Schutz B. A first course in general relativity

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143 6.1 Differentiable manifolds and tensors

Differential structure

We shall really only consider ‘differentiable manifolds’. These are spaces that are con-

tinuous and differentiable. Roughly, this means that in the neighborhood of each point

in the manifold it is possible to deﬁne a smooth map to Euclidean space that preserves

derivatives of scalar functions at that point. The surface of a sphere is differentiable

everywhere. That of a cone is differentiable except at its apex. Nearly all manifolds

of use in physics are differentiable almost everywhere. The curved spacetimes of GR

certainly are.

The assumption of differentiability immediately means that we can deﬁne one-forms

and vectors. That is, in a certain coordinate system on the manifold, the members of the set

{φ

,α

}are the components of the one-form

dφ; and any set of the form {aφ

,α

+ bψ

,α

}, where

a and b are functions, is also a one-form ﬁeld. Similarly, every curve (with parameter, say,

λ) has a tangent vector



V deﬁned as the linear function that takes the one-form

dφ into the

derivative of φ along the curve, dφ/dλ:



dφ,



V=



dφ) =∇



φ = dφ/dλ. (6.1)

Any linear combination of vectors is also a vector. Using the vectors and one-forms so

deﬁned, we can build up the whole set of tensors of type





, just as we did in SR. Since

we have not yet picked out any





tensor to serve as the metric, there is not yet any

correspondence between forms and vectors. Everything else, however, is exactly as we had

in SR and in polar coordinates. All of this comes only from differentiability, so the set of

all tensors is said to be part of the ‘differential structure’ of the manifold. We will not have

much occasion to use that term.

Review

It is useful here to review the fundamentals of tensor algebra. We can summarize the

following rules.

(1) A tensor ﬁeld deﬁnes a tensor at every point.

(2) Vectors and one-forms are linear operators on each other, producing real numbers. The

linearity means:

˜p, a



V + b



W=a˜p,



V+b˜p,



W,

a ˜p +b˜q,



V=a˜p,



V+b˜q,



V,

where a and b are any scalar ﬁelds.

(3) Tensors are similarly linear operators on one-forms and vectors, producing real

numbers.

(4) If two tensors of the same type have equal components in a given basis, they have

equal components in all bases and are said to be identical (or equal, or the same). Only

tensors of the same type can be equal. In particular, if a tensor’s components are all

zero in one basis, they are zero in all, and the tensor is said to be zero.

144 Curved manifolds

(5) A number of manipulations of components of tensor ﬁelds are called ‘permissible

tensor operations’ because they produce components of new tensors:

(i) Multiplication by a scalar ﬁeld produces components of a new tensor of the

same type.

(ii) Addition of components of two tensors of the same type gives components of a

new tensor of the same type. (In particular, only tensors of the same type can be

added.)

(iii) Multiplication of components of two tensors of arbitrary type gives components

of a new tensor of the sum of the types, the outer product of the two tensors.

(iv) Covariant differentiation (to be discussed later) of the components of a tensor of

type





gives components of a tensor of type



M+1



(v) Contraction on a pair of indices of the components of a tensor of type





pro-

duces components of a tensor of type



N−1

M−1



. (Contraction is only deﬁned between

an upper and lower index.)

(6) If an equation is formed using components of tensors combined only by the permissible

tensor operations, and if the equation is true in one basis, then it is true in any other.

This is a very useful result. It comes from the fact that the equation (from (5) above) is

simply an equality between components of two tensors of the same type, which (from

(4)) is then true in any system.

6.2 Riemannian manifolds

So far we have not introduced a metric on to the manifold. Indeed, on certain manifolds a

metric would be unnecessary or inconvenient for whichever problem is being considered.

But in our case the metric is absolutely fundamental, since it will carry the information

about the rates at which clocks run and the distances between points, just as it does in

SR. A differentiable manifold on which a symmetric





tensor ﬁeld g has been singled

out to act as the metric at each point is called a Riemannian manifold. (Strictly speak-

ing, only if the metric is positive-deﬁnite – that is, g(



V) > 0 for all



V = 0 – is it

called Riemannian; indeﬁnite metrics, like SR and GR, are called pseudo-Riemannian.

