Torsional Newton-Cartan gravity from a pre-newtonian expansion of GR

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1 Torsional Newton-Cartan gravity from a pre-newtonian expansion of GR Dieter Van den Bleeken arxiv.org/submit/ Supported by Bog azic i University Research Fund Grant nr 17B03P1 SCGP 10th March 2017

2 Gravity in various regimes c LT 1 c LT 1 k (1) µν = 1 c 4 T µν Φ = ρ G µν = G N c 4 T µν? G N L 5 T 3 G N L 5 T 3

3 Outline Overview Expanding the relativistic geometry TNC geometry Expanding the Einstein equations TTNC gravity Remarks An example Point particle motion physics?

4 Overview Everyday gravity: Newton+corrections Post-Newtonian Expansion Standard approach: start in most convenient coordinates Covariant approach Newton-Cartan + corrections (Kunzle 76, Ehlers 81, Dautcourt 90) (Dautcourt 96, Tichy-Flanagan 11)

5 I will revisit the covariant approach outcome physical interpretation Overview Relax an assumption on the relativistic Levi-Civita connection Clean separation of the expansion of the fields and of the eoms TTNC gravity torsion appears at lower order than the Newtonian potential: pre-newtonian non-relativistic, strong gravity regime extends Newtonian physics to include strong gravitational time dilation

6 Large c expansion Expanding the geometry 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)

7 Large c expansion Expanding the geometry 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) Constraint g µν (c)g νρ (c) = δ ρ µ

8 Large c expansion Expanding the geometry 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) Constraint g µν (c)g νρ (c) = δ ρ µ At first order g µν = h µν with h µν τ ν = 0

9 Expanding the geometry Some technology to solve invertibility at higher order τ ν τ ν + h µρ h ρν = δ ν µ τ ρ τ σ h ρσ = 0 We introduced unphysical fields remove redundancy by gauge symmetry δ χ τ µ = h µρ χ ρ δ χ h µν = τ µ χ ν + τ ν χ µ Decompose all metric coefficients using the projectors τ µ ν = τ ν τ ν h µ ν = h µρ h ρν

10 Result Expanding the geometry Independent fields τ µ, h µν, C µ, β µν, B µ, γ µν (-2) g µν = τ µ τ ν g µν = h µν g µν = 2τ (µ C ν) + h µν (2) g µν = τ µ τ ν + 2τ (µ h ν)λ C λ + β µν (2) g µν = B µ τ ν + τ µ B ν C µ C ν h µρ h νσ β ρσ (4) g µν = (h ρσ C ρ C σ 2τ ρ C ρ ) τ µ τ ν +2τ (µ h ν)ρ ( B ρ + (C σ τ σ )C ρ + h ρλ C σ β σλ) +γ µν

11 Expanding the geometry Physical metric is boost invariant ˆτ µ = τ µ h µν C ν ĥ µν = h µν + 2τ (µ C ν) + 2ˆΦτ µ τ ν ˆΦ = τ ρ C ρ hρσ C ρ C σ ˆβ µν = β µν + h µρ h µσ C ρ C σ ˆB µ = B µ + h µρ β ρσ C σ τ µ ( β ρσ C ρ C σ + (τ ρ C ρ h ρσ C ρ C σ ) 2) C µ (τ ρ C ρ h ρσ C ρ C σ ) ˆγ µν = γ µν + 2h ρ(µ h ν)σ C ρ ( ˆBσ 2ˆΦC σ )

12 Expanding the geometry Physical metric is boost invariant LO (-2) g µν = τ µ τ ν g µν = h µν NLO g µν = 2ˆΦτ µ τ ν + ĥµν (2) g µν = ˆτ µˆτ ν + ˆβ µν NNLO (2) g µν = τ µ ˆBν + τ ν ˆBµ ĥµρĥνσ ˆβ ρσ (4) g µν = 2ˆΦˆτ µˆτ ν + 2ˆτ (µ h ν)ρ ˆBρ + ˆγ µν

13 Expanding diffeomorphisms Remember the expansion ansatz g µν (c) = i= 1 (2i) g µν c 2i Left invariant by diffeos of the form ξ µ (c) = i=0 (2i) ξ µ c 2i Tensor coefficients are tensors under ξ µ Diffeo s of the effective theory The (2i) ξ µ generate additional gauge transformations

14 Expanding diffeomorphisms Remember the expansion ansatz g µν (c) = i= 1 (2i) g µν c 2i Left invariant by diffeos of the form ξ µ (c) = i=0 (2i) ξ µ c 2i Tensor coefficients are tensors under ξ µ Diffeo s of the effective theory Γ λ µν transforms as a connection other coefficients are tensors

15 Expanding metric compatibility Unique Levi-Civita connection µ g νρ = 0 µ g νρ = 0 At NLO µ h νλ = (-2) Γ ν µρ µ (τ ν τ λ ) = (-2) Γ ρ µν (2) g ρλ (-2) Γ λ µρ g ρλ (-2) Γ ρ µλ (2) g ρν g ρν

