General Relativity

Summary Sheet I. Tensor Analysis, Geodesics, Covariant Derivatives

Tensor Analysis

The basics of general tensor analysis are very similar to those of special relativity, with the Lorentztransformation Λα

β replaced by general transition functions from xα→x′α, ∂x′α/∂xβ .A scalar transforms according to

φ′(x′) = φ(x)

where a single x denotes the four coordinates xα.A contravariant vector transforms according to

V ′α(x′) =∂x′α

∂xβV β(x)

Note that the coordinates themselves, xα, are not vectors under general coordinate transformations (althoughinfinitesimal changes in coordinates dxα are).A covariant vector transforms according to

U ′

β(x′) =∂xα

∂x′βUα(x)

Similar transformation laws hold for more general tensors, e.g.,

T ′α

βγ =∂x′α

∂xµ

∂xν

∂x′β

∂xρ

∂x′γT µ

νρ

The Kronecker delta, δβα = diag(1, 1, 1, 1) is a second rank tensor, whose components take the same

numerical value in all coordinate systems.Basic rules of tensor algebra follow easily. As in special relativity, linear combinations, outer products andcontractions of tensors give tensors, but the partial derivative of a tensor is not a tensor.The metric tensor gαβ is a symmetric second rank tensor that is non-singular. This means that detgαβ 6=0, and thus its inverse exists, and is denoted gαγ , where

gαγgαβ = δγβ

The metric may be used to raise and lower indices,

Vβ = gαβV α, Uα = gαβUβ

The inner product between vectors is defined as before, as

U · V = gαβUαV β

Geodesic Motion

Particles and light rays in curved space satisfy the geodesic equation,

F µ =d2xα

dλ2+ Γα

βγ

dxβ

dλ

dxγ

dλ= 0

where the connection is given by

Γαβγ =

1

2gαµ (gµβ,γ + gµγ,β − gβγ,µ)

1

The connection is not a tensor. It transforms according to the rule:

Γ′αβγ =

∂x′α

∂xµ

∂xν

∂x′β

∂xρ

∂x′γΓµ

νρ −∂xµ

∂x′β

∂xν

∂x′γ

∂2x′α

∂xµ∂xν

The connection is symmetric on its lower indices, Γαβγ = Γα

γβ. It is often written so that the upper index sitsin between the lower pair. However, we will never have need to raise and lower the indices on the connection(although this is done in some textbooks), so the location of the upper index in relation to the lower onesdoes not matter.

The second derivative term in F µ is also not a tensor, and transforms in such a way as to cancel theinhomogeneous term in the transformation law of the connection, hence F µ is in fact a tensor.

The quantity gαβ xαxβ , where a dot denotes differentiation with respect to λ, is constant along thegeodesic, hence gives a first integral of the equation of motion. In particular,

gαβ

dxα

dλ

dxβ

dλ=

{

−1, for particles;0, for light rays.

where the parameter λ is proper time s for particles.The geodesic equation may be derived using the calculus of variations: the functional

S[xα(λ)] =

∫

dλ L(xα, xα)

is extremized by the path xα(λ) obeying the Euler-Lagrange equation

d

dλ

(

∂L

∂xα

)

−∂L

∂xα= 0

The geodesic equation, for particles, then follows by taking

L =(

−gαβxαxβ)

1

2

and the geodesics have the interpretation as the paths that extremize the total distance S[xα(λ)] betweentwo points. However, it is readily shown that the geodesic equation may also be derived by taking

L = gαβ xαxβ

In practice, the geodesics for a given metric are always derived by using this form of L in the Euler-Lagrangeequations. Furthermore, the connections Γα

βγ may be read off from the equations for the geodesics, and thisis generally easier than calculating them directly.The Newtonian limit of the geodesic equation for particles involves low velocities and static, weak fields.Let

∣

∣dx/dt∣

∣ << 1 and write gαβ = ηαβ +hαβ, where ηαβ is the Minkowski space metric, |hαβ | << 1 and hαβ

is independent of time. Then the geodesic equation implies that

d2t

ds2= 0,

d2xi

ds2−

1

2h00,i

(

dt

ds

)2

= 0

Comparing with the Newtonian equation of motion for a particle in gravitational potential φ,

d2x

dt2+ ∇φ = 0

implies that g00 = −1 − 2φ in the Newtonian limit. This fixes one component of the metric in terms of theNewtonian potential in the limit of weak, static, gravitational fields.

