4+1 Formalism for a Local Metric with Parameterized Evolution Martin

4+1 Formalism for a Local Metric with Parameterized Evolution

Martin Land

Hadassah College, Jerusalem http://cs.hac.ac.il/staﬀ/martin International Association for Relativistic Dynamics (IARD) http://www.iard-relativity.org

IARD 2020 Prague, Czech Republic

Martin Land — IARD 2020 4+1 Formalism May 2020 1 / 34 Stueckelberg-Horwitz-Piron (SHP) Formalism Electrodynamics Relativity Maxwell’s equations −−−−−−−−−−−−−→ Lorentz group Einstein Manifest covariance ←−−−−−−−−−−−−− Tensor formulation Minkowski & Fock µν µ µ µ ∂ν F (x) = eJ (x) x˙ = dx /ds µ µν 2 µ ν Mx¨ = eF (x)x˙ν ds = −ηµνdx dx Pair creation/annihilation −−−−−−−−−−−−−→ External evolution parameter Stueckelberg E = p0 = Mx˙0 < 0 x˙µ = dxµ/dτ ds2 = −x˙2(τ)dτ2 Covariant canonical dynamics ←−−−−−−−−−−−−− Unconstrained phase space Horwitz & Piron 2 [p − eA(x)] µ ∂K ∂K K = + V(x) x˙ = p˙µ = − µ 2M ∂pµ ∂x τ-dependent gauge ﬁeld −−−−−−−−−−−−−→ Evolving block universe Horwitz, Sa’ad, Aµ(x) −→ aµ(x, τ) Arshansky, Land M → M(τ) d V(x) −→ −ea (x, τ) (particle + ﬁeld masses) = 0 5 dτ

Martin Land — IARD 2020 4+1 Formalism May 2020 2 / 34 Stueckelberg-Horwitz-Piron (SHP) Formalism Covariant canonical mechanics with parameterized evolution

8D unconstrained phase space =⇒ τ 6= proper time

dxµ xµ(τ), x˙µ(τ) x˙µ = λ, µ, ν,... = 0, 1, 2, 3 dτ Canonical electrodynamics with scalar Hamiltonian (K = total mass)

1 d ∂L ∂L L = Mg (x)x˙µx˙ν + e x˙µa (x, τ) + e a (x, τ) 0 = − 2 µν µ 5 dτ ∂x˙µ ∂xµ

2 [p − ea(x, τ)] µ ∂K ∂K K = − ea5(x, τ) x˙ = p˙ µ = − µ 2M ∂pµ ∂x

Extend SHP to include pseudo-5D metric describing τ-evolution of spacetime

1 L = Mg (x, τ)x˙αx˙ β α, β, γ, δ = 0, 1, 2, 3, 5 x5 = c τ 2 αβ 5

Martin Land — IARD 2020 4+1 Formalism May 2020 3 / 34 SHP — Geometry and Evolution 4D block universe M(τ) at each τ Physical event xµ(τ) in SHP Irreversible occurrence at time τ

 xµ(τ ) occurs after xµ(τ )  2 1  µ µ τ2 > τ1 =⇒ x (τ2) cannot change x (τ1)   No grandfather paradox Evolution 4D block universe M(τ) occurs at τ Inﬁnitesimally close 4D block universe M(τ + dτ) occurs at τ + dτ

M(τ) −−−−−−−−−−−−−−−−−−−−−−−−−→ M(τ + dτ) Hamiltonian K generates evolution in τ ) scalar K =⇒ No conﬂict with general diﬀeomorphism invariance external τ

Martin Land — IARD 2020 4+1 Formalism May 2020 4 / 34 Geometry and Trajectory Standard approach to motion in general relativity Two neighboring events in spacetime manifold M (instantaneous displacement) 2 µ ν 2 Interval δx = gµνδx δx = (x2 − x1) Invariance of interval — geometrical statement about M Trajectory Map arbitrary parameter ζ to sequence of events xµ(ζ) Timelike interval between any two events ⇒ take ζ → s (proper time) In 4D block universe M Trajectory — sequence of instantaneous timelike displacements “Motion” appears as displacements in x0(s) Path length −→ Lagrangian 2 µ ν µ ν 2 δx = gµνδx δx = gµνx˙ x˙ δs g = (−, +, +, +)

