Schutz B. A first course in general relativity

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133 5.4 Christoffel symbols and the metric

So it works, almost magically. But it is important to realize that it is not magic: it fol-

lows directly from the facts that g

αβ,μ

= 0 in Cartesian coordinates and that g

αβ;μ

are the

components of the same tensor ∇g in arbitrary coordinates.

Perhaps it is useful to pause here to get some perspective on what we have just done.

We introduced covariant differentiation in arbitrary coordinates by using our understand-

ing of parallelism in Euclidean space. We then showed that the metric of Euclidean

space is covariantly constant: Eq. (5.71). When we go on to curved (Riemannian)

spaces we will have to discuss parallelism much more carefully, but Eq. (5.71) will

still be true, and therefore so will all its consequences, such as those we now go on to

describe.

Calculating the Christoffel symbols from the metric

The vanishing of Eq. (5.72) leads to an extremely important result. We see that Eq. (5.72)

can be used to determine g

αβ,μ

in terms of 

αβ

. It turns out that the reverse is also true,

that 

αβ

can be expressed in terms of g

αβ,μ

. This gives an easy way to derive the Christof-

fel symbols. To show this we ﬁrst prove a result of some importance in its own right: in

any coordinate system 

αβ

≡ 

βα

. To prove this symmetry consider an arbitrary scalar

ﬁeld φ. Its ﬁrst derivative ∇φ is a one-form with components φ

,β

. Its second covariant

derivative ∇∇φ has components φ

,β;α

and is a





tensor. In Cartesian coordinates these

components are

,β,α

∂

∂x

∂

∂x

and we see that they are symmetric in α and β, since partial derivatives commute. But if a

tensor is symmetric in one basis it is symmetric in all bases. Therefore

,β;α

= φ

,α;β

(5.73)

in any basis. Using the deﬁnition, Eq. (5.63)gives

,β,α

− φ

,μ



βα

= φ

,α,β

− φ

,μ



αβ

in any coordinate system. But again we have

,α,β

= φ

,β,α

in any coordinates, which leaves



αβ

,μ

= 

βα

,μ

for arbitrary φ. This proves the assertion



αβ

= 

βα

in any coordinate system. (5.74)

134 Prefacetocurvature

We use this to invert Eq. (5.72) by some advanced index gymnastics. We write three

versions of Eq. (5.72) with different permutations of indices:

αβ,μ

= 

αμ

νβ

+ 

βμ

αν

αμ,β

= 

αβ

νμ

+ 

μβ

αν

−g

βμ,α

=−

βα

νμ

− 

μα

βν

We add these up and group terms, using the symmetry of g, g

βν

= g

νβ

αβ,μ

+ g

αμ,β

− g

βμ,α

= (

αμ

− 

μα

νβ

+ (

αβ

− 

βα

νμ

+ (

βμ

+ 

μβ

αν

In this equation the ﬁrst two terms on the right vanish by the symmetry of ,Eq.(5.74),

and we get

αβ,μ

+ g

αμ,β

− g

βμ,α

= 2g

αν



βμ

We are almost there. Dividing by 2, multiplying by g

αγ

(with summation implied on α)

and using

αγ

αν

≡ δ

gives

αγ

αβ,μ

+ g

αμ,β

− g

βμ,α

) = 

βμ

. (5.75)

This is the expression of the Christoffel symbols in terms of the partial derivatives of the

components of g. In polar coordinates, for example,



rθ

αθ

αr,θ

+ g

αθ,r

− g

rθ ,α

Since g

rθ

= 0 and g

θθ

= r

−2

,wehave



rθ

θr,θ

+ g

θθ,r

− g

rθ ,θ

)

θθ,r

), r =

This is the same value for 

rθ

, as we derived earlier. This method of computing 

βμ

so useful that it is well worth committing Eq. (5.75) to memory. It will be exactly the same

in curved spaces.

