Useful Formulae in Riemannian Geometry - Perimeter Institute
Useful Formulae in Riemannian Geometry - Perimeter Institute
Useful Formulae in Riemannian Geometry - Perimeter Institute
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1. Curvature tensors<br />
Conventions, Def<strong>in</strong>itions, Identities, and Other <strong>Useful</strong> <strong>Formulae</strong><br />
Consider a d + 1 dimensional manifold M with metric g µν . The covariant derivative on M that is metriccompatible<br />
with g µν is ∇ µ .<br />
Christoffel<br />
Riemann Tensor<br />
Ricci Tensor<br />
Schouten Tensor<br />
Weyl Tensor<br />
Γ λ µν = 1 2 gλρ (∂ µ g ρν + ∂ ν g µρ − ∂ ρ g µν ) (1)<br />
R λ µσν = ∂ σ Γ λ µν − ∂ ν Γ λ µσ + Γ κ µνΓ λ κσ − Γ κ µσΓ λ κν (2)<br />
S µν = 1<br />
d − 1<br />
R µν = δ σ λ Rλ µσν (3)<br />
(<br />
R µν − 1 )<br />
2 d g µνR<br />
∇ ν S µν = ∇ µ S ν ν (5)<br />
C λ µσν = R λ µσν + g λ ν S µσ − g λ σ S µν + g µσ S λ ν − g µν S λ σ (6)<br />
(4)<br />
Commutators of Covariant Derivatives<br />
[∇ µ , ∇ ν ] A λ = R λσµν A σ (7)<br />
[∇ µ , ∇ ν ] A λ = R λ σµνA σ (8)<br />
Bianchi Identity<br />
∇ κ R λµσν − ∇ λ R κµσν + ∇ µ R κλσν = 0 (9)<br />
∇ ν R λµσν = ∇ µ R λσ − ∇ λ R µσ (10)<br />
∇ ν R µν = 1 2 ∇ µR (11)<br />
Bianchi Identity for Weyl<br />
∇ ν C λµσν = (d − 2)<br />
∇ λ ∇ σ C λµσν = d − 2<br />
d − 1<br />
(∇ µ S λσ − ∇ λ S µσ<br />
)<br />
(12)<br />
[<br />
∇ 2 R µν − 1<br />
2d g µν∇ 2 R − d − 1 ( ) d + 1<br />
2d<br />
∇ µ∇ ν R − Rµ λ R νλ (13)<br />
d − 1<br />
+ C λµσν R λσ (d + 1)<br />
+<br />
d(d − 1) R R µν + 1<br />
d − 1 g µν<br />
(R λσ R λσ − 1 )]<br />
d R2<br />
1
2. Euler Densities<br />
Let M be a manifold with dimension d + 1 = 2n an even number. Normalized so that χ(S 2n ) = 2.<br />
Euler Number<br />
Euler Density<br />
Curvature Two-Form<br />
Examples<br />
E 2n =<br />
∫<br />
χ(M) = d 2n x √ g E 2n (14)<br />
∫M<br />
= e 2n<br />
(15)<br />
M<br />
1<br />
(8π) n Γ(n + 1) ɛ µ 1 ...µ 2n<br />
ɛ ν1 ...ν 2n<br />
R µ 1 µ 2 ν 1 ν 2 . . . R<br />
µ 2n−1 µ 2n ν 2n−1 ν 2n (16)<br />
e 2n<br />
=<br />
1<br />
(4π) n Γ(n + 1) ɛ a 1 ...a 2n<br />
R a 1 a 2 ∧ . . . ∧ R<br />
a 2n−1 a 2n (17)<br />
R a b = 1 2 Ra b c d ec ∧ e d (18)<br />
E 2 = 1<br />
8π ɛ µνɛ λρ R µνλρ (19)<br />
= 1<br />
4π R<br />
E 4 = 1<br />
128π 2 ɛ µνλρ ɛ αβγδ R µναβ R λργδ (20)<br />
= 1<br />
32π 2 (<br />
R µνλρ R µνλρ − 4 R µν R µν + R 2)<br />
= 1<br />
32π 2 Cµνλρ C µνλρ − 1 ( d − 2<br />
8π 2 d − 1<br />
) (<br />
R µν R µν − d + 1<br />
4 d R2 )<br />
3. Hypersurfaces<br />
