General Relativity Solutions to Exercises by Sergei Winitzki

Problem sets - General Relativity Solutions by Sergei Winitzki Last modified: 2008-09-23

Contents 1 Coordinates and 1-forms 1.1 Invertible transformations . . . . . . . . 1.2 Examples of coordinate transformations 1.3 Basis in tangent space . . . . . . . . . . 1.4 Differentials of functions as 1-forms . . . 1.5 Basis in cotangent space . . . . . . . . . 1.6 Linearly independent 1-forms . . . . . . 1.7 Transformation law for 1-forms . . . . . 1.8 Examples of transformations . . . . . . 1.9 Supplementary problem sheet . . . . . .

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2 2 2 3 3 4 4 4 5 5

2 Tensors 2.1 Definition of tensor product . . . 2.3 Example of tensor . . . . . . . . 2.5 Contraction of tensor indices . . 2.8 Examples of spaces with a metric 2.9 Supplementary problem sheet . .

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7 7 8 8 8 9

3 The 3.1 3.2 3.3 3.4 3.5

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10 10 11 11 11 11

4 Geodesics and curvature 4.1 Geodesics . . . . . . . . . . . . . . . 4.1.1 First derivation . . . . . . . 4.1.2 Second derivation . . . . . . . 4.2 Commutator of covariant derivatives 4.3 Parallel transport . . . . . . . . . . . 4.4 Riemann tensor . . . . . . . . . . . . 4.5 Lorentz transformations . . . . . . .

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12 12 12 13 13 13 14 15

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15 15 15 16 16 17

6 The gravitational field 6.1 Degrees of freedom . . . . . . . . . . . . . . . . 6.2 Spherically symmetric spacetime . . . . . . . . 6.2.1 Straightforward solution . . . . . . . . . 6.2.2 Solution using conformal transformation 6.3 Equations of motion . . . . . . . . . . . . . . .

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18 18 18 18 21 22

Christoffel symbol Transformations 1 . . . Transformations 2 . . . Covariant derivatives . . The Leibnitz rule . . . . Locally inertial reference

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5 Gravitation theory applied 5.1 Redshift . . . . . . . . . . . . . . . . . 5.2 Energy-momentum tensor 1 . . . . . . 5.3 Energy-momentum tensor 2 . . . . . . 5.4 Weak gravity . . . . . . . . . . . . . . 5.5 Equations of motion from conservation

1

7 Weak gravitational fields 7.1 Gravitational bending of light 7.2 Einstein tensor for weak field 7.3 Gravitational perturbations I 7.4 Gravitational perturbations II

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23 23 24 24 25

8 Gravitational radiation I 8.1 Gauge invariant variables . . . . . . . . . 8.2 Detecting gravitational waves . . . . . . . 8.2.1 Using distances between particles . 8.2.2 Using geodesic deviation equation 8.3 Poisson equation . . . . . . . . . . . . . . 8.4 Metric perturbations 1 . . . . . . . . . . . 8.5 Metric perturbations 2 . . . . . . . . . . .

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27 27 27 27 28 28 28 29

9 Gravitational radiation II 9.1 Projection of the matter tensor . . . . . . . . . . 9.2 Matter sources . . . . . . . . . . . . . . . . . . . 9.3 Energy-momentum tensor of gravitational waves 9.4 Power of emitted radiation . . . . . . . . . . . . .

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30 30 30 31 32

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10 Derivation: gravitational waves in flat spacetime A GNU Free Documentation License A.0 Applicability and definitions . . . . A.1 Verbatim copying . . . . . . . . . . A.2 Copying in quantity . . . . . . . . A.3 Modifications . . . . . . . . . . . .

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34

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35 36 36 36 37

This is a complete set of solutions to the exercises offered in the lecture course “General Relativity” (Allgemeine Relativitätstheorie) taught in the winter semester of 2006-2007 by Prof. V. F. Mukhanov at the Department of Physics, Ludwig-Maximilians-University, in Munich, Germany. A good textbook corresponding to the level of this course: General Relativity: An Introduction for Physicists by M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby (Cambridge University Press, 2006). Many of the exercises and all of the solutions were authored by Dr. Sergei Winitzki. This file contains only the solutions; the problem statements are available separately. This text is copyrighted by Sergei Winitzki (2008) and distributed under the GNU Free Documentation License. In particular, it is permitted to print, copy, and distribute the entire text (or parts thereof) at no charge. The text may be modified at will, and modified versions may be distributed again under the terms of the same license. The license also stipulates that the entire machine-readable and human-editable source code for this text should be made available when the text is distributed in any form. The full text of the license is in Appendix A. The full LATEX source for this text is attached inside the PDF file.

1 1.1

Coordinates and 1-forms Invertible transformations

The inverse function theorem guarantees that the equations ξ α = ξ α (x) are solvable near a point x0 if det ∂ξ α (x)/∂xβ 6= 0 at x0 . Under this condition, the coordinate transformation is invertible at x0 . Note: we are inverting not just one function ξ = ξ(x), but we are determining x from a system of n equations, say ξ α (x) = C α , where C α are n given values.

1.2

Examples of coordinate transformations

1. a) Since x = u 1 + v 2 + u3 /3, it is clear that x has range (−∞, +∞) for any fixed value of v as u varies in the range (−∞, +∞). Similarly, y has the range (−∞, +∞). To verify that the coordinate system (x, y) covers the entire plane, it is sufficient to show that x has the full range at every fixed value of y. It is sufficient to consider y0 > 0 (else change v → −v). At fixed y = y0 > 0, we have y0 = v + vu2 + v 3 /3 and thus the admissible values of u are from −∞ to +∞,

2

3 while the admissible values of v are from 0 to v = vmax such that y0 = vmax + 31 vmax . Then s y0 − v − 13 v 3 y0 − v − 31 v 3 (we have ≥ 0 for 0 < v < vmax ), u=± v v s 1 3 y − v − v y0 − v − 13 v 3 0 3 x = ± 1 + v2 + 3v v s y0 − v − 31 v 3 2 8 2 y0 + v + . =± 3 9 3v v

We have now expressed x as a function of v, i.e. x = x(v). When v varies from 0 to vmax , x(v) varies from ±∞ to 0. Since x(v) is nonsingular for v > 0, it follows that x has the full range. Therefore, the coordinates (x, y) cover the entire two-dimensional plane. b) The coordinate transformation is nonsingular if det

∂ (x, y) 6= 0. ∂ (u, v)

Compute: det

∂x ∂u ∂x ∂v

∂y ∂u ∂y ∂v

1 + u2 + v 2 2uv 2uv 1 + u2 + v 2 2 = 1 + 2 u2 + v 2 + u2 − v 2 > 0.

= det

(1)

Therefore there are no singular points. 2. a) To determine the range, first consider φ = 0. Then x = r sinh θ, y = 0, z = r cosh θ. It is clear that z 2 − x2 = r2 . Since r ≥ 0, the coordinates (x,p y, z) cover only the domain |z| > |x|. With arbitrary φ, it is clear that the coordinates (x, y, z) cover the domain |z| > x2 + y 2 . b) Compute the determinant:   sinh θ cos φ r cosh θ cos φ −r sinh θ sin φ det  sinh θ sin φ r cosh θ sin φ r sinh θ cos φ  = r2 sinh θ. cosh θ r sinh θ 0

The coordinates are singular if r = 0 or θ = 0. c) The singularity at r = 0 is due to the fact that the set {r = 0, θ, φ} corresponds to a single point p x = y = z = 0. This is similar to the singularity of the spherical coordinates at r = 0. Points along the cone |z| = x2 + y 2 are not covered because they correspond to θ → ∞, r → 0. The singularity at θ = 0, r 6= 0 is due to the fact that the set {r, θ = 0, φ} corresponds to the point x = y = 0, z = r at fixed r 6= 0. This is similar to the polar coordinate singularity. 3. a) To determine the range, note that r sin θ ≥ 0 for the given range of θ and r. However, this is immaterial since the factors cos φ and sin φ will make x, y cover the full range (−∞, +∞). The coordinates (x, y, z) are a slight modification of the standard spherical coordinates. These coordinates cover the whole space (x, y, z). b) Compute the determinant:   − sin θ cos φ −r cos θ cos φ r sin θ sin φ det  − sin θ sin φ −r cos φ sin φ −r sin θ cos φ  = r2 sin θ. cos θ −r sin θ 0 This is nonzero unless r = 0 or θ = 0. c) The singularities are completely analogous to those in the spherical coordinates.

1.3

Basis in tangent space

Suppose that the vectors eα = ∂x∂α are linearly dependent, then there exist constants cα , not all zero, such that the vector cα eα equals zero. Act with this vector on the coordinate function x1 : cα eα x1 = cα

∂ 1 x = c1 . ∂xα

By assumption, cα eα = 0, therefore c1 = 0. It follows that every cα equals zero, contradicting the assumption.