The tensorial nature of 

βμ

Since e

is a vector, ∇e

is a





tensor whose components are 

αβ

.Hereα is ﬁxed and

μ and β are the component indices: changing α changes the tensor ∇e

, while changing

μ or β changes only the component under discussion. So it is possible to regard μ and

135 5.5 Noncoordinate bases

β as component indices and α as a label giving the particular tensor referred to. There is

one such tensor for each basis vector e

. However, this is not terribly useful, since under

a change of coordinates the basis changes and the important quantities in the new sys-

tem are the new tensors ∇e



which are obtained from the old ones ∇e

in a complicated

way: they are different tensors, not just different components of the same tensor. So the set



αβ

in one frame is not obtained by a simple tensor transformation from the set 



of another frame. The easiest example of this is Cartesian coordinates, where 

βμ

≡ 0,

while they are not zero in other frames. So in many books it is said that 

αβ

are not

components of tensors. As we have seen, this is not strictly true: 

αβ

are the (μ, β) com-

ponents of a set of





tensors ∇e

. But there is no single





tensor whose components

are 

αβ

, so expressions like 

αβ

are not components of a single tensor, either. The

combination

,α

+ V



μα

is a component of a single tensor ∇



5.5 Noncoordinate bases

In this whole discussion we have generally assumed that the non-Cartesian basis vectors

were generated by a coordinate transformation from (x, y)tosome(ξ , η). However, as we

shall show below, not every ﬁeld of basis vectors can be obtained in this way, and we

shall have to look carefully at our results to see which need modiﬁcation (few actually do).

We will almost never use non-coordinate bases in our work in this course, but they are

frequently encountered in the standard references on curved coordinates in ﬂat space, so

we should pause to take a brief look at them now.

Polar coordinate basis

The basis vectors for our polar coordinate system were deﬁned by

e



= 



e

where primed indices refer to polar coordinates and unprimed to Cartesian. Moreover,

we had





= ∂x

/∂x



where we regard the Cartesian coordinates {x

} as functions of the polar coordinates {x



We found that

e



·e



≡ g



= δ



i.e. that these basis vectors are not unit vectors.

136 Prefacetocurvature

Polar unit basis

Often it is convenient to work with unit vectors. A simple set of unit vectors derived from

the polar coordinate basis is:

e

ˆr

=e

, e

e

, (5.76)

with a corresponding unit one-form basis

˜ω

ˆr

dr, ˜ω

= r

dθ. (5.77)

The student should verify that

e

ˆα

·e

≡ g

ˆα

= δ

ˆα

˜ω

ˆα

·˜ω

≡ g

ˆα

= δ

ˆα



(5.78)

so these constitute orthonormal bases for the vectors and one-forms. Our notation, which

is fairly standard, is to use a ‘caret’ or ‘hat’, ˆ, above an index to denote an orthornormal

basis. Now, the question arises, do there exist coordinates (ξ , η) such that

e

ˆr

=e

∂x

∂ξ

e

∂y

∂ξ

e

(5.79a)

and

e

=e

∂x

∂η

e

∂y

∂η

e

? (5.79b)

If so, then {e

ˆr

, e

} are the basis for the coordinates (ξ, η) and so can be called a coordinate

basis; if such (ξ , η) can be shown not to exist, then these vectors are a noncoordinate basis.

The question is actually more easily answered if we look at the basis one-forms. Thus, we

seek (ξ , η) such that

˜ω

ˆr

dξ = ∂ξ/∂x

dx +∂ξ/∂y

dy,

˜ω

dη = ∂η/∂x

dx +∂η/∂y

dy.

⎫

⎬

⎭

(5.80)

Since we know ˜ω

ˆr

and ˜ω

in terms of

dr and

dθ, we have, from Eqs. (5.26) and (5.27),

˜ω

ˆr

dr = cos θ

dx +sin θ

dy,

˜ω

= r

dθ =−sin θ

dx +cos θ

dy.



(5.81)

(The orthonormality of ˜ω

ˆr

and ˜ω

are obvious here.) Thus if (ξ, η)exist,wehave

∂η

∂x

=−sin θ,

∂η

∂y

= cos θ . (5.82)

If this were true, then the mixed derivatives would be equal:

∂

∂y

∂η

∂x

∂

∂x

∂η

∂y

. (5.83)

This would imply

∂

∂y

(−sin θ) =

∂

∂x

(cos θ) (5.84)

137 5.5 Noncoordinate bases

∂

∂y



√

+ y

)



∂

∂x



√

+ y

)



= 0.

This is certainly not true. Therefore ξ and η do not exist: we have a noncoordinate basis.

(If this manner of proof is surprising, try it on

dr and

dθ themselves.)

