chapter 3 covariance and tensor...

Chapter 3

Covariance and Tensor Notation

The termcovarianceimplies a formalism in which thelaws of physics maintain the same form under a specifiedset of transformations.

EXAMPLE:Lorentz covarianceimplies equations that areconstructed in such a way that they do not change theirform under Lorentz transformations (three boosts betweeninertial systems and three rotations).a

aAn inertial system is a frame of reference in which Newton’s first law of motionholds. Thus, for example, rotating frames and accelerated frames are not inertial.An inertial system is therefore in uniform translational motion with respect to anyother inertial frame.

43

44 CHAPTER 3. COVARIANCE AND TENSOR NOTATION

3.1 Covariance and Tensor Notation

• We shall be concerned generically with a transformation be-tween one set of spacetime coordinates, denoted by

x≡ xµ = (x0,x1,x2,x3)

and a new set

x′µ = x′µ(x) µ = 0,1,2,3

wherex = xµ denotes the original (untransformed) coordinates.

This notation is an economical form of

x′µ = ξ µ(x1,x2,x3,x4) (µ = 1,2, . . .)

where the single-valued, continuously differentiablefunctions ξ µ assign a new (primed) coordinate(x′1,x′2,x′3,x′4) to a point of the manifold with oldcoordinates(x1,x2,x3,x4). This transformation may beabbreviated tox′µ = ξ µ(x) and, even more tersely, tox′µ = x′µ(x).

• Coordinates are justlabels, so laws of physics cannot dependon them. This implies that the systemx′µ is not privileged andtherefore this transformation should beinvertible.

• Notice carefully that we are talking about thesame pointde-scribed in two different coordinate systems.

3.1. COVARIANCE AND TENSOR NOTATION 45

As a minimum, we must consider the transformations of

• Fields

• Derivatives of fields

• Integrals of fields.

The first two are necessary to formulate equations of mo-tion, and the latter enter into various conservation laws.

To facilitate this, we shall introduce a setof mathematical quantities calledtensorsthatare a generalization of the idea of scalars andvectors to more components.


3.1.1 Scalar Transformation Law

Simplest possibility: a field has a single component (mag-nitude) at each point that is unchanged by the transforma-tion

ϕ ′(x′) = ϕ(x).

Quantities such asϕ(x) that are unchanged under the co-ordinate transformation are calledscalars.

EXAMPLE: Value of the temperature at dif-ferent points on the surface of the Earth.


3.1.2 Vectors

The gradient of a scalar fieldϕ(x) obeys

∂ϕ(x)∂x′µ

= ∑ν

∂ϕ(x)∂xν

∂xν

∂x′µ≡

∂ϕ(x)∂xν

∂xν

∂x′µ

(All partial derivatives understood to be evaluated at samepointP.)

Einstein summation convention:

• An index that is repeated, once as a superscript and once as asubscript, implies a summation over that index.

• Such an index is a dummy index that is removed by the summa-tion and should not appear on the other side of the equation.

• A repeated (dummy) index may be replaced by any other indexnot already in use without altering equation:AαBα = Aβ Bβ .

• A superscript (subscript) in a denominator counts as a subscript(superscript) in a numerator.

• Greek indices (α,β , . . .) denote the full set of spacetime indicesrunning over0, 1, 2, 3.

• Roman indices (i, j, . . .) denote the indices1, 2, 3running onlyover the spatial coordinates.

• Placement of indices matters: generallyxα andxα will be differ-ent quantities.

• At all stages of manipulating equations, the indices on thetwosides of an equation (including their up or down placement) mustmatch.


We can classify tensors according to a notationtmn , wheren is the

number of lower indices andm is the number of upper indices

• Thus a scalar is a tensor of typet00, since it carries no indices.

• The sum ofn andm is the rank of the tensor. A scalar is a tensorof rank zero.

There are two kinds of rank-1 tensors, having the index pattern t01 and

t10, respectively. The first is called acovariant vector:

COVARIANT VECTOR: A tensor having a transforma-tion law that mimics that of the scalar field gradient,

A′µ(x′) =

∂xν

∂x′µAν(x) (covariant vector)

is of typet01 and is termed acovariant vectoror 1-form.

ECONOMY OF NOTATION:The preceding equation really meansfour equations:

A′µ =

∂x0

∂x′µA0+

∂x1

∂x′µA1 +

∂x2

∂x′µA2+

∂x3

∂x′µA3 (µ = 0,1,2,3)

each containing four terms. It is equivalent to the matrix equation

A′0

A′1

A′2

A′3

=

∂x0/∂x′0 ∂x1/∂x′0 ∂x2/∂x′0 ∂x3/∂x′0

∂x0/∂x′1 ∂x1/∂x′1 ∂x2/∂x′1 ∂x3/∂x′1

∂x0/∂x′2 ∂x1/∂x′2 ∂x2/∂x′2 ∂x3/∂x′2

∂x0/∂x′3 ∂x1/∂x′3 ∂x2/∂x′3 ∂x3/∂x′3

A0

A1

A2

A3

.