p µ ν 1 µ ν Lconstrained = −gµνx˙ x˙ Lunconstrained = 2 gµνx˙ x˙

Martin Land — IARD 2020 4+1 Formalism May 2020 5 / 34 Geometry and Evolution Pseudo-5D metric Two neighboring events: xµ(τ) ∈ M(τ) x¯µ(τ + δτ) ∈ M(τ + δτ) Distance dxµ = x¯µ(τ + δτ) − xµ(τ) ' x¯µ(τ) + x¯˙µ(τ)δτ − xµ(τ) = δxµ + x¯˙µδτ Squared interval (referred to x coordinates) 2 µ ν µ ν µ ν 2 α β dx = gµνδx δx + gµνx¯˙ δx δτ + gµνx¯˙ x¯˙ δτ ' gαβ (x, τ) δx δx Contributions to interval α,β=0,1,2,3,5 µ ν gµν δx δx Geometrical interval between two events at τ Expresses symmetries of spacetime manifold M 5 5 g55 δx δx Dynamical interval between events in M(τ) and M(τ + δτ) Expresses symmetries of evolution generated by Hamiltonian K

Martin Land — IARD 2020 4+1 Formalism May 2020 6 / 34 Example in space 1 2 Particle in 2D space — expanding disk with radius R(τ) = 2 gτ Points on expanding disk

q = R (τ)(cos θ, sin θ) q¯ = R¯ (τ + δτ) cos θ¯, sin θ¯

Distance dq = q¯ − q ≈ (δR + δτ R (τ)) Rˆ + Rδθ θˆ

Geometrical distances: δθ = θ¯ − θ δR = R¯ (τ) − R(τ)

Dynamical distance: δτ R (τ) = R (τ + δτ) − R (τ) = gτδτ Interval 2 2 2 2 2 2 2 a b dq = δR + R δθ + g τ δτ + 2gτδRδτ = gabδζ δζ Pseudo-3D metric  1 0 1  2 δζ = (δR, δθ, δτ) gab =  0 R 0  1 0 gτ

Martin Land — IARD 2020 4+1 Formalism May 2020 7 / 34 Example in space Equations of motion Lagrangian 1 ˙ a ˙ b 1 ˙ 2 2 ˙ 2 ˙ 2 2 L = 2 Mgabζ ζ = 2 M R + R θ + 2gτR + g τ

d 2  0 = M R˙ + gτ − MRθ˙  dτ  `2 −→ ¨ = − MR 3 Mg d  MR 0 = MR2θ˙ −→ MR2θ˙ = `  dτ Qualitative result Particle at edge of disk sees force F = `2/MR3 − Mg Particle moves at edge of disk Mg > `2/MR3 =⇒ As if attracted by gravitational force Appears as “external” force FR = −Mg −→ Enters through evolution of circular geometry

Martin Land — IARD 2020 4+1 Formalism May 2020 8 / 34 Canonical Mechanics in General 5D Spacetime Lagrangian 1 L = Mg (xµ, x5)x˙αx˙ β λ, µ, ν = 0, 1, 2, 3 α, β, γ = 0, 1, 2, 3, 5 2 αβ Euler-Lagrange −→ geodesic equations Dx˙γ 1 0 = = x¨γ + Γγ x˙αx˙ β Γγ = gγδ ∂ g + ∂ g − ∂ g Dτ αβ αβ 2 α δβ β δα δ βα Canonical momentum ∂L 1 p = = Mg x˙ β −→ x˙α = gαβ p α ∂x˙α αβ M β Conserved Hamiltonian 1 dK Dx˙ β K = x˙α p − L = gαβ p p = Mg x˙α = 0 α 2M α β dτ αβ Dτ Poisson bracket dK ∂K 1 ∂gαβ = {K, K} + = p p = 0 dτ ∂τ 2M α β ∂τ

Martin Land — IARD 2020 4+1 Formalism May 2020 9 / 34 Break 5D symmetry −→ 4D+1

5 Constrain non-dynamical scalar x ≡ c5τ

1 α β 1 µ ν µ 1 2 L = 2 Mgαβ(x, τ)x˙ x˙ = 2 Mgµν x˙ x˙ + Mc5 gµ5x˙ + 2 Mc5 g55 Euler-Lagrange −→ geodesic equations

