Gravitation: Curvature - An Introduction to General Relativitylaguna.gatech.edu/Gravitation/notes/Chapter03.pdf · Gravitation:Curvature An Introduction to General Relativity Pablo

Gravitation: CurvatureAn Introduction to General Relativity

Pablo Laguna

Center for Relativistic AstrophysicsSchool of Physics

Georgia Institute of Technology

Notes based on textbook: Spacetime and Geometry by S.M. CarrollSpring 2013

Pablo Laguna Gravitation: Curvature

Curvature

Covariant Derivatives

Parallel Transport and Geodesics

The Reimann Curvature Tensor

Symmetries and Killing Vectors

Maximally Symmetric Spacetimes

Geodesic Deviation


Introduction

The metric defines the geometry in a manifold.

Curvature in a manifold depends on the metric, but how.

The form of the metric is strongly dependent on the coordinate system used.

We need a formal definition of curvature.

Curvature plays a central role in general relativity:

a measure of localspacetime curvature

=

a measure of localmatter energy density


Gaussian Curvature

In R3, the Gaussian curvature is defined as the product of the principal curvatures; that is, K = κ1κ2. Its value isintrinsic to the surface and does not depend on the embedding.


Covariant Derivatives

Recall: The partial derivative of a tensor is not in general a tensor.

We need to have a generalization of equations such that ∂νTµν = 0 to be tensorial, so they are invariantunder coordinate transformations.

In flat space in inertial coordinates, ∂µ is a map from (k, l) tensors to (k, l + 1) that is linear and obey theLeibniz rule.

Covariant Derivative∇:

∇ is a map from (k, l) tensor fields to (k, l + 1) tensor fields

∇(T + S) = ∇T +∇S

Leibniz (product) rule: ∇(T ⊗ S) = (∇T )⊗ S + T ⊗ (∇S) .


Covariant Derivative of Vectors

Consider v(xα) and v(xα + dxα) such that dxα = tα ε with tα defining the direction of the covariantderivative.

Parallel transport the vector v(xα + tα ε) back to the point xα and call it v‖(xα)

Covariant Derivative:

∇tv(xα) = limε→0

v‖(xα)− v(xα)

ε

In a local inertial frame:(∇tv)α = tβ∂βvα

Thus,∇βvα = ∂βvα

Notice: The above expression is not valid in curvilinear coordinates. In general,

vα‖ (xδ) = vα(xδ + εtδ) +Γαβγvγ (xδ)(εtβ )

component changes basis vector changes

Therefore,

∇βvα = ∂βvα + Γαβγvγ


Transformation properties of Γαβγ

Since the covariant derivative yields tensors,

∇µ′Vν′ =

∂xµ

∂xµ′∂xν′

∂xν∇µVν .

thus

∇µ′Vν′ = ∂µ′V

ν′ + Γν′µ′λ′V

λ′

=∂xµ

∂xµ′∂xν′

∂xν∂µVν +

∂xµ

∂xµ′Vν

∂

∂xµ∂xν′

∂xν+ Γν

′µ′λ′

∂xλ′

∂xλVλ

and∂xµ

∂xµ′∂xν′

∂xν∇µVν =

∂xµ

∂xµ′∂xν′

∂xν∂µVν +

∂xµ

∂xµ′∂xν′

∂xνΓνµλVλ

therefore

Γν′µ′λ′

∂xλ′

∂xλVλ +

∂xµ

∂xµ′Vλ

∂

∂xµ∂xν′

∂xλ=∂xµ

∂xµ′∂xν′

∂xνΓνµλVλ

and finally

Γν′µ′λ′ =

∂xµ

∂xµ′∂xλ

∂xλ′∂xν′

∂xνΓνµλ −

∂xµ

∂xµ′∂xλ

∂xλ′∂2xν

′

∂xµ∂xλ.

Notice: the connection coefficients are not the components of a tensor.