CONTRAVARIANT VECTORS: A differential trans-forms like

dx′µ =∂x′µ

∂xν dxν ,

which suggests a second rank-1 transformation rule

A′µ(x′) =∂x′µ

∂xν Aν(x) (contravariant vector).

A tensor that behaves in this way is of typet10 and is termed

acontravariant vector.


Therefore, we expect the possibility of two rank-1 tensors:

1. Covariant vectors (1-forms), which carry a lower in-dex and transform like the gradient of a scalar:

A′µ(x′) =

∂xν

∂x′µAν(x) (covariant vector)

2. Contravariant vectors, which carry an upper indexand transform like the coordinate differential:

A′µ(x′) =∂x′µ

∂xν Aν(x) (contravariant vector).

In the general case they must be distinguished(by placement—upper or lower—of their in-dices).


3.1.3 Scalar Product

Covariant and contravariant vectors permit defining a scalar product

A·B≡ AµBµ = AµBµ

This transforms as a scalar because from

A′µ(x′) =

∂xν

∂x′µAν(x) A′µ(x′) =

∂x′µ

∂xν Aν(x).

we have that

A′·B′ = A′µB′µ =

∂xν

∂x′µAν

∂x′µ

∂xα Bα =∂xν

∂x′µ∂x′µ

∂xα AνBα

=∂xν

∂xα AνBα = δ να AνBα

= AαBα = A·B,

where we have introduced theKronecker deltathrough

δ µν =

∂x′µ

∂x′ν=

∂xµ

∂xν =

{

1 µ = ν0 µ 6= ν

Eliminating indices by summing over them in tensor prod-ucts is calledcontraction. The scalar product has no tensorindices left and is said to befully contracted.


3.1.4 Rank-2 Tensors

We may distinguish three kinds of rank-2 tensors according to thetransformation laws

T ′µν =

∂xα

∂x′µ∂xβ

∂x′νTαβ

T ′νµ =

∂xα

∂x′µ∂x′ν

∂xβ Tβα

T ′µν=

∂x′µ

∂xα∂x′ν

∂xβ Tαβ

This may easily be generalized to tensors of any rank.

• Covariant Tensors:carry only lower indices

• Contravariant Tensors:carry only upper indices

• Mixed Tensors:carry both upper and lower indices

EXAMPLE: the Kronecker deltaδ νµ is a mixed tensor of rank 2.

Memory aid:

• Each upper indexµ on left side requires right-side“factor” of form ∂x′µ/∂xν (prime in numerator).

• Each lower indexν on left side requires right-side“factor” of form ∂xµ/∂x′ν (prime in denominator).

• “Position of index = position of primed coordinate”


NOTE:Not all quantities carrying indices are tensors;it isthe transformation laws that define the tensors.

NOTE: We employ a standard shorthand by using “a ten-sorTµν ” to mean “a tensor with componentsTµν ”.


3.1.5 Metric Tensor

A rank-2 tensor of particular importance is the metric tensor gµν be-cause it is associated with the line element

ds2 = gµνdxµdxν .

It is symmetric (gµν = gνµ) and satisfies the usual rank-2 tensor trans-formation law

g′µν =∂xα

∂x′µ∂xβ

∂x′νgαβ

The contravariant metric tensorgµν is defined by the requirement

gµαgαν = δ νµ .

(Thus,gµν andgµν are matrix inverses.)

We may use contraction with the metric tensor to raise andlower tensor indices; for example

Aµ = gµνAν Aµ = gµνAν .

Thus, the scalar product of vectors may also be expressedas

A·B = gµνAµBν ≡ AνBν .


3.1.6 Antisymmetric 4th-Rank Tensor

A rank-4 tensorεαβγδ of particular importance may beintroduced by the requirement that

• ε0123= 1

• εαβγδ be completely antisymmetric in the exchangeof any two indices (thus it must vanish if any twoindices are the same).

This tensor is commonly called thecompletely antisym-metric 4th-rank tensoror theLevi–Civita symbol.


3.2 Symmetric and Antisymmetric Tensors

The symmetry of tensors under exchanging pairs of indices isoftenimportant.

• An arbitrary rank-2 tensor can always be decomposed into asymmetric and antisymmetric part:

Tαβ = 12(Tαβ +Tβα)+ 1

2(Tαβ −Tβα),

where the first term is clearly symmetric and the second termantisymmetric under exchange of indices.