In textbooks that deal with vector calculus in curvilinear coordinates, almost all use the

unit orthonormal basis rather than the coordinate basis. Thus, for polar coordinates, if a

vector has components in the coordinate basis PC,



V −→

(a, b) ={V



}, (5.85)

then it has components in the orthonormal basis PO



V −→

(a, rb) ={V

ˆα

}. (5.86)

So if, for example, the books calculate the divergence of the vector, they obtain, instead of

our Eq. (5.56),

∇·V =

∂

∂r

(rV

ˆr

) +

∂

∂θ

. (5.87)

The difference between Eqs. (5.56) and (5.87) is purely a matter of the basis for



General remarks on noncoordinate bases

The principal differences between coordinate and noncoordinate bases arise from the fol-

lowing. Consider an arbitrary scalar ﬁeld φ and the number

dφ(e

), where e

is a basis

vector of some arbitrary basis. We have used the notation

dφ(e

) = φ

,μ

. (5.88)

Now, if e

is a member of a coordinate basis, then

dφ(e

) = ∂φ/∂x

and we have, as

deﬁned in an earlier chapter,

,μ

∂φ

∂x

: coordinate basis. (5.89)

But if no coordinates exist for {e

}, then Eq. (5.89) must fail. For example, if we let

Eq. (5.88) deﬁne φ

, ˆμ

, then we have

∂φ

∂θ

. (5.90)

In general, we get

∇

ˆα

φ ≡ φ

,ˆα

= 

ˆα

∇

φ = 

ˆα

∂φ

∂x

(5.91)

for any coordinate system {x

} and noncoordinate basis {e

ˆα

}. It is thus convenient to con-

tinue with the notation, Eq. (5.88), and to make the rule that φ

,μ

= ∂φ/∂x

only in a

coordinate basis.

138 Prefacetocurvature

The Christoffel symbols may be deﬁned just as before

∇

e

ˆα

= 

ˆμ

ˆα

e

ˆμ

, (5.92)

but now

∇

= 

∂

∂x

, (5.93)

where {x

} is any coordinate system and {e

} any basis (coordinate or not). Now, however,

we cannot prove that 

ˆμ

ˆα

= 

ˆμ

ˆα

, since that proof used φ

,ˆα,

= φ

β,ˆα

, which was true

in a coordinate basis (partial derivatives commute) but is not true otherwise. Hence, also,

Eq. (5.75)for

αβ

in terms of g

αβ,γ

applies only in a coordinate basis. More general

expressions are worked out in Exer. 20, § 5.8.

What is the general reason for the nonexistence of coordinates for a basis? If {˜ω

¯α

} is a

coordinate one-form basis, then its relation to another one {

} is

˜ω

¯α

= 

¯α

∂x

¯α

∂x

. (5.94)

The key point is that 

¯α

, which is generally a function of position, must actually be the

partial derivative ∂x

¯α

/∂x

everywhere. Thus we have

∂

∂x



¯α

∂

¯α

∂x

∂

¯α

∂x

∂

∂x



¯α

. (5.95)

These ‘integrability conditions’ must be satisﬁed by all the elements 

¯α

in order for ˜ω

¯α

to be a coordinate basis. Clearly, we can always choose a transformation matrix for which

this fails, thereby generating a noncoordinate basis.

Noncoordinate bases in this book

We shall not have occasion to use such bases very often. Mainly, it is important to under-

stand that they exist, that not every basis is derivable from a coordinate system. The algebra

of coordinate bases is simpler in almost every respect. We may ask why the standard treat-

ments of curvilinear coordinates in vector calculus, then, stick to orthonormal bases. The

reason is that in such a basis in Euclidean space, the metric has components δ

αβ

,sothe

form of the dot product and the equality of vector and one-form components carry over

directly from Cartesian coordinates (which have the only orthonormal coordinate basis!).

In order to gain the simplicity of coordinate bases for vector and tensor calculus, we have

to spend time learning the difference between vectors and one-forms!

5.6 L o ok i ng a h ea d

The work we have done in this chapter has developed almost all the notation and concepts

we will need in our study of curved spaces and spacetimes. It is particularly important that

the student understands §§ 5.2–5.4 because the mathematics of curvature will be developed

139 5.8 Exercises

by analogy with the development here. What we have to add to all this is a discussion of

parallelism, of how to measure the extent to which the Euclidean parallelism axiom fails.

