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Chapter 2 Curvilinear Coordinates and General Tensors Flipbook PDF
Chapter 2 Curvilinear Coordinates and General Tensors 2.1 Curvilinear Coordinates We devote this chapter to the developm
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Chapter 2
Curvilinear Coordinates and General Tensors
2.1 Curvilinear Coordinates We devote this chapter to the development of four-dimensional geometry in arbitrary curvilinear coordinates. We shall deal with field quantities. A field quantity has the same nature at all points of space. Such a quantity will be disturbed by the curvature. If we take a point quantity Q (or one of its components if it has several), we can differentiate with respect to any of the four coordinates. We write the result ∂Q = Q ,µ . ∂xµ A subscript preceded by a comma will always denote a derivative in this way. We put the index downstairs in order that we may maintain a balancing of the indexes in the general equations. We can see this balancing by noting that the change in Q, when we move from the point x µ to a neighboring point x µ + d x µ , is δ Q = Q ,µ δ x µ
(2.1)
where summation over µ is understood. The repeated indexes appearing once in the lower and once in the upper position are automatically summed over; this is Einstein’s summation convention. Let us consider the transformation from one coordinate system x 0 , x 1 , x 2 , x 3 to another x 0 , x 1 , x 2 , x 3 : x µ = f µ (x 0 , x 1 , x 2 , x 3 )
(2.2)
where the f µ are certain functions. When we transform the coordinates, their differentials transform according to the relation dxµ =
∂ x µ ν dx . ∂ x ν
(2.3)
Any set of four quantities Aµ (µ = 0, 1, 2, 3) which, under coordinate change, transform like the coordinate differentials, is called a contravariant vector: 19
20
2 Curvilinear Coordinates and General Tensors
Aµ =
∂ x µ ν A . ∂ x ν
(2.4)
If φ is somewhat scalar, under a coordinate change, the four quantities ∂φ/∂ x µ transform according to the formula ∂φ ∂ x ν ∂φ = . ∂xµ ∂ x µ ∂ x ν
(2.5)
Any set of four quantities Aµ (µ = 0, 1, 2, 3) that, under a coordinate transformation, transform like the derivatives of a scalar is called a covariant vector: Aµ =
∂ x ν A ∂xµ ν
(2.6)
From the two contravariant vectors Aµ and B µ we may form the 16 quantities = 0, 1, 2, 3). These 16 quantities form the components of a contravariant tensor of the second rank: Any aggregate of 16 quantities T µν that, under a coordinate transformation, transform like the products of two contravariant vectors Aµ B ν (µ, ν
T µν =
∂ x µ ∂ x ν αβ T ∂ x α ∂ x β
(2.7)
is a contravariant tensor of rank two. We may also form a covariant tensor of rank two from two covariant vectors, which transform according to the formula Tµν =
∂ x α ∂ x β T . ∂ x µ ∂ x ν αβ
(2.8)
µ
Similarly, we can form a mixed tensor Tν of order two that transforms as follows Tνµ =
∂ x µ ∂ x β α T . ∂ x α ∂ x ν β
(2.9)
We may continue this process and multiply more than two vectors together, taking care that their indexes are all different. In this way we can construct tensors of higher rank. The total number of free indexes of a tensor is called its rank (or order). We may set a subscript equal to a superscript and sum over all values of this index, which results in a tensor having two fewer free indexes than the original one. This process is called contraction. For example, if we start with a fourth-order tensor T µ νρ σ , one way of contracting it is to put σ = ρ, which gives the secondρ rank tensor T µ νρ , having only 16 components, arising from the four values of µ ρ and ν. We could contract again to get the scalar T µ µρ with just one component. It is easy to show that the inner product of contravariant and covariant vectors, Aµ B µ , is an invariant, that is, independent of the coordinate system A µ B µ =
∂ x α ∂ x µ ∂xα β A B = A B β = δβα Aα B β = Aα B α . ∂ x µ ∂ x β α ∂xβ α
2.1 Curvilinear Coordinates
21
The square of the line element in curvilinear coordinates is a quadratic form in the differentials d x µ : ds 2 = gµν d x µ d x ν . Since the contracted product of gµν and the contravariant tensor d x µ d x ν is a scalar, the gµν forms a covariant tensor: ds 2 = g αβ d x α d x β = gαβ d x α d x β Now, d x α = (∂ x α /(∂ x µ )d x µ , so that g αβ or
∂ x α ∂ x β µ ν d x d x = gµν d x µ d x ν ∂xµ ∂xν
∂ x α ∂ x β g αβ µ − gµν d x µ d x ν = 0. ∂x ∂xν
The above equation is identically zero for arbitrary d x µ , so we have gµν =
∂ x α ∂ x β g ∂ x µ ∂ x ν αβ
(2.10)
that is, gµν is a covariant tensor of rank two. It is called the metric tensor or the fundamental tensor. The metric tensor is locally Minkoskian. So far, covariant and contravariant vectors have no direct connection with each other except that their inner product is an invariant. A space in which covariant and contravariant vectors exist separately is called affine. Physical quantities are independent of the particular choice of the mode of description, that is, independent of the possible choices of contravariance or covariance. In metric space, contravariant and covariant vectors can be converted into each other with the help of the fundamental tensor gµν . For example, we can get the covariant vector Aµ from the contravariant vector Aν Aµ = gµν Aν . (2.11) Since the determinant |g| does not vanish, these equations can be solved for Aν in terms of the Aµ . Let the result be Aν = g µν Aµ
(2.12)
By combining the two transformations (2.11) and (2.12), we have Aµ = gµν g να Aα . Since the equation must hold for any four quantities Aµ , we can infer gµν g να = δµ α .
(2.13)
In other words, g µν is the inverse of gµν and vice versa, that is g µν =
M µν |g|
where M µν is the minor of the element gµν .
(2.14)
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2 Curvilinear Coordinates and General Tensors
Equation (2.11) may be used to lower any upper index occurring in a tensor. Similarly, (2.12) can be used to raise any downstairs index. It is necessary to remember the position from which the index was lowered or raised, because when we bring the index back to its original site, we do not want to interchange the order of indexes, in general T µν = T νµ . Two tensors, Aµν and B µν , are said to be reciprocal to each other if Aµν B να = δα µ .