• For completely symmetric and completely antisymmetric rank-2tensors we have

Tαβ = ±Tβα Tαβ = ±Tβα T βα = ±T β

α ,

where the plus sign holds if the tensor is symmetric and the mi-nus sign if it is antisymmetric.

• More generally, we say that a tensor of rank two or higher issymmetricin any two of its indices if exchanging those indicesleaves the tensor invariant andantisymmetric(sometimes termedskew-symmetric) in any two indices if it changes sign upon switch-ing those indices.

3.2. SYMMETRIC AND ANTISYMMETRIC TENSORS 57

• It is common to denote symmetrizing and antisymmetrizing op-erations on tensor indices by a bracket notation in which() in-dicates symmetrization and[ ] indicates antisymmetrization overindices included in the brackets.

• For example, symmetrization over all indices for a rank-N co-variant tensorTα ,β ,...ω corresponds to

T(α ,β ,...ω) ≡1

N!(Sum over permutations on indicesα,β , . . .ω)

and antisymmetrization over all indices ofTα ,β ,...ω correspondsto

T[α ,β ,...ω ] ≡1N!

(±Sum over permutations on indicesα,β , . . .ω) ,

where the notation± indicates that the terms of the sum havea plus sign if they correspond to an even number of index ex-changes and a negative sign if they correspond to an odd numberof index exchanges.

• Thus, for rank-2 contravariant tensors we may write

T(αβ ) = 12(T

αβ +Tβα) T [αβ ] = 12(T

αβ −Tβα)


• In the more general case one may be interested in symmetrizingor antisymmetrizing over only a subset of indices for higher-ranktensors.

• If the indices are contiguous the above notation suffices withonly the indices to be symmetrized or antisymmetrized includedin the brackets.

• In the event that indices to be symmetrized or antisymmetrizedare not adjacent to each other, the preceding notation may beextended by using vertical brackets to exclude indices fromthesymmetrization or antisymmetrization.

Example: The expression

Tα [β |γ |δ ] =12(Tαβγδ −Tαδγβ )

corresponds to a rank-4 covariant tensor thathas been antisymmetrized in its second andfourth indices only.


3.2.1 Invariant Integration

Change of volume elements for spacetime integration:

d4x = det

(∂x∂x′

)

d4x′

wheredet(∂ (x)/∂ (x′)) is the Jacobian determinant of the transforma-tion between the coordinates.

Example: For(x,y) ↔ (u,v) the Jacobian is the determinant of(

∂x/∂u ∂x/∂v

∂y/∂u ∂y/∂v

)

or its inverse.

The metric tensor transforms as

g′µν =∂xα

∂x′µ∂xβ

∂x′νgαβ (Triple matrix product)

Therefore, since:

determinant of a product= product of determinants

the determinant of the metric tensorg≡ detgµν transforms as

g′ = det

(∂x∂x′

)

det

(∂x∂x′

)

g → det

(∂x∂x′

)

=

√

|g′|√

|g|

which gives when inserted into the first equation√

|g|d4x =√

|g′|d4x′,

(|g| becauseg can be negative in 4-D spacetime).


This is a scalar expression and thus its value is indepen-dent of coordinates. It follows that an integral of the form

I =∫

ϕ(x)√

|g|d4x

with fixed boundaries is a scalar ifϕ(x) is a scalar.

In integrals we shall employ

dV =√

|g|d4x

as an invariant volume element.


Example: The metric for a 2-dimensional spherical sur-face (2-sphere) is specified by the line element

ds2 = gi j dxidxj ,

which is explicitly in spherical coordinates

dℓ2 = R2dθ2+R2sin2θdϕ2.

This may be written as the matrix equation

dℓ2 = (dθ dϕ)

(

R2 0

0 R2sin2θ

)

︸︷︷︸gi j

(

dθdϕ

)

.

The area of the 2-sphere may then be expressed as the“invariant volume integration”

A=∫√

|g|d2x =∫ 2π

0dϕ∫ π

0

√

detgi j dθ

=∫ 2π

0dϕ∫ π

0R2sinθdθ = 4πR2.

where the metric tensorgi j is the2×2 matrix in the pre-ceding equation for the line element.

In this 2-dimensional example the sign of thedeterminant is positive, so no absolute valueis required under the radical.