Let Σ ⊂ M be a d dimensional hypersurface whose embedd<strong>in</strong>g is described locally by an outward-po<strong>in</strong>t<strong>in</strong>g, unit<br />
normal vector n µ . Rather than keep<strong>in</strong>g track of the signs associated with n µ be<strong>in</strong>g either spacelike or timelike,<br />
we will just assume that n µ is spacelike. Indices are lowered and raised us<strong>in</strong>g g µν and g µν , and symmetrization<br />
of <strong>in</strong>dices is implied when appropriate.<br />
First Fundamental Form / Induced Metric on Σ<br />
h µν = g µν − n µ n ν (21)<br />
Projection onto Σ<br />
⊥ T µ ... ν ... = h µ λ . . . hσ ν . . . T λ ... σ ... (22)<br />
Second Fundamental Form / Extr<strong>in</strong>sic Curvature of Σ<br />
K µν = ⊥(∇ µ n ν ) = h λ<br />
µ h σ<br />
ν ∇ λ n σ = 1 2 £ nh µν (23)<br />
Trace of Extr<strong>in</strong>sic Curvature<br />
K = ∇ µ n µ (24)<br />
2
‘Acceleration’ Vector<br />
a µ = n ν ∇ ν n µ (25)<br />
Surface-Form<strong>in</strong>g Normal Vectors<br />
n µ =<br />
1<br />
√<br />
g νλ ∂ ν α∂ λ α ∂ µα ⇒ ⊥∇ [ µ n ν ] = 0 (26)<br />
Covariant Derivative on Σ compatible with h µν<br />
D µ T α ... β ... = ⊥ ∇ µT α ... β ... ∀ T = ⊥ T (27)<br />
Intr<strong>in</strong>sic Curvature of (Σ, h)<br />
[D µ , D ν ] A λ = R λ σµνA σ ∀ A λ = ⊥ A λ (28)<br />
Gauss-Codazzi<br />
Projections of the Ricci tensor<br />
Decomposition of the Ricci scalar<br />
⊥R λµσν = R λµσν − K λσ K µν + K µσ K νλ (29)<br />
(<br />
⊥ R λµσν n λ) = D ν K µσ − D σ K µν (30)<br />
(<br />
⊥ R λµσν n λ n σ) = − L n K µν + Kµ λ K λν + D µ a ν − a µ a ν (31)<br />
⊥ (R µν ) = R µν + D µ a ν − a µ a ν − L n K µν − K K µν + 2K λ<br />
µ K ν λ (32)<br />
⊥ (R µν n µ ) = D µ K µν − D ν K (33)<br />
R µν n µ n ν = − L n K − K µν K µν + D µ a µ − a µ a µ (34)<br />
R = R − K 2 − K µν K µν − 2 L n K + 2 D µ a µ − 2 a µ a µ (35)<br />
Lie Derivatives along n µ £ n K µν = n λ ∇ λ K µν + K λν ∇ µ n λ + K µλ ∇ ν n λ (36)<br />
4. Sign Conventions for the Action<br />
⊥ (£ n F µ ... ν ...) = £ n F µ ... ν ... ∀ ⊥F = F (37)<br />
These conventions follow We<strong>in</strong>berg, keep<strong>in</strong>g <strong>in</strong> m<strong>in</strong>d that he def<strong>in</strong>es the Riemann tensor with a m<strong>in</strong>us sign<br />
relative to our def<strong>in</strong>ition. They are appropriate when us<strong>in</strong>g signature (−, +, . . . , +). The d + 1-dimensional<br />
Newton’s constant is 2κ 2 = 16πG d+1 . The sign on the boundary term follows from our def<strong>in</strong>ition of the extr<strong>in</strong>sic<br />
curvature.<br />
Gravitational Action<br />
I G = 1<br />