Covariant differentiation of 1-forms

A possibility is:∇µων = ∂µων + Γλµνωλ

To find Γλµν we need that∇

commutes with contractions: ∇µ(Tλλρ) = (∇T )µλλρ

reduces to the partial derivative on scalars: ∇µφ = ∂µφ

Then

∇µ(ωλVλ) = (∇µωλ)Vλ + ωλ(∇µVλ)

= (∂µωλ)Vλ + ΓσµλωσVλ + ωλ(∂µVλ) + ωλΓλµρVρ

But

∇µ(ωλVλ) = ∂µ(ωλVλ)

= (∂µωλ)Vλ + ωλ(∂µVλ)

Therefore0 = ΓσµλωσVλ + ΓσµλωσVλ .

Since ωσ and Vλ are completely arbitrary,Γσµλ = −Γσµλ .


Consequently:

Covariant differentiation of Vectors

∇βvα = ∂βvα + Γαβγvγ

Covariant differentiation of 1-forms

∇µων = ∂µων − Γλµνωλ

Covariant differentiation of general Tensors

∇σTµ1µ2···µk ν1ν2···νl = ∂σTµ1µ2···µk ν1ν2···νl+Γµ1σλ

Tλµ2···µk ν1ν2···νl + Γµ2σλ

Tµ1λ···µk ν1ν2···νl + · · ·

−Γλσν1Tµ1µ2···µk

λν2···νl− Γλσν2

Tµ1µ2···µkν1λ···νl

− · · ·


Properties of Γαµν:Γαµν has n3 components; that is, 64 in 4-dimensions.

Γαµν is called connection because it helps to transport tensors.

Sµνλ = Γλµν − Γλµν is a tensor. Recall

Γν′µ′λ′ =

∂xµ

∂xµ′∂xλ

∂xλ′∂xν′

∂xνΓνµλ −

∂xµ

∂xµ′∂xλ

∂xλ′∂2xν

′

∂xµ∂xλ

Γν′µ′λ′ =

∂xµ

∂xµ′∂xλ

∂xλ′∂xν′

∂xνΓνµλ −

∂xµ

∂xµ′∂xλ

∂xλ′∂2xν

′

∂xµ∂xλ

then

Sµ′λ′ν′ =

∂xµ

∂xµ′∂xλ

∂xλ′∂xν′

∂xνSµλ

ν

If Γλµν is a connection, Γλνµ is also a connection. Thus, the torsion tensor is defined by

Tµνλ = Γλµν − Γλνµ = 2Γλ[µν] .

A spacetime metric gµν induces a unique connection if Γαµν is (1) torsion-free: Γλµν = Γλ(µν) and (2)metric compatible: ∇ρgµν = 0.

Metric compatibility implies that

∇ρgµν = 0

gµλ∇ρVλ = ∇ρ(gµλVλ) = ∇ρVµ


Christoffel symbols

From metric compatibility:

∇ρgµν = ∂ρgµν − Γλρµgλν − Γλρνgµλ = 0

∇µgνρ = ∂µgνρ − Γλµνgλρ − Γλµρgνλ = 0

∇νgρµ = ∂νgρµ − Γλνρgλµ − Γλνµgρλ = 0

Subtract the second and third from the first,

∂ρgµν − ∂µgνρ − ∂νgρµ + 2Γλµνgλρ = 0 .

Multiply by gσρ to get

Christoffel Symbols

Γσµν =1

2gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν )


The Christoffel symbols vanish in flat space in Cartesian coordinates

The Christoffel symbols do not vanish in flat space in curvilinear coordinates.

For example, if ds2 = dr2 + r2dθ2, it is not difficult to show that Γrθθ = −r and Γθθr = 1/r

At any one point p in a spacetime (M, gµν ), it is possible to find a coordinate system for which Γσµν = 0(recall local flatness)

Very useful property:∇µVµ = ∂µVµ + Γ

µµλ

Vλ

but

Γµµλ

=1

2gµρ(∂µgλρ + ∂λgρµ − ∂ρgµλ)

=1

2gµρ∂λgρµ =

1

2∂λ ln |g| =

1√|g|∂λ

√|g|

then

∇µVµ = ∂µVµ + Vµ1√|g|∂µ

√|g|

thus

∇µVµ =1√|g|∂µ(

√|g|Vµ)