(2.15)
A tensor is called symmetric with respect to two contravariant or two covariant indexes if its components remain unaltered on interchange of the indexes. For exµνα νµα ample, if Aβγ = Aβγ , the tensor is symmetric in µ and ν. If a tensor is symmetric with respect to any two contravariant and any two covariant indexes, it is called symmetric. A tensor is called skew-symmetric with respect to two contravariant or two covariant indexes if its components change sign upon interchange of the indexes. Thus, µνα νµα if Aβγ = −Aβγ , the tensor is skew-symmetric in µ and ν. If a tensor is symmetric (or skew-symmetric) with respect to two indexes in one coordinate system, it remains symmetric (skew-symmetric) with respect to those two indexes in any other coordinate system. It is easy to prove this. For example, if B αβ is symmetric, B αβ = B βα , then B αβ =
∂ x α ∂ x β γ δ ∂ x α ∂ x β δγ B = B = B βα γ δ ∂x ∂x ∂xγ ∂xδ
(2.16)
i.e., the tensor remains symmetric in the primed coordinate system. Every tensor can be expressed as the sum of two tensors, one of which is symmetric and the other skew-symmetric in a pair of covariant or contravariant indices. Consider, for example, the tensor B αβ . We can write it as B αβ = 1/2(B αβ + B βα ) + 1/2(B αβ − B βα )
(2.17)
with the first term on the right-hand side symmetric and the second term skewsymmetric. By similar reasoning the result is seen to be true for any tensor. It is obvious that the sum or difference of two or more tensors of the same rank and type (i.e., same number of contravariant indices and same number of covariant indices) is also a tensor of the same rank and type. Thus if Aλ µν and Bλ µν are tensors, then Cλ µν = Aλ µν + Bλ µν and Dλ µν = Aλ µν − Bλ µν are also tensors. µ.... A given quantity Nνρ... with various up and down indexes may or may not be a tensor. We can test whether it is a tensor or not by using the quotient law, which can be stated as follows: Suppose we have a quantity X and we do not know whether it is a tensor or not. If an inner product of X with an arbitrary tensor is a tensor, then X is also a tensor.
For example, let X = Pλµν , and Aλ is an arbitrary contravariant vector; if Pλµν Aλ = Q µν is a tensor, then Pλµν is a contravariant tensor of rank 3. We can prove this explicitly:
2.2 Parallel Displacement and Covariant Differentiation
Aλ Pλµν =
∂ x α ∂ x β γ A P γ αβ . ∂xµ ∂xν
but Aγ = Hence, Aλ Pλµν =
23
∂ x γ λ A . ∂xλ
∂ x α ∂ x β ∂ x γ λ A P γ αβ . ∂xµ ∂xν ∂xλ
This equation must hold for all values of Aλ , so we have Pλµν =
∂ x α ∂ x β ∂ x γ P , ∂ x µ ∂ x ν ∂ x λ γ αβ
(2.18)
showing that Pλµν is a contravariant tensor of rank 3. µ.... For a nontensor Nνρ... we can raise and lower indexes by the same rules as for a tensor. Thus, for example ρ = Nηαµ . (2.19) g αν Nνη
2.2 Parallel Displacement and Covariant Differentiation In this section and the following three sections we will give the full of apparatus of differential geometry. The reader may be in danger of being overwhelmed by algebra, but to simplify mathematics is not to make it simple, either. Please do not despair; just relax and try to enjoy it. We have seen that a covariant vector is transformed according to the formula Ai =
∂ x k A ∂ xi k
(2.20)
where the coefficients are functions of the coordinates. So vectors at different points transform differently. Because of this fact, d Ai is not a vector, since it is the difference of two vectors located at two infinitesimally separated points of space-time. We can easily verify this directly from (2.20) d Ai =
2 k ∂ x k ∂ x d A + A dx j k k ∂ xi ∂ xi ∂ x j
which shows that d Ai does not transform at all like a vector. The same also applies to the differential of a contravariant vector. When using curvilinear coordinates, a differential can be obtained only when the two vectors to be subtracted from each other are located at the same point in spacetime. In order to do so, we must what we call parallel displace one of the vectors to the point where the other vector is located, after which we determine the difference of two vectors, which now refer to one and the same point in space-time.
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2 Curvilinear Coordinates and General Tensors
x2 -
The concept of parallel displacement of a vector is very clear in Cartesian coordinates: displace a vector parallel to itself so that both its length and orientation are unchanged. We can extend the idea of parallel displacement of a vector to curved spaces in a consistent way. This requires us to assume that there always exist Galilean coordinates in the immediate vicinity of a point in space-time; in such a coordinate system the idea of an infinitesimal parallel displacement of a vector works. In other words, a vector can be transported parallel to itself without changing its length and orientation. We illustrate, in Fig. 2.1, with the example of a curved two-dimensional surface in a three-dimensional Euclidean space. During the infinitesimal parallel displacement of two vectors, Aµ and B µ , the angle between them clearly remains unchanged, and so the inner (scalar) product of two vectors, Aµ B µ , does not change under parallel displacement. For arbitrary coordinates we define the operation of infinitesimal parallel displacement of a vector Aµ from Point P to a neighboring Point Q to be one that leaves the inner product with an arbitrary vector B µ invariant. Parallel displacements are independent of the paths taken on a Euclidean plane (a flat surface), as shown in Fig. 2.2a. On a curved surface, however, we will obtain a different final result on the path taken (Figure 2.2b). We can transfer a vector continuously along a path by the process of parallel displacement. In curvilinear coordinates, the components of a vector would be expected
ax
is
Q(xQ1 ,xQ2)
Parallel displacement in galilean coordinate
Curve
p(
x1 p, xp2)
Curve x1
- a xis
→ A
Fig. 2.1 Parallel displacement in curvilinear coordinates.
A2 A1 = A2
A1
D
1 C A
B
A 1
2
2
(a)
(b)
Fig. 2.2 Parallel transport around a closed curve.
2.2 Parallel Displacement and Covariant Differentiation
25
to change under a parallel displacement, unlike the case of a Cartesian coordinate. Therefore, if Aµ are the components of a contravariant vector at the point P(x µ ), and Aµ + d Aµ the components at a neighboring point Q(x µ + d x µ ), where Aµ + d Aµ = Aµ +
∂ Aµ σ dx ∂xσ
(2.21)
an infinitesimal parallel displacement of Aµ from P to Q would produce a variation of its components, δAµ . δAµ should be a linear function of the coordinate differentials and the components Aµ . We write it in the form ν δ Aν = − αβ Aα d x β
(2.22)
ν are certain functions of the coordinates and are called Christoffel where the αβ symbols of the second kind. Their form depends on the coordinate system. It will ν = 0. From this be proved in Section 4.4 that in a Galilean coordinate system αβ ν it is already clear that the quantities αβ do not form a tensor, since a tensor that is equal to zero in one coordinate system is equal to zero in every other one. In a ν vanish over all of space. curved space it is impossible to make all the αβ The vector resulting from parallel displacement from Point P to Point Q is Aµ + δAµ . Subtraction of these two quantities gives us µ ∂A µ µ µ µ α DA = dA − δ A = + σ α A d x σ . (2.23) ∂xσ
We would expect the difference d Aµ − δ Aµ to be a vector since it is the difference of two vectors at the same point; the quantity ∂ Aµ + σµα Aα ∂xσ then is a mixed tensor called the covariant derivative of Aµ and written µ
A;σ =
∂ Aµ + σµα Aα . ∂xσ
(2.24)
From δ(Aµ Aµ ) = 0 it follows, using (2.22), that α Aα d x β . δ Aµ = µβ
From this and a similar procedure that leads to (2.23) and (2.24), we obtain the covariant derivative of Aµ : Aµ;σ =
∂ Aµ ∂xσ
α − µσ Aα .
