − 2 Z d d − 1 x √− g T 0 0 E = − 16 π G d 16 π G d 0 Z d d − 2 x p V d − 2 := det ¯ g ij r 0 E = = M at d = 4 2 G 4 x 0 = constant hypersurface is space-like even inside the horizon.

cf. Komar energy Komar energy for BH diverges for non-zero cosmological constant. But it is ill-defined at d=3. This choice reproduces known results. our result Our definition is much more robust and universal. 4 Λ r d − 1 d d − 2 x p� g r [0 ξ r ] = lim  � Z c cV d − 2 ( d � 3) r d − 3 E Komar = � 0 16 π G d 16 π G d ( d � 2)( d � 1) r →∞ r →∞ E = ( d − 2) V d − 2 r d − 3 E Komar = c ( d − 3) V d − 2 r d − 3 0 0 at Λ = 0 16 π G d 16 π G d c = d − 2 d − 3

I-2. BTZ black hole Killing vectors Angular momentum d=3 AdS Energy EMT Bandaos-Teitelboim-Zanelli, PRL69(1992)1849. 1 ds 2 = − f ( r ) dt 2 + f ( r ) dr 2 + r 2 ( d φ − ω ( r ) dt ) 2 , f ( r ) = r 2 L 2 − 2 G N M θ ( r ) + G 2 N J 2 ω ( r ) = G N J 4 r 2 , 2 r 2 , ξ µ T := − δ µ 0 , ξ S := δ µ φ ∂ r ( r 3 ω r ) δ ( r ) M 0 = − M T 0 T 0 φ = − 16 π G N 8 π r r Z 2 π Z 0 = M dr r T 0 E = − d φ 4 π 0 Z 2 π Z φ = J dr r T 0 P φ = d φ 8 0

stationary spherically symmetric metric Einstein equation solution to OV equation EOS with Oppenheimer-Volkoff equation I-3. Compact star with perfect fluid Oppenheimer-Volkoff, PR55(1939)374. ds 2 = − f ( r )( dx 0 ) 2 + h ( r ) dr 2 + r 2 ˜ g ij dx i dx j T 0 T r T i j = δ i 0 = − ρ ( r ) , r = P ( r ) , j P ( r ) r d − 1 − dP ( r ) = G d M ( r ) ⇢ ✓ 2 Λ ◆� ( P ( r ) + ρ ( r )) h ( r ) d − 3 + 8 π P ( r ) − r d − 2 ( d − 2) M ( r ) ( d − 1) G d dr 2 Λ r 2 h ( r ) = k − 2 G d M ( r ) 1 − r d − 3 ( d − 2)( d − 1) Z r 8 π dss d − 2 ρ ( s ) , M ( r ) = M (0) = 0 d − 2 0 P = P ( ρ )

Schwarzschild metric outside star Schwarzschild metric with m = M ( R ) 2 Λ r 2 h ( r ) = k − 2 G d M ( R ) 1 f ( r ) = − r d − 3 ( d − 2)( d − 1) r ∞ R ρ ( r ) = P ( r ) = 0 r > R, ρ ( r ) radius of compact star R from P ( r = R ) = 0

Energy of a compact star Killing vector cf. Angus-Cho-Park, Eur. Phys. JC 78 (2018) no.6, 500 corrections due to a structure inside star gravitational mass felt by distant objects (estimation by quasi-local energy) A similar result can be found in their appendix. conserved energy ξ µ = − δ µ 0 Z R Z ∞ Z p p d d − 2 x f ( r ) h ( r ) r d − 2 ρ ( r ) | g | T 0 E = − 0 = V d − 2 dr 0 0 Z R dM ( r ) 8 π = ( d − 2) V d − 2 f ( r ) h ( r ) dM ( r ) d − 2 r d − 2 ρ ( r ) = p dr 8 π dr 0 Z R " # = ( d − 2) V d − 2 M ( r ) d M ( R ) − dr log | f ( r ) h ( r ) | 8 π 2 0

Physical meaning leading correction term (Newtonian limit) Komar energy misses the gravitational interaction energy. cf. Komar energy correction term represents the gravitational interaction energy ! Our definition is physically more sensible. gravitational interaction energy at d=4 Z R dr rM ( r ) ρ ( r ) + · · · ∆ E ' � G d V d − 2 0 Z R d d − 1 x d d − 1 y ⇢ ( ~ x ) ⇢ ( ~ y ) U 4 := − G d Z = 4 ⇡ G 4 dr rM ( r ) ⇢ ( r ) 2 | ~ y | x − ~ 0 E Komar = c ( d − 3) V d − 2 M ( R ) 8 π

constant density star of the same mass M due to the energy conservation, unless an extra energy correction fixed is released. Size of corrections Correction could be large ! Gas of a total mass M (with negligible interaction) can not becomes a compact ρ ( r ) = ρ 0 d = 4 , Λ = 0 M BH = 4 π R 3 ρ 0 R ≥ R min = 9 GM BH 4 3 ✓ R  ◆� 3 q 0 − R 2 sin − 1 r 2 R (3 r 2 0 − R 2 ) − 3 r 2 r 2 E = M BH − πρ 0 0 := 0 8 π G 4 ρ 0 r 0 ' � 68% of M BH at R = R min

II. Conserved charge without symmetry S. Aoki, T. Onogi and S. Yokoyama, “Charge conservation, Entropy, and Gravitation”, arXiv:2010.07660[gr-qc].

If the Killing vector is absent (no symmetry), fix direction A solution exists (at least locally in t). simultaneous linear ODE initial value is given on a “conservation condition” for v hypersurface at fixed t. sufficient condition in general 1st order linear PDE Charge Z p d d − 1 x | g | T 0 Q [ v ]( t ) = ν v ν , Σ t dQ [ v ] � � d d − 1 ⃗ | g | T µ ν ∇ µ v ν . = x 6 = 0 dt Σ t ν ∇ µ v ν = 0 , T µ A µ ν ∂ ν v µ + B µ v µ = 0 , ν := T µ ν , B µ = T α β Γ β A µ α µ A µ ∂ µ v + Bv = 0 , v µ = v δ µ µ 0 dx µ = A µ ( x ) , dt dv ( t ) = − B ( x ) v ( t ) . dt