This is a distinction that we won’t bother to make.) It is important to understand that

in picking out a metric we ‘add’ structure to the manifold; we shall see that the met-

ric completely deﬁnes the curvature of the manifold. Thus, by our choosing one metric

g the manifold gets a certain curvature (perhaps that of a sphere), while a different g



would give it a different curvature (perhaps an ellipsoid of revolution). The differen-

tiable manifold itself is ‘primitive’: an amorphous collection of points, arranged locally

like the points of Euclidean space, but not having any distance relation or shape spec-

iﬁed. Giving the metric g gives it a speciﬁc shape, as we shall see. From now on we

shall study Riemannian manifolds, on which a metric g is assumed to be deﬁned at

every point.

(For completeness we should remark that it is in fact possible to deﬁne the notion of

curvature on a manifold without introducing a metric (so-called ‘afﬁne’ manifolds). Some

145 6.2 Riemannian manifolds

texts actually approach the subject this way. But since the metric is essential in GR, we

shall simply study those manifolds whose curvature is deﬁned by a metric.)

The metric and local flatness

The metric, of course, provides a mapping between vectors and one-forms at every point.

Thus, given a vector ﬁeld



V(P) (which notation means that



V depends on the posi-

tion P, where P is any point), there is a unique one-form ﬁeld

V(P) = g(



V(P), ).

The mapping must be invertible, so that associated with

V(P) there is a unique



V(P).

The components of g are called g

αβ

; the components of the inverse matrix are called

αβ

. The metric permits raising and lowering of indices in the same way as in SR,

which means

= g

αβ

In general, {g

αβ

} will be complicated functions of position, so it will not be true that there

would be a simple relation between, say, V

and V

in an arbitrary coordinate system.

Since we wish to study general curved manifolds, we have to allow any coordinate

system. In SR we only studied Lorentz (inertial) frames because they were simple. But

because gravity prevents such frames from being global, we shall have to allow all

coordinates, and hence all coordinate transformations, that are nonsingular. (Nonsingu-

lar means, as in § 5.2, that the matrix of the transformation, 



≡ ∂x



/∂x

, has an

inverse.) Now, the matrix (g

αβ

) is a symmetric matrix by deﬁnition. It is a well-known

theorem of matrix algebra (see Exer. 3, § 6.9) that a transformation matrix can always

be found that will make any symmetric matrix into a diagonal matrix with each entry

on the main diagonal either +1, −1, or zero. The number of +1 entries equals the

number of positive eigenvalues of (g

αβ

), while the number of −1 entries is the number

of negative eigenvalues. So if we choose g originally to have three positive eigenval-

ues and one negative, then we can always ﬁnd a 



to make the metric components

become



) =

⎛

⎜

⎝

−1000

0100

0010

0001

⎞

⎟

⎠

≡ (η

αβ

). (6.2)

From now on we will use η

αβ

to denote only thematrixinEq.(6.2), which is of course the

metric of SR.

There are two remarks that must be made here. The ﬁrst is that Eq. (6.2) is only possi-

ble if we choose (g

αβ

) from among the matrices that have three positive and one negative

eigenvalues. The sum of the diagonal elements in Eq. (6.2) is called the signature of the

metric. For SR and GR it is +2. Thus, the fact that we have previously deduced from phys-

ical arguments that we can always construct a local inertial frame at any event, ﬁnds its

mathematical representation in Eq. (6.2), that the metric can be transformed into η

αβ

146 Curved manifolds

that point. This in turn implies that the metric has to have signature +2 if it is to describe

a spacetime with gravity.

The second remark is that the matrix 



that produces Eq. (6.2) at every point may

not be a coordinate transformation. That is, the set {˜ω



= 



} may not be a coor-

dinate basis. By our earlier discussion of noncoordinate bases, it would be a coordinate

transformation only if Eq. (5.95) holds:

∂



∂x

∂



∂x

In a general gravitational ﬁeld this will be impossible, because otherwise it would imply

the existence of coordinates for which Eq. (6.2) is true everywhere: a global Lorentz frame.

However, having found a basis at a particular point P for which Eq. (6.2) is true, it is

possible to ﬁnd coordinates such that, in the neighborhood of P,Eq.(6.2) is ‘nearly’ true.

This is embodied in the following theorem, whose (rather long) proof is at the end of this

section. Choose any point P of the manifold. A coordinate system {x

}can be found whose

origin is at P and in which:

αβ

) = η

αβ

+ 0[(x

)

]. (6.3)

That is, the metric near P is approximately that of SR, differences being of second order in

the coordinates. From now on we shall refer to such coordinate systems as ‘local Lorentz

frames’ or ‘local inertial frames’. Eq. (6.3) can be rephrased in a somewhat more precise

way as:

αβ

(P) = η

αβ

for all α, β; (6.4)

∂

∂x

αβ

(P) = 0 for all α, β, γ ; (6.5)

but generally

∂

∂x

αβ

(P) = 0

for at least some values of α, β, γ , and μ if the manifold is not exactly ﬂat.

