Riemann Tensor Components

8/20/2019 Riemann Tensor Components

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Appendix: Mathematical Miscellany

1. Vector Products The dot and cross products are often introduced via trigonometric functions and/or matrixoperations, but they also arise quite naturally from simple considerations of Pythagoras'theorem. Given two points a and b in the three-dimensional vector space with Cartesiancoordinates (ax,ay,az) and (bx,by,bz) respectively, the squared distance between these two

points is

If (and only if) these two vectors are perpendicular, the distance between them is thehypotenuse of a right triangle with edge lengths equal to the lengths of the two vectors, sowe have

if and only if a and b are perpendicular. Equating these two expressions and cancelingterms, we arrive at the necessary and sufficient condition for a and b to be perpendicular

This motivates the definition of the left hand quantity as the "dot product" (also called thescalar product) of the arbitrary vectors a = (ax,ay,az) and b = (bx,by,bz) as the scalar

quantity

At the other extreme, suppose we seek an indicator of whether or not the vectors a and b are parallel. In any case we know the squared length of the vector sum of these two vectors is

We also know that S = |a| + |b| if and only if a and b are parallel, in which case we have

Equating these two expressions for S2, canceling terms, and squaring both sides gives thenecessary and sufficient condition for a and b to be parallel


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Expanding these expressions and canceling terms, this becomes

Notice that we can gather terms and re-write this equality as

Obviously a sum of squares can equal zero only if each term is individually zero, which of course was to be expected, because two vectors are parallel if and only if their componentsare in the same proportions to each other, i.e.,

which represents the vanishing of the three terms in the previous expression. Thismotivates the definition of the cross product (also known as the vector product) of twovectors a = (ax,ay,az) and b = (bx,by,bz) as consisting of those three components, ordered

symmetrically, so that each component is defined in terms of the other two components of the arguments, as follows

By construction, this vector is null if and only if a and b are parallel. Furthermore, noticethat the dot products of this cross product and each of the vectors a and b are identicallyzero, i.e.,

As we saw previously, the dot product of two vectors is 0 if and only if the vectors are

perpendicular, so this shows that a b is perpendicular to both a and b. There is, however,

an arbitrary choice of sign, which is conventionally resolved by the "right-hand rule". It

can be shown that if is the angle between a and b, then ab is a vector with magnitude

|a||b|sin() and direction perpendicular to both a and b, according to the right-hand rule.

Similarly the scalar ab equals |a||b|cos().

2. Differentials In Chapter 5.2 we gave an intuitive description of differentials such as dx and dy asincremental quantities, but strictly speaking the actual values of differentials are arbitrary,


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because only the ratios between them are significant. Differentials for functions of multiplevariables are just a generalization of the usual definitions for functions of a single variable.For example, if we have z = f(x) then the differentials dz and dx are defined as arbitraryquantities whose ratio equals the derivative of f(x) with respect to x. Consequently we have

dz/dx = f '(x) where f '(x) signifies the partial derivative z/x, so we can express this in theform

In this case the partial derivative is identical to the total derivative, because this f isentirely a function of the single variable x.

If, now, we consider a differentiable function z = f(x,y) with two independent variables, wecan expand this into a power series consisting of a sum of (perhaps infinitely many) terms

of the form Axmyn. Since x and y are independent variables we can suppose they are eachfunctions of a parameter t, so we can differentiate the power series term-by-term, withrespect to t, and each term will contribute a quantity of the form

where, again, the differentials dx,dy,dz,dt are arbitrary variables whose ratios only are

constrained by this relation. The coefficient of dy/dt is the partial derivative of Axmyn, withrespect to y, and the coefficient of dx/dt is the partial with respect to x, and this will applyto every term of the series. So we can multiply through by dt to arrive at the result

The same approach can be applied to functions of arbitrarily many independent variables.

3. Differential Operators The standard differential operators are commonly expressed as formal "vector products"

involving the ("del") symbol, which is defined as

where ux, uy, uz are again unit vectors in the x,y,z directions. The scalar product of with

an arbitrary vector field V is called the divergence of V, and is written explicitly as


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The vector product of with an arbitrary vector field V is called the curl , given explicitly by

Note that the curl is applied to a vector field and returns a vector, whereas the divergence is

applied to a vector field but returns a scalar. For completeness, we note that a scalar fieldQ(x,y,z) can be simply multiplied by the operator to give a vector, called the gradient , asfollows

Another common expression is the sum of the second derivatives of a scalar field withrespect to the three directions, since this sum appears in the Laplace and Poisson equations.Using the "del" operator this can be expressed as the divergence of the gradient (or the "div

grad") of the scalar field, as shown below.