 µ µ λ σ µ σ 2 µ α x¨ + Γ x˙ x˙ + 2c5Γ x˙ + c Γ = 0 Dx˙ α α β γ  λσ 5σ 5 55 0 = = x¨ + Γβγx˙ x˙ −→ Dτ 5  x¨ = c˙5 ≡ 0

Symmetry-broken connection

µ 1 µν 5 Γαβ = 2 g ∂βgνα + ∂αgνβ − ∂νgαβ Γαβ ≡ 0 Hamiltonian

µ 1 µν µ 1 2 K = pµx˙ − L = 2M g pµ pν − Mc5 gµ5x˙ − 2 Mc5 g55

dK ∂K = = − 1 Mx˙µx˙ν ∂ g − Mc x˙µ ∂ g − 1 Mc2 ∂ g dτ ∂τ 2 τ µν 5 τ µ5 2 5 τ 55

Martin Land — IARD 2020 4+1 Formalism May 2020 10 / 34 Matter Non-thermodynamic dust Number of events per spacetime volume = n(x, τ) Particle mass density = ρ(x, τ) = Mn(x, τ) 5-component event current = jα (x, τ) = ρ(x, τ)x˙α(τ) = Mn(x, τ)x˙α(τ) Continuity equation α µ ∇α j = ∇µ j + ∂τρ Mass-energy-momentum tensor ( Tµν = ρx˙µx˙ν ∇ Tαβ = 0 Tαβ = ρx˙αx˙ β −→ β 5β β T = c5j

Einstein equations 1 8πG G = R − Rg = T αβ αβ 2 αβ c4 αβ

Martin Land — IARD 2020 4+1 Formalism May 2020 11 / 34 Weak Field Approximation

Small perturbation to ﬂat metric

2 αβ gαβ = ηαβ + hαβ −→ ∂γgαβ = ∂γhαβ hαβ ≈ 0 h ' η hαβ

¯ 1 Deﬁne hαβ = hαβ − 2 ηαβh −→ Einstein equations

16πG γ γ T = ∂ ∂γh¯ + ∂α∂γh¯ − ∂γ∂γh¯ − ∂α∂ h¯ c4 αβ β α β αβ β ¯ αλ Impose gauge condition ∂λh = 0 −→ wave equation ! 16πG η T = − γ h¯ = − µ + 55 2 h¯ 4 αβ ∂ ∂γ αβ ∂ ∂µ 2 ∂τ αβ c c5 Solve with principal part Green’s function

0 − |x−x | 0 4G Z Tαβ t c , x , τ h¯ (x, τ) = d3x0 αβ c4 |x − x0|

Martin Land — IARD 2020 4+1 Formalism May 2020 12 / 34 Post-Newtonian Approximation

Point source X = (cT(τ), 0) in co-moving frame where T˙ = 1 + α (τ) /2 and α2 ≈ 0

c2 T00 = mc2T˙ 2δ3 (x) ρ (t − T (τ)) Tαi = 0 T55 = 5 T00 ≈ 0 c2 Writing M(τ) = m ρ (t − T (τ)) a slowly varying density function

4GM c2 h¯00 (x, τ) = T˙ 2 h¯ αi (x, τ) = 0 h¯55 (x, τ) = 5 h¯00 ≈ 0 c2R c2

µ 1 µν µ 1 µ0 Γ00 = − η ∂νh00 Γ50 = η ∂τ h00 2 2c5 Equations of motion for a test particle in spherical coordinates putting θ = π/2

2GM α (τ) t¨ = (∂ h ) t˙ + ˙x · (∇h ) t˙2 ≈ 1 + α˙ (τ) t˙ τ 00 00 c2R 2  L 2  R˙ ˙ + R ¨ = −→ ˙ = c  2 φ φ 0 φ 2 ¨x = (∇h ) t˙2 −→ MR 2 00 L2 GM  R¨ − = − t˙2T˙ 2 M2R3 R2

Martin Land — IARD 2020 4+1 Formalism May 2020 13 / 34 Post-Newtonian Evolution

Solution to t equation neglecting R˙ /c 1 and ∂τρ ≈ 0

2GM 1 1 2GM t˙ = exp α + α2 −→ t˙2T˙ 2 ' 1 + 1 + α c2R 4 2 c2R

Taking α = 0 recovers t˙ = 1 for Newtonian case

Using solution to t equation in radial equation

d 1 1 L2 GM 1 GM d R˙ 2 + − 1 + α (τ) = − α (τ) dτ 2 2 M2R2 R 2 2R dτ

dK LHS is where K is particle Hamiltonian = particle mass dτ In source rest frame, T˙ 6= 1 ⇒ ∆energy without ∆momentum ⇒ ∆mass