1.4

Differentials of functions as 1-forms d(x) = dx, d (xy) = xdy + ydx, d 4x2 y + x3 z = 8xy + 3x2 z dx + 4x2 dy + x3 dz,

3

d x2 = 2xdx,

d (x + y) = dx + dy, p xdx + ydy d 3 x2 + y 2 = 3 p . x2 + y 2

Now let us compute dh by first finding d (arctan (x ± y)) = d arctan

2x x2 − y 2 − 1

dx ± dy

1 + (x ± y)2

, "

2dx 4x (xdx − ydy) = − 2 2 − y2 − 1 4x2 x 1 + (x2 −y2 −1)2 (x2 − y 2 − 1) −2 x2 + y 2 + 1 dx + 4xydy = . (x2 − y 2 − 1)2 + 4x2 1

#

Adding these together and noting that 2 2 2 1 + (x − y) = x2 − y 2 − 1 + 4x2 , 1 + (x + y) we get

2x dh(x, y) = d arctan (x + y) + arctan(x − y) + arctan 2 x − y2 − 1

= 0.

This means that h(x, y) is a constant. By using the tangent sum rule, we can easily show that h(x, y) = 0.

1.5

Basis in cotangent space

Note that the relation

dxα ,

∂ ∂xβ

= δ αβ

is the definition of how the 1-form dxα acts on vectors ∂/∂xβ . Now, it is clear that any 1-form is decomposed as a linear combination of the 1-forms dx1 , ..., dxn . It remains to show that all these forms are linearly independent. If this were not so, there would exist a linear combination cα dxα = 0 such that not all cα = 0. Act with this on a vector ∂/∂x1 and obtain ∂ ∂ 0 = 0, 1 = cα dxα , 1 = c1 . ∂x ∂x Therefore c1 = 0. Similarly, we find that every other cα = 0, which contradicts the assumption.

1.6

Linearly independent 1-forms

1. Two 1-forms d(ex cos y), d(ex sin y) are linearly independent for every x, y because d (ex cos y) = ex cos ydx − ex sin ydy, d (ex sin y) = ex sin ydx + ex cos ydy, and the following determinant is always nonzero, x e cos y det ex sin y

−ex sin y ex cos y

= e2x 6= 0.

2. Two 1-forms (1 + y)dx − 2xydy, 8dx are linearly independent if the following determinant is nonzero, 1 + y −2xy det = 16xy. 8 0 This happens for xy 6= 0. 3. Three 1-forms dx + dy, dx + dz, dy + dz are always linearly independent. 4. Three 1-forms dx − dy, dy − dz, dz − dx are always linearly dependent (their sum is zero).

1.7

Transformation law for 1-forms

The transformation law for 1-forms, d˜ xα =

∂x ˜α β dx , ∂xβ

under a coordinate transformation xα → x ˜α = x ˜α (xβ ), is merely a different interpretation of the definition of the 1-form α α ˜ (x) is considered a scalar function in the coordinates xβ . d˜ x (see Problem 1.4), where x

4

1.8

Examples of transformations

a) First compute dx and dy: dx = 1 + u2 + v 2 du + (2uv) dv, dy = (2uv) du + 1 + u2 + v 2 dv.

Then it is easy to compute xdx + ydy,etc. For instance,

2 (du + dv) 1 + (u + v) dx + dy 1 =− d 2 . 2 =− x+y 2 2 (x + y) (u + v) 1 + 31 (u + v)

b) The component transformation matrix is given in Eq. (1).

1.9

Supplementary problem sheet

1A Tangent plane If the tangent plane is at angle α with the horizontal, then the acceleration is g sin α (from elementary mechanics). Since 0 ≤ α < π2 , we need to maximize α or, equivalently, tan α, which equals ∂z rh 1 2 −2 = 2 e− 2 r σ , ∂r σ

r≡

p x2 + y 2 .

1

The maximum of ∂z/∂r is at r0 = σ. For example, a point with maximum acceleration is x0 = σ, y0 = 0, z0 = −he− 2 . The tangent plane at a point (x0 , y0 , z0 ) is given by the equation nx (x − x0 ) + ny (y − y0 ) + nz (z − z0 ) = 0, where (nx , ny , nz ) are the components of the normal vector, 1 h ∂z ∂z , , 1 = e− 2 , 0, 1 . (nx , ny , nz ) = ∂x ∂y σ x0 ,y0 ,z0

Therefore the equation of the tangent plane is

x − 12 z e + = 0. σ h

1B Induced metric The induced metric is found by taking the expression for the bulk metric, g = dx2 + dy 2 + dz 2 , and computing dx, dy, dz through the forms dθ and dφ: dx = 2a sin θ cos θ cos φdθ − a sin2 θ sin φdφ,

dy = 2a sin θ cos θ sin φdθ + a sin2 θ cos φdφ, dz = 2a cos 2θdθ.

Therefore

g = dx2 + dy 2 + dz 2 = a2 dθ2 + a2 (sin θ)4 dφ2 .

The metric is degenerate at θ = 0 and θ = π2 . The singularities at these points are not merely coordinate singularities that disappear when choosing a different coordinate system; but the reason is subtle. To figure out the nature of these singularities, let us visualize the surface in a neighborhood of θ = 0. The y = 0 section of the surface corresponds to sin φ = 0, so φ = 0 or φ = π. Then √ 1 − 1 − 4z 2 1 − cos 2θ =± . x = ± sin2 θ, y = sin θ cos θ, x = ± 2 2 This is a union of two circles touching at x = z = 0. Hence, the surface is a torus with zero inner radius, i.e. intersecting itself at x = y = z = 0. The rotational symmetry around the z axis leads to a “cusp” at θ = 0: the surface has a sharp corner and the metric cannot be made smooth and nondegenerate by any choice of local coordinates. The situation near θ = π2 is similar.

5

1C Embedded waves The surface is defined by sin v y= √ , 2 − sin u

cos v , x= √ 2 − sin u 1. For y = 0, we have v = 0 and thus

cos u z=√ . 2 − sin u

1 cos u x= √ ,z = √ . 2 − sin u 2 − sin u

To visualize this line in the (x, z) plane, we eliminate u from these equations and find sin u =

√ 1 2− , x

√ 2 x − 2 + z 2 = 1.

√ Therefore the line is a circle of radius 1 centered at x = 2, z = 0 . This circle does not intersect the z axis since √ √ −1 by multiplying with cos v and sin v. Therefore, 2 > 1. Now we see that (x, y) is obtained from 2 − sin u the full surface is a rotation surface, where we need to use the x coordinate as the radius. Therefore, the figure in the (x, z) plane needs to be rotated around the z axis. The resulting surface is a torus. It may be described by the equation p √ 2 x2 + y 2 − 2 + z 2 − 1 = 0. Note that

p 1 x2 + y 2 = √ > 0. 2 − sin u

Also

√ √ √ p √ 1 − 2 2 − sin u 2 sin u − 1 2 2 √ x +y − 2= = √ . 2 − sin u 2 − sin u

2. Since the surface is now given by an equation of the form F (x, y, z) = 0, the normal vector (up to a constant factor C) can be found as ! p p √ √ ∂F ∂F ∂F x2 + y 2 − 2 x2 + y 2 − 2 x, 2 p y, 2z . (nx , ny , nz ) = C =C 2 p , , ∂x ∂y ∂z x2 + y 2 x2 + y 2 Multiplying by C ≡

p x2 + y 2 (this factor is chosen for simplicity), we have p √ p √ p (nx , ny , nz ) = x x2 + y 2 − 2 , y x2 + y 2 − 2 , z x2 + y 2 .

1 2

Expressed through the coordinates (u, v), this becomes nα = (nx , ny , nz ) =

! √ 2 sin u − 1 2 sin u − 1 cos u cos v √ 2 , sin v √ 2 , √ 2 . 2 − sin u 2 − sin u 2 − sin u √

The equation of the tangent plane at point x0 is

where nα must be computed at x = x0 .

nα xα − xα (0) = 0,

3. We compute

Now we can expand

∂x cos v cos u = √ 2 , ∂u 2 − sin u sin v cos u ∂y = √ 2 , ∂u 2 − sin u √ ∂z 1 − 2 sin u = √ 2 , ∂u 2 − sin u dx =

∂x sin v , = −√ ∂v 2 − sin u cos v ∂y , = √ ∂v 2 − sin u ∂z = 0. ∂v

∂x ∂x du + dv, ∂u ∂v

6

etc.

Therefore the induced metric is 

g = dx2 + dy 2 + dz 2 =  du2 + dv 2 = √ 2 . 2 − sin u

√

cos u 2 − sin u

2

!2

+

!2  √ dv 2 1 − 2 sin u 2 √ 2  du + √ 2 2 − sin u 2 − sin u

The vector V a = (cos v, sin v) is not a unit vector because cos2 v + sin2 v 1 g(V, V ) = √ 2 = √ 2 6= 1. 2 − sin u 2 − sin u

The Cartesian components of the vectors ∂/∂u, ∂/∂v are found from ∂ ∂x(u, v) ∂ ∂y(u, v) ∂ ∂z(u, v) ∂ = + + , ∂u ∂u ∂x ∂u ∂y ∂u ∂z

∂ ∂x(u, v) ∂ ∂y(u, v) ∂ ∂z(u, v) ∂ = + + . ∂v ∂v ∂x ∂v ∂y ∂v ∂z

Therefore, the vector V a has the following Cartesian components, ∂ ∂ sin v cos v V = cos v − √ +√ 2 − sin u ∂x 2 − sin u ∂y

! √ cos v cos u ∂ sin v cos u ∂ 1 − 2 sin u ∂ + sin v + √ + √ √ 2 2 2 2 − sin u ∂x 2 − sin u ∂y 2 − sin u ∂z ! √ √ ! sin2 v cos u + 2 cos2 v − sin u cos2 v ∂ sin u + cos u − 2 ∂ + = sin v cos v √ √ 2 2 ∂x ∂y 2 − sin u 2 − sin u √ 1 − 2 sin u ∂ + sin v √ . 2 2 − sin u ∂z

This vector is within the tangent plane because nα V α = 0, nα V

α

√

√ ! 2 sin u − 1 sin u + cos u − 2 = cos v √ √ 2 sin v cos v 2 2 − sin u 2 − sin u ! √ √ √ sin2 v cos u + 2 cos2 v − 2 sin u cos2 v 2 sin u − 1 + sin v √ √ 2 2 2 − sin u 2 − sin u √ cos u 1 − 2 sin u + √ 2 sin v √ 2 2 − sin u 2 − sin u

(after simplification) = 0.