(2.25)
The tensor character of (2.24) and (2.25) can be established formally by showing that they obey the required transformation laws. This will require us first to establish α . It is not difficult to do this, but very tedious, the transformation laws for the µσ and so we shall not do it in this book.
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2 Curvilinear Coordinates and General Tensors
To obtain the contravariant derivative, we raise the index that denotes differentiation, µ (2.26) Aµ;σ = g σ α A;α . α = 0, and so covariant differentiation reduces to ordiIn Galilean coordinates, µσ nary differentiation. We may also obtain the covariant derivative of a tensor by determining the change in the tensor under an infinitesimal parallel displacement. For example, let us consider any arbitrary tensor T µν expressible as a product of two contravariant vectors Aµ B ν . Under infinitesimal parallel displacement
δ(Aµ B ν ) = Aµ δ B ν + B ν δ Aµ = −Aµ ν αβ B α d x β − B ν µ βσ Aσ d x β . By virtue of the linearity of this transformation we also have δ Aµν = − Aµβ ν βα + Aβν µ βα d x α . Substituting this in D Aµν = d Aµν − δ Aµν = Aµν ;α d x α we get the covariant derivative of the tensor T µν in the form T µν ;α =
∂ T µν + µ βα T βν + ν βα T µβ . ∂xα
(2.27)
In similar fashion we obtain the covariant derivative of the mixed tensor T µ ν and the covariant tensor Tνµ in the form ∂ T µν − β να T µ β + µ βα T β ν , ∂xα ∂ Tµν = − β µα Tβν − β να Tµβ . ∂xα
T µ ν;α =
(2.28)
Tµν;α
(2.29)
One can similarly determine the covariant derivative of a tensor of arbitrary rank. In doing this one finds the following rule of covariant differentiation: ..... with respect to x µ , you To obtain the covariant derivative of the tensor T..... ..... µ ..... ) a term add to the ordinary derivative ∂ T..... /∂ x for each covariant index ν(T..ν.. α ..... ..ν.. ν ..α.. − µν T..α.. , and for each contravariant index ν(T..... ) a term + αµ T..... . The covariant derivative of the metric tensor gµν is zero. To show this we note that the relation D Aµ = gµν D Aν is valid for the vector D Aµ as for any vector. On the other hand, we have Aµ = gµν Aν , so that D Aµ = D(gµν Aν ) = gµν D Aν + Aν Dgµν . Comparing with D Aµ = gµν D Aν , we have Aν Dgµν = 0. But the vector Aν is arbitrary, so
2.3 Symmetry Properties of the Christoffel Symbols
27
Dgµν = 0. Therefore, the covariant derivative is gµν;α = 0.
(2.30)
Thus, gµν may be considered as a constant during covariant differentiation. The covariant derivative of a product can be found by the same rule as for ordinary differentiation of products. In doing this we must consider the covariant derivative of a scalar φ as an ordinary derivative, that is, as the covariant vector φk = ∂φ/∂ x k , in accordance with the fact that for a scalar δφ = 0, and hence Dφ = dφ. For example (Aµ Bν );α = Aµ;α Bν + Aµ Bν;α .
(2.31)
2.3 Symmetry Properties of the Christoffel Symbols ν is symmetric in the subscripts. If δAν is a coordinate differWe now show that αβ ν ential d x , then (2.22) becomes ν δ(d x ν ) = − αβ dxαdxβ.
(2.32)
Next, we return to the local Cartesian coordinate system by the transformations x α = f α (x 1 , x 2 , . . . .) (2.33) x α = ϕ α (x 1 , x 2 , . . . . . .) where the primed coordinates are local Cartesian coordinates. From (2.33) we obtain dxα =
∂ f α β dx . ∂ x β
(2.34)
Under a parallel displacement, δ(d x β ) = 0, so that from (2.34) we have δ(d x ν ) =
∂2 f ν ∂ 2 f ν ∂ϕ δ ∂ϕ γ α β d x δ d x γ = δ γ dx dx . δ γ ∂x ∂x ∂x ∂x ∂xα ∂xβ
Comparing this with (2.32) we obtain ν αβ =−
∂ 2 f ν ∂ϕ δ ∂ϕ γ . ∂ x δ ∂ x γ ∂ x α ∂ x β
(2.35)
ν is also The right-hand side is clearly symmetric in the indexes α and β, so that αβ symmetric in α and β.
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2 Curvilinear Coordinates and General Tensors
2.4 Christoffel Symbols and the Metric Tensor It is very useful to express the ’s in terms of the metric tensors. Let Aµ be any contravariant vector, Aµ = gµν Aν a covariant vector. From the definition of parallel displacement δ(Aµ Aµ ) = 0, we have δ(Aµ Aµ ) = gµν (x µ + d x µ )[Aν + δ Aν ][Aµ + δ Aµ ] − gµν (x µ )Aν Aµ = 0. Carrying out these operations gives us ∂gµν ∂xα
Aµ Aν d x α + gµν Aµ δ Aν + gµν Aν δ Aµ = 0.
Making use of (2.22) to eliminate δAµ and δAν gives us ∂gµν ∂xα
β β − gνβ να − gνβ µα = 0.
(2.36)
β
Now, να is symmetric in the lower indexes ν and α, and this symmetry allows permutation of ν and α to obtain ∂gµα
β β − gµβ να − gαβ µν = 0.