For convenience, this operation is often written as 2, and is called the Laplacian operator.All the above operators apply to 3-vectors, but when dealing with 4-vectors in Minkowskispacetime the analog of the Laplacian operator is the d'Alembertian operator

4. Differentiation of Vectors and Tensors The easiest way to understand the motivation for the definitions of absolute and covariantdifferentiation is to begin by considering the derivative of a vector field in three-dimensional Euclidean space. Such a vector can be expressed in either contravariant or covariant form as a linear combination of, respectively, the basis vectors u1, u2, u3 or the

dual basis vectors u1

, u2

, u3

, as follows

where Ai are the contravariant components and Ai are the covariant components of A, and

the two sets of basis vectors satisfy the relations


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where gij and gij are the covariant and contravariant metric tensors. The differential of A

can be found by applying the chain rule to either of the two forms, as follows

If the basis vectors ui and ui have a constant direction relative to a fixed Cartesian frame,

then dui = dui = 0, so the second term on the right vanishes, and we are left with the

familiar differential of a vector as the differential of its components. However, if the basisvectors vary from place to place, the second term on the right is non-zero, so we must notneglect this term if we are to allow curvilinear coordinates.

As we saw in Part 2 of this Appendix, for any quantity Q = f(x) and coordinate xi we have

so we can substitute for the three differentials in (1) and re-arrange terms to write theresulting expressions as

Since these relations must hold for all possible combinations of dxi , the quantities inside parentheses must vanish, so we have the following relations between partial derivatives

If we now let Ai

j and A

ij denote the projections of the ith components of (2a) and (2b)

respectively onto the jth basis vector, we have

and it can be verified that these are the components of second-order tensors of the typesindicated by their indices (superscripts being contravariant indices and subscripts being

covariant indices). If we multiply through (using the dot product) each term of (2a) by ui,

and each term of (2b) by ui, and recall that uiu j =

i j, we have


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For convenience we now define the three-index symbol

which is called the Christoffel symbol of the second kind. Although the Christoffel symbolis not a tensor, it is very useful for expressing results on a metrical manifold with a given

system of coordinates. We also note that since the components of uiu j are constants (either

0 or 1), it follows that (uiu j)/xk = 0, and expanding this partial derivative by the chain

rule we find that

Therefore, equations (3) can be written in terms of the Christoffel symbol as

These are the covariant derivatives of, respectively, the contravariant and covariant forms of the vector A. Obviously if the basis vectors are constant (as in Cartesian or obliquecoordinate systems) the Christoffel symbols vanish, and we are left with just the first termson the right sides of these equations. The second terms are needed only to account for thechange in basis with position of general curvilinear coordinates.

It might seem that these definitions of covariant differentiation depend on the fact that weworked in a fixed Euclidean space, which enabled us to assign absolute meaning to thecomponents of the basis vectors in terms of an underlying Cartesian coordinate system.However, it can be shown that the Christoffel symbols we've used here are the same as theones defined in Section 5.4 in the derivation of the extremal (geodesic) paths on a curvedmanifold, wholly in terms of the intrinsic metric coefficients gij and their partial derivatives

with respect to the general coordinates on the manifold. This should not be surprising,considering that the definition of the Christoffel symbols given above was in terms of the

basis vectors u j and their derivatives with respect to the general coordinates, and noting that


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the metric tensor is just gij = uiu j . Thus, with a bit of algebra we can show that

in agreement with Section 5.4. We regard equations (4) as the appropriate generalization of differentiation on an arbitrary Riemannian manifold essentially by formal analogy with theflat manifold case, by the fact that applying this operation to a tensor yields another tensor,and perhaps most importantly by the fact that in conjunction with the developments of Section 5.4 we find that the extremal metrical path (i.e., the geodesic path) between two

points is given by using this definition of "parallel transport" of a vector pointed in thedirection of the path, so the geodesic paths are locally "straight". Of course, when we allow curved manifolds, some new phenomena arise. On a flatmanifold the metric components may vary from place to place, but we can still determinethat the manifold is flat, by means of the Riemann curvature tensor described in Section5.7. One consequence of flatness, obvious from the above derivation, is that if a vector istransported parallel to itself around a closed path, it assumes its original orientation when itreturns to its original location. However, if the metric coefficients vary in such a way thatthe Riemann curvature tensor is non-zero, then in general a vector that has been transported

parallel to itself around a closed loop will undergo a change in orientation. Indeed, Gaussshowed that the amount of deflection experienced by a vector as a result of being parallel-transported around a closed loop in a 2-dimensional manifold is exactly proportional to theintegral of the curvature over the enclosed region. The above definition of covariant differentiation immediately generalizes to tensors of any

order. In general, the covariant derivative of a mixed tensor T consists of the ordinary partial derivative of the tensor itself with respect to the coordinates xk , plus a terminvolving a Christoffel symbol for each contravariant index of T, minus a term involving aChristoffel symbol for each covariant index of T. For example, if r is a contravariant indexand s is a covariant index, we have