2 2.1

Tensors Definition of tensor product

a),b) A direct calculation using the property dxi , ∂/∂xk = δki gives: ∂ hω1 , v1 i = dx + 2ydy, 3 = 3, ∂x

etc.

The results: T (v1 , v1 ) = 0,

T (v1 , v2 ) = 6x.

c) d) First, show that the sum of two linear functions is again a linear function: If A(x) and B(x) are linear functions, i.e. if A(x + λy) = A(x) + λA(y) and likewise for B, then A + B obviously has the same property. Now, since a tensor is defined as a multi-linear function, it is clear that tensors form a vector space.

7

2.3

Example of tensor

a) An obvious example of such T is the vector product, T (u, v) = u× v, defined in three-dimensional space. To determine the rank of T , we need to represent T as a multilinear number-valued function of some number of 1-forms and vectors, e.g. A(f1 , ..., fr , v1 , ..., vs ). It is clear that T (v1 , v2 ) itself is not such a function because it has vector values instead of scalar (number) values. So we need to add a 1-form to the list of arguments. We can define A(f1 , v1 , v2 ) = hf1 , T (v1 , v2 )i and then it’s clear that A is multilinear. Therefore T is a tensor of rank 1+2. α b) The calculation may go as follows. We need to determine the components Tβγ such that α

α β γ [T (u, v)] = Tβγ u v .

So we rewrite the given definition of T (u, v) in the index notation, e.g. like this: α

[T (u, v)] = 2εα βγ uβ v γ − uα nβ v β . α Now we would like to move uβ v γ out of the brackets and so determine Tβγ . However, the expression above contains α β β γ u v instead of u v . Therefore we rename the index β to γ and also introduce a Kronecker symbol δβα , so as to rewrite identically uα nβ v β = uβ v γ nγ δβα .

Therefore α [T (u, v)] = 2εα βγ uβ v γ − uβ v γ nγ δβα = 2εα βγ − nγ δβα uβ v γ , α Tβγ = 2εα βγ − nγ δβα .

2.5

Contraction of tensor indices

a) Using the definition of a tensor as a multilinear function, it is easy to show that linear combinations of tensors are also multilinear functions. Tensor products and contractions are also multilinear. The arguments are much simpler than the proof of tensor transformation law for components. b) Contracting two lower indices, e.g. Tααβ , gives components of a quantity which is not a tensor because these components do not transform correctly under changes of basis. If Tααβ were a tensor it would transform as ∂xλ T˜ααγ = Tµµλ . ∂x ˜γ However, this does not agree with the contraction of the tensor Tαβγ , which transforms as ∂xλ ∂xµ ∂xν Tλµν . T˜αβγ = ∂x ˜α ∂ x ˜β ∂ x ˜γ The contraction over α = β yields X

T˜ααγ =

α

X ∂xλ ∂xµ ∂xν ∂xλ T = 6 Tµµλ . λµν ∂x ˜α ∂ x ˜α ∂ x ˜γ ∂x ˜γ α

c) Calculation gives because εα αγ = 0 and δαα = 3.

2.8

α Tαγ = 2εα αγ − nγ δαα = −3nγ

Examples of spaces with a metric

a) We perform the calculation in components, g(u, v) = uα v α − εαβγ εα λµ uβ nγ v λ nµ . We would like to write g(u, v) = gαβ uα v β , where gαβ are the components of the metric tensor. Using the known identity for the ε-symbol, εαβγ εα λµ = δβλ δγµ − δβµ δγλ , we find g(u, v) = uα v α − (δβλ δγµ − δβµ δγλ ) uβ nγ v λ nµ = uα v α − uλ v λ nµ nµ + uλ nλ vµ nµ . 8

Denote n2 ≡ nµ nµ ≡ g(n, n), and then we need to relabel indices such that v α uβ can be moved outside the parentheses. The result is g(u, v) = δαβ − n2 δαβ + nα nβ v α uβ . Therefore

gαβ = δαβ − n2 δαβ + nα nβ .

To analyze the conditions under which det gαβ 6= 0, we can choose an orthonormal basis such that nα is parallel to the first basis vector. Then the components of the vector n in this basis are (|n| , 0, 0) and the matrix gαβ has the following simple form:   1 0 0 . 0 gαβ =  0 1 − n2 2 0 0 1−n 2 Then it is clear that det gαβ = 1 − n2 . Therefore, the matrix gαβ is nondegenerate if n2 6= 1. b) Similar calculations give 3 + n2 0 gαβ = . 0 3 Therefore the determinant of gαβ is always nonzero. c) Considerations are analogous to b), except that the size of the matrix gαβ is larger. d) The metric is (dx)2 + (dy)2 + (dz)2 , and we need to express dx, dy, dz through du and dv. A calculation gives g = dx2 + dy 2 + dz 2 = R2 sinh2 u + cosh2 u du2 + R2 cosh2 u dv 2 .

2.9

Supplementary problem sheet

2A Vector equations a) The equation contains two given vectors Aα and Bα . The solution xα can be found as a linear combination of Aα , Bα , and the cross product εαβγ Aβ B γ with unknown coefficients. Using vector notation, we have x = αA + βB + γ (A × B) . Substituting this expression into the given equation, kx + x × A = B, and using the known identity (A × B) × C = B (A · C) − A (B · C) ,

(2)

we find A (kα − A · B) + B (kβ + A · A − 1) + (A × B) (kγ − β) = 0. On purpose, we write this equation in the form of linear combination of the three vectors A, B, and A × B. Since we are considering the generic case, these three vectors are independent and so each of the coefficients above must be zero: kα − A · B = 0,

kβ + A · A − 1 = 0,

kγ − β = 0.

Solving this system of equations, we find (assuming k 6= 0 in the generic case) α=

A·B , k

β=

1−A·A , k

γ=

1−A·A . k2

b) We have in vector notation x × A = B, Multiply ×C:

x · C = k.

(x × A) × C = B × C.

Simplify using the identity (2), A (x · C) − x (A · C) = kA − x (A · C) = B × C. Therefore

kA − B × C . A·C c), d) The equations have the form xα Mβα = Aβ , where Mβα is a matrix and Aβ is a known vector. The solution is x = M −1 A, where M −1 is the inverse matrix (it exists in the generic case). x=

9

2B Tensor equations a) Since the vectors Aα and Bα are a basis in two-dimensional space (they are nonzero and orthogonal), then the symmetric tensor X αβ can be written generally as X αβ = f Aα Aβ + g Aα B β + Aβ Aα + hB α B β ,

where the coefficients f, g, h are unknown. It remains to determine these coefficients. Using Aα B α = 0 and denoting 2 Aα Aα ≡ |A| , we get the system of equations X αβ Aα = B β

2

2

⇒ f |A| Aβ + g |A| B β = B β , 2

2

X αβ δαβ = 0 ⇒ f |A| + h |B| = 0. The result is g = |A|−2 , f = h = 0, so X αβ =

Aα B β + B α Aβ |A|

2

.

b) An antisymmetric tensor X αβ in three dimensions can be always expressed as X αβ = εαβγ uγ , where uγ is an unknown vector that we need to determine. We can now rewrite the conditions on X αβ in a vector form, X αβ Aα = B β X αβ Bα = 0

⇒ u×A =B ⇒ u×B= 0

It follows that u is parallel to B and then the condition u × A = B leaves the only solution u = 0, and therefore X αβ = 0 is the only admissible solution.

2C Degeneracy of the metric a) The metric can be written in the basis {dx, dy} as the matrix y2 1 + x2 g= . 1 + x2 0 2 The determinant of this matrix is − 1 + x2 which is always nonzero. b) In the basis where Aα is parallel to the first basis vector, the vector Aα has components (A, 0, 0, 0, ...) and therefore the metric gαβ has the form   −r2 0 0 ...   0 1 0   .  0 0 1   .. .. . .

The metric is degenerate if r = 0 (i.e. at the origin).