Source transfers mass to perturbed metric ﬁeld hαβ

Particle mass not generally conserved −→ particle absorbs mass from hαβ Taking α = 0 recovers nonrelativistic motion with conserved Hamiltonian

Martin Land — IARD 2020 4+1 Formalism May 2020 14 / 34 3+1 Formalism in General Relativity Time evolution formalism for Einstein equations Formulate ﬁeld equations as initial value problem Decompose 10 components of spacetime metric into: 6 component space metric 3 component (spatial) shift vector 1 component (time) lapse Decompose 10 components of Einstein equations into:

6 component partial diﬀerential equation (PDE) of second order in ∂t 4 constraints on space metric, its 1st t-derivative, energy-momentum Based on mathematics of embedded hypersurfaces Darmois (1927), Lichnerowicz (1939), Choquet-Bruhat (1952) Intrinsic structure of hypersurface Extrinsic structure imposed by embedding hypersurface in larger manifold Arnowitt, Deser, Misner (1962) ADM Hamiltonian formulation of GR Intrinsic and extrinsic structure −→ canonical conjugate ﬁeld variables 4+1 formalism generalizes 3+1 framework as presented in: Gourgoulhon (2007), Bertschinger (2005), ADM (1962), Isham (1992), Blau (2020)

Martin Land — IARD 2020 4+1 Formalism May 2020 15 / 34 Schematic Outline of 3+1 Formalism Deﬁne a foliation of 4D spacetime Choose scalar ﬁeld t(x) on spacetime Hypersurface = simultaneous spacetime points x sharing t(x) = constant Choose t(x) to identify spacelike hypersurfaces (3D space manifold at time t) Each vector V tangent to each hypersurface is spacelike (V2 > 0 ) Vector n normal to hypersurface is timelike (n2 = −1 ) Projection operators split tangent and normal components of 4D objects

Induced space metric γij is spacelike projection of spacetime metric gµν i Spacetime metric gµν −→ γij, (normal) lapse N, and (tangent) shift N Spacelike projection of 4D covariant derivative compatible with γij Spacelike projection of 4D curvature −→ usual 3D (intrinsic) curvature Timelike projection of 4D curvature −→ extrinsic curvature of t-evolving space Einstein equations Project Einstein equations onto spacelike hypersurface and timelike normal 10 components of Einstein equations split into two groups

6 second order PDEs describing t-evolution of space metric γij 4 non-evolving constraints on initial conditions

Martin Land — IARD 2020 4+1 Formalism May 2020 16 / 34 4+1 Formalism in SHP General Relativity Schematic Outline I: The Embedding

5D pseudo-spacetime coordinates X = (x, c5τ) ∈ M5 = M × R

Manifold M5 an admixture of 4D spacetime geometry and τ-evolution

In ﬂat pseudo-spacetime gαβ → ηαβ = diag (−1, 1, 1, 1, σ) where σ = ±1

Natural foliation of 5D pseudo-spacetime M5 −→ spacetime hypersurface ∑τ Time function ( ) −→ = ∈ M ( ) = ( ) − = τ X ∑τ0 X 5 S X τ X τ0 0 Vector normal to hypersurface is -like ( 2 = = ± ) n ∑τ0 τ n σ 1 Projection operators split tangent and normal components of 5D objects

Induced spacetime metric γµν is projection onto M of 5D metric gαβ µ 5D metric gαβ −→ γµν, lapse N, and (tangent) shift N

Spacetime projection of 5D covariant derivative compatible with γµν

Spacetime projection of 5D curvature −→ usual 4D (intrinsic) curvature Rµνλρ

τ projection of 5D curvature −→ extrinsic curvature of τ-evolution Kαβ

Martin Land — IARD 2020 4+1 Formalism May 2020 17 / 34 4+1 Formalism in SHP General Relativity Schematic Outline II: Einstein Equations

5D Einstein equations 1 8πG G = R − Rg = T αβ αβ 2 αβ c4 αβ Bianchi relation αβ ∇αG = 0 5 constraints on solution to ﬁeld equations