It's convenient to remember that each Christoffel symbol in this expression has the index of

xk in one of its lower positions, and also that the relevant index from T is carried by thecorresponding Christoffel symbol at the same level (upper or lower), and the remainingindex of the Christoffel symbol is a dummy that matches with the relevant index position inT. One very important result involving the covariant derivative is known as Ricci's Theorem.The covariant derivative of the metric tensor is gij is


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If we substitute for the Christoffel symbols from equation (5), and recall that

we find that all the terms cancel out and we're left with gij,k = 0. Thus the covariant

derivative of the metric tensor is identically zero, which is what prompted Einstein to

identify it with the gravitational potential, whose divergence vanishes, as discussed inSection 5.8.

5. Notes on Curvature Derivations Direct substitution of the principal q values into the curvature formula of Section 5.3 gives asomewhat complicated expression, and it may not be obvious that it reduces to theexpression given in the text. Even some symbolic processors seem to be unable toaccomplish the reduction. So, to verify the result, recall that we have

where m = (ca)/b. The roots of the quadratic in q are

and of course qq' = 1. From the 2nd equation we have q2 = 1 + 2mq, so we can substitute

this into the curvature equation to give

Adding and subtracting c in the numerator, this can be written as

Now, our assertion in the text is that this quantity equals (a+c) + b . If we subtract2c from both of these quantities and multuply through by 1 + mq, our assertion is

Since q = m + the right hand term in the square brackets can be written as

bq bm, so we claim that


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Expanding the right hand side and cancelling terms and dividing by m gives

Now we multiply by the conjugate quantity q' to give

The quantities bq' cancel, and we are left with m = (c a)/b, which is the definition of m.Of course the same derivation applies to the other principle curvature if we swap q and q'. Section 5.3 also states that the Gaussian curvature of the surface of a sphere of radius R is

1/R 2. To verify this, note that the surface of a sphere of radius R is described by x2 + y2 +

z2 = R 2, and we can consider a point at the South pole, tangent to a plane of constant z.

Then we have

Taking the negative root (for the South Pole), factoring out R, and expanding the radical

into a power series in the quantity (x2 + y2) / R 2 gives

Without changing the shape of the surface, we can elevate the sphere so the South pole is just tangent to the xy plane at the origin by adding R to all the z values. Omitting all powers of x and y above the 2nd, this gives the quadratic equation of the surface at this point

Thus we have z = ax2 + bxy + cx2 where

from which we compute the curvature of the surface


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as expected.

6. Odd Compositions It's interesting to review the purely formal constraints on a velocity composition law (suchas discussed in Section 1.8) to clarify what distinguishes the formulae that work from thosethat don't. Letting v12, v23, and v13 denote the pairwise velocities (in geometric units)

between three co-linear particles P1, P2, P3, a composition formula relating these speeds

can generally be expressed in the form

where f is some function that transforms speeds into a domain where they are simplyadditive. It's clear that f must be an "odd" function, i.e., f(-x) = -f(x), to ensure that thesame composition formula works for both positive and negative speeds. This rules out

transforms such as f(x) = x2, f(x) = cos(x), and all other "even" functions.

The general "odd" function expressed as a power series is a linear combination of odd powers, i.e.,

so we can express any such function in terms of the coefficients [c1,c3,...]. For example, if

we take the coefficients [1,0,0,...] we have the simple transform f(x) = x, which gives theGalilean composition formula v13 = v12 + v23. For another example, suppose we

"weight" each term in inverse proportion to the exponent by using the coefficients [1, 1/3,

1/5, 1/7,...]. This gives the transform

leading to Einstein's relativistic composition formula

From the identity atanh(x) = ln[(1+x)/(1x)]/2 we also have the equivalent multiplicativeform

which is arguably the most natural form of the relativistic speed composition law. Thevelocity parameter p = (1+v)/(1-v) also gives very natural expressions for other observables

as well, including the relativistic Doppler shift, which equals , and the spacetimeinterval between two inertial particles each one unit of proper time past their point of


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intersection, which equals p1/4 p-1/4. Incidentally, to give an equilateral triangle in

spacetime, this last equation shows that two particles must have a mutual speed of =0.745...