2 κ<br />
∫M<br />
2 d d+1 x √ ( )<br />
g R − 2 Λ + 1 ∫<br />
κ 2<br />
= 1<br />
2 κ<br />
∫M<br />
2 d d+1 x √ g<br />
∂M<br />
(<br />
R + K 2 − K µν K µν − 2 Λ<br />
d d x √ γ K (38)<br />
)<br />
(39)<br />
3
Gauge Field Coupled to Particles<br />
I M = − 1 4<br />
M<br />
− ∑ n<br />
+ ∑ n<br />
∫<br />
M<br />
d d+1 x √ g F µν F µν (40)<br />
m n<br />
∫<br />
e n<br />
∫<br />
(<br />
dp<br />
dp dxµ n(p)<br />
dp<br />
−g µν (x n (p)) dxµ n(p)<br />
dp<br />
dx ν ) 1/2<br />
n(p)<br />
(41)<br />
dp<br />
A µ (x n (p)) (42)<br />
Gravity M<strong>in</strong>imally Coupled to a Gauge Field<br />
∫<br />
I = d d+1 x √ [ 1<br />
g<br />
2 κ 2 (R − 2 Λ) − 1 ]<br />
4 F µν F µν + 1 ∫<br />
κ 2 d d x √ γ K (43)<br />
5. Hamiltonian Formulation<br />
The canonical variables are the metric h µν on Σ and its conjugate momenta π µν . The momenta are def<strong>in</strong>ed<br />
with respect to evolution <strong>in</strong> the spacelike direction n µ , so this is not the usual notion of the Hamiltonian as the<br />
generator of time translations.<br />
∂M<br />
Bulk Lagrangian Density<br />
L M = 1 (<br />
)<br />
2 κ 2 K 2 − K µν K µν + R − 2 Λ<br />
(44)<br />
Momentum Conjugate to h µν<br />
π µν = ∂L M<br />
∂ (£ n h µν ) = 1<br />
2 κ 2 (<br />
h µν K − K µν) (45)<br />
Momentum Constra<strong>in</strong>t<br />
H µ = 1 κ 2 ⊥ (nν G µν ) = 2 D ν π µν = 0 (46)<br />
Hamiltonian Constra<strong>in</strong>t<br />
6. Conformal Transformations<br />
H = − 1 κ 2 nµ n ν G µν = 2 κ 2 (<br />
π µν π µν − 1<br />
d − 1 π2 )<br />
+ 1<br />
2 κ 2 (<br />
R − 2 Λ<br />
)<br />
= 0 (47)<br />
The dimension of spacetime is d + 1. Indices are raised and lowered us<strong>in</strong>g the metric g µν and its <strong>in</strong>verse g µν .<br />
Metric<br />
Christoffel<br />
ĝ µν = e 2 σ g µν (48)<br />
̂Γ λ µν = Γ λ µν + Θ λ µν (49)<br />
Θ λ µν = δ λ µ ∇ ν σ + δ λ ν ∇ µ σ − g µν ∇ λ σ (50)<br />
4
Riemann Tensor<br />
̂R λ µρν = R λ µρν + δ λ ν ∇ µ ∇ ρ σ − δ λ ρ ∇ µ ∇ ν σ + g µρ ∇ ν ∇ λ σ − g µν ∇ ρ ∇ λ σ (51)<br />
+ δ λ ρ ∇ µ σ ∇ ν σ − δ λ ν ∇ µ σ ∇ ρ σ + g µν ∇ ρ σ ∇ λ σ − g µρ ∇ ν σ ∇ λ σ (52)<br />
(<br />
)<br />
+ g µρ δ λ ν − g µν δ λ ρ ∇ α σ ∇ α σ (53)<br />
Ricci Tensor<br />
̂R µν = R µν − g µν ∇ 2 σ − (d − 1) ∇ µ ∇ ν σ + (d − 1) ∇ µ σ ∇ ν σ (54)<br />
− (d − 1) g µν ∇ λ σ ∇ λ σ (55)<br />
Ricci Scalar<br />
̂R ( )<br />
= e −2 σ R − 2 d ∇ 2 σ − d (d − 1) ∇ µ σ ∇ µ σ<br />
(56)<br />
Schouten Tensor<br />
Ŝ µν = S µν − ∇ µ ∇ ν σ + ∇ µ σ ∇ ν σ − 1 2 g µν∇ λ σ ∇ λ σ (57)<br />
Weyl Tensor<br />
Ĉ λ µρν = C λ µρν (58)<br />
Normal Vector<br />