3 3.1

The Christoffel symbol Transformations 1

We are considering a flat space where Euclidean coordinates exist. Suppose {xα } and {˜ xα } are two coordinate systems, α while {ξ } is the standard Euclidean coordinate system. The Christoffel symbols are defined as ∂ 2ξν ∂xα ∂xβ ∂ 2ξν = ∂x ˜α ∂ x ˜β

Γµαβ = ˜µ Γ αβ

∂xµ , ∂ξ ν ∂x ˜µ . ∂ξ ν

˜ and Γ can be found as follows. Assuming that the functions xα (˜ The relationship between Γ x) and also x ˜α (x) are known, we may express the partial derivative operators using the chain rule, ∂x ˜α ∂ ∂ = , ∂xµ ∂xµ ∂ x ˜α ∂ ∂xα ∂ = . µ ∂x ˜ ∂x ˜µ ∂xα

10

Also we can express ∂x ˜µ ∂x ˜µ ∂xλ = . ∂ξ ν ∂xλ ∂ξ ν ˜ µ (when Γµ is known) as follows, Therefore we can calculate Γ αβ αβ ˜µ ∂xλ ˜µ = ∂x Γ αβ ∂xλ ∂ξ ν

∂xγ ∂ ∂x ˜α ∂xγ

∂xδ ∂ ∂x ˜β ∂xδ

ξν

∂x ˜µ ∂xλ ∂xγ ∂xδ ∂ 2 ξ ν ∂ 2 xδ ∂ξ ν = + β γ δ ∂xλ ∂ξ ν ∂ x ˜α ∂ x ˜β ∂xγ ∂xδ ∂x ˜ ∂x ˜ ∂x ∂ 2 xδ ∂x ˜µ ∂xγ ∂xδ λ Γ + γδ ∂xλ ∂ x ˜α ∂ x ˜β ∂x ˜β ∂ x ˜γ γ δ 2 λ µ ∂ x ∂x ˜ ∂x ∂x λ Γ + β γ = ∂xλ ∂ x ˜α ∂ x ˜β γδ ∂ x ˜ ∂x ˜

=

∂x ˜µ ∂xλ ∂ξ ν ∂xλ ∂ξ ν ∂xδ ∂x ˜µ . ∂xλ

(3)

Note that the Euclidean coordinate system ξ ν is not needed to determine the transformation of Γ.

3.2

Transformations 2

Consider a vector field uµ . Assume that ∇ν uµ obeys the correct transformation law for rank (1,1) tensors, µ ˜α ˜ α u˜β ∂x ∂ x ∇ν uµ = ∇ , ∂x ˜β ∂xν

and substitute

∂ µ u + Γµαν uα , ∂xν ∂ µ ˜µ α ˜ νu u ˜ + Γαν u ˜ . ∇ ˜µ ≡ ∂x ˜ν ∇ν uµ ≡

˜ Note that We can now express Γ through Γ. ∂x ˜α ∂xβ = δγα ∂xβ ∂ x ˜γ because the matrices are ∂ x ˜α /∂xβ and ∂xβ /∂ x ˜γ are inverse to each other. The result is ˜ρ ∂xλ ˜ν ∂ x ˜ρ ∂ 2 xλ ∂ x ˜ν ∂ x ˜ µνρ ∂ x + . Γλαβ = Γ ∂xα ∂xβ ∂ x ˜µ ∂x ˜ν ∂ x ˜ρ ∂xα ∂xβ

3.3

Covariant derivatives

The rule is that every upper index gets a +Γ and every lower index gets a −Γ. Each term with Γ replaces one index in the original tensor by one of the indices in Γ. Therefore we can write the answer as β λβ αλ T αβ γδµ;ν = ∂ν T αβ γδµ + Γα γδµ + Γλν T γδµ λν T

− Γλγν T αβ λδµ − Γλδν T αβ γλµ − Γλµν T αβ γδλ

3.4

The Leibnitz rule

Perform an explicit calculation, Aα;γ B β + Aα B β;γ = B β ∂γ Aα − Γλαγ Aλ + Aα ∂γ B β + Γβλγ B λ ; Aα B β ;γ = ∂γ Aα B β − Γλαγ Aα B β + Γβλγ Aα B λ .

This proves the required property.

3.5

Locally inertial reference frame

In this problem (unlike problem 3.1) the coordinate system {ξ α } is not a flat Euclidean coordinate system, but it is just a coordinate system which is like Euclidean at one point. Now we want to use the formula (3), which will enable us to ˜ in the coordinate system ξ, given the Christoffel symbol Γ in the original coordinate compute the Christoffel symbol Γ system {xα }. To use that formula, we need to compute some derivatives. Denoting {˜ xα } ≡ {ξ α }, we find ∂x ˜α µ µ α α µ ν α α µ = δ + x − x δ Γ = δ + x − x β β (0)µν β (0) Γ(0)µβ . (0) ∂xβ 11

The inverse derivative, ∂xα /∂ x ˜β , can be found by inverting this matrix; the result can be found simply by assuming that ∂xα µ α µ 2 = δ + x − x Aα , β µβ + O (x − x(0) ) (0) ∂x ˜β where Aα µβ is an unknown matrix. So up to quadratic terms we find ∂xα µ 2 α µ Γα . = δ − x − x β (0)µβ + O (x − x(0) ) (0) ∂x ˜β Therefore ∂ 2 xλ = −Γλ(0)βγ + O x − x(0) . ∂x ˜β ∂ x ˜γ Finally, we find ˜µ ∂ 2 xλ ∂ x ˜µ ∂xγ ∂xδ λ ˜µ = ∂x Γ + Γ γδ αβ ∂xλ ∂ x ˜α ∂ x ˜β ∂x ˜β ∂ x ˜γ ∂xλ µ µ = Γαβ − Γ(0)αβ + O x − x(0) .

At x = x0 we have Γµαβ = Γµ(0)αβ . It follows that the new Christoffel symbol is equal to zero at x = x0 . ˜ + ..., Alternatively, one can use the transformation law for the Christoffel symbol in the inverse direction, Γ = Γ... i.e. one denotes {xα } ≡ {ξ α }, {˜ xα } ≡ {xα }. This has the advantage that only derivatives ∂ 2 ξ/∂x∂x need to be computed, and not the derivatives ∂ 2 x/∂ξ∂ξ. Since all derivatives only need to be evaluated at x = x0 , the first-order derivatives ∂x/∂ξ at x = x0 can be found as the inverse matrix to ∂ξ α /∂xβ = δβα , i.e. ∂xα /∂ξ β = δβα . This considerably simplifies the calculations.

4 4.1

Geodesics and curvature Geodesics

(a) Note that d/ds is the ordinary (not “covariant”) derivative in the direction of uα . The geodesic equation can be rewritten for the 1-form uα as duα uγ uα;γ = 0 = − Γβαγ uβ uγ . ds An explicit formula for Γβαγ yields 1 (gβα,γ + gβγ,α − gαγ,β ) uβ uγ . 2 Note that gβα,γ − gαγ,β is antisymmetric in (β ↔ γ). Therefore these terms will cancel after a contraction with uβ uγ . The remaining term yields 1 Γβαγ uβ uγ = gαβ,γ uβ uγ . 2 b) We give two derivations; the first one is direct and the second one uses the property (a). Γβαγ uβ uγ = Γβ αγ uβ uγ =

4.1.1

First derivation

Note that

d gαβ uα uβ = uγ gαβ uα uβ ,γ ds where we must use an ordinary derivative instead of the covariant derivative (according to the definition of d/ds). So we find uγ gαβ uα uβ ,γ = gαβ,γ uα uβ uγ + 2gαβ uα uγ uβ,γ .

Now we need to simplify an expression containing uβ,γ . By assumption, the derivative of the vector field uα satisfies d α β γ γ α α β γ u + Γα βγ u u = u u,γ + Γβγ u u = 0 ds 1 αλ = uγ uα (gλµ,γ + gλγ,µ − gµγ,λ ) uγ uµ . ,γ + g 2 Therefore 1 αλ uα uγ uα (gλµ,γ + gλγ,µ − gµγ,λ ) uγ uµ ,γ = − uα g 2 1 1 = − uλ (gλµ,γ + gλγ,µ − gµγ,λ ) uγ uµ = − gλµ,γ uλ uµ uγ . 2 2 What remains is a straightforward computation: uγ gαβ uα uβ ,γ = gαβ,γ uα uβ uγ + 2gαβ uα uγ uβ,γ = gαβ,γ uα uβ uγ − gλµ,γ uλ uµ uγ = 0.

12

4.1.2

Second derivation

We write

d d gαβ uα uβ = g αβ uα uβ = uγ g αβ ,γ uα uβ + 2g αβ uα,γ uβ uγ . ds ds Now we need to express uα,γ uγ ≡ duα /ds. To do that, we use the property derived in (a), duα 1 = gβγ,αuβ uγ , ds 2 and find

dg αβ dg αβ dgλγ λ γ d g αβ uα uβ = uα uβ + g αβ gλγ,α uλ uβ uγ = uα uβ + u u . ds ds ds ds It remains to express the derivative of g αβ through the derivative of gαβ . We use the identity d d α g αβ gβγ = δ = 0, ds ds γ

thus

dgβγ αβ dg αβ dgλµ αλ βµ dg αβ gβγ = − g ⇒ =− g g . ds ds ds ds

Therefore

dg αβ dgλµ αλ βµ dgλµ λ µ uα uβ = − g g uα uβ = − u u , ds ds ds

and thus

4.2

d g αβ uα uβ = 0. ds

Commutator of covariant derivatives

First compute α α λ uα ;β = u,β + Γβλ u , α α λ uα ;βγ = u,β + Γβλ u

=

uα ,βγ

+

,γ

λ Γα βλ,γ u

µ µ α λ λ − Γµβγ uα + Γα ,µ + Γµλ u µγ u,β + Γβλ u

µ µ µ λ α α λ α + Γα βλ u,γ + Γµγ u,β + Γµγ Γβλ u − Γβγ u;µ .