This measure is the famous Riemann tensor.

5.7 Further reading

The Eötvös and Pound–Rebka–Snider experiments, and other experimental fundamentals

underpinning GR, are discussed by Dicke (1964), Misner et al. (1973), Shapiro (1980),

and Will (1993, 2006). See Hoffmann (1983) for a less mathematical discussion of the

motivation for introducing curvature. For an up-to-date review of the GPS system’s use of

relativity, see Ashby (2003).

The mathematics of curvilinear coordinates is developed from a variety of points of view

in: Abraham and Marsden (1978), Lovelock and Rund (1990), and Schutz (1980b).

5.8 E xe rc is e s

1 Repeat the argument that led to Eq. (5.1) under more realistic assumptions: suppose a

fraction ε of the kinetic energy of the mass at the bottom can be converted into a photon

and sent back up, the remaining energy staying at ground level in a useful form. Devise

a perpetual motion engine if Eq. (5.1) is violated.

2 Explain why a uniform external gravitational ﬁeld would raise no tides on Earth.

3 (a) Show that the coordinate transformation (x, y) → (ξ , η) with ξ = x and η = 1

violates Eq. (5.6).

(b) Are the following coordinate transformations good ones? Compute the Jacobian and

list any points at which the transformations fail.

(i) ξ = (x

+ y

)

1/2

, η = arctan(y/x);

(ii) ξ = ln x, η = y;

(iii) ξ = arctan(y/x), η = (x

+ y

)

−1/2

4 A curve is deﬁned by {x = f (λ), y = g(λ), 0  λ  1}. Show that the tangent vector

(dx/dλ,dy/dλ) does actually lie tangent to the curve.

5 Sketch the following curves. Which have the same paths? Find also their tangent vectors

where the parameter equals zero.

(a) x = sin λ, y = cos λ;(b)x = cos(2π t

), y = sin(2πt

+ π); (c) x = s, y = s +4;

(d) x = s

, y =−(s −2)(s +2); (e) x = μ, y = 1.

6 Justify the pictures in Fig. 5.5.

7 Calculate all elements of the transformation matrices 



and 



for the trans-

formation from Cartesian (x, y) – the unprimed indices – to polar (r, θ )–theprimed

indices.

140 Prefacetocurvature

8 (a) (Uses the result of Exer. 7.) Let f = x

+ y

+ 2xy, and in Cartesian coordinates



V → (x

+ 3y, y

+ 3x),



W → (1, 1). Compute f as a function of r and θ, and ﬁnd

the components of



V and



W on the polar basis, expressing them as functions of r

and θ.

(b) Find the components of

df in Cartesian coordinates and obtain them in polars

(i) by direct calculation in polars, and (ii) by transforming components from

Cartesian.

one-forms

V and

W associated with



V and



W. (ii) Obtain the polar components of

V and

W by transformation of their Cartesian components.

9 Draw a diagram similar to Fig. 5.6 to explain Eq. (5.38).

10 Prove that ∇



V, deﬁned in Eq. (5.52), is a





tensor.

11 (Uses the result of Exers. 7 and 8.) For the vector ﬁeld



V whose Cartesian compo-

nents are (x

+ 3y, y

+ 3x), compute: (a) V

,β

in Cartesian; (b) the transformation









,β

to polars; (c) the components V



;ν



directly in polars using the

Christoffel symbols, Eq. (5.45), in Eq. (5.50); (d) the divergence V

,α

using your results

in (a); (e) the divergence V



;μ



using your results in either (b) or (c); (f) the divergence



;μ



using Eq. (5.56) directly.

12 For the one-form ﬁeld ˜p whose Cartesian components are (x

+ 3y, y

+ 3x), com-

pute: (a) p

α,β

in Cartesian; (b) the transformation 







α,β

to polars; (c) the

components p



;ν



directly in polars using the Christoffel symbols, Eq. (5.45), in

Eq. (5.63).

13 For those who have done both Exers. 11 and 12, show in polars that g



;ν



= p



;ν



14 For the tensor whose polar components are (A

= r

, A

rθ

= r sin θ , A

θr

= r cos θ ,

θθ

= tan θ ), compute Eq. (5.65) in polars for all possible indices.