̂n µ = e −σ n µ ̂n µ = e σ n µ (59)<br />
Extr<strong>in</strong>sic Curvature<br />
̂K ( )<br />
µν = e σ K µν + h µν n λ ∇ λ σ<br />
̂K ( )<br />
= e −σ K + d n λ ∇ λ σ<br />
(60)<br />
(61)<br />
7. Small Variations of the Metric<br />
Consider a small perturbation to the metric of the form g µν → g µν + δg µν . All <strong>in</strong>dices are raised and lowered<br />
us<strong>in</strong>g the unperturbed metric g µν and its <strong>in</strong>verse. All quantities are expressed <strong>in</strong> terms of the perturbation to<br />
the metric with lower <strong>in</strong>dices, and never <strong>in</strong> terms of the perturbation to the <strong>in</strong>verse metric. As <strong>in</strong> the previous<br />
sections, ∇ µ is the covariant derivative on M compatible with g µν and D µ is the covariant derivative on a<br />
hypersurface Σ compatible with h µν .<br />
Inverse Metric<br />
g µν → g µν − g µα g νβ δg αβ + g µα g νβ g λρ δg αλ δg βρ + . . . (62)<br />
Square Root of Determ<strong>in</strong>ant<br />
(<br />
√ √ g → g 1 + 1 )<br />
2 gµν δg µν + . . .<br />
(63)<br />
5
Variational Operator<br />
δ(g µν ) = δg µν δ 2 (g µν ) = δ(δg µν ) = 0 (64)<br />
)<br />
δ(g µν ) = −g µα g νβ δg αβ δ 2 (g µν ) = δ<br />
(−g µλ g νρ δg λρ = 2 g µα g νβ g λρ δg αλ δg βρ (65)<br />
F(g + δg) = F(g) + δF(g) + 1 2 δ 2 F(g) + . . . + 1 n! δ n F(g) + . . . (66)<br />
Christoffel (All Orders)<br />
δ Γ λ µν = 1 2 gλρ ( ∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν<br />
)<br />
δ 2 Γ λ µν = −g λα g ρβ δg αβ<br />
(<br />
∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν<br />
)<br />
δ n Γ λ µν = n 2 δ n−1 ( g λρ) ( ∇ µ δg ρν + ∇ ν δg µρ − ∇ ρ δg µν<br />
)<br />
(67)<br />
(68)<br />
(69)<br />
Riemann Tensor<br />
δ R λ µσν = ∇ σ δ Γ λ µν − ∇ ν δ Γ λ µσ (70)<br />
Ricci Tensor<br />
δ R µν = ∇ λ δ Γ λ µν − ∇ ν δ Γ λ µλ (71)<br />
= 1 )<br />
(∇ λ ∇ µ δg λν + ∇ λ ∇ ν δg µλ − g λρ ∇ µ ∇ ν δg λρ − ∇ 2 δg µν (72)<br />
2<br />
Ricci Scalar<br />
δ R = −R µν δg µν + ∇ µ ( ∇ ν δg µν − g λρ ∇ µ δg λρ<br />
)<br />
(73)<br />
Surface Form<strong>in</strong>g Normal Vector<br />
δ n µ = 1 2 n µ n ν n λ δg νλ = 1 2 δg µν n ν + c µ (74)<br />
c µ = 1 2 n µ n ν n λ δg νλ − 1 2 δg µν n ν = − 1 2 h λ<br />
µ δg λν n ν (75)<br />
Extr<strong>in</strong>sic Curvatures<br />
δ K µν = 1 2 nα n β δg αβ K µν − n µ c λ h λν − n ν c λ h µλ + 1 ( )<br />
2 δg λρ n ρ n µ K λ ν + n ν Kµ<br />
λ<br />
− 1 2 h µ λ hν ρ n α ( )<br />
∇ λ δg αρ + ∇ ρ δg λα − ∇ α δg λρ<br />
(76)<br />
δ K = − 1 2 Kµν δg µν − 1 2 nµ ( ∇ ν δg µν − g νλ ∇ µ δg νλ<br />
)<br />
+ D µ c µ (77)<br />
6