Project Einstein equations onto spacetime hypersurface and τ-like normal Decompose 15 components of ﬁeld equations into two groups 10 unconstrained second order PDEs

Describe τ-evolution of spacetime metric γµν and extrinsic curvature Kµν n o 5 non-evolving constraints on initial conditions: γµν , ∂τγµν , Tαβ

Martin Land — IARD 2020 4+1 Formalism May 2020 18 / 34 Foliation of 5D Pseudo-Spacetime M5 = M × R Embedding ( xµ ∈ M, µ = 0, 1, 2, 3 Injective mapping Φ : M → M with 5 α X = (x, c5τ) ∈ M5, α = 0, 1, 2, 3, 5 4D hypersurface — implicitly deﬁned submanifold = ∈ M ( ) = where ( ) = ( ) − = 5 − ∑τ0 X 5 S X 0 S X τ X τ0 X /c5 τ0

Rank 4 Jacobian ∂Xα Eα = −→ E = ∂ = ∂/∂xµ as basis for tangent space of ∑ µ ∂xµ µ µ τ0 τ0

Unit normal to ∑τ0 ( · = α = −1/2 n Eµ nαEµ 0 n = σ g55 ∂ S (X) −→ α α 2 αβ n = g nαnβ = σ Induced metric on M α β ∂X ∂X β ds2 = g dXαdXβ = g dxµdxν = γ dxµdxν −→ γ = g Eα E αβ αβ ∂xµ ∂xν µν µν αβ µ ν

Martin Land — IARD 2020 4+1 Formalism May 2020 19 / 34 Projection Operators Projections ( Ak = σ (A · n) n A ∈ Tx (M5) = tangent space of M5 −→ A⊥ = A − σ (A · n) n Normal Projection Operator

γβ 2 2 β β Nαβ = σnαnβ Nαγ N = σ n nαn = Nα Tangent Projection Operator

α αβ α β γβ β β β Pαβ = gαβ − σnαnβ Pβ = g − σn n PαγP = Pα = δα − σnαn Completeness Relation α α α δβ = Pβ + σn nβ

Projector Pµν restricted to ∑τ is metric γµν

α α β α µ α V ∈ Tx (M5) −→ V⊥ = Pβ V ∈ Tx (∑τ) −→ ∃ v ∈ Tx (M) such that V⊥ = v Eµ

α β α β α β γµν = gαβEµEν = Pαβ + σnαnβ EµEν = PαβEµEν = Pµν

Martin Land — IARD 2020 4+1 Formalism May 2020 20 / 34 Decomposition of the Metric Time evolution ( α hypersurface ∑τ = {X | τ (X) = τ0} Embedding Xα (x, τ) −→ 0 α α trajectory X (τ) = {X (x, τ) | x = x0} General trajectory: normal component ∈ and tangent component ∈ ∑τ0+δτ ∑τ0 ∂Xα τ → τ + δτ ⇒ Xα + δτ = Xα + Nnα + NµEα δτ ∂τ µ x0 Lapse function N and Shift vector Nµ Spacetime shift ∂Xα Xα → Xα + δxµ = Xα + Eαδxµ ∂xµ µ 5D interval τ0 2 α β h α α µ µ i h β β ν ν i ds = gαβdX dX = gαβ Nn c5dτ + Eµ (N c5dτ + dx ) Nn c5dτ + Eν (N c5dτ + dx )

2 α α β Decomposition of metric using n = σ nαEµ = 0 γµν = gαβEµEν  1 1  " # µν + µ ν − µ γ σ 2 N N σ 2 N γµν Nµ αβ  N N  gαβ = 2 µ ν g =   Nµ σN + γ N N  1 1  µν −σ Nµ σ N2 N2

Martin Land — IARD 2020 4+1 Formalism May 2020 21 / 34 Intrinsic Geometry

Covariant derivatives — compatibility — curvature h i On γ M5 gβγ −→ Γσβγ −→ ∇α gβγ = 0 −→ ∇β, ∇α Xδ = XγRδαβ λ On M γµν −→ Γµνλ −→ Dµ γνλ = 0 −→ Dν, Dµ Xρ = XλRρµν

Projected Covariant Derivative ¯ ⊥ γ δ γ δ For V ∈ M5 deﬁne ∇αVβ = Pα ∇γ PβVδ = Pα Pβ∇γVδ