Parallel TransportAs mentioned before, covariant differentiation involves computing how tensor change.However, tensor are maps from vectors and 1-forms to real numbers at a given point.Question: What is behind the changes computed by∇ acting on tensors?Answer: ∇ gives the instantaneous rate of change of a tensor field in comparison to what the tensor wouldbe if it were parallel transported.The results of parallel transporting a tensor is path dependent.

q

p

keep vectorconstant


Keeping the Tensor Constant

In flat spacetime, the constancy of a tensor along a curve xµ(λ) can be states as( d

dλT)µ1µ2···µk

ν1ν2···νl =dxσ

dλ∂σTµ1µ2···µk ν1ν2···νl = 0

Define the directional derivative as:D

dλ=

dxµ

dλ∇µ

Notice that this derivative is a map from (k, l) tensors to (k, l) . Thus, the parallel transport condition or equation ofparallel transport is defined as:

( D

dλT)µ1µ2···µk

ν1ν2···νl ≡dxσ

dλ∇σTµ1µ2···µk ν1ν2···νl = 0

For a vector, this equation takes the form

d

dλVµ + Γµσρ

dxσ

dλVρ = 0

If the connection Γµσρ is metric compatible, then

D

dλgµν =

dxσ

dλ∇σgµν = 0


Theorem: Given to vectors Vµ and Wµ that are parallel-transported along a curve xα(λ), the inner productgµνVµWν is preserved along this curve.

Proof:

D

dλ(gµνVµWν ) =

( D

dλgµν

)VµWν + gµν

( D

dλVµ)

Wν + gµνVµ( D

dλWν

)= 0

Lemma: The norm of vectors and orthogonality are preserved under parallel transport


Geodesics

A geodesic generalizes the notion of a straight line in Euclidean space to curved space.

Definition 1: A geodesic is the path of shortest distance.

Definition 2: A geodesic is the path that parallel transports its own tangent vector.

Definitions 1 and 2 are equivalent if the connection is the Christoffel connection.

From Definition 2, let dxµ/dλ be the tangent vector to a path xµ(λ) The condition that dxµ/dλ be paralleltransported is

D

dλ

dxµ

dλ= 0 ,

or alternatively

d2xµ

dλ2+ Γµρσ

dxρ

dλ

dxσ

dλ= 0 .

This is called the geodesic equation.

Since in Euclidean space in Cartesian coordinates Γµρσ = 0, the geodesic equation becomes d2xµ/dλ2 = 0,which is the equation for a straight line.


From Definition 1, consider a time-like curve xα(λ) and its corresponding proper time functional

τ =

∫dτ =

∫ √−gµνdxµdxν =

∫ (−gµν

dxµ

dλ

dxν

dλ

)1/2

dλ =

∫ √−f dλ

where we have introduced the following definition: f ≡ gµν dxµdλ

dxνdλ . The extrema of this functional will give us the

shortest-distance path. That is,

δτ =

∫δ√−f dλ = −

∫ 1

2(−f )−1/2

δf dλ = 0

Without loss of generality, we can select dxα/dλ to be the 4-velocity vector Vα; that is, λ = τ and f = −1.Therefore the stationary points of τ =

∫dτ are equivalent to the stationary points of

I =1

2

∫f dτ =

1

2

∫gµν

dxµ

dτ

dxν

dτdτ

Consider now changes in the proper time under infinitesimal variations of the path,

xµ → xµ + δxµ

gµν → gµν + δxσ∂σgµν .

then

δI =1

2

∫ (∂σgµν

dxµ

dτ

dxν

dτδxσ + 2 gµν

dxµ

dτ

d(δxν )

dτ

)dτ


Consider the last term

∫ (gµν

dxµ

dτ

d(δxν )

dτ

)dτ = −

∫ (gµν

d2xµ

dτ2+

dgµνdτ

dxν

dτ

)δxν dτ

= −∫ (

gµνd2xµ

dτ2+ ∂σgµν

dxσ

dτ

dxν

dτ

)δxν dτ

The δI becomes:

δI = −∫ [

gµσd2xµ

dτ2+

1

2

(−∂σgµν + ∂νgµσ + ∂µgνσ

) dxµ

dτ

dxν

dτ

]δxσdτ = 0

which yields

gµσd2xµ

dτ2+

1

2

(−∂σgµν + ∂νgµσ + ∂µgνσ

) dxµ

dτ

dxν

dτ= 0

or

d2xµ

dτ2+ Γσµν

dxµ

dτ

dxν

dτ= 0


Properties of Geodesics

The geodesic equation is a generalization of Newton’s law f = ma for the case f = 0.

For the Lorentz force case, in general relativity

d2xµ

dτ2+ Γµρσ

dxρ

dτ

dxσ

dτ=

q

mFµν

dxν

dτ

The transformation τ → λ = aτ + b, for some constants a and b, leaves the geodesic equation invariant.λ is called an affine parameter

Notice: The demand that the tangent vector be parallel-transported constrains the parametrization of thecurve.

For a general parametrization,

d2xµ

dα2+ Γµρσ

dxρ

dα

dxσ

dα= f (α)

dxµ

dα,

where

f (α) = −(

d2α

dλ2

)( dα

dλ

)−2


Properties of Geodesics

In a spacetime with Lorentzian metric, the character (timelike/null/spacelike) of the geodesic never changes.

For time-like curves with Uα = dxα/dτ the 4-velocity, the geodesic equation reads Uλ∇λUµ = 0 .

In terms of the 4-momentum pµ = m Uµ the geodesic equations reads pλ∇λpµ = 0

For a null geodesic the proper time parameter τ vanishes. There is no preferred choice of affine parameterin that case. One can for instance pick the affine parameter λ such that pα = dxα/dλ

The energy of a particle (time-like or null) is the given by E = −pµUµ


The Riemann Curvature Tensor

Recall that in flat space:

Parallel-transport around a closed loop leaves a vector unchanged.

Covariant derivatives of tensors commute.

Initially parallel geodesics remain parallel.

How do these properties get modified by the presence of curvature and how can we quantify those changes?

Recall also that:

Parallel-transport of a vector around a closed loop in a curved space will lead to a transformation of thevector.

The resulting transformation depends on the total curvature enclosed by the loop.

Goal: to have a local description of the curvature at each point. Such description is provided by the Riemanncurvature tensor.


Consider the following situation: The parallel-transport of a vector Vµ around the loop in the figure below.

(0, 0)

B

( a, 0)

( a, b)(0, b)

Aµ

B

Aµ

The change δVµ experience by Vµ as it is parallel-transported and returned to the starting point must be

proportional to Vµ

depend on Aµ and Bµ

anti-symmetric in Aµ and Bµ to indicate the direction followed in the loop

thus

δVρ = (δa)(δb)AνBµRρσµνVσ

where Rρσµν is a (1, 3) tensor known as the Riemann or curvature tensor. Notice:

Rρσµν = −Rρσνµ


Commutator of two covariant derivatives: it measures the difference between parallel transporting the tensor firstone way and then the other, versus the opposite ordering.

µ

µ

That is:

[∇µ,∇ν ]Vρ = ∇µ∇νVρ −∇ν∇µVρ

= ∂µ(∇νVρ)− Γλµν∇λVρ + Γρµσ∇νVσ − (µ↔ ν)

= ∂µ∂νVρ + (∂µΓρνσ)Vσ + Γρνσ∂µVσ − Γλµν∂λVρ − ΓλµνΓρλσ

Vσ

+Γρµσ∂νVσ + ΓρµσΓσνλVλ − (µ↔ ν)

= (∂µΓρνσ − ∂νΓρµσ + Γρµλ

Γλνσ − Γρνλ

Γλµσ)Vσ − 2Γλ[µν]∇λVρ

= RρσµνVσ − Tµνλ∇λVρ

where

Riemann Tensor

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + Γρµλ

Γλνσ − Γρνλ

Γλµσ


Important to notice:

The antisymmetry of Rρσµν in µν is obvious.