α Since we want to compute the commutator uα ;βγ − u;γβ , we can omit the terms that are symmetric in (β ↔ γ). These terms are the following: µ λ α uα Γα Γµβγ uα ,βγ , βλ u,γ + Γµγ u,β , ;µ .

The remaining terms are

µ α λ α λ uα ;βγ = Γβλ,γ u + Γµγ Γβλ u + (symmetric in β ↔ γ) ,

which yields

µ µ α α λ α α α λ Γα uα λβ,γ − Γλγ,β + Γλβ Γγµ − Γλγ Γβµ = u Rλγβ . ;βγ − u;γβ = u

4.3

Parallel transport

The parallel-transported vector can be represented by a 1-form Aα (s) such that dAα − Γβαg Aβ uγ = 0, ds

uγ ≡

dxγ . ds

However, the closed curve is assumed to cover only a very small neighborhood of one point x0 , so we can approximate Aα by a constant, Aα (x0 ), along the curve. Therefore I I I dAα δAα = ds = Γβαγ (x)Aβ (x)dxγ ≈ Aβ (x0 ) Γβαγ (x)dxγ . ds Now, in a locally inertial system at x0 we have Γβαγ (x0 ) = 0. Therefore we can Taylor expand Γβαγ (x) near x0 as Γβαγ (x) = xλ − xλ0 Γβαγ,λ + O (x − x0 )2 .

Therefore

δAα ≈ Aβ (x0 )

I

Γβαγ,λ xλ − xλ0 dxγ ≈ Aβ (x0 )Γβαγ,λ (x0 )

I

xλ dxγ ,

where we have again approximated Γβαγ,λ (x) by its value at x = x0 , and also used the identity I I I d xγ xλ = 0 = xγ dxλ + xλ dxγ . 13

H

dxγ = 0. Further,

Therefore we may rewrite δAα ≈

4.4

Aβ Γβαγ,λ

1 x dx = Aβ Γβαγ,λ − Γβαλ,γ 2 I 1 β xλ dxγ . = Aβ Rαλγ 2

I

λ

γ

I

xλ dxγ

Riemann tensor

a) It is more convenient to consider the fully covariant tensor Rαβγδ . This tensor has the following symmetries, Rαβγδ = −Rβαγδ = −Rαβδγ , Rαβγδ + Rβγαδ + Rγαβδ = 0,

(4) (5)

Rαβγδ = Rγδαβ .

(6)

However, it is known that the property (6) follows from (4)-(5), therefore it is sufficient to consider these two properties [note that (5) does not follow from (4), (6)]. Let us first consider the property (4). For fixed γ, δ, we have that Rαβγδ is an antisymmetric n × n matrix (indices α, β). This matrix has 12 n(n − 1) independent components. Likewise for fixed α, β. Therefore, the number of independent components of Rαβγδ is reduced to 2 1 n(n − 1) . N1 = 2

Now we use the property (5). Let us see whether the property (5) is nontrivial at fixed δ. If α = γ, then the property (5) becomes Rαβαδ + Rβααδ + Rααβδ = 0 (no summation), which is already a consequence of (4). Likewise for β = γ or for α = β. Therefore, the property (5) is a new constraint only if all three indices α, β, γ are different (i.e. α 6= β, α 6= γ, β 6= γ). Suppose that α, β, γ are different. There are N2 =

1 n(n − 1)(n − 2) 6

choices of such α, β, γ. Therefore, for each δ = 1, ..., n we obtain N2 additional constraints. Finally, let us check that every such constraint is nontrivial for every δ (even if δ is equal to one of α, β, γ). Suppose δ = α, then (5) becomes Rαβγα + Rβγαα + Rγαβα = 0 (no summation). This is a nontrivial constraint (equivalent to Rαβαγ = Rαγαβ ). Therefore, the number of constraints is nN2 , and thus the total number of independent components of Rαβγδ is n2 n2 − 1 . N = N1 − nN2 = 12 b) Weinberg, Chapter 6, §8 c) There is only one independent component of Rαβγδ in two dimensions. For instance, we can choose R1212 as the independent parameter. Then we can express the Ricci tensor as Rαβ = g λµ Rλαµβ . Calculating component by component, we find R11 = g 22 R1212 , R = g αβ Rαβ Note that the matrix

R12 = −g 12 R1212 , R22 = g 11 R1212 , = 2g 11 g 22 − 2g 12 g 21 R1212 = 2gR1212 .

g 22 −g 12

−g 12 g 11

is equal to the inverse matrix to g αβ (which is gαβ ), multiplied by the determinant det g αβ ; since det g αβ = 1/g, we have 22 g −g 12 = ggαβ . −g 12 g 11 Therefore

1 1 Gαβ = Rαβ − gαβ R = ggαβ R1212 − gαβ 2gR1212 = 0. 2 2

14

4.5

Lorentz transformations

A Lorentz transformation is represented by a matrix Λα β such that β Λα λ Λµ gαβ = gλµ .

Consider an infinitesimal Lorentz transformation, Λαλ = δ αλ + εH αλ . The number of parameters in Lorentz transformations is the same as the number of parameters in H αβ . The condition for H αβ is (δ αλ + εH αλ ) δ βµ + εH βµ gαβ = gλµ .

Disregarding terms of order ε2 , we find

0 = δ βµ H αλ gαβ + δλα H βµ gαβ = Hµλ + Hλµ . Therefore, Hλµ is an antisymmetric n × n matrix, which has 12 n(n − 1) independent components. For n = 4 we get 6 components. These can be interpreted as three spatial rotations and three Lorentz rotations (boosts).

5 5.1

Gravitation theory applied Redshift

In the weak field limit, the Newtonian gravitational potential near a mass M is Φ=

GM , r

while the component g00 of the metric is

2Φ . c2 (We write the units explicitly.) Therefore the redshift factor z(r) at distance r from the center of the Earth is r 2GM GM z(r) = 1 + 2 ≈ 1 + 2 . c r c r g00 = 1 +

To compare the redshift factors at the surface of the Earth, denote by RE the radius of the Earth. We know that the gravitational acceleration at the surface is m GM gE = 2 ≈ 9.81 2 . RE s 2 Therefore, it is convenient to express GM = gE RE . For a vertical distance L between sender and receiver, we find z(RE ) gE RE L RE 1 + gE RE c−2 gE RE = 1+ 2 1− = . ≈1+ RE 2 −2 z(RE + L) c R + L c (RE + L) 1 + gE RE c RE +L E

Since in our problem L ≪ RE , we may approximate z(RE ) L gE L ≈ 1 + 2 ≈ 1 + 1.1 16 . z(RE + L) c 10 m

5.2

Energy-momentum tensor 1

In the nonrelativistic limit, we may disregard gravitation; gαβ = ηαβ . The EMT of an ideal fluid is T αβ = −pη αβ + (p + ρ) uα uβ , where uα is the 4-velocity vector of the fluid motion. In the nonrelativistic limit, uα ≈ (1, ~v), where ~v is the 3-vector of velocity and |~v | ≪ 1 in the units where c = 1. The conservation law is 0 = T αβ ,β = −p,α + (p + ρ),β uα uβ + (p + ρ) uα uβ ,β + (p + ρ) uα ,β uβ . Let us simplify this expression by introducing the time derivative along the fluid flow, d ≡ uα ∂α . dt 15

Then we find 0 = −p,α + uα p˙ + ρ˙ + (p + ρ) uβ ,β + (p + ρ) u˙ α .

(7)

ρ˙ + (p + ρ) uβ ,β = 0.

(8)

α

Contracting with uα and using uα u˙ = 0, we find

This is the relativistic continuity equation. Using this equation, we find from Eq. (7) that 0 = −p,α + uα p˙ + (p + ρ) u˙ α .

(9)

Now let us apply the nonrelativistic limit, uα ≈ (1, ~v), to Eqs. (8) and (9). In our notation, for any quantity X we have ∂ d X˙ ≡ X ≡ X + (~v · ∇) X. dt ∂t The continuity equation (8) gives dρ + (p + ρ) div~v = 0. dt This is the ordinary, nonrelativistic continuity equation.1 Finally, Eq. (9) gives ~ + ~v p˙ + (p + ρ) ~v˙ = 0. ∇p (Note that p,j = −∇j p.) This is the Euler equation, d~v 1 = dt p+ρ

5.3

~ − ~v dp . −∇p dt

Energy-momentum tensor 2

Compute the covariant derivative, T

αβ

;α

1 = Φ Φ − g αβ Φ;λ Φ;λ 2 =Φ

;α

;α

;β

;α Φ

;β

;α

+Φ Φ

;β

;α

= Φ;α ;α Φ;β .

−g

;α αβ

Φ;λ Φ;λα

Here we used Φ;αβ = Φ;βα which follows from Γλαβ = Γλβα (note that Φ is a scalar; covariant derivatives do not commute when applied to vectors!) and also the property Φ;α ;β X α = Φ;αβ X α , which is due to gαβ;µ = 0. Therefore, we get

Φ;α ;α Φ;β = 0;

;β this entails Φ;α ;α = 0 (since Φ;β = 0 is a weaker condition than Φ;α = 0 then also Φ;α ;α = 0, so it is ;α = 0, i.e. if Φ sufficient to write the latter).