0 0 α0 β γ using ¯ 0 0 0 0 0 0 ∇αPβγ = −σPα Pβ Pγ ∇α nγ nβ + nβ ∇α nγ = 0 ¯ Associate Dα = ∇α on ∑τ −→ Dµ on M α α γ γ through Dµ = EµDα = EµPα ∇γ = Eµ ∇γ

Projected Curvature R¯ γ δαβ h i On ¯ ¯ γ ∑τ γβγ −→ Γσβγ −→ Dα γβγ = 0 −→ Dβ, Dα Xδ = XγRδαβ

Martin Land — IARD 2020 4+1 Formalism May 2020 22 / 34 Extrinsic Geometry Extrinsic Curvature

Curvature of M as manifold embedded in M5 as ∑τ

Gradient ∇γnδ in M5 γ δ Extrinsic curvature = projection of gradient onto ∑τ −→ Kαβ = −Pα Pβ ∇γnδ

2 β δ Using n = σ −→ ∇αnβ n = 0 −→ ∇αnβ ∈ ∑τ −→ Pβ (∇γnδ) = ∇γnβ

γ γ Kαβ = −Pα ∇γnβ = −∇αnβ + σnα n ∇γnβ

55 −1/2 Using nβ = σ g ∂βS (X) = σN∇βS (X)

nβ nγ 1 Expanding nγ∇ n = nγ (∇ N) + nγ N∇ = −σ D N γ β γ N β N N β 1 K = −∇ n − n D N αβ α β α N β

Martin Land — IARD 2020 4+1 Formalism May 2020 23 / 34 Extrinsic Geometry

Evolution of Hypersurface ∑τ

Time Evolution of ∑τ ∂Xα ∂Xα δXα = δx5 = δτ −→ Eα = (∂ )α = Nnα + NµEα x5 ∂τ 5 5 µ ∂ x0 x0

α α 2 2 Deﬁne m = Nn ⇒ m = σN −→ ∂5 = m + N 1 Using K = −∇ n − n D N αβ α β α N β

∇βmα = N∇βnα + nα∇β N = −NKβα − nβDα N + nα∇β N

Lie derivative of γµν along ∂5

L5 = Lm + LN

γ γ γ Using Lm γαβ = m ∇γγαβ + γγβ∇αm + γαγ∇βm = −2NKαβ Evolution equation for spacetime metric

L5 γαβ − LN γαβ = −2NKαβ −→ L5 γµν − LN γµν = −2NKµν

Martin Land — IARD 2020 4+1 Formalism May 2020 24 / 34 Decomposition of the Riemann Tensor Gauss-Codazzi Relations

α0 α0 α0 α α0 α0 α Using δα = Pα + σnαn Eµ Pα = Eµ nαEµ = 0 to expand

0  α β λ δ α0 β γ δ0 λ E E E E P P P 0 P R 0 0 0 0 = R  µ ν γ σ α β γ δ δ α β γ σµν  0 0 0 0  0 0 γ = α β γ δ γ −→ α β λ γ δ α0 β γ λ Rδαβ δα δβ δγ0 δδ R 0 0 0 Eµ Eν Eγ Pγ0 n Pα Pβ R 0 0 = σNR5µν δ α β  δα β  0  αµ β δ β γ α0 2 µ  E E P 0 n P n R 0 = N R ν αα β δβ γ 5ν5 where γ δ α β because of the symmetries of the Riemann tensor Rδαβ n n n = 0 Gauss relation

Decomposing projected Ricci equation for V tangent to ∑τ h i ¯ γ µ ¯ µ µ µ Dβ, Dα Vδ = VγRδαβ −→ R νλρ = R νλρ − σ KλKρν − Kρ Kλν Codazzi relation Decomposing Ricci equation for unit normal n

h i γ γ γ0 5 1 ∇ , ∇ n = R 0 n −→ R = σ D K − D K β α γ αβ µνλ N λ νµ ν λµ

Martin Land — IARD 2020 4+1 Formalism May 2020 25 / 34 Evolution of the Extrinsic Curvature