(2.37)
∂gνα β β − gνβ µα − gαβ µν = 0. ∂xµ
(2.38)
∂xν
Similarly, we write
γ
Solving (2.36), (2.37), (2.38) for µα , we obtain γ = µα
∂gµα ∂gνα 1 γ ν ∂gνµ . g + − 2 ∂xα ∂xµ ∂xν
(2.39)
The Christoffel symbol of the first kind is ν,µα
∂gνα ∂gνα 1 ∂gνµ . = + − 2 ∂xα ∂xµ ∂xν
(2.40)
It is often written as [µα, ν]. Clearly ν,µα = ν,αµ . The Christoffel symbols are also known as the affine connections. The Christoffel symbols all vanish in Galilean coordinates, as the metric tensors are all constants in Galilean coordinates. The equation gµν;σ = 0 can be used to offer an alternative derivation of (2.39) and (2.40). We write in accordance with the general definition (2.29): gµν;α =
∂gµν ∂xα
β β − gβν µα − gµβ να =
∂gµν ∂xα
− ν,µα − µ,να = 0.
2.5 Geodesics
29
From this we have, permuting the indexes µ, ν, α: ∂gµν
∂xα ∂gαµ
= ν,µα + µ,να .
= µ,να + α,µν . ∂xν ∂gνα = −α,νµ − ν,αµ . ∂xµ
Taking half the sum of these equations and remembering that µ,να = µ,αν , we find ∂gµα ∂gνα 1 ∂gµν µ,να = + − . (2.41) 2 ∂xα ∂xν ∂xµ µ
From this we have for the symbols να = g νβ µ,να µ να
1 = g µβ 2
∂gβν ∂xα
+
∂gβα ∂xν
∂g − να ∂xβ
.
(2.42)
A coordinate system in which the Christoffel symbols vanish at Point P is called a geodesic coordinate system, and Point P is said to be the pole.
2.5 Geodesics As an application of the notion of parallel displacement and covariant differentiation, let’s consider the geodesic equation. A geodesic is the curve defined by the requirement that each element of it is a parallel displacement of the preceding element. We shall see later that the world line of a point-like particle not acted upon by any forces, except gravitation, is a time-like geodesic. If we take a point with coordinates x µ and move it along a path, we then have µ x as a function of some parameter s. There is a tangent vector t µ = d x µ /ds at each point of the path. As we go along the path the vector t µ gets shifted by parallel displacement: we shift the initial position from x µ to x µ + t µ ds, and then shift the vector t µ to this new position by parallel displacement, then shift the point again in the direction fixed by the new t µ , and so on. If we are given the initial point and the initial value of the vector t µ , not only can the path be determined but also the parameter s along it. A path produced in this way is called a geodesic. We get the geodesic equations by applying (2.22) with Aµ = t µ
or
σ dt ν ν µ dx + µσ = 0, t ds ds
(2.43)
σ µ d2xν ν dx dx = 0. + µσ ds ds ds 2
(2.44)
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2 Curvilinear Coordinates and General Tensors
If the vector t µ is initially a null vector, it always remains a null vector and the path is called a null geodesic. If the vector t µ is initially time-like (i.e., t µ t µ > 0), it is always time-like and we have a time-like geodesic. If t µ is initially space-like (t µ t µ < 0), it is always space-like and we have a space-like geodesic. For a time-like geodesic we may multiply the initial t µ by a factor so as to make its length unity. This requires only a change in the scale of s. The vector t µ now always has a unit length. It is simply the velocity vector u µ = d x µ /dτ , and the parameter s has becomes the proper time τ . (2.43) becomes du µ ν uµuσ = 0 + µσ dτ
(2.43a)
and (2.44) becomes
σ µ d2xν ν dx dx + µσ = 0. (2.44a) 2 dτ dτ dτ We make the physical assumption that the world line of a particle not acted on by any forces, except gravitation, is a time-like geodesic. Note that the terms ν u µ u σ in (2.43a) may be interpreted as the gravitational forces, and the com−m µσ ponents of the metric tensor gµν play the role of the classical gravitational potential (as the Christoffel symbol is proportional to the derivatives of the metric tensor; see [2.39] and [2.40]). Choosing a local Galilean frame in which gµν = constants, the Christoffel symν = 0, and du µ /dτ = 0. Therefore, the gravitational forces can be locally bols µσ eliminated, and the geodesic equations can be locally reduced to the special relativistic equations of motion, in agreement with the equivalence. The path of a light ray is a null geodesic. It is fixed by (2.44) referring to some parameter s along the path. The proper time τ cannot now be used because dτ vanishes.
2.6 The Stationary Property of Geodesics We now examine the stationary property of geodesics. A geodesic that is not a null geodesic joining two points P and Q has a stationary value compared with the interval (line element) measured along another neighboring curve joining P and Q. This property holds good for a straight line in flat space and, in that case, it is also true that the straight line gives the shortest interval from one point to another. In curved space, the geodesic is no longer a straight line because space-time is no longer flat, and the particle motion is not rectilinear and uniform, in general. However, we can show that the geodesic is a path of extreme length. (We will not enquire whether or not the geodesic in a curved space gives the minimum or maximum value of the interval between any of its points). To show this, we demonstrate that the relations which must be satisfied to give a stationary value to the integral gλµ d x λ d x µ (2.45) s=
2.6 The Stationary Property of Geodesics
31
are simply the equations of geodesics (2.44) of the previous section. Let us first introduce a parameter α and write (2.45) as s=
αQ
αP
gλµ
dxλ dxµ dα dα
1/2 dα
where α varies from point to point of the geodesic curve described by the relations which we are seeking for, and we write it as x µ = f µ (α).