2

Covariant Derivatives

The partial derivative of a tensor yields an object that is not a tensor under general coordinate transfor-mations. We therefore define a new type of derivative, the covariant derivative, defined so as to map tensorsinto tensors.The covariant derivative of a vector is defined by

∇αV µ ≡ V µ;α = ∂αV µ + Γµ

αβV β

The covariant derivative of a covector is defined by

∇αUµ ≡ Uµ;α = ∂αUµ − ΓβαµUβ

The covariant derivative of a scalar is just the partial derivative

∇αφ = ∂αφ

The covariant derivative of a general tensor has the form

∇µT αβγ = ∂µT α

βγ + ΓαµσT σ

βγ − ΓσµβT α

σγ − ΓσµγT α

βσ

When taking two covariant derivatives, start by calculating the outer derivative first, e.g.,

∇µ∇νVα = ∂µ(∇νVα) − Γσµν(∇σVα) − Γσ

µα(∇νVσ)

and then insert the formula for the covariant derivative of a vector.The covariant derivative has the following properties. It is linear,

∇α(aA + bB) = a∇αA + b∇αB

where a and b are numerical constants, and A and B are tensors of any rank; it obeys the Leibniz rule

∇α(AB) = (∇αA)B + A(∇αB)

for any tensors A, B; it commutes with contractions,

(

T αβ;γ

)

α=β= T α

α;γ

The covariant derivative has the important property that the metric is covariantly constant

∇αgβγ = 0 = ∇αgβγ

Likewise the Kronecker delta,

∇αδβγ = 0

The first relation is the natural analogue of the relation, ∂αηβγ = 0 in special relativistic equations. It hasthe important consequence that special relativistic laws of physics may be successfully turned into generallycovariant laws by replacing ηαβ with gαβ and ∂µ with ∇µ.The geodesic equation may be written in terms of the covariant derivative as,

Uβ∇βUα = 0

where Uα = dxα/ds is the velocity 4-vector.

3

Curvature Tensor

The covariant derivative commutes on scalars, which means that it has the property,

∇α∇βφ = ∇β∇αφ

This is not true on higher rank tensors. On vectors,

∇µ∇νV α −∇ν∇µV α = RαβµνV β

whereRβ

αµν = ∂µΓβνα − ∂νΓβ

µα + ΓγναΓβ

µγ − ΓγµαΓβ

νγ

is called the Riemann curvature tensor.In flat space, and in Cartesian coordinates (t, x, y, z), gαβ = ηαβ = constant, so the connections all vanish

and the Riemann tensor is zero. Because it is a tensor, this means that it must also vanish in any other setof coordinates in flat space. Hence the Riemann tensor provides a coordinate independent characterizationof flatness. In fact, it may be shown that a space is flat (i.e. it has a global set of coordinates (t, x, y, z) inwhich the metric gαβ is the Minkowsk metric ηαβ) if and only if the Riemann tensor is zero.The Riemann tensor has the following symmetries. Let Rαβµν = gαγRγ

βµν , then

Rαβµν = −Rαβνµ

Rαβµν = −Rβαµν

Rαβµν = Rµναβ

It also obeys the cyclic identity

Rαβµν + Rανβµ + Rαµνβ = 0

These relations are proved by considering a point P in the space, and choosing locally inertial coordinatesclose to that point, so that gαβ = ηαβ and gαβ,γ = 0 at P (this is the general relativistic version of choosinglocal coordinates such that g = ∇φ = 0 in Newtonian gravity). Then at P , in those coordinates,

Rαβµν =1

2(gαν,βµ − gνβ,αµ − gαµ,βν + gµβ,αν)

The above symmetry relations are then easily proved. The symmetry relations are tensor equations, so ifthey are true at P in one coordinate system they are true at P in any coordinate system. Furthermore, thepoint P is arbitrary, so the relations hold generally.The Riemann tensor is a 4th rank tensor, so has N 4 components in N dimensions (= 256 if N = 4). However,the above relations reduce this number to N 2(N2 − 1)/12 independent components (= 20 if N = 4).The Riemann tensor also obeys the Bianchi identity,

Rαβµν;σ + Rαβσµ;ν + Rαβνσ;µ = 0

The Ricci tensor Rµν is defined byRµν = Rα

µαν

and is symmetric, Rµν = Rνµ. The Ricci scalar R is defined by

R = gµνRµν

The Einstein tensor Gαβ is defined by

Gαβ = Gβα = Rαβ −1

2gαβR

4

and contracting the Bianchi identities leads to the property

∇αGαβ = 0

Absolute Derivatives

Consider a curve xα(λ), where λ is a parameter. Then the absolute derivative of a vector V α along thecurve is defined by

DV α

dλ=

dxβ

dλ∇βV α =

dV α

dλ+ Γα

βγ

dxβ

dλV γ

It clearly takes tensors into tensors, and moreover, it reduces to the ordinary derivative, dV α/dλ, in a locallyinertial frame. Its properties follow readily from the properties of the covariant derivative, e.g.,

Dφ

dλ=

dφ

dλ,

D

dλgµν = 0

and so on.

Geodesic Deviation

Consider a geodesic xα(s), and a neighbouring geodesic, xα(s) + ξα(s). Then it may be shown that

D2ξµ

ds2= xαxβξγRµ

αβγ

This is the equation of geodesic deviation and describes the extent to which neighbouring geodesicsconverge or diverge as a result of non-uniformities in the gravitational field. It clearly exposes the Riemanntensor as responsible tidal effects. The analagous Newtonian equation is

d2(δxi)

dt2= −Eijδxj

where

Eij =∂2φ

∂xi∂xj

hence the combination −xαxβRµαβγ in general relativity is analagous to the Newtonian “tidal tensor” Eij .

5