Decompose Ricci equation for unit normal n

α α α δ ∇β∇γn − ∇γ∇βn = Rδβγn

Project once onto n and twice onto ∑τ 0 0 γ0 β α0 α0 γ0 β α0 δ 0 ∇ 0 ∇ 0 − ∇ 0 ∇ 0 = 0 0 Pαα n Pβ β γ n γ β n Pαα n Pβ Rδβ γ0 n 1 Use K = −∇ n − n D N αβ α β α N β 0 γ0 β α0 δ γ 1 γ δ ε to obtain P 0 n P R 0 n = −K K + D D N + P P n ∇ K αα β δβ γ0 αγ β N β α α β ε γδ

Lie derivative of Kαβ γ γ γ (L5 − LN) Kαβ = Lm Kαβ = m ∇γKαβ + Kγβ∇αm + Kαγ∇βm

α α α α Using ∇βm = −NKβ − nβD N + n ∇β N and combining the above leads to 0 α0 β 1 1 δ P P R 0 0 = σ L K + σ D D N + R¯ − σKK + σ2K K α β α β N m αβ N α β αβ αβ α βδ

Martin Land — IARD 2020 4+1 Formalism May 2020 26 / 34 Decomposition of the Einstein Equations

Evolution Equation for Kαβ Einstein equations 1 8πG 8πG 1 R − g R = T −→ R = T − g T T = gαβT αβ 2 αβ c4 αβ αβ c4 αβ 2 αβ αβ

Decompose Tαβ

0 0 β0 0 0 α α β Tαβ = Tα0 β Pα + σn nα Pβ + σn nβ = Sαβ + 2σnα pβ + nαnβκ where 0 β0 0 β0 α 0 α 0 Sαβ = Pα Pβ Tα0 β ←→ Tµν pβ = −n Pβ Tα0 β ←→ T5µ α β κ = n n Tαβ ←→ T55 T = S + σκ

0 α0 β 1 1 Combining 0 0 with Pα Pβ Tα0 β − 2 gα0 β T = Sαβ − 2 γαβ (S + σκ) Lm Kαβ leads to

(L5 − LN) Kµν = −DµDν N λ 8πG 1 +N −σR¯ µν + KKµν − 2K K + σ Sµν − γ (S + σκ) µ νλ c4 2 µν

Martin Land — IARD 2020 4+1 Formalism May 2020 27 / 34 Decomposition of the Einstein Equations Constraint Equations Projecting Einstein equations twice onto normal n 1 8πG 1 8πG R − g R nαnβ = T nαnβ −→ R nαnβ − σR = κ αβ 2 αβ c4 αβ αβ 2 c4 Contracting indices in Gauss relation

α β 2 αδ R − 2σRαβn n = R¯ − σ K − K Kαδ Hamiltonian Constraint Combining above leads to

2 µν 16πG R¯ − σ K − K Kµν = −σ κ c4 Momentum Constraint

Projecting Einstein equations onto ∑τ and normal n 0 0 α β 1 α β 8πG 8πG n P R 0 − g 0 R = n P T 0 = − p β αβ 2 αβ β c4 αβ c4 β Combining with Codazzi relation leads to

µ 8πG DµK − DνK = pν ν c4

Martin Land — IARD 2020 4+1 Formalism May 2020 28 / 34 Decomposition of the Einstein Equations Summary of Diﬀerential Equations in 4+1 Formalism Evolution equation for spacetime metric

1 Lτ γµν − LN γµν = −2NKµν c5 Evolution equation for extrinsic curvature

1 Lτ − LN Kµν = −DµDν N c5 λ 8πG 1 +N −σR¯ µν + KKµν − 2K K + σ Sµν − γ (S + σκ) µ νλ c4 2 µν Hamiltonian Constraint

2 µν 16πG R¯ − σ K − K Kµν = −σ κ c4 Momentum Constraint µ 8πG DµK − DνK = pν ν c4

Martin Land — IARD 2020 4+1 Formalism May 2020 29 / 34 Evolution Versus Constraints Bianchi relation 1 ∇ Gαβ = ∇ Rαβ − gαβR = 0 α α 2 In D dimensions symmetric tensor Gαβ has D(D + 1)/2 components αβ G has 15 components on M5 5 components of Bianchi relation −→ 10 independent components + 5 constraints Order of τ derivative 8πG Einstein equations G = T −→ 2nd order in τ derivatives of g αβ c4 αβ αβ n o Initial conditions for evolution PDE = gαβ , ∂τ gαβ , Tαβ Bianchi relation