The derivation depends only connection (no mention of the metric was made).

Thus, the definition is true for any connection, whether or not it is metric compatible or torsion free.

The action of [∇ρ,∇σ ] can be generalized to a tensor of arbitrary rank:

[∇ρ,∇σ ]Xµ1···µk ν1···νl = − Tρσλ∇λXµ1···µk ν1···νl

+Rµ1λρσXλµ2···µk ν1···νl + Rµ2

λρσXµ1λ···µk ν1···νl + · · ·

−Rλν1ρσXµ1···µkλν2···νl

− Rλν2ρσXµ1···µkν1λ···νl

− · · ·

One can the view the torsion tensor and the curvature tensors as

T (X , Y ) = ∇X Y −∇Y X − [X , Y ]

R(X , Y )Z = ∇X∇Y Z −∇Y∇X Z −∇[X,Y ]Z

where∇X = Xµ∇µ. That is

RρσµνXµYνZσ = Xλ∇λ(Yη∇ηZρ)− Yλ∇λ(Xη∇ηZρ)

− (Xλ∂λYη − Yλ∂λXη)∇ηZρ


Theorem:

If the components of the metric are constant in some coordinate system, the Riemann tensor will vanish.

If the Riemann tensor vanishes we can always construct a coordinate system in which the metriccomponents are constant.

Proof:

Part 1: If in some coordinate system ∂σgµν = 0, then Γρµν = 0 and ∂σΓρµν = 0; thus Rρσµν = 0.

Part 2: see textbook


Properties of the Riemann TensorHow many independent components does the Riemann tensor have?

In principle, it has n4 independent components in n-dimensions.

The anti-symmetry in the last two indices implies there are only n(n − 1)/2 independent values these lasttwo indices can take on, there are n3(n − 1)/2 independent components.

Consider Rρσµν = gρλRλσµν in Riemann:

Rρσµν = gρλ(∂µΓλνσ − ∂νΓλµσ)

=1

2gρλgλτ (∂µ∂νgστ + ∂µ∂σgτν − ∂µ∂τ gνσ − ∂ν∂µgστ − ∂ν∂σgτµ + ∂ν∂τ gµσ)

=1

2(∂µ∂σgρν − ∂µ∂ρgνσ − ∂ν∂σgρµ + ∂ν∂ρgµσ)

then

Rρσµν = −Rσρµν

Rρσµν = Rµνρσ

Rρ[σµν] = 0

Since these are tensorial equations, they are also true in any coordinate system.

There are 1/12n2(n2 − 1)independent components in the Riemann tensor. In 4-dimensions, there are 20independent components.


The Bianchi Identity

Consider the covariant derivative of the Riemann tensor, evaluated in Riemann normal coordinates:

∇λRρσµν = ∂λRρσµν

=1

2∂λ(∂µ∂σgρν − ∂µ∂ρgνσ − ∂ν∂σgρµ + ∂ν∂ρgµσ) .

and consider

∇λRρσµν +∇ρRσλµν +∇σRλρµν

=1

2(∂λ∂µ∂σgρν − ∂λ∂µ∂ρgνσ − ∂λ∂ν∂σgρµ + ∂λ∂ν∂ρgµσ

+∂ρ∂µ∂λgσν − ∂ρ∂µ∂σgνλ − ∂ρ∂ν∂λgσµ + ∂ρ∂ν∂σgµλ+∂σ∂µ∂ρgλν − ∂σ∂µ∂λgνρ − ∂σ∂ν∂ρgλµ + ∂σ∂ν∂λgµρ)

= 0 .

Since Rρσµν = −Rσρµν then

Bianchi identity

∇[λRρσ]µν = 0 .


The Ricci Tensor

Ricci Tensor

Rµν = Rλµλν .

Because of Rρσµν = RµνρσRµν = Rνµ ,

Also

Ricci scalar

R = Rµµ = gµνRµν .


The Weyl Tensor

Weyl Tensor

Cρσµν = Rρσµν −2

(n − 2)

(gρ[µRν]σ − gσ[µRν]ρ

)+

2

(n − 1)(n − 2)Rgρ[µgν]σ .