5.4

Weak gravity

A very short solution is to write R00 directly through Γ and note that only Γα 00,α comes in (if we disregard terms of second order). Then compute Γα explicitly through Φ. (Assume that g = 0.) We may disregard terms of order ΓΓ αβ,0 00 because Γ is of order Φ, and also we may raise and lower indices using ηµν instead of gµν . (This is somewhat heuristic; see below.) The calculation goes like this: α R00 = Γα 00,α − Γ0α,0 + O( Γ Γ ), 1 αβ Γα (g0β,0 + gβ0,0 − g00,β ) = −η αβ Φ,β , 00 = η 2

therefore (using Φ,0 = 0) R00 = −η αβ Φ,αβ = −Φ = −Φ,00 + Φ,11 + Φ,22 + Φ,33 = ∆Φ. Here is another, somewhat more comprehensive solution. In the weak field limit, we write gµν = ηµν + hµν . 1 Note that in the usual, nonrelativistic continuity equations as they are written in most books, there is no p + ρ - just ρ. This is so because in most cases the matter is nonrelativistic, so p ≪ ρ and p + ρ ≈ ρ. This is, however, not true for relativistic matter, such as photons (electromagnetic radiation) for which p = 13 ρ.

16

Then we only compute everything up to first order in hµν . Therefore, we may raise and lower indices using the Minkowski metric ηαβ instead of gαβ . Note: the Newtonian limit does not determine the components of gµν except for g00 = 1 + 2Φ. The actual metric gµν is equal to ηµν plus a small first-order deviation, hµν , but this deviation cannot be expressed just through the Newtonian potential Φ ≡ 21 h00 . In principle, one needs to solve the full Einstein equations to find hµν ; in other words, one needs to determine other, “post-Newtonian potentials” and not just the Newtonian potential Φ. However, when one only wants to compute effects of gravitation on motion of slow bodies, only g00 is necessary. So it is sufficient to compute just the Newtonian potential Φ. But e.g. trajectories of light rays cannot be computed accurately in the Newtonian limit (because light does not move slowly). To compute trajectories of light rays, one needs all components of hµν , not just h00 . Let us do the computation through hµν in a more general way. First we compute the Christoffel symbol and the Ricci tensor: 1 λγ η (hαγ,ν + hνγ,α − hαν,γ ) ; ⇒ Γ ∼ O(Φ); 2 = Γλαβ,λ − Γλαλ,β + ΓΓ − ΓΓ ≈ Γλαβ,λ − Γλαλ,β .

Γλαν = λ Rαβ = Rαλβ

(We may disregard the ΓΓ terms since they are second order in Φ.) Now we compute (again up to first order in Φ) 1 1 λγ η (hαγ,λβ + hλγ,αβ − hαλ,γβ ) = η λγ hλγ,αβ , 2 2 1 = η λγ (hαγ,βλ + hβγ,αλ − hαβ,γλ − hλγ,αβ ) . 2

Γλαλ,β = Rαβ

Let us now compute just the component R00 , recalling that hµν is time-independent (so hµν,0 = 0) and h00 = 2Φ: 1 λγ 1 η (h0γ,0λ + h0γ,0λ − h00,γλ − hλγ,00 ) = − η λγ h00,γλ 2 2 = −η λγ Φ,λγ = ∆Φ.

R00 =

5.5

Equations of motion from conservation law

We would like to rewrite the covariant conservation law T µν ;ν = 0 through ordinary derivatives. The given relations are useful; let’s derive them first. ∂ 1 ∂ √ 1 −g = − √ gg αβ gαβ,ν g =− √ ∂xν 2 −g ∂xν 2 −g 1√ = −gg αβ gαβ,ν ; 2 1 ∂ √ 1 1 Γµµν = g µα (gµα,ν + gνα,µ − gνµ,α ) = g µα gµα,ν = √ −g. 2 2 −g ∂xν Now rewrite the covariant derivative of T µν explicitly: √ 1 µν T µν ;µ = T,µ + Γµαµ T αν + Γναµ T µα = √ −gT µν ,ν + Γναβ T αβ . −g

Apply this to the given T µν :

0=T

µν

;µ

dxµ dxν (4) σ σ δ (x − x (s)) ds ds ds ,µ Z β α m0 dx dx ν +√ Γ (x)δ (4) (xσ − xσ (s)). ds −g ds ds αβ

m0 = √ −g

Z

Since in the first term the dependence on xσ is only through δ (4) , we can rewrite i i dxµ ∂ h (4) α d h (4) α α α δ (x − x (s)) δ (x − x (s)) = − ds ∂xµ ds

(this is easily understood if read from right to left) and then integrate by parts, Z Z d2 xν dxν d (4) σ σ δ (x − x (s)) = − ds 2 δ (4) (xσ − xσ (s)). ds ds ds ds Finally, 0=

√

−g µν T ;µ = m0

Z

ds

β α d2 xν ν dx dx δ (4) (xσ − xσ (s)). + Γ αβ ds2 ds ds

This is a function of xσ which should equal zero everywhere. Therefore, the integrand should vanish for every value of s, β α d2 xν ν dx dx + Γ = 0. αβ ds2 ds ds

17

Remark: in general, equations of motion do not follow from conservation law, but they do follow if there is only one field. (e.g. one fluid, or one scalar field, or some number of point particles). The situation in ordinary mechanics is similar: e.g. the equation of motion for a particle follow from the conservation of energy only if the motion is in one dimension: E= 0=

mv 2 + V (x) = const, 2

dE = (mv˙ + V ′ (x)) v dt

⇒ mv˙ = −V ′ (x).

However, equations of motion do not follow from conservation of energy if there is more than one degree of freedom. Similarly, equations of motion for say two scalar fields Φ, Ψ do not follow from the conservation of their combined Tµν . These fields have two different equations of motion, and one cannot hope to derive them from a single conservation law.

6

The gravitational field

6.1

Degrees of freedom

The electromagnetic field is described by a 4-vector potential Aµ (x). This would give 4 degrees of freedom. However, there is also a gauge symmetry, Aµ → Aµ + φ,µ , where φ(x) is an arbitrary function of spacetime. Using this gauge symmetry, we may e.g. set the component A0 (x) = 0. Then only three functions of spacetime (A1 , A2 , A3 ) are left. Hence the electromagnetic field has 3 degrees of freedom. There are additional gauge symmetries involving functions φ(x) that do not depend on time. Since these functions φ(x) are functions only of three arguments, they do not change the number of degrees of freedom.

6.2 6.2.1

Spherically symmetric spacetime Straightforward solution

A direct computation listing all the possible Christoffel symbols and components of the Ricci tensor is certainly straightforward but very long. Here is a way to compute the curvature tensor without writing individual components. Since the metric has a diagonal form, let us denote gαβ = ηαβ Aα ,

g αβ = η αβ

1 Aα

(no summation!),

(10)

where Aα ≡ eN , eL , r2 , r2 sin2 θ

(11)

is a fixed array of four functions. For this calculation, we do not use the Einstein summation convention any more; every summation will be written explicitly. However, we make heavy use of the fact that ηαβ 6= 0 only if α = β, and that ηλλ = η λλ . At the end of the calculation of the Ricci tensor Rαβ , we shall substitute the known functions Aα and use the resulting simplifications. We begin with the calculation of the Christoffel symbols, Γλαβ =

X1 µ

=

2

η λµ

1 [ηαµ Aµ,β + ηβµ Aµ,α − ηαβ Aα,µ ] Aλ

1 Aλ,α 1 Aα,λ 1 λ Aλ,β + δβλ − ηλλ ηαβ . δα 2 Aλ 2 Aλ 2 Aλ

(12)

Note that the summation over µ results in setting λ = µ due to η λµ , and that we have relations such as ηλλ ηαλ = δαλ and δαβ ηαα = ηαβ , which hold without summation. For convenience, we rewrite Eq. (12) as Aα 1 λ λ λ δ Bλ,β + δβ Bλ,α − ηλλ ηαβ Bα,λ , Γαβ = 2 α Aλ where we defined the auxiliary function Bα ≡ ln Aα . As a check, we compute the “trace” of the Christoffel symbols and compare with the known formula, # " X X √ 1 1X Aα λ Γαλ = Bα,α + Bλ,α − Bα,α = Bλ,α = ln −g ,α . 2 Aα 2 λ

λ

Let us also denote for brevity C ≡ ln

√

−g =

λ

1X Bλ ; 2 λ

18

X λ

Γλαλ = C,α .

