Nonrelativistic, non-newtonian gravity

Átírás

1 Nonrelativistic, non-newtonian gravity Dieter Van den Bleeken Bog azic i University based on arxiv: and work in progress with C ag ın Yunus IPM Tehran 27th May 2016

2 Nonrelativistic, non-newtonian gravity Gravity in various regimes c LT 1 c LT 1 k (1) µν = 1 c 4 T (0) µν Φ = ρ G µν = G N c 4 T µν? G N L 5 T 3 G N L 5 T 3

3 Nonrelativistic, non-newtonian gravity Gravity in various regimes c LT 1 c LT 1 k (1) µν = 1 c 4 T (0) µν Φ = ρ G µν = G N c 4 T µν? G N L 5 T 3 G N L 5 T 3 [Dautcourt] Newton-Cartan gravity

4 Motivation Outline From NG towards GR NC gravity/geometry From GR to NC NC NG dimensional reduction in NC From GR to TTNC Example Legend NG = Newtonian Gravity NC = Newton-Cartan GR = General Relativity TT = Twistless Torsional

5 Motivation Outline From NG towards GR NC gravity/geometry From GR to NC [Cartan; Trautman; Kunzl] Review [Dautcourt; Tichy, Flanagan] NC NG dimensional reduction in NC From GR to TTNC New Related work [Afshar, Bergshoeff, Hartong et al.] Example Legend NG = Newtonian Gravity NC = Newton-Cartan GR = General Relativity TT = Twistless Torsional

6 Motivation Isn t GR our best theory of gravity? our most complicated theory of gravity all aproximation schemes are welcome

7 Motivation Isn t GR our best theory of gravity? our most complicated theory of gravity all aproximation schemes are welcome Why nonrelativistic? c is quite large in SI units more recently condensed matter on non-flat backgrounds non-relativistic holography

8 Motivation Isn t GR our best theory of gravity? our most complicated theory of gravity all aproximation schemes are welcome Why nonrelativistic? c is quite large in SI units more recently condensed matter on non-flat backgrounds non-relativistic holography Why non-newtonian? analytic description of strong gravity! isn t this an empty set? actually, no... is this a physically relevant regime? wasn t this done 50 years ago? please let me know!

9 From NG towards GR c LT 1 c LT 1 k (1) µν = 1 c 4 T (0) µν Φ = ρ G N L 5 T 3 G µν = G N c 4 T µν? G N L 5 T 3

10 From NG towards GR Geometrize Newtonian gravity a = Φ geodesic equation Γ i 00 = i Φ Φ = ρ curvature condition R 00 (Γ) = ρ

11 From NG towards GR Geometrize Newtonian gravity a = Φ geodesic equation Γ i 00 = i Φ Φ = ρ curvature condition R 00 (Γ) = ρ covariantize this: Newton-Cartan geometry/gravity

12 Newton-Cartan geometry/gravity Geometry fields: τ µ, h µν, Γ λ µν constraints: τ µ h µν = 0, h [µν] = 0, Γ λ [µν] = 0 µ τ ν = 0 µ h νλ = 0 h λ(µ R ν) λρσ(γ) = 0

13 Newton-Cartan geometry/gravity Geometry fields: τ µ, h µν, Γ λ µν constraints: τ µ h µν = 0, h [µν] = 0, Γ λ [µν] = 0 µ τ ν = 0 µ h νλ = 0 h λ(µ R ν) λρσ(γ) = 0 Gravity EOM: R µν (Γ) = τ µ τ ν ρ

14 Newton-Cartan geometry/gravity Geometry fields: τ µ, h µν, Γ λ µν constraints: τ µ h µν = 0, h [µν] = 0, Γ λ [µν] = 0 µ τ ν = 0 µ h νλ = 0 h λ(µ R ν) λρσ(γ) = 0 Gravity EOM: R µν (Γ) = τ µ τ ν ρ Manifest 1+3 dim coordinate invariance Equal to Newtonian gravity when τ µ = δ 0 µ, h µ0 = 0, h ij = δ ij, Γ j 0i = 0

15 From GR to Newton-Cartan c LT 1 c LT 1 k (1) µν = 1 c 4 T (0) µν Φ = ρ G N L 5 T 3 G µν = G N c 4 T µν R µν (Γ) = ρτ µ τ ν G N L 5 T 3

16 From GR to Newton-Cartan Large c expansion g µν (c) = i= 1 Ansatz: (2i) g µν c 2i (-2) g µν = τ µ τ ν g µν (c) = i=0 (2i) g µν c 2i (natural via x 0 = ct)