16 Expanding metric compatibility Unique Levi-Civita connection µ g νρ = 0 µ g νρ = 0 At NLO µ h νλ = (-2) Γ ν µρ µ (τ ν τ λ ) = (-2) Γ ρ µν Can be massaged into (2) g ρλ (-2) Γ λ µρ g ρλ (-2) Γ ρ µλ (2) g ρν g ρν (nc) µ h νλ = 0 and (nc) µ τ ν = 0

17 Expanding metric compatibility Unique Levi-Civita connection µ g νρ = 0 µ g νρ = 0 Can be massaged into (nc) µ h νλ = 0 and (nc) µ τ ν = 0 Where (nc) Γ λ µν = 1 2 hλρ ( µ h ρν + ν h µρ ρ h µν ) +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] )

18 Setup Expanding the Einstein equations R µν = 8πG N T µν, T µν = c 4 ( T µν 1 D 2 g µνg ρσ T ρσ Given the expansion of the metric (2i) R µν = R µν c 2i Consistency T µν = i= 2 i= 2 (2i) T µν c 2i In full generality (-4) T µν 0 we will restrict to the case ) (-4) T µν = 0 (assumption)

19 +8πG N T µν Expanding the Einstein equations Result: TTNC gravity LO h κλ h ρσ [κ τ ρ] [λ τ σ] = 0 twistless torsion: [µ τ ν] = τ [µ â µ] NLO τ µ τ ν h λρ D λ â ρ = 8πG N (-2) T µν NNLO (nc) R µν = ĥµ ρ ĥ σ ν D ρ â σ K µρ τ ν h ρσ â σ τ µ τ ν (h ρσ â ρ σ (ˆΦ + 1 ) 2 ˆβ) D ρ ( ˆβ ρσ â σ )

20 Remarks Gauge symmetries Diffeos, boosts and lower order diffeos. δ Λˆτ µ = h µρ D ρ Λ δ Λ ĥ µν = 2τ (µ ĥ ν) ρ D ρ Λ δ Λ ˆΦ = ˆτ ρ D ρ Λ δ Λ â µ = τ µ â ρ h ρσ D σ Λ δ ζ ˆΦ = âρ ζ ρ δ Λ ˆβµν = 2h µρ h νσ ˆKρσ Λ δ ˆβµν (nc) ρ(µ ζ = 2h ρ ζ ν)

21 Remarks Gauge symmetries Diffeos, boosts and lower order diffeos. Higher order fields B µ and γ µν do not appear in the TTNC eoms β µν can be removed from the TTNC eoms by gaugefixing the lower order diffeos we are left with exactly the field content and symmetries of standard TTNC

22 Remarks Gauge symmetries Diffeos, boosts and lower order diffeos. Higher order fields In case of vanishing torsion need (-2) T µν = 0 we are left with exactly the field content and symmetries of standard TTNC eoms reduce to standard NC part of lower order diffeos remain unfixed: become Bargmann U(1)!

23 Remarks Gauge symmetries Diffeos, boosts and lower order diffeos. Higher order fields we are left with exactly the field content and symmetries of standard TTNC In case of vanishing torsion eoms reduce to standard NC Physical interpretation torsion corresponds to non-trivial timelike warpfactor of order c 2 effective non-relativistic theory capturing strong gravitational time dilaton

24 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 )

25 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

26 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

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

28 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

29 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

30 Starting point Probe particle motion I understand the math but not the physics V µ = i= 1 (i) V µ c i

31 Result Probe particle motion I understand the math but not the physics (-1) V λ (nc) (-1) λ V µ = (τ ρ V ρ ) 2 h µλ â λ ( ) (-1) V λ (nc) λ V µ + V λ (nc) (-1) λ V µ = τ ρ V ρ (-1) â σ V σ ˆτ µ (1) + 2τ σ V σ h µλ â λ ( (-1) V λ (nc) (1) λ V µ + V λ (nc) λ V µ + V (1) λ (nc) (-1) λ V µ (1) ) = τ ρ τ σ V ρ V (1) σ + 2 V ρ V (2) σ h µλ â λ ( +τ ρ V ρ V σ + 2 V (1) (ρ (-1) V σ) ) ( τ σ ˆβµλâ λ + â σ ˆτ µ) ( ) + V (-1) ρ V (-1) 1 σ (nc) ˆβµλ σ ˆτ µ ˆKρσ 2ĥρλ when â µ = 0 reduces to (-1) V µ = 0 V λ (nc) λ V µ = 0

32 Summary & outlook Punch line: the large c expansion of GR can be extended beyond the Newtonian regime to include strong time dilation effects. The effective theory describing this is TTNC. Open questions: physical interpretation of point particle motion relation to other work on TTNC is this regime relevant in Nature? is this expansion of practical use in such situations?

33 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?

34 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)