(2.46)
Any other neighboring curve joining P and Q has equations of the form x¯ λ = x λ + εy λ = f λ (α) + εy λ (α) where y λ = 0 at the end points P and Q, i.e., at α = αP and α = αQ , and ε is a small quantity whose square and higher powers are negligibly small. If s¯ is the line element along the neighboring curve joining Pand Q, then s¯ =
αQ
αP
gλµ (x) ¯
d x¯ λ d x¯ µ dα dα
1/2 dα
and therefore, neglecting all powers of ε higher than the first,
1/2 ∂gλµ d x λ σ dxλ dxµ dy λ d x µ gλµ +ε y + 2gλµ s¯ − s = dα dα dα d x σ dα dα dα αP 1/2 αO dxλ dxµ dα, − gλµ dα dα αP
αQ
which can be reduced to αQ ∂gλµ d x λ σ dy λ d x µ dα 1 y + 2gλµ ds. s¯ − s = ε 2 αP d x σ dα dα dα ds Note that ds = gλµ d x λ d x µ . We can simplify the calculation, if s = 0, by assuming that the parameter α is identical with s itself measured along the geodesic. Then dα/ds = 1 and αQ ∂gλµ d x λ σ dy λ d x µ 1 y ds, + 2g s¯ − s = ε λµ 2 αP d x σ dα dα ds with the x λ , y λ now regarded as functions of s. Integration of the second term in the last equation by parts yields αQ ∂gλµ d x λ d x µ d dxµ d x µ λ sQ 1 σ −2 gσ µ y ds + ε gλµ y . s¯ − s = ε 2 αP d x σ ds ds ds ds ds sP
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2 Curvilinear Coordinates and General Tensors
But the functions y λ vanish at sP and s Q ; hence the integrated term is zero. Therefore, if the interval is to have a stationary value for the geodesic curve compared with neighboring curves, s¯ − s must be zero for any choice of the function y λ . This is possible only if the coefficient of each y σ in the integrand is separately zero, and therefore the differential equations of the geodesic are the n equations 1 ∂gλµ d x λ d x µ dxµ d gσ µ − = 0 (σ = 1, 2, . . ., n). (2.47) ds ds 2 ∂ x σ ds ds Now, d ds
gσ µ
= gσ µ
dxµ ds
− gσ µ
∂gµσ d x λ d x µ d2xµ + 2 ∂ x λ ds ds ds
d2xµ 1 dxλ dxµ (g . + + g ) µσ,λ λσ,µ 2 ds ds ds 2
Thus, the equations of the geodesic may be written d2xµ 1 dxλ dxµ + (gµσ,λ + gλσ,µ − gλµ,σ ) 2 2 ds ds ds d2xµ dxλ dxµ = 0. + σ,λµ = gσ µ ds ds ds 2
gσ µ
(2.48)
Multiplying this by g τ σ and summing over σ , the result is λ µ d2xτ µ dx dx = 0, + λµ ds ds ds 2
(τ = 1, 2, . . ., n),
(2.49)
which is simply the standard form of (2.44) for geodesics. The above work shows that we may use the stationary condition as the definition of a geodesic, except in dealing with the propagation of light. In that case, we have null geodesics, so the deduction as given above cannot be applied because ds vanish throughout.
2.7 The Curvature Tensor In a flat space, if we perform two (ordinary) differentiations in succession their order does not matter. However, this does not, in general, hold for covariant differentiation in a curved space, except for a scalar φ. For the case of a scalar, we have α α φ;α = φ,µ,ν − µν φ,σ . φ;µ;ν = (φ;µ ),ν − µν
(2.50)
α is symmetric in the lower indexes µ and ν, so the order of differentiation Since µν does not matter.
2.7 The Curvature Tensor
33
Now if we take a contravariant vector Aµ and apply covariant differentiations twice to it, we will find the order of differentiation is very important. First, the covariant differentiation of Aµ gives a mixed tensor µ
A;ν =
∂ Aµ µ α + αν A ∂xν
Covariant differentiation of this mixed tensor gives µ
A;ν;β =
∂ µ µ α µ A;ν + αβ Aα;ν − νβ A;α ∂xβ
or µ
A;ν;β =
µ
α ∂ αν ∂ 2 Aµ µ µ α µ ∂A + αν + Aα + Aα;ν αβ − A;α βν . β ν β ∂x ∂x ∂x ∂xβ
(2.51)
Interchanging β and ν, we obtain µ
A;β;ν =
µ
α ∂ αβ ∂ 2 Aµ µ ∂A µ α µ + αβ ν + Aα + Aα;β αν − A;α βν ν β ∂x ∂x ∂x ∂xν
Subtracting (2.52) from (2.51), we get
µ µ ∂ αβ γ µ µ µ µ α ∂ αν γ µ − + αν γβ − αβ γ ν = Aα Rανβ A;ν;β − A;β;ν = A ∂xβ ∂xν where µ
Rανβ = µ
µ
(2.52)
(2.53)
µ
µ ∂ αβ ∂ αν γ µ γ − + αν γβ − αβ γµν . β ∂x ∂xν
(2.54)
µ
Since A;ν;β − A;β;ν and Aα are tensors, Rανβ must be the component of a tensor, by the quotient law. It is called the curvature tensor or the Riemann tensor, and it depends solely on the Christoffel symbols and their derivatives. In flat space all gµν can be transformed into constants (rectangular or Galilean coordinates) and µ all Christoffel symbols vanish; hence Rανβ = 0. Being a tensor equation, it holds in all coordinate systems (Cartesian, oblique, or curvilinear). In a curved space, µ Rανβ will not vanish. Therefore it is a measure of the curvature of space. From (2.54) it follows that the curvature tensor is antisymmetric in the indices ν and β: µ µ (2.55) Rανβ = −Rαβν . Furthermore, it is easy to verify that the following identity is valid α α α Rβγδ + Rδβγ + Rγδβ = 0. µ
(2.56)
In addition to the mixed curvature tensor Rανβ , we can also use the covariant curvature tensor η (2.57) Rαβγδ = gαη Rβγδ .
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2 Curvilinear Coordinates and General Tensors
From the transformation law for Rαβγδ Rαβγδ
∂ 2 gβγ ∂ 2 gαγ ∂ 2 gβδ ∂ 2 gαδ + − − ∂xβ∂xγ ∂ xα∂ xδ ∂xβ∂xδ ∂ xα∂ xγ µ µ ν ν +gµν βγ αδ . − βδ αγ
1 = 2
(2.58)
It is not difficult to derive this transformation law, but it is very tedious; hence we simply give it here without derivation. From (2.58) we see the following symmetry properties: Rαβγδ = −Rβαγδ = −Rαβδγ , Rαβγδ = Rγδαβ (2.59) i.e., the tensor is antisymmetric in each of the index pairs α, β, and γ, δ and is symmetric under the interchange of the two pairs with each other. Thus, all components Rαβγδ , in which α = β or γ = δ are zero. As for Rαβγδ we also have the identity Rαβγδ + Rαδβγ + Rαγδβ = 0.