Notice:

The Ricci tensor and the Ricci scalar contain information about “traces” of the Riemann tensor. The Weyltensor is the Riemann tensor with “all of its contractions removed. ”

All possible contractions of Cρσµν vanish, while it retains the symmetries of the Riemann tensor:

Cρσµν = C[ρσ][µν] ,

Cρσµν = Cµνρσ ,Cρ[σµν] = 0 .

The Weyl tensor is only defined in three or more dimensions, and in three dimensions it vanishes identically.

For n ≥ 4 it satisfies a version of the Bianchi identity,

∇ρCρσµν = −2(n − 3)

(n − 2)

(∇[µRν]σ +

1

2(n − 1)gσ[ν∇µ]R

).

The Weyl tensor is that it is invariant under conformal transformations. That is, Cρσµν for some metric gµνand Cρσµν for a metric gµν = Ω2(x)gµν are the same. So the Weyl tensor is a.k.a. the conformaltensor..


The Einstein Tensor

Contract twice the Bianchi identity

∇λRρσµν +∇ρRσλµν +∇σRλρµν = 0

to get

0 = gνσgµλ(∇λRρσµν +∇ρRσλµν +∇σRλρµν )

= ∇µRρµ −∇ρR +∇νRρν ,

or

∇µRρµ −1

2∇ρR = 0

Define

Einstein Tensor

Gµν = Rµν −1

2R gµν ,

to get

Contracted Bianchi Identity

∇µGµν = 0


ExampleConsider the two-sphere, with metric

ds2 = a2(dθ2 + sin2θ dφ2) ,

where a the radius of the sphere. Then coefficients are

Γθφφ = − sin θ cos θ

Γφθφ

= Γφφθ

= cot θ .

The only non vanishing component of the Reimann tensor is

Rθφθφ = ∂θΓθφφ − ∂φΓθθφ + ΓθθλΓλφφ − ΓθφλΓλθφ

= (sin2θ − cos2

θ)− (0) + (0)− (− sin θ cos θ)(cot θ)

= sin2θ .

Thus

Rθφθφ = gθλRλφθφ= gθθRθφθφ= a2 sin2

θ .

From Rµν = gαβRαµβν one gets

Rθθ = gφφRφθφθ = 1Rθφ = Rφθ = 0

Rφφ = gθθRθφθφ = sin2θ .

and the Ricci scalar

R = gθθRθθ + gφφRφφ =2

a2.


Example cont.

The Ricci scalar for a two-dimensional manifold completely characterizes the curvature

For this example it is a constant over the two-sphere.

The manifold is maximally symmetric

In any number of dimensions, the curvature of a maximally symmetric space satisfies (for some constant a)

Rρσµν = a−2(gρµgσν − gρνgσµ) ,


Symmetries and Killing Vectors

A manifold M possesses a symmetry if the geometry is invariant under a certain transformation that maps M intoitself.

Isometry: a symmetry of the metric.

Example: 4-dimensional Minkowski space-time. The metric

ds2 = ηµνdxµ dxν

has the symmetries

xµ → xµ + aµ translations

xµ → Λµ νxν Lorentz transformation

Notice:∂σ∗gµν = 0 ⇒ xσ∗ → xσ∗ + aσ∗ is a symmetry


Recall the geodesic equation

0 = pν∇νpµ

= pν∂νpµ − Γδµν pν pδ = mdxν

dτ∂νpµ − Γδµν pν pδ

= mdpµdτ−

1

2gδλ

(∂µgνλ + ∂νgµλ − ∂λgµν

)pν pδ

= mdpµdτ−

1

2

(∂µgνλ + ∂νgµλ − ∂λgµν

)pν pλ

= mdpµdτ−

1

2∂µgνλpν pλ

therefore

4-Momentum Conservation

∂σ∗gµν = 0 along a geodesic ⇒dpσ∗

dτ= 0


Consider the following coordinate transformation on the 4-dimensional Minkowski metric: z = zx . Then

ds2 = ηµνdxµ dxν = −dt2 + (1 + z2)dx2 + dy2 + x2dz2 + 2xzdx dz

Where did the translational symmetries go? We need a method to define symmetries invariant under coordinatetransformations.