The existence of local Lorentz frames is merely the statement that any curved space has

a ﬂat space ‘tangent’ to it at any point. Recall that straight lines in ﬂat spacetime are the

world lines of free particles; the absence of ﬁrst-derivative terms (Eq. (6.5)) in the metric

of a curved spacetime will mean that free particles are moving on lines that are locally

straight in this coordinate system. This makes such coordinates very useful for us, since the

equations of physics will be nearly as simple in them as in ﬂat spacetime, and if constructed

by the rules of § 6.1 will be valid in any coordinate system. The proof of this theorem is at

the end of this section, and is worth studying.

147 6.2 Riemannian manifolds

Lengths and volumes

The metric of course gives a way to deﬁne lengths of curves. Let dx be a small vector

displacement on some curve. Then dx has squared length ds

= g

αβ

. (Recall that

we call this the line element of the metric.) If we take the absolute value of this and take its

square root, we get a measure of length: dl ≡|g

αβ

1/2

. Then integrating it gives

l =

along

curve

αβ

1/2

(6.6)

αβ

dλ

1/2

dλ, (6.7)

where λ is the parameter of the curve (whose endpoints are λ

and λ

). But since the

tangent vector



V has components V

= dx

/dλ, we ﬁnally have:

l =



V ·



1/2

dλ (6.8)

as the length of the arbitrary curve.

The computation of volumes is very important for integration in spacetime. Here, we

mean by ‘volume’ the four-dimensional volume element we used for integrations in Gauss’

law in § 4.4. Let us go to a local Lorentz frame, where we know that a small four-

dimensional region has four-volume dx

, where {x

} are the coordinates which

at this point give the nearly Lorentz metric, Eq. (6.3). In any other coordinate system {x



}

it is a well-known result of the calculus of several variables that:

∂(x

, x

)

∂(x



, x



, x



, x



)



, (6.9)

where the factor ∂()/∂( ) is the Jacobian of the transformation from {x



} to {x

},as

deﬁned in § 5.2:

∂(x

, x

)

∂(x



, x



, x



, x



)

= det

⎛

⎜

⎝

∂x

/∂x



∂x

/∂x



···

∂x

/∂x



⎞

⎟

⎠

= det(



). (6.10)

This would be a rather tedious way to calculate the Jacobian, but there is an easier way

using the metric. In matrix terminology, the transformation of the metric components is

(g) = ()(η)()

, (6.11)

where (g) is the matrix of g

αβ

,(η)ofη

αβ

, etc., and where ‘T’ denotes transpose. It follows

that the determinants satisfy

det (g) = det () det (η) det (

). (6.12)

148 Curved manifolds

But for any matrix

det () = det (

), (6.13)

and we can easily see from Eq. (6.2) that

det (η) =−1. (6.14)

Therefore, we get

det (g) =−[det ()]

. (6.15)

Now we introduce the notation

g := det (g



), (6.16)

which enables us to conclude from Eq. (6.15) that

det (



) = (−g)

1/2

. (6.17)

Thus, from Eq. (6.9) we get

= [−det (g



)]

1/2



= (−g)

1/2



. (6.18)

This is a very useful result. It is also conceptually an important result because it is the ﬁrst

example of a kind of argument we will frequently employ, an argument that uses locally

ﬂat coordinates to generalize our ﬂat-space concepts to analogous ones in curved space. In

this case we began with dx

= d

x in a locally ﬂat coordinate system. We argue

that this volume element at P must be the volume physically measured by rods and clocks,

since the space is the same as Minkowski space in this small region. We then ﬁnd that the

value of this expression in arbitrary coordinates {x



} is Eq. (6.18), (−g)

1/2



, which is

thus the expression for the true volume in a curved space at any point in any coordinates.

We call this the proper volume element.

It should not be surprising that the metric comes into it, of course, since the metric

measures lengths. We only need remember that in any coordinates the square root of the

negative of the determinant of (g

αβ

) is the thing to multiply by d

x to get the true, or proper,

volume element.