We proceed to the computation of the Ricci tensor. We use the formula (with Landau-Lifshitz sign conventions) X X µ λ Rαβ = Γλαβ,λ − Γλαλ,β + Γλµ Γαβ − Γλαµ Γµβλ . λ

λ,µ

We now compute the necessary terms: X

Γλαβ,λ =

λ

λ

X

X 1 1 1 Bα,αβ + Bβ,αβ − ηαβ ηλλ 2 2 2

Aα Bα,λ Aλ

, ,λ

Γλαλ,β = C,αβ ,

λ

X

Γµλµ Γλαβ

λ,µ

1 λ Aα 1 1 λ Bα,λ = C,λ δα Bλ,β + δβ Bλ,α − ηλλ ηαβ 2 2 2 Aλ λ # " X 1 Aα = C,α Bα,β + C,β Bβ,α − ηαβ C,λ ηλλ Bα,λ , 2 Aλ X

λ

X λ,µ

Γλαµ Γµλβ

Aλ Aα 1X λ µ µ λ Bα,λ δλ Bµ,β + δβ Bµ,λ − ηµµ ηλβ Bλ,µ δα Bλ,µ + δµ Bλ,α − ηλλ ηαµ = 4 Aλ Aµ λ,µ Aα 1X (here set λ = α) Bα,µ Bα,µ δαµ Bµ,β + δβµ Bµ,α − ηµµ ηαβ = 4 µ Aµ 1X Aµ + (here set λ = µ) Bµ,µ Bµ,α Bµ,β + δβµ Bµ,µ − ηµµ ηµβ 4 µ Aµ Aλ Aα 1X (here set µ = α) Bα,λ δλα Bα,β + δβα Bα,λ − ηαα ηλβ Bλ,α ηλλ ηαα − 4 Aλ Aα λ X 1 1 1 Aα = Bα,α Bα,β + Bα,β Bβ,α − ηαβ Bα,µ Bα,µ ηµµ 4 4 4 Aµ µ 1X 1 1 + Bµ,α Bµ,β + Bβ,α Bβ,β − Bβ,α Bβ,β 4 µ 4 4 X 1 1 Aα 1 Bα,λ Bα,λ + Bα,β Bβ,α ηλλ − Bα,α Bα,β − ηαβ 4 4 Aλ 4 λ X 1 1X 1 Aα = Bα,β Bβ,α − ηαβ Bα,µ Bα,µ + ηµµ Bµ,α Bµ,β . 2 2 Aµ 4 µ µ

Finally, we put all the terms together: Rαβ =

1 1 Bα + Bβ − C 2 2

,αβ

X 1 ηλλ − ηαβ Aα 2 λ

"

Bα,λ Aλ

Bα,λ + C,λ Aλ ,λ

#

1 1 1 1X + C,α Bα,β + C,β Bβ,α − Bα,β Bβ,α − Bµ,α Bµ,β . 2 2 2 4 µ We can simplify this expression by considering separately diagonal and off-diagonal components: # " X Bα,λ 1 Bα,λ + C,λ Rαα = (Bα − C),αα − ηαα Aα ηλλ 2 Aλ ,λ Aλ λ 1X 1 Bµ,α Bµ,α ; + C,α − Bα,α Bα,α − 2 4 µ ! X 1 1 Rαβ = Bα + Bβ − Bλ + (C,α Bα,β + C,β Bβ,α − Bα,β Bβ,α ) 2 2 λ

−

1X 4

µ

Bµ,α Bµ,β .

,αβ

(only for α 6= β)

19

Now we need to simplify this expression further by using the specific form of the metric (10)-(11). We have At = eN , Ar = eL , Aθ = r2 , Aφ = r2 sin2 θ; Bt = N, Br = L, Bθ = 2 ln r, Bφ = 2 ln r + 2 ln sin θ; √ N +L + 2 ln r + ln sin θ; C = ln −g = 2 N˙ + L˙ N ′ + L′ 2 C,t = , C,r = + , C,θ = cot θ, C,φ = 0. 2 2 r Note that the term 1 2

Bα + Bβ −

X λ

Bλ

!

,αβ

always vanishes when α 6= β. We find (after some omitted algebra):   ! ln r2 sin2 θ ,λ ln r2 sin2 θ ,λ 1 2 2 X  Rφφ = r sin θ + C,λ ηλλ  2 Aλ Aλ λ ,λ ′ ′ 1 1 N −L + 2 e−L − 2 ; = −r2 sin2 θ 2r r r ′ N − L′ 1 Rθθ = 1 − r2 e−L + 2 ; 2r r Rtφ = Rrφ = Rθφ = Rtθ = 0; 1 1 2 12 1 2 cot θ = 0; Rrθ = C,θ Bθ,r − Bφ,r Bφ,θ = cot θ − 2 4 2 r 4r ! 1 N˙ + L˙ ′ 1 1 ˙ ′ L˙ 2 ˙ N ′ + L′ Rtr = L − N˙ L′ − N˙ N ′ − LL = ; N + + 2 2 2 r 4 4 r ! ˙ ˙ L′ − N ′ 2 1 −N ′′ ¨ + L − N L˙ + 1 L′ ; + 2 + N′ + eL−N L Rrr = 2 r 4 2 2 r ¨ −L N˙ − L˙ N ′ − L′ 1 2 Rtt = + L˙ + eN −L N ′′ + + N′ . 2 4 2 2 r Finally, we compute the Ricci scalar, R=

X

ηλλ

λ

1 Rλλ Aλ

¨ N˙ − L˙ L − + L˙ 2 4

!

′ 1 N − L′ 2 + e−L N ′′ + N′ + 2 2 r ! ′ ′ −N ′′ 1 −N ¨ L˙ − N˙ ˙ 2 1 ′ −L ′L − N −e L+ L + 2 +N + L − e 2 r 4 r 2 2 ′ ′ 1 N − L′ N − L′ 1 1 1 −L + 2 + + 2 e−L − 2 − 2 +e r 2r r 2r r r ! ˙ − L˙ N N ′ − L′ 2 L′ − N ′ −N ¨ + L˙ −L =e − e−L −2 − 2. − N ′′ + N ′ 2 r 2 r

=e

−N

Hence, the nonzero components of the Einstein tensor are Gtt = eN −L

L′ eN + 2; r r

L˙ ; r 2 N′ eL = 2 + − 2; r r r ! −N ′ ′ ′ N˙ − L˙ N ′′ 1 N − L′ 2e ′L − N 2 −L ¨ ˙ +r −L + L ; + 2 +− +N − = −r e 2r r 2 4 2 2 ! " # ′ ′ ′ ′ ˙ − L˙ 1 2 2 N − L N L − N 2 ¨ + L˙ = − r sin θ e−N −L − e−L − . + 2 − N ′′ + N ′ 2 2 r r 2

Gtr = Grr Gθθ Gφφ

20

(13)

6.2.2

Solution using conformal transformation

A more clever way to reduce the amount of computation is to notice that the metric gµν is simplified after a conformal transformation,  −2 L  r e 0   −r−2 eN . gµν = r2 hµν = r2    1 0 sin2 θ

The metric hµν separates into the r − t components and the θ − φ components. We shall first compute the Ricci tensor for the metric hµν and then determine how Rαβ changes under a conformal transformation. The calculation of Rαβ for the metric hµν is much simpler because hµν is a direct (“block”) sum of two metrics defined on 2-dimensional spaces. It is clear that Rαβ will also be a direct sum of the corresponding two-dimensional Ricci tensors. Let us first compute the Ricci tensor for a diagonal metric γab = diag eA , eB in two dimensions; set A ≡ A1 , B ≡ A2 , and indices a, b, c,... range from 1 to 2. For a two-dimensional metric γab , we know that the Ricci tensor is proportional to γab , namely (see Problem 4.4c) Rab =

1 γab R, 2

R ≡ γ ab Rab = 2 (det γab ) R1212 .

So it is sufficient to compute say R11 , R11 = Γa11,a − Γa1a,1 + Γaba Γb11 − Γb1a Γa1b , and afterwards we will have Rab = γab

R11 ; γ11

R = γ ab Rab = 2

R11 . γ11

The necessary Christoffel symbols are found as (no implicit summation from now on!) Γa1b =

X1 c

Γ112 =

2

γ ac (γ1c,b + γbc,1 − γ1b,c ) =

1 A,2 ; 2

Γ212 =

1 B,1 ; 2

1 A1,b δ1a + δba Ab,1 − δb1 eA1 −Aa A1,a ; 2

1 1 1 Γa11 = A1,1 δ1a − eA1 −Aa A1,a ; Γ111 = A,1 ; Γ211 = − eA−B A,2 . 2 2 2 X 1 1 1 a Γba = ∂b ln (det γcd ) = (A1,b + A2,b ) = (A + B),b . 2 2 2 a Then the component R11 of the Ricci tensor is

R11 = Γa11,a − Γa1a,1 + Γaba Γb11 − Γb1a Γa1b 1 1 1 = Γ111,1 + Γ211,2 − (A + B),11 + (A + B),1 Γ111 + (A + B),2 Γ211 − Γ111 Γ111 − 2Γ112 Γ211 − Γ212 Γ212 2 2 2 1 1 1 A−B 1 1 1 1 1 1 1 e A,2 ,2 − (A + B),11 + (A + B),1 A,1 − (A + B),2 eA−B A,2 − A,1 A,1 + A,2 eA−B A,2 − B,1 B,1 = A,11 − 2 2 2 2 2 2 2 4 2 4 1 1 1 1 = − B,11 − eA−B A,22 − A,2 (A − B),2 eA−B + B,1 (A − B),1 . 2 2 4 4 We also find (note the symmetry apparent in this formula; this shows that at least this is not obviously wrong) R=2

R11 1 1 = −e−A B,11 − e−B A,22 − A,2 (A − B),2 e−B − e−A B,1 (B − A),1 . γ11 2 2

Well, we are not going to finish this calculation here, anyway. But this is roughly how it goes. Let us at least derive the useful formula below. The relationship between the curvature tensors under a conformal transformation is found as follows. First we define the conformally transformed metric for convenience as follows, g˜αβ = e2Ω gαβ . λ Then the Christoffel symbols receive a correction which is a tensor Bαβ , λ ˜ λαβ = Γλαβ + Bαβ Γ ;

λ Bαβ ≡ δαλ Ω,β + δβλ Ω,α − gαβ Ω,λ .