17 From GR to Newton-Cartan Large c expansion g µν (c) = i= 1 Ansatz: (2i) g µν c 2i (-2) g µν = τ µ τ ν g µν (c) = i=0 (2i) g µν c 2i (natural via x 0 = ct) Solving invertibility leads to (0) g µν = h µν (0) g µν = 2τ (µ C ν) + h µν (2) g µν = τ µ τ ν + 2τ (µ h ν)λ C λ where τ µ τ ν + h µλ h λν = δ µ ν

18 Large c expansion From GR to Newton-Cartan g µν (c) = i= 1 (2i) g µν c 2i g µν (c) = i=0 (2i) g µν c 2i Ansatz: (-2) g µν = τ µ τ ν (natural via x 0 = ct) Solving invertibility leads to (0) g µν = h µν (0) g µν = 2τ (µ C ν) + h µν (2) g µν = τ µ τ ν + 2τ (µ h ν)λ C λ where τ µ τ ν + h µλ h λν = δ µ ν Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ

19 From GR to Newton-Cartan (0) (0) g µν = τ µ τ ν g µν = h µν (0) (2) g µν = 2τ (µ C ν) + h µν g µν = τ µ τ ν + 2τ (µ h ν)λ C λ Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ Expand Einstein equations (for perfect fluid) (-4) R µν = R (-2) µν = 0 automatic (0) R µν = (0) T µν R µν (Γ) = ρ τ µ τ ν

20 Newton - Cartan Newtonian Gravity c LT 1 c LT 1 k (1) µν = 1 c 4 T (0) µν Φ = ρ G N L 5 T 3 G µν = G N c 4 T µν R µν (Γ) = ρτ µ τ ν G N L 5 T 3

21 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0

22 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0 Standard argument: 1) In 3d Ricci flat = flat 2) C i is harmonic, hence constant 3) NC=NG Conclusion: nonrelativistic non-newtonian gravity doesn t exist

23 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0 Standard argument: 1) In 3d Ricci flat = flat 2) C i is harmonic, hence constant 3) NC=NG Or does it?

24 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0 Standard argument: 1) In 3d Ricci flat = flat 2) C i is harmonic, hence constant 3) NC=NG Or does it? What about higher dimensions?

25 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0 Standard argument: 1) In 3d Ricci flat = flat 2) C i is harmonic, hence constant 3) NC=NG What about other energy-momentum? Or does it? What about higher dimensions?

26 Newton - Cartan Newtonian Gravity The dof s of NC manifest themselves after partial gauge-fixing Diff 4 Diff 3 (t) R ij = 0 j K ji = 2h jk [i ḣ j]k i G i = 1 2 hij ḧ ij + 1 4ḣijḣij 1 4 K ijk ij + 4πρ where G i = i Φ Ċi Φ = C 0 Standard argument: 1) In 3d Ricci flat = flat 2) C i is harmonic, hence constant 3) NC=NG What about other energy-momentum? Or does it? What about higher dimensions? dimensional reduction

27 Newton - Cartan Newtonian Gravity dimensional reduction We showed c GR NC KK KK [arxiv: ] EMD c NCMD Here is NCMD i i Ω = 1 4 Ω3 F ij F ij i ( Ω 3 E i) = 1 2 Ω3 K ij F ij i ( Ω 3 F ij) = 0 ΩR ij = i j Ω Ω3 F i k F jk j ( ΩK j i ) = Ω i (h jk ḣ jk ) + j (Ωḣij) 2 i Ω + Ω 3 F ij E j i ( ΩG i ) = 1 4 Ω (ḣij ḣ ij K ij K ij + 2h ij ḧ ij ) Ω3 E i E i + Ω

28 From GR to Twistless Torsional NC c LT 1 c LT 1 k (1) µν = 1 c T (0) 4 µν Φ = ρ G N L 5 T 3 G µν = G N c 4 T µν R µν (Γ) = ρτ µ τ ν G N L 5 T 3

29 From GR to Twistless Torsional NC c LT 1 c LT 1 k (1) µν = 1 c T (0) 4 µν Φ = ρ G N L 5 T 3 G µν = G N c 4 T µν R µν (Γ) = ρτ µ τ ν G N L 5 T 3

30 From GR to Twistless Torsional NC Large c expansion Solving invertibility Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ

31 From GR to Twistless Torsional NC Large c expansion Solving invertibility Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ +τ λ [µ τ ν] + h λρ ( C µ [ν τ ρ] + C ν [µ τ ρ] C ρ [µ τ ν] ) Torsional Newton - Cartan geometry

32 From GR to Twistless Torsional NC Large c expansion Solving invertibility Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ +τ λ [µ τ ν] + h λρ ( C µ [ν τ ρ] + C ν [µ τ ρ] C ρ [µ τ ν] ) Expanding Einstein equations (-4) R µν = 0 τ [µ ν τ λ] = 0 [µ τ ν] = τ [µ a ν] Twistless Torsional NC geometry