Perhaps it would be helpful to quote an example from three dimensions. Here proper

volume is (g)

1/2

, since the metric is positive-deﬁnite (Eq. (6.14) would have a + sign). In

spherical coordinates the line element is dl

= dr

+ r

dθ

+ r

sin

θ dφ

, so the metric is

) =

⎛

⎝

10 0

0 r

00r

sin

⎞

⎠

. (6.19)

Its determinant is r

sin

θ,so(g)

1/2



sin θ dr dθ dφ, (6.20)

which we know is the correct volume element in these coordinates.

149 6.2 Riemannian manifolds

Proof of the local-flatness theorem

Let {x

} be an arbitrary given coordinate system and {x



} the one which is desired: it

reduces to the inertial system at a certain ﬁxed point P. (A point in this four-dimensional

manifold is, of course, an event.) Then there is some relation

= x



), (6.21)





= ∂x

/∂x



. (6.22)

Expanding 



in a Taylor series about P (whose coordinates are x



)givesthe

transformation at an arbitrary point x near P:





(x) = 



(P) +(x



− x



)

∂



∂x



(P)



− x



)(x



− x



)

∂





∂x



∂x



(P) +···,

= 



+ (x



− x



)

∂

∂x



∂x



− x



)(x



− x



)

∂

∂x



∂x



∂x



+···. (6.23)

Expanding the metric in the same way gives

αβ

(x) =g

αβ

+ (x



− x



)

∂g

αβ

∂x



− x



)(x



− x



)

∂

αβ

∂x



∂x



+···. (6.24)

We put these into the transformation,



= 







αβ

, (6.25)

to obtain



(x) =







αβ

+ (x



− x



)[







αβ,γ



+ 



αβ

∂

/∂x



∂x



+ 



αβ

∂

/∂x



∂x



]



− x



)(x



− x



)[···]. (6.26)

Now, we do not know the transformation, Eq. (6.21), but we can deﬁne it by its Taylor

expansion. Let us count the number of free variables we have for this purpose. The matrix





has 16 numbers, all of which are freely speciﬁable. The array {∂

/∂x



∂x



}

has 4 ×10 = 40 free numbers (not 4 ×4 ×4, since it is symmetric in γ



and μ



The array {∂

/∂x



∂x



∂x



} has 4 ×20 = 80 free variables, since symmetry on all

150 Curved manifolds

rearrangements of λ



, γ



and μ



gives only 20 independent arrangements (the general

expression for three indices is n(n +1)(n +2)/3!, where n is the number of values each

index can take, four in our case). On the other hand, g

αβ

, g

αβ,γ



and g

αβ,γ



are all

given initially. They have, respectively, 10, 10 × 4 = 40, and 10 ×10 = 100 independent

numbers for a fully general metric. The ﬁrst question is, can we satisfy Eq. (6.4),



= η



? (6.27)

This can be written as



= 







αβ

. (6.28)

By symmetry, these are ten equations, which for general matrices are independent. To

satisfy them we have 16 free values in 



. The equations can indeed, therefore, be sat-

isﬁed, leaving six elements of 



unspeciﬁed. These six correspond to the six degrees

of freedom in the Lorentz transformations that preserve the form of the metric η



. That

is, we can boost by a velocity v (three free parameters) or rotate by an angle θ around a

direction deﬁned by two other angles. These add up to six degrees of freedom in 



that leave the local inertial frame inertial.

The next question is, can we choose the 40 free numbers ∂



/∂x



in Eq. (6.26)in

such a way as to satisfy the 40 independent equations, Eq. (6.5),



,μ



= 0? (6.29)

Since 40 equals 40, the answer is yes, just barely. Given the matrix 



, there is one

and only one way to arrange the coordinates near P such that 



,γ



has the right values

to make g



,μ



= 0. So there is no extra freedom other than that with which to make

local Lorentz transformations.

The ﬁnal question is, can we make this work at higher order? Can we ﬁnd 80 numbers





,γ



which can make the 100 numbers g



,μ



= 0? The answer, since 80 <

100, is no. There are, in the general metric, 20 ‘degrees of freedom’ among the second

derivatives g



,μ



. Since 100 − 80 = 20, there will be in general 20 components that

cannot be made to vanish.

Therefore we see that a general metric is characterized at any point P not so much by its

value at P (which can always be made to be η

αβ

), nor by its ﬁrst derivatives there (which

can be made zero), but by the 20 second derivatives there which in general cannot be made

to vanish. These 20 numbers will be seen to be the independent components of a tensor

which represents the curvature; this we shall show later. In a ﬂat space, of course, all 20

vanish. In a general space they do not.