Define K = ∂σ∗ such that ∂σ∗ gµν = 0.

This is equivalent to Kµ = (∂σ∗ )µ = δµσ∗ .

If K is the generator of an isometry, then pσ∗ = Kνpν = Kνpν is invariant along the K -direction

That isdpσ∗

dτ= 0 ⇐⇒ pµ∇µ(Kνpν ) = 0

Expanding the last equation

0 = pµ∇µ(Kνpν )

= pµKν∇µpν + pµpν∇µKν

= pµpν∇µKν = pµpν∇(µKν)

Therefore∇(µKν) = 0 =⇒ pµ∇µ(Kνpν ) = 0

Killing’s Equation

∇(µKν) = 0


Theorem: If a vector K is a Killing vector, it is always possible to find a coordinate system in which K = ∂σ∗ . Ifthere are more than one Killing vector. In general it is not possible to find coordinates for which all vectors are of theform K = ∂σ∗ .

Notice: For a Killing vector, the Riemann equation

∇[µ∇ν]Kρ = Rρ σµνKσ

takes the form∇µ∇νKρ = Rρ νµσKσ

thus∇µ∇νKµ = RνσKσ

From the contracted Bianchi identity

0 = Kν(∇µRµν −

1

2∇νR

)= 2 Kν∇µRµν − Kµ∇µR

= 2∇µ(KνRµν )− 2 Rµν∇µKν − Kµ∇µR

= 2∇µ(∇λ∇µKλ)− 2 Rµν∇(µKν) − Kµ∇µR

= 2∇µ∇λ∇µKλ − 2 Rµν∇(µKν) − Kµ∇µR

= Kµ∇µR


Killing vectors in flat space

Translations:

Xµ = (1, 0, 0)

Yµ = (0, 1, 0)

Zµ = (0, 0, 1)

Rotations:

Rµ = (−y, x, 0)

Sµ = (z, 1,−x)

Tµ = (0,−z, y)


Geodesic Deviation Equation

The notion of parallel does not extend naturally from flat to curved spaces.

Instead, construct a one-parameter family of non-intersecting geodesics, γs(t); namely, for each s ∈ Rthere is a geodesic γs with affine parameter t .

The collection γs(t) defines a smooth two-dimensional surface.

Chose s and t to be the coordinates in this surface.

The entire surface is the set of points xµ(s, t) ∈ M.

There are two natural vector fields:

Tµ =∂xµ

∂ttangent vectors

Sµ =∂xµ

∂sdeviation vectors

t

s

T

S

( )s tµ

µ


Define

Vµ = (∇T S)µ = Tρ∇ρSµ relative velocity of geodesics

aµ = (∇T V )µ = Tρ∇ρVµ relative acceleration of geodesics

Since S and T are basis vectors adapted to a coordinate system, [S, T ] = 0

Since we are considering vanishing torsion, [S, T ] = 0 impies that

Sρ∇ρTµ − Tρ∇ρSµ = 0

Thus

aµ = Tρ∇ρ(Tσ∇σSµ)

= Tρ∇ρ(Sσ∇σTµ)

= (Tρ∇ρSσ)(∇σTµ) + TρSσ∇ρ∇σTµ

= (Sρ∇ρTσ)(∇σTµ) + TρSσ(∇σ∇ρTµ + RµνρσTν )

= (Sρ∇ρTσ)(∇σTµ) + Sσ∇σ(Tρ∇ρTµ)− (Sσ∇σTρ)∇ρTµ + RµνρσTνTρSσ

= RµνρσTνTρSσ .

Therefore

Geodesic deviation equation

aµ =D2

dt2Sµ = RµνρσTνTρSσ ,


Documents

Gravitation: Curvature - An Introduction to General Relativitylaguna.gatech.edu/Gravitation/notes/Chapter03.pdf · Gravitation:Curvature An Introduction to General Relativity Pablo