(2.60)
A fourth-rank tensor has 44 = 256 components. However, because of the above symmetries, the number of algebraically independent components of Rαβγδ is only 20. By contracting the curvature tensor, we get the symmetric Ricci tensor: λ = Rµσ . Rσµ = Rσµλ
(2.61)
According to (2.54), we have Rσ µ =
∂ σλ µ ∂xλ
−
∂ σλ λ ν λ + σλ µ λν − σν λ µν . ∂xµ
(2.62)
This tensor is clearly symmetric: Rσµ = Rµσ . Finally, contracting Rσµ we obtain R = g σµ Rσµ = g σλ g µβ Rσµλβ,
(2.63)
which is called the scalar curvature of the space. One of the most important tensors in the study of gravitation is the Einstein tensor, defined by 1 (2.64) G µν = Rµν − gµν R = G νµ . 2 The Einstein tensor G µν is purely geometric in character, being built up from gµν and their first and second derivatives. And it is linear in the second derivatives of gµν . It can be shown that the covariant divergence of G µν vanishes identically (we leave it as a problem). This property will be used later to formulate the gravitational field equations. There is apparent obscurity surrounding the physical meaning of the Riemann tensor; a simple example of calculating the curvature of space doesn’t elucidate
2.7 The Curvature Tensor
35
the physical meaning of the Riemann tensor. This may be helpful to visualize the Riemann tensor. Let us consider the two-dimensional space on the surface of a sphere of radius a. The metric of this two-dimensional space is ds 2 = a 2 (dθ 2 + sin2 θ dφ 2 ). The covariant and contravariant metric tensors are 2 0 0 a 1/a 2 µν , g = gµν = 0 a 2 sin2 θ 0 1/(a 2 sin2 θ ) and g = a 4 sin2 θ. The Christoffel symbols are given by (2.42). By direction calculation, we find that the only non-zero symbols are 1 θα ∂gφα ∂gαφ ∂gφφ 1 φα ∂gθα ∂gφα ∂gθφ φ θ + − , and θφ = g + − φφ = g 2 ∂φ ∂φ ∂α 2 ∂φ ∂θ ∂α where α takes the values θ and φ; these become 1 φφ ∂gφφ g = cot θ. ∂θ 2 ∂θ Let us first calculate the Ricci tensor (the contracted Riemann tensor) that is given by (2.62). The non-zero components are θ φφ = −g θθ
∂gφφ
φ
= − sin θ cos θ,
φ
Rθθ = θφ θφ +
φ
θφ =
∂ cot θ 1 = −1 = cot2 θ − ∂θ sin2 θ
and Rφφ = − sin2 θ. The Riemann scalar curvature R of the space is given by R = g θθ Rθθ + g φφ Rφφ = −2/a 2 . If the reader is familiar with the analytical geometry of surfaces, then recall that R here is equivalent to 2 R= ρ1 ρ2 where ρ1 and ρ2 are the two principal radii of curvature of the surfaces. On the sphere these radii coincide and are equal to the spherical radius. The Riemann scalar curvature thus bears a simple relationship to the radius curvature of the twodimensional space on the spherical surface. Proceeding similarly, we find, from (2.57), the only independent component of the Riemann tensor θ
θ ∂ φφ ∂ θφ θ θ α θ α − + θα φφ − φα φθ = a 2 sin2 θ, Rθφθφ = gθθ Rφθφ = gθθ ∂θ ∂φ and by (2.59) we have other non-zero components Rφθφθ = −Rθφφθ = −Rφθθφ = Rθφθφ = a 2 sin2 θ.
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2 Curvilinear Coordinates and General Tensors
2.8 The Condition for Flat Space If space is flat, we may choose a Cartesian coordinate system. Then the gµν are all constant and all Christoffel symbols vanish; hence, from (2.54), µ
Rανβ = 0 µ
in this coordinates system. But if Rανβ = 0 in one coordinate system, the components are zero in all coordinate systems. Hence if a space is flat, the Riemann curvature tensor must vanish, which is a necessary condition for a space to be flat. µ The converse is a sufficient condition: if Rανβ = 0, the space is flat. We now proceed to prove it. µ If Rανβ = 0, and if we can find a coordinate system for which its metric tensor is constant, then the space is flat. To this end, let us take vector Aµ located at Point x and parallel displace it to Point x + d x, and then parallel displace it to Point µ x + d x + δx. If the Riemann curvature tensor Rανβ is zero, the result would be the same if we had displace Aµ from x to x + δx and then to x + δx + d x. That is, the displacement is independent of the path. Thus we can displace the vector to a distant point and the result we get is independent of the path to the distant point. If we displace the vector Aµ at x to all points by parallel displacement, we will get a vector field that satisfies Aµ;ν = 0: Aµ;ν =
∂ Aµ ∂xν
σ − µν Aσ = 0,
or
σ Aµ,ν = µν Aσ .
If Aµ is the gradient of a scalar S, Aµ = ∂ S/∂ x µ = S,µ , then the above equation becomes S,µν = σ µν S,σ . Because σ µν = σ νµ , S,µν = S,νµ the above equations can be integrated. Now let us take four independent scalars satisfying the last equations and let them to be the coordinates y α of a new coordinate system. Thus, we have α = σ µν y,σ , y,µν α = ∂ 2 y α /∂ x µ ∂ x ν . where y,µν Let us go back to Eq. (10) that now takes the form
gµλ (x) = gαβ (y)
∂ yα ∂ yβ . ∂xµ ∂xλ
Differentiation of this equation with respect to x ν yields 2 α ∂gαβ (y) ∂ y α ∂ y β ∂gµλ ∂ y ∂ yβ ∂ yα ∂ 2 yβ = + gαβ (y) + µ λ ν ∂xν ∂xν ∂xµ ∂xλ ∂ x µ∂ x ν ∂ x λ ∂x ∂x ∂x ∂gαβ (y) ∂ y α ∂ y β σ α β α σ β = + g (y) y , y , + y , y , µν αβ σ λ µ λν σ ∂xν ∂xµ ∂xλ α β ∂gαβ (y) ∂ y ∂ y σ σ + gαλ (x) µν + gµσ (x) λν . = ∂xν ∂xµ ∂xλ
2.9 Geodesic Deviation
37
The last two terms can be rewritten in terms of the Christoffel symbols of the first kind: σ σ gαλ (x) µν + gµσ (x) λν = λ,µν + µ,λν = ∂gµλ /∂ x ν = gµλ,ν
where the Christoffel symbols are given by (2.40). Combining this with the last equation, we obtain ∂gαβ (y) ∂ y α ∂ y β = 0. ∂xν ∂xµ ∂xλ It follows that ∂gαβ (y) = 0. ∂xν Thus, the metric tensor of the new system of coordinates is constant, and the Christroffel symbols all vanish identically. In other words, we have flat space.
2.9 Geodesic Deviation We can get a good insight into the nature of a space just by examining the problem of geodesic deviation. For example, consider two nearby freely falling particles which travel on paths x µ (τ) and x µ (τ) + δx µ (τ). The equations of motion are then given by d2xµ dxν dxλ µ =0 + (x) νλ dτ dτ dτ 2
(2.65)
and d2 µ d d µ [x + δx µ ] + νλ (x + δx) [x ν + δx ν ] [x λ + δx λ ] = 0. 2 dτ dτ dτ
(2.66)
Evaluating the difference between these equations to first order in δx µ gives µ
ν ρ ∂ νλ ρ d x ν d x λ d 2 δx µ µ dx dx + 2 =0 + δx νλ ∂xρ dτ dτ dτ dτ dτ 2
(2.67)
or, in terms of covariant derivatives along the curves x µ (τ), ν ρ D2 λ λ µ dx dx . δx = R δx νµρ dτ dτ Dτ 2
(2.68)
This (2.68) is called the equation of geodesic deviation. Although a freely falling particle appears to be at rest in a coordinate system falling with the particle, a pair of nearby freely falling particles will exhibit a relative motion that can reveal the presence of a gravitational field to an observer who falls with the particles. The effect of the right-hand side of (2.68) becomes negligible when the separation between particles is much less than the characteristic dimensions of the field. This indicates clearly that the local inertial frames are only locally applicable; otherwise, the principle of equivalence will be violated.