6.3 Covariant differentiation

We now look at the subject of differentiation. By deﬁnition, the derivative of a vector ﬁeld

involves the difference between vectors at two different points (in the limit as the points

come together). In a curved space the notion of the difference between vectors at different

151 6.3 Covariant differentiation

points must be handled with care, since in between the points the space is curved and the

idea that vectors at the two points might point in the ‘same’ direction is fuzzy. However, the

local ﬂatness of the Riemannian manifold helps us out. We only need to compare vectors

in the limit as they get inﬁnitesimally close together, and we know that we can construct

a coordinate system at any point which is as close to being ﬂat as we would like in this

same limit. So in a small region the manifold looks ﬂat, and it is then natural to say that the

derivative of a vector whose components are constant in this coordinate system is zero at

that point. In particular, we say that the derivatives of the basis vectors of a locally inertial

coordinate system are zero at P.

Let us emphasize that this is a deﬁnition of the covariant derivative. For us, its justiﬁca-

tion is in the physics: the local inertial frame is a frame in which everything is locally like

SR, and in SR the derivatives of these basis vectors are zero. This deﬁnition immediately

leads to the fact that in these coordinates at this point, the covariant derivative of a vector

has components given by the partial derivatives of the components (that is, the Christoffel

symbols vanish):

:β

= V

.β

at P in this frame. (6.30)

This is of course also true for any other tensor, including the metric:

αβ;γ

= g

αβ,γ

= 0atP.

(The second equality is just Eq. (6.5).) Now, the equation g

αβ;γ

= 0istrueinoneframe

(the locally inertial one), and is a valid tensor equation; therefore it is true in any basis:

αβ:γ

= 0 in any basis. (6.31)

This is a very important result, and comes directly from our deﬁnition of the covariant

derivative. Recalling § 5.4, we see that if we have 

αβ

= 

βα

, then Eq. (6.31) leads to

Eq. (5.75)forany metric:



μν

αβ

βμ,ν

+ g

βν,μ

− g

μν,β

). (6.32)

It is left to Exer. 5, § 6.9, to demonstrate, by repeating the ﬂat-space argument now in

the locally inertial frame, that 

βα

is indeed symmetric in any coordinate system, so that

Eq. (6.32) is correct in any coordinates. We assumed at the start that at P in a locally inertial

frame, 

μν

= 0. But, importantly, the derivatives of 

μν

at P in this frame are not all

zero generally, since they involve g

αβ,γμ

. This means that even though coordinates can be

found in which 

μν

= 0 at a point, these symbols do not generally vanish elsewhere. This

differs from ﬂat space, where a coordinate system exists in which 

μν

= 0 everywhere.

So we can see that at any given point, the difference between a general manifold and a ﬂat

one manifests itself in the derivatives of the Christoffel symbols.

152 Curved manifolds

Eq. (6.32) means that, given g

αβ

, we can calculate 

μν

everywhere. We can therefore

calculate all covariant derivatives, given g. To review the formulas:

;β

= V

,β

+ 

μβ

, (6.33)

α;β

= P

α,β

− 

αβ

, (6.34)

αβ

;γ

= T

αβ

,γ

+ 

μγ

μβ

+ 

μγ

αμ

. (6.35)

Divergence formula

Quite often we deal with the divergence of vectors. Given an arbitrary vector ﬁeld V

, its

divergence is deﬁned by Eq. (5.53),

;α

= V

,α

+ 

μα

. (6.36)

This formula involves a sum in the Christoffel symbol, which, from Eq. (6.32), is



μα

αβ

βμ,α

+ g

βα,μ

− g

μα,β

)

αβ

βμ,α

− g

μα,β

) +

αβ

αβ,μ

. (6.37)

This has had its terms rearranged to simplify it: notice that the term in parentheses is

antisymmetric in α and β, while it is contracted on α and β with g

αβ

, which is symmetric.

The ﬁrst term therefore vanishes (see Exer. 26(a), § 3.10) and we ﬁnd



μα

αβ

αβ,μ

. (6.38)

Since (g

αβ

)istheinversematrixof(g

αβ

), it can be shown (see Exer. 7, § 6.9) that the

derivative of the determinant g of the matrix (g

αβ

)is

,μ

= gg

αβ

βα,μ

. (6.39)

UsingthisinEq.(6.38), we ﬁnd



μα

= (

√

− g)

,μ

√

− g. (6.40)

Then we can write the divergence, Eq. (6.36), as

;α

= V

,α

√

− g

(

√

− g)

,α

(6.41)