33 From GR to Twistless Torsional NC Large c expansion Solving invertibility Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ +τ λ [µ τ ν] + h λρ ( C µ [ν τ ρ] + C ν [µ τ ρ] C ρ [µ τ ν] ) Expanding Einstein equations (-4) R µν = 0 τ [µ ν τ λ] = 0 [µ τ ν] = τ [µ a ν] (-2) R µν = τ µ τ ν D ρ a ρ D µ = (nc) µ a µ

34 From GR to Twistless Torsional NC Large c expansion Solving invertibility Metric compatibility & finite Levi-Civita implies that µ τ ν = 0 µ h νλ = 0 K µν = 2 [µ C ν] Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν )+τ λ (µ τ ν) +h λρ τ (µ K ν)ρ +τ λ [µ τ ν] + h λρ ( C µ [ν τ ρ] + C ν [µ τ ρ] C ρ [µ τ ν] ) Expanding Einstein equations (-4) R µν = 0 τ [µ ν τ λ] = 0 [µ τ ν] = τ [µ a ν] (-2) R µν = τ µ τ ν D ρ a ρ D µ = (nc) µ a µ (0) ) R µν = R (nc) µν +D (ĥρν ρ â µ 1 (nc) (nc) 2aρ ρ ĥ µν a ρ τ (µ ν)ˆτ ρ τ µτ ν a 2ˆτ 2

35 Schwarzschild ( ds 2 = c 2 1 2m ) c 2 r Example dt 2 + ( 1 2m ) 1 c 2 dr 2 + r 2 dω 2 r = c 2 dt 2 + 2m r dt2 + dr 2 + r 2 dω 2 + O(c 2 )

36 Schwarzschild ( ds 2 = c 2 1 2m ) c 2 r Example dt 2 + ( 1 2m ) 1 c 2 dr 2 + r 2 dω 2 r = c 2 dt 2 + 2m r dt2 + dr 2 + r 2 dω 2 + O(c 2 ) τ 0 = 1, τ i = 0 h ij = δ ij, h µ0 = 0 C 0 = m r, C i = 0

37 Schwarzschild ( ds 2 = c 2 1 2m ) c 2 r Example dt 2 + ( 1 2m ) 1 c 2 dr 2 + r 2 dω 2 r = c 2 dt 2 + 2m r dt2 + dr 2 + r 2 dω 2 + O(c 2 ) τ 0 = 1, τ i = 0 h ij = δ ij, h µ0 = 0 C 0 = m r, C i = 0 This solves NC eom Φ = C 0 = m r Newtonian gravity of point mass dτ = 0 a µ = 0 no torsion

38 Example Schwarzschild (extremely massive) ( ds 2 = c 2 1 2M ) ( dt M r r ) 1 dr 2 + r 2 dω 2

39 Example Schwarzschild (extremely massive) ( ds 2 = c 2 1 2M ) ( dt M r r ) 1 dr 2 + r 2 dω 2 τ µ = ( 1 2M r ) 1/2 δ t µ, h µν = ( ( 0 1 2M ) r 0 0 r r 2 sin 2 θ ) C µ = 0

40 Example Schwarzschild (extremely massive) ( ds 2 = c 2 1 2M ) ( dt M r r ) 1 dr 2 + r 2 dω 2 τ µ = ( 1 2M r ) 1/2 δ t µ, h µν = ( ( 0 1 2M ) r 0 0 r r 2 sin 2 θ ) C µ = 0 This solves TTNC eom Φ = C 0 = 0 vanishing Newtonian potential! dτ 0 a µ = M r 2 ( 1 2M r curved spatial geometry! ) 1 δ r µ non-vanishing torsion! Nonrelativistic non-newtonian gravity

41 Summary & outlook Two punch lines: nonrelativistic gravity is more than a Newtonian potential Appearance of TTNC out of GR In progress/to do: precise relation with bottom up TTNC add energy momentum gauge fixed version precise physical interpretation real world applications?

42 Gaussian normal coordinates There always exist coordinates such that (on a patch) ds 2 = c 2 dσ 2 + g ij dx i dx j doesn t that imply we can always remove the torsion?

43 Gaussian normal coordinates There always exist coordinates such that (on a patch) ds 2 = c 2 dσ 2 + g ij dx i dx j doesn t that imply we can always remove the torsion? The transformation to GN coordinates is not compatible with (standard) large c expansion Schwarzschild (extremely massive) ds 2 = c 2 dσ 2 + 2M r(σ, ρ) dρ2 + r(σ, ρ) 2 dω 2 = c 2 dσ 2 + c 4/3 ( 9 2 M ) 2/3 σ 4/3 dω 2 + O(c 1/3 ) r(σ, ρ) = ( ) 2/ M(cσ + ρ)