7. Relativistic Form of Kepler’s Third Law

One might wonder why, in the derivation of the equation for planetary orbits presented in

Section 6.2, we apparently made use of only two of the three geodesic equations, excludingequation (8). Recall that we integrated equations (7) and (9), and then substituted the resultsinto the line element, which then gave the equation of motion represented by (10).However, given all but one of the geodesic equations along with the line element, theremaining geodesic equation is redundant. To see this, we can differentiate equation (10)

with respect to and divide through by 2(dr/d) to give

We would arrive at this same equation if we inserted the squared derivatives of thecoordinates into equation (8), so we have consistency with all of the geodesic equations.Another point to be clarified is whether the analog of Kepler’s third law implicit in thisequation is consistent with the claim in Section 5.5 that Kepler’s third law is formally valid

in Schwarzschild spacetime. Letting = d/d denote the proper angular speed, we have h

= r 2, and the above equation can be written as

Obviously if = 0 this gives the "proper" analog of Newton's inverse-square law equating

radial gravitational acceleration to –m/r 2. With non-zero value of the term 2r corresponds to the Newtonian centripetal acceleration which, if we defined the tangential

velocity v = r, would equal the classical v2/r. This term serves to offset the inward pull of

gravity, but in the relativistic version we find not 2r but 2r(13m/r). For a circular orbit

we would have d2r/dt2 = 0 and hence

This may seem inconsistent with the claim in Section 5.5 that the Keplerian relation m =

2r 3 is exactly valid for circular orbits in Schwarzschild spacetime. However, in that

section we defined as the derivative of with respect to coordinate time, whereas here we

have taken to be the derivative of with respect to proper time. To see that this is

consistent, divide through the line element by (dt)2 and note that dr/d = 0 for a circular orbit, so we have


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8. Independent Components of the Curvature Tensor As shown in Section 5.7, the fully covariant Riemann curvature tensor at the origin of

Riemann normal coordinates, or more generally in terms of any “tangent” coordinatesystem with respect to which the first derivatives of the metric coefficients are zero, has thesymmetries

These symmetries imply that although the curvature tensor in four dimensions has 256components, there are only 20 algebraic degrees of freedom. To prove this, we first notethat the anti-symmetry in the first two indices and in the last two indices implies that all thecomponents of the form R aaaa, R aabb, R aabc, R abcc, and all permutations of R aaab are zero,

because they equal the negation of themselves when we transpose either the first two or thelast two indices. The only remaining components with fewer than three distinct indices areof the form R abab and R abba, but these are the negatives of each other by transposition of

the last two incides, so we have only six independent components of this form (which is thenumber of ways of choosing two of four indices). The only non-zero components with

exactly three distinct indices are of the forms R abac = R baac = R abca = R baca, so we have

twelve independent components of this form (because there are four choices for theexcluded index, and then three choices for the repeated index). The remaining componentshave four distinct indices, but each component with a given permutation of indices actuallydetermines the values of eight components because of the three symmetries and anti-

symmetries of order two. Thus, on the basis of these three symmetries there are only 24/8 =3 independent components of this form, which may be represented by the three componentsR 1234, R 1342, and R 1432. However, the skew symmetry implies that these three

components sum to zero, so they represent only two degrees of freedom. Hence we canfully specify the Riemann curvature tensor (with respect to “tangent” coordinates) by givingthe values of the six components of the form R abab, the twelve components of the form

R abac, and the values of R 1234 and R 1342, which implies that the curvature tensor (with

respect to any coordinate sytem) has 6 + 12 + 2 = 20 algebraic degrees of freedom. The same reasoning can be applied in any number of dimensions. For a manifold of Ndimensions, the number of independent non-zero curvature components with just twodistinct indices is equal to the number of ways of choosing 2 out of N indices. Also, thenumber of independent non-zero curvature components with 3 distinct indices is equal tothe number of ways of choosing the N-3 excluded indices out of N indices, multiplied by 3for the number of choices of the repeated index. This leaves the components with 4 distinctindices, of which there are 4! times the number of ways of choosing 4 of N indices, butagain each of these represents 8 components because of the symmetries and anti-symmetries. Also, these components can be arranged in sets of three that satisfy the three-way skew symmetry, so the number of independent components of this form is reduced by afactor of 2/3. Therefore, the total number of algebraically independent components of the


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curvature tensor in N dimensions is

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