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2 Curvilinear Coordinates and General Tensors
2.10 Laws of Physics in Curved Spaces The laws of physics must be valid in all coordinate systems. If they are expressed as tensor equations, whenever they involve the derivative of a field quantity, it must be a covariant derivative. Even if we are working with flat space (which means neglecting the gravitational field) and we are using curvilinear coordinates, we must write our equations in terms of covariant derivatives if we want them to hold in all coordinate systems. As for the problem of generalizing a particular physics law from the flat Minkowski space to a general curved space, there is not a unique solution at all. In fact, the problem is highly complicated, since in general there will be an interaction between the space (the gµν ) and the physical phenomenon whose laws we are trying to formulate. But if the object under consideration does not appreciably influence the gµν , that is, if the gµν are determined by objects much more massive than the object under consideration, we may then consider the gµν as given functions of the spacetime variables, gµν (x σ ). In this case the geometry is rigidly determined and the effect of the physical object under study on the geometrical structure may be neglected. Under these circumstances, we may take over the special relativistic laws by substituting √ d → D, ∂µ → Dµ , d → −gd where
∂ Aν , D A = Aµ ;ν . ∂xµ µ As an example, consider the motion of a free particle. Its time track in Minkowski space is characterized by the equations ∂µ Aν =
d 2 x i /ds 2 = 0, i = 0, 1, 2, 3.
(2.69)
These equations imply a straight line in the four-dimensional Minkowski space, which in turn corresponds to a uniform rectilinear motion in three-dimensional space. These equations can be derived from the stationary condition that the integral ∫ ds, taken along the motion between two points P and Q is stationary if one makes a small variation of the path keeping the end points fixed: Q ds = 0 (2.70) δ P
where the variation vanishes at the end points P and Q. If the particle is subject to the action of a gravitational field, its equation of motion is no longer a straight line, because the spacetime is curved. But the particle still follows a stationary trajectory, the geodesic. As shown above, the geodesic equation can be obtained from the same variational principle (2.70), provided that the Minkowski metric is replaced by the curved space metric gµν and ds 2 = gµν d x µ d x ν . Instead of computing explicitly the variation in (2.70), we can simply obtain the geodesic equations as the covariant generalization of (2.69):
2.11 The Metric Tensor and the Classical Gravitational Potential
39
D 2 x i /Ds2 = 0, which is equivalent to α d2 x i /ds2 + µν (d x µ /ds)(d x ν /ds) = 0,
the geodesic equations.
2.11 The Metric Tensor and the Classical Gravitational Potential The presence of a gravitational field modifies the structure of spacetime. Any gravitational field is just a change in the metric of space-time, as determined by the metric tensor gµν . Through the geodesic equations of motion, we can now provide the expressions governing the union of geometry and gravitation. To this end, let us compare the Newtonian equation of motion of a particle in a gravitational field and its geodesic equations of motion in a curved-space geometry: d 2 x α /ds 2 + ∂φ/∂ x α = 0
(2.71)
d 2 x α /ds 2 + α µν (d x µ /ds)(d x ν /ds) = 0
(2.72)
where φ is the Newtonian gravitational potential. These two equations have a fundamental similarity in that both are independent of the mass of the moving body under consideration. Thus, both equations satisfy the principle of equivalence. Now since the derivative d2 x α /ds 2 is the four-acceleration of the particle, the quantity – m α µν u µ u ν may be interpreted as the gravitational force, and then the components of the metric tensor gµν play the role of the Newtonian gravitational potential φ (as the Christoffel symbols are constructed from the derivatives of the gµν ). We must first show that this interpretation is consistent with the Newtonian equations of motion; namely, we must show that in the limit of ordinary velocities the geodesic equations reduce to the Newtonian equations. To see this, let the velocity d x α /dt c. √ Then ds 2 = g00 c2 dt 2 , and ds = g00 cdt, so that d 2 x α /dt 2 + α 00 c2 = 0 where
α 00 = ∂g00 /∂ x α .
From this we see that in this limit g00 = K + 2φ/c2 . Since in flat space g00 = 1, we have K = 1 and (2.73) g00 = 1 + 2φ/c2 This shows that the identification postulated above is plausible, i.e., the metric tensor gµν plays the role of Newtonian gravitational potential. We should be careful to note that the physical content of the two equations, (2.71 and 2.72), are entirely different. In Newton’s equation we have a field φ, which
40
2 Curvilinear Coordinates and General Tensors
causes the motion. The particle is under a force and its velocity changes in time. In the geodesic equations, on the other hand, there is no physical agent such as φ. The particle follows a geodesic that is determined by the geometry of the space-time. This change in interpretation is actually a conceptual simplification, since inertia and gravitation are unified and the concept of external force is eliminated from the theory of gravitation.