αβ 1 5β µβ αβ ∇αG = 0 −→ ∂τ G = −∇µG + {Christoﬀel symbols} × G c5 nd 5β st RHS at most 2 order in ∂τ =⇒ G at most 1 order in ∂τ G5β −→ 5 constraints on initial conditions

αβ 1 5β Bianchi relation: G = 0 ⇒ ∂τ G = 0 −→ constraint is conserved τ0 c5 τ0

Martin Land — IARD 2020 4+1 Formalism May 2020 30 / 34 Static Schwarzschild-like Geometry

Metric for Tαβ = 0 −→ Sµν = pµ = κ = 0 2 2 2 2 2 2 2 2 2 2 2 ds = −c Bdt + Adr + r dθ + r sin θdφ + σWc5dτ where GM B (r) = A−1 (r) = 1 − Φ (r) = 1 − W = W (x, τ) 0 rc2 " # γµν Nµ γµν (r) 0 gαβ = 2 µ ν = 2 Nµ σN + γµν N N 0 σN (x, τ) √ with ¯ = for Schwarzschild spacetime metric and = R 0 √ N W Dynamical Equations −→ 4D wave equation for W

(∂5 − LN) γµν = −2NKµν −→ ∂τγµν = −2NKµν −→ Kµν = 0 q ¯ λ µν ∂5Kµν = −DµDν N + N −σRµν + KKµν − 2Kµ Kνλ −→ γ DµDν W (x, τ) = 0

Constraints trivially satisﬁed for R¯ = Kµν = pν = κ = 0

2 µν 16πG µ 8πG R¯ − σ K − K Kµν = −σ κ DµK − DνK = pν c4 ν c4

Martin Land — IARD 2020 4+1 Formalism May 2020 31 / 34 Perturbation Around Schwarzschild Geometry Admit variable mass

2 −1 M (τ) = M [1 + α (τ)] α 1 −→ B = A = 1 − Φ0 [1 + α (τ)]

Take W = 1 4D connection τ-dependent but retains unperturbed form −→ R¯ = 0 2 2 Dynamical equations (neglecting terms in α and Φ0) Metric evolution 1 Φ α˙ (τ) 1 = − −→ = − = − 0 diag ∂5γµν 2NKµν Kµν ∂τγµν 1, 2 , 0, 0 2c5 2c5 B

µ µλ Φ0α˙ (τ) µ Kν = γ Kλν = − diag (−1, 1, 0, 0) −→ K = Kµ = 0 2c5

µ 2 2 Curvature evolution, using R¯ = 0, N = 1, N = 0, and Kµν ∝ α ≈ 0

1 1 1 8πG 1 ∂ K = − Φ α¨ (τ) diag 1, , 0, 0 = σ S − γ (S + σκ) c τ µν 2 0 2 4 µν µν 5 2c5 B c 2

Martin Land — IARD 2020 4+1 Formalism May 2020 32 / 34 Perturbing Energy Momentum Tensor Hamiltonian constraint −→ no perturbing mass density

2 µν 16πG 2 2 R¯ − σ K − K Kµν = −σ κ −→ κ ∝ α Φ ≈ 0 c4 0 Momentum constraint −→ perturbing mass current into r direction

µ µ λ µ µ λ pν = DµKν − DνK = ∂µKν + Kν Γλµ − KλΓνµ

1 0 µ µ λ 0 µ 0 0 1 1 p0 = ∂rK0 + K0Γ0µ − KλΓ0µ = K0Γ0µ − K0Γ00 − K1Γ01 = 0

1 λ µ µ λ 1 1 p1 = ∂rK1 + K1 Γλµ − KλΓ1µ = − Φ0α˙ (τ) 2 c5r Evolution equation −→ perturbing energy density and momentum density in r direction

1 c2 16πG 1 α¨ (τ) Φ diag 1, , 0, 0 = −σ 5 S − γ S 0 B2 c2 c2 µν 2 µν

!−1 c2 16πG S = S = −σ 5 Φ α¨ (τ) S = S = S = 0 00 11 c2 c2 0 22 33

Martin Land — IARD 2020 4+1 Formalism May 2020 33 / 34 Thank You For Your Patience

Slides and preprints: http://cs.hac.ac.il/staﬀ/martin

Martin Land — IARD 2020 4+1 Formalism May 2020 34 / 34