15 For the vector whose polar components are (V

= 1, V

= 0), compute in polars

all components of the second covariant derivative V

;μ;ν

. (Hint: to ﬁnd the second

derivative, treat the ﬁrst derivative V

;μ

as any





tensor: Eq. (5.66).)

16 Fill in all the missing steps leading from Eq. (5.74)toEq.(5.75).

17 Discover how each expression V

,α

and V



μα

separately transforms under a change

of coordinates (for 

μα

, begin with Eq. (5.44)). Show that neither is the standard tensor

law, but that their sum does obey the standard law.

18 Verify Eq. (5.78).

19 Verify that the calculation from Eq. (5.81)toEq.(5.84), when repeated for

dr and

dθ,

shows them to be a coordinate basis.

20 For a noncoordinate basis {e

}, deﬁne ∇

e

−∇

e

:= c

μν

e

and use this in place

of Eq. (5.74) to generalize Eq. (5.75).

21 Consider the x −t plane of an inertial observer in SR. A certain uniformly accelerated

observer wishes to set up an orthonormal coordinate system. By Exer. 21, § 2.9,his

world line is

t(λ) = a sinh λ, x(λ) = a cosh λ, (5.96)

where a is a constant and aλ is his proper time (clock time on his wrist watch).

141 5.8 Exercises

(a) Show that the spacelike line described by Eq. (5.96) with a as the variable parameter

and λ ﬁxed is orthogonal to his world line where they intersect. Changing λ in

Eq. (5.96) then generates a family of such lines.

(b) Show that Eq. (5.96) deﬁnes a transformation from coordinates (t, x) to coordi-

nates (λ, a), which form an orthogonal coordinate system. Draw these coordinates

and show that they cover only one half of the original t − x plane. Show that the

coordinates are bad on the lines |x|=|t|, so they really cover two disjoint quadrants.

This observer will do a perfectly good job, provided that he always uses Christoffel

symbols appropriately and sticks to events in his quadrant. In this sense, SR admits

accelerated observers. The right-hand quadrant in these coordinates is sometimes

called Rindler space, and the boundary lines x =±t bear some resemblance to the

black-hole horizons we will study later.

22 Show that if U

∇

= W

, then U

∇

= W

Curved manifolds

6.1 Differentiable manifolds and tensors

The mathematical concept of a curved space begins (but does not end) with the idea of a

manifold. A manifold is essentially a continuous space which looks locally like Euclidean

space. To the concept of a manifold is added the idea of curvature itself. The introduction

of curvature into a manifold will be the subject of subsequent sections. First we study the

idea of a manifold, which we can regard as just a fancy word for ‘space’.

Manifolds

The surface of a sphere is a manifold. So is any m-dimensional ‘hyperplane’ in an n-

dimensional Euclidean space (m  n). More abstractly, the set of all rigid rotations of

Cartesian coordinates in three-dimensional Euclidean space will be shown below to be

a manifold. Basically, a manifold is any set that can be continuously parametrized. The

number of independent parameters is the dimension of the manifold, and the parameters

themselves are the coordinates of the manifold. Consider the examples just mentioned.

The surface of a sphere is ‘parametrized’ by two coordinates θ and φ.Them-dimensional

‘hyperplane’ has m Cartesian coordinates, and the set of all rotations can be parametrized

by the three ‘Euler angles’, which in effect give the direction of the axis of rotation (two

parameters for this) and the amount of rotation (one parameter). So the set of rotations is a

manifold: each point is a particular rotation, and the coordinates are the three parameters. It

is a three-dimensional manifold. Mathematically, the association of points with the values

of their parameters can be thought of as a mapping of points of a manifold into points of the

Euclidean space of the correct dimension. This is the meaning of the fact that a manifold

looks locally like Euclidean space: it is ‘smooth’ and has a certain number of dimensions.

It must be stressed that the large-scale topology of a manifold may be very different from

Euclidean space: the surface of a torus is not Euclidean, even topologically. But locally the

correspondence is good: a small patch of the surface of a torus can be mapped 1–1 into the

plane tangent to it. This is the way to think of a manifold: it is a space with coordinates,

that locally looks Euclidean but that globally can warp, bend, and do almost anything (as

long as it stays continuous).