2.12 Some Useful Calculation Tools We conclude this chapter by giving a number of useful aids in manipulation of tensor quantities. First of all, it is very helpful to bear in mind that the covariant derivative of the metric tensor gµν is zero. That is, the metric tensor gµν may be considered as a constant during covariant differentiation. We now derive an expression for the contracted Christoffel symbol µ αµ that will be very useful later on. From (2.39) we have ∂gνµ ∂gαµ ∂gνα 1 µ . αµ = g µν + − 2 ∂xµ ∂xα ∂xν Changing the positions of the dummy indexes µ and ν in the first term and remembering gµν = gνµ , we see that the first and third terms then cancel each other, so that ∂gµν 1 µ αµ = g µν , (2.74) 2 ∂xα which can be simplified. To do this we calculate the differential dg of the determinant g made up from the components of the metric tensor gµν ; dg can be obtained by taking the differential of each component of the tensor gµν and multiplying it by its coefficient in the determinant, i.e., by the corresponding minor: dg = dgµν M µν where M µν is the minor of the component gµν . Now, g µν = Thus,
M µν , g
M µν = g µν g
dg = gg µν dg µν = −g gµν dg µν
The expression on the far right of the above equation follows from d(gµν g µν ) = d(δµµ ) = d(4) = 0. We then have
∂gµν ∂g ∂g µν µν = g g = −g g . µν ∂xα ∂xα ∂xα
(2.75)
2.12 Some Useful Calculation Tools
41
The use of (2.75) enables us to write (2.74) in the form µ αµ =
√ 1 µν ∂gµν 1 ∂g ∂ ln −g g = = . 2 ∂xα 2g ∂ x α ∂xα
(2.76)
This expression is very useful. First consider the covariant divergence Aµ ;µ : Aµ ;µ =
∂ Aµ µ + αµ Aα . ∂xµ µ
Substituting the expression (2.76) for αµ , we obtain µ
A;µ =
√ √ ∂ Aµ 1 ∂ −g Aµ α ∂ ln −g + A = . √ ∂xµ ∂xα −g ∂xµ
(2.77)
We now consider the covariant divergence of a contravariant tensor of the second µν rank T;α . From (2.27), we have T µν ;α =
∂ T µν µ µ + βν T βν + βα T µβ . ∂xν
Changing the positions of the dummy indexes µ and β in the third term on the righthand side, we obtain T µν ;ν =
∂ T µν β µ + βν T βν + νβ T µν . ∂xν
Substituting the expression (2.76) for β νβ , we obtain T µν ;ν = or
√ ∂ T µν ∂ ln −g µν µ βν + T + T βν ∂xν ∂xν
∂ √ 1 ( −gT µν ) + µβν T βν . T µν ;ν = √ −g ∂ x ν
(2.78)
Similarly, for a mixed tensor, (2.28) leads to ∂ β√ 1 β β Tα;β = √ T −g − ασ Tβα . −g ∂ x β α
(2.79)
For an antisymmetric tensor F βν = −F νβ , then µ
µ
µ
βν F βν = νβ F νβ = − βν F βµ .
(2.80)
The expression in the middle is obtained by interchanging the dummy indexes β and ν, and the expression on the far right follows from the F βν = −F νβ , and µ νβ = µ βν . From (2.80) it follows that µ
βν F βµ = 0.
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2 Curvilinear Coordinates and General Tensors
Thus, for an antisymmetric tensor F αβ , the last term of (2.78) vanishes and the covariant divergence is √ ∂ 1 αβ (F αβ −g). F;β = √ −g ∂ x β
(2.81)
For a symmetric tensor Sαβ , rearrangement of the last term of (2.79) gives √ ∂ 1 1 ∂gβν βν β Sα;β = √ (Sαβ −g) − S . β −g ∂ x 2 ∂xα
(2.82)
In Cartesian coordinates, the curl ∂ Aµ ∂xν
−
∂ Aν ∂xµ
is an antisymmetric tensor. In curvilinear coordinates this tensor is Aµ;ν − Aν;µ : ρ ρ Aµ;ν − Aν;µ = Aµ,ν − µν Aρ − (Aν,µ − νµ Aρ )
Since ρ µν = ρ νµ , we have Aµ;ν − Aν;µ =
∂ Aµ ∂xν
−
∂ Aν . ∂xµ
(2.83)
This result may be stated: covariant curl equals ordinary curl, but it holds only for a covariant vector. For a contravariant vector we could not form the curl because the suffixes would not balance. Finally, we transform to curvilinear coordinates the sum ∂ 2ϕ ∂ xα ∂ x α of the second derivatives of a scalar ϕ. In curvilinear coordinates this sum goes over ;α . But covariant differentiation of a scalar reduces to ordinary differentiation: to ϕ;α ϕ;α = ∂ϕ/∂ x α . Raising the index α, we have ϕ ;α = g αβ ∂ϕ/∂ x β , which is a contravariant tensor of rank one. Using formula (2.77), we find √ ∂ 1 ;α αβ ∂ϕ ϕ;α = √ . −g g −g ∂ x α ∂xβ
(2.84)
In view of Eq. (77), Gauss’ theorem for transformation of the integral of a vector over a hypersurface into an integral over a four-volume can now be written as √ √ Aα −g d Sα = ∫ Aα ;α −g d. (2.85)
2.13 Problems
43
2.13 Problems 2.1. Write the terms in each of the following indicated sums a . a jk x k
b . A pq B qr
c.gr s = g jk
∂x j ∂xk ∂ xr ∂ xs
2.2. Write the transformation law for the following tensors: a . Ai jk
b . B pq i jk
2.3. A quantity A( j, k, m, n), which is a function of the coordinates x i , transforms to another coordinate system x¯ i according to the rule A( p, q, r, s) =
∂ x j ∂ xq ∂ xr ∂ xs A( j, k, m, n). ∂x p ∂xk ∂xm ∂xn
Is the quantity a tensor? If so, write the tensor in suitable notation and give the covariant and contravariant rank. 2.4. Show that the property of symmetry (or antisymmetry) with respect to indexes of a tensor is invariant under coordinate transformation. 2.5. A covariant tensor has components x y, 2y − z 2 , x z in rectangular coordinates; find its covariant components in spherical coordinates. 2.6. Prove that the contraction of the outer product of the two tensors A p and B p is a scalar. 2.7. Determine the Christoffel symbols of the second kind in rectangular and cylindrical coordinates. 2.8. The line element on the surface of a sphere of radius a in Euclidean space is given by ds 2 = a 2 (dθ 2 + sin2 θ dφ 2 ). For this space calculate i kl , i, k, l = 1, 2 (with θ = x1 , φ = x2 ). 2.9. Find the covariant derivative of Ai j B km n with respect to x q . 2.10. Prove that ∂ √ 1 k −g A ∇·A = √ −g ∂ x k p
and
∂ 1 ∇ φ=√ −g ∂ x k 2
√
∂φ −gg ∂ xr kr
2.11. Determine the force acting on a particle in a constant gravitational field. 2.12. Prove that the covariant divergence of the Einstein tensor vanishes. 2.13. The distance s between two points on a curve x µ = x µ (λ) is given by 2 dxα dxβ . s = ds = gαβ dλ dλ 1
.
44
2 Curvilinear Coordinates and General Tensors
Show that the necessary condition that s be an extremum is that ∂L d ∂L − =0 ν ∂x dλ ∂ x˙ ν where L = gαβ
dxα dxβ dxν , and x˙ ν = . dλ dλ dλ
2.14. Show that great circles drawn on the surface of a sphere are geodesics. 2.15. Show that, with (2.43), in a Euclidean space geodesics are straight lines.
References Dirac PAM (1975) General Theory of Relativity. (John Wiley & Sons, New York) Landau LD, Lifshitz EM (1975) The Classical Theory of Fields. (Pergamon Press, Oxford UK)
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