3.2.2 Covariant Derivatives

Let us now consider the derivatives of tensor quantities. First intro-duce two common compact notations for partial derivatives

ϕ,µ ≡∂ϕ(x)∂xµ ∂µϕ ≡

∂ϕ(x)∂xµ

The derivative of a scalar is a covariant vector and scalars and theirderivatives are well-defined tensors. But, for the derivative of a co-variant vector, by using the rule for the derivative of a product

A′µ ,ν ≡

∂A′µ

∂x′ν=

∂∂x′ν

(

Aα∂xα

∂x′µ

)

︸︷︷︸

A′µ

=∂Aα∂x′ν

∂xα

∂x′µ+Aα

∂ 2xα

∂x′ν∂x′µ

=∂Aα

∂xβ∂xβ

∂x′ν︸︷︷︸

chain rule

∂xα

∂x′µ+Aα

∂ 2xα

∂x′ν∂x′µ

= Aα ,β∂xβ

∂x′ν∂xα

∂x′µ︸︷︷︸

Tensor

+ Aα∂ 2xα

∂x′ν∂x′µ︸︷︷︸

Not a tensor

In curved spacetime it is not possible to transform awaythe second term globally:Partial differentiation of tensorsis NOT a covariant operation in curved spacetime (for ten-sors of rank 1 or higher) .


But if we introduce theChristoffel symbolsΓλαβ and require that they

obey(not a tensor—see the 2nd derivatives!)

Γ′λαβ = Γκ

µν∂xµ

∂x′α∂xν

∂x′β∂x′λ

∂xκ +∂ 2xµ

∂x′α∂x′β∂x′λ

∂xµ ,

we may show that (Exercise),

(

A′µ ,ν −Γ′λ

µνA′λ

)

︸︷︷︸

≡B′µν

=(

Aα ,β −Γκαβ Aκ

)

︸︷︷︸

≡Bαβ

∂xα

∂x′µ∂xβ

∂x′ν.

For the quantity in parentheses this is the transformation law for arank-2 covariant tensor:

B′µν = Bαβ

∂xα

∂x′µ∂xβ

∂x′ν.

This suggests that we define thecovariant derivativeof a vector as

Aµ ;ν ≡

tensor︷︸︸︷

Aµ ,ν︸︷︷︸

not tensor

− ΓλµνAλ︸︷︷︸

not tensorwhere a subscript comma denotes ordinary partial differentiation anda subscript semicolon denotes covariant differentiation with respectto the variables following it.

(We shall also employ the notationDµ to denote an operator that takesthe covariant derivative with respect toxµ .)

The covariant derivative of a covariant vector then trans-forms as a covariant tensor of rank 2, even though neitherof its terms is a tensor.


Likewise, we can introduce the covariant derivative of acontravariant vector

Aλ;µ = Aλ

,µ +ΓλαµAα ,

and the covariant derivatives of the three possible rank-2tensors through

Aµν ;λ = Aµν ,λ −Γαµλ Aαν −Γα

νλ Aµα

Aµν ;κ = Aµ

ν ,κ +ΓµακAα

ν −ΓανκAµ

α

Aµν;κ = Aµν

,κ +ΓµακAαν +Γν

ακAµα

(which are rank-3 tensors), and so on.

Heuristic rule for constructing the covariant derivative ofa tensor having any rank:

• Form the ordinary partial derivative of the tensor

• Add one Christoffel symbol term having the sign andform for a covariant vector for each lower index ofthe tensor

• Add one Christoffel symbol term having the sign andform for a contravariant vector for each upper indexof the tensor


Most rules for partial differentiation carry over with suitable general-ization for covariant differentiation.

Example: Covariant derivative of product

(AµBν);λ = Aµ ;λ Bν +AµBν ;λ .

which is the usual (Leibniz rule) result.

The most important exception concerns the properties ofsuccessive covariant differentiations. Although partialderivative operators normally commute,covariant deriva-tive operators generally do not commute with each other.

One important consequence of covariant differentiation(Exercise):

Dαgµν = gµν ;α = 0,

Some implications:

• Raising and lowering index by contraction withgµνcommutes with covariant differentiation.

• This will allow in the Einstein field equations avac-uum energy term(accelerated expansion and dark en-ergy).


3.2.3 Invariant Equations

The properties of tensors elaborated above ensure thatanyequation will be invariant under general coordinate trans-formations provided that it equates tensors having thesame upper and lower indices.

EXAMPLES:

• If the quantitiesTµν andUµ

ν each transform as mixed rank-2 ten-sors andTµ

ν = Uµν in thex coordinate system, then in thex′ co-

ordinate systemT ′µν = U ′µ

ν .

• An equation that equates any tensor to zero is invariant undergeneral coordinate transformations.

• Equations such asTνµ = 10 or Tµ = Uµ generally are not valid

in all coordinate systems because they equate tensors of differentkinds.

chapter 3 covariance and tensor...

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