The Riemann and the Ricci tensors are defined (in Landau-Lifshitz sign convention) by Rλαµβ = Γλαβ,µ − Γλαµ,β + Γλµν Γναβ − Γλβν Γναµ ,

Rαβ = Rλαλβ = Γλαβ,λ − Γλλα,β + Γλλν Γναβ − Γλβν Γναλ . 21

˜λ ˜λ The same relation holds for R αµβ and Rαβ through Γαβ (note that these relations do not involve the metric gαβ explicitly). Therefore λ λ λ λ ν λ ν λ ν λ ν λ ν λ ν ˜λ R αµβ − R αµβ = Bαβ,µ − Bαµ,β + Bµν Γαβ + Γµν Bαβ + Bµν Bαβ − Bβν Γαµ − Γβν Bαµ − Bβν Bαµ ; λ λ λ ν λ ν λ ν λ ν ˜ αβ − Rαβ = Bαβ,λ R − Bλα,β + Bλν Γναβ + Γλλν Bαβ + Bλν Bαβ − Bβν Γναλ − Γλβν Bαλ − Bβν Bαλ .

We shall only compute the expression for the Ricci tensor Rαβ . As a preparation, we compute λ Bλα = δλλ Ω,α ≡ N Ω,α ,

where N = δλλ = gαβ g αβ is the number of spacetime dimensions. We shall always raise and lower indices using the original metric gαβ . So we compute term by term, ˜ αβ − Rαβ = δ λ Ω,β + δ λ Ω,α − gαβ Ω,λ − N Ω,αβ + N Ω,ν Γν + Γλ δ ν Ω,β + δ ν Ω,α − gαβ Ω,ν R α β β α αβ λν ,λ + N Ω,ν δαν Ω,β + δβν Ω,α − gαβ Ω,ν − δβλ Ω,ν + δνλ Ω,β − gβν Ω,λ Γναλ − Γλβν (δαν Ω,λ + δλν Ω,α − gαλ Ω,ν ) − δβλ Ω,ν + δνλ Ω,β − gβν Ω,λ (δαν Ω,λ + δλν Ω,α − gαλ Ω,ν ) λ ,ν ,λ ν λ λ = 2Ω,αβ − gαβ Ω,λ ,λ − gαβ,λ Ω − N Ω,αβ + N Ω,ν Γαβ + Γλα Ω,β + Γλβ Ω,α − gαβ Γλν Ω

+ 2N Ω,α Ω,β − N gαβ Ω,ν Ω,ν − Γναβ Ω,ν − Γλλα Ω,β + gβν Γναλ Ω,λ − Γλαβ Ω,λ − Γλλβ Ω,α + gαλ Γλβν Ω,ν − (2 + N ) Ω,α Ω,β + 2gαβ Ω,λ Ω,λ

λ ,ν ,ν ,λ = −gαβ Ω,λ ,λ − gαβ Γλν Ω − N gαβ Ω,ν Ω + 2gαβ Ω,λ Ω h i λ ,ν = − (N − 2) Ω,αβ − Ω,ν Γναβ + (N − 2) Ω,α Ω,β − gαβ (N − 2) Ω,λ Ω,λ + Ω,λ + Γ Ω λν ,λ ν ,λ ν ,λ ,λ + gβν Γαλ Ω + gαν Γβλ Ω − gαβ,λ Ω .

Now we note that some of the Γ terms can be absorbed into covariant derivatives, and also that the terms in the last bracket cancel, gβν Γναλ Ω,λ + gαν Γνβλ Ω,λ − gαβ,λ Ω,λ = 0,

so the resulting formula can be written more concisely as

˜ αβ − Rαβ = (N − 2) [Ω,α Ω,β − Ω;αβ ] − gαβ (N − 2) Ω,λ Ω,λ + Ω;λ ;λ . R

The modified Ricci scalar is

˜ = g˜αβ R ˜ αβ = e−2Ω g αβ Rαβ + e−2Ω g αβ (N − 2) [Ω,α Ω,β − Ω;αβ ] − gαβ (N − 2) Ω,λ Ω,λ + Ω;λ ;λ R = e−2Ω R + e−2Ω (N − 2) [Ω,α Ω,α − Ω;α ;α ] − N (N − 2) Ω,λ Ω,λ + Ω;λ ;λ = e−2Ω {R − (N − 2) (N − 1) Ω,α Ω,α − 2 (N − 1) Ω;α ;α } .

The Einstein tensor is modified as follows, ˜ αβ = R ˜ αβ − 1 g˜αβ R ˜ = Rαβ + (N − 2) [Ω,α Ω,β − Ω;αβ ] − gαβ (N − 2) Ω,λ Ω,λ + Ω;λ ;λ G 2 1 − gαβ [R − (N − 2) (N − 1) Ω,α Ω,α − 2 (N − 1) Ω;α ;α ] 2 (N − 2) (N − 3) ,α ;α = Gαβ + (N − 2) [Ω,α Ω,β − Ω;αβ ] + gαβ Ω,α Ω + (N − 2) Ω ;α . 2 Note that there is no change in Gαβ in two dimensions (since the Einstein tensor is always equal to zero).

6.3

Equations of motion

The equation for the covariant component u1 (s) is

Using the metric gαβ

du1 1 ∂ − uα uβ (gαβ ) = 0. ds 2 ∂r n o ˙ φ˙ , where ˙ ≡ d/dλ, we find ˙ r, = diag f, −1/f, −r2, −r2 sin2 θ , where f ≡ 1 − rg /r, and uα = t, ˙ θ, 1 d −f −1 r˙ − dλ 2

df ˙2 d t − dr dr

1 r˙ 2 − 2rθ˙2 − 2rφ˙ 2 sin2 θ = 0. f

To derive this equation from other equations given in the lecture, we transform in a clever way the expression i d d h ˙2 0= f t − f −1 r˙ 2 − r2 θ˙2 − r2 φ˙ 2 sin2 θ . K= dλ dλ 22

(14)

Namely, we try to separate terms of the form

For example,

d dλ

(uα ) out of the terms of the form

d d d d u1 u1 = g11 u1 u1 = 2u1 g11 u1 − u1 u1 g11 dλ dλ dλ dλ d d d f t˙2 = 2t˙ f t˙ − t˙2 f, dλ dλ dλ

We find

d dλ

(uα uα ) in the following way,

(no summation!).

etc.

i d d h ˙2 f t − f −1 r˙ 2 − r2 θ˙2 − r2 sin2 θφ˙ 2 K= dλ dλ df d d d 2 ˙ ˙2 d d r θ +θ f t˙ − t˙2 − 2r˙ f −1 r˙ + r˙ 2 f −1 − 2θ˙ r2 = 2t˙ dλ dλ dλ dλ dλ dλ d 2 2 ˙ ˙2 d 2 2 ˙ r sin θφ + φ r sin θ . − 2φ dλ dλ

0=

Now we substitute the given equations (2)-(4), and also evaluate derivatives of the metric, e.g. df /dλ = f ′ r, ˙ so d f′ ˙ 2 φ˙ 2 sin θ cos θ + θ˙2 2rr˙ + φ˙ 2 2r sin2 θr˙ + 2φ˙ 2 r2 θ˙ sin θ cos θ 0 = −t˙2 f ′ r˙ − 2r˙ f −1 r˙ − r˙ 2 2 r˙ − 2θr dλ f d f′ 2 2 2 2 −1 ′ ˙2 ˙ ˙ = r˙ −2f r˙ − f t − 2 r˙ + 2rθ + 2rφ sin θ . dλ f

This is obviously equivalent to Eq. (14). Note: the reason one of the equations follows from other equations is that the equation uα uα = const is a consequence of the four geodesic equations, uβ uα ;β = 0, and the fact that gαβ;µ = 0. Therefore, when we consider the four geodesic equations and the equation uα uα = const, any one of these five equations is a consequence of four others.

7 7.1

Weak gravitational fields Gravitational bending of light

In the lecture it was shown that the trajectory of a light ray in polar coordinates satisfies the equation 1 2GM 3 rg d2 1 + = , rg ≡ ≈ 3km, dφ2 r r 2 r2 c2 where M is the mass of the Sun. Introduce an auxiliary variable v(φ) ≡ r−1 and solve the equation v ′′ + v =

3 rg v 2 2

perturbatively, assuming that v is small, v(φ) = v0 (φ) + v1 (φ) + ... The unperturbed solution is v0 (φ) =

1 cos φ, R0

where R0 is the distance of closest approach to the Sun. Then v1′′ + v1 =

3 rg 3 rg cos2 φ = (1 + cos 2φ) . 2 2 R0 4 R02

The solution is found with undetermined coefficients, v1 (φ) = A + B cos 2φ,

A=

3 rg , 4 R02

B=−

1 rg . 4 R02

The total deflection angle is found as δ = φ1 − φ2 − π, where φ1,2 are fixed by the condition v(φ) = 0. We find a quadratic equation s 2R R R02 0 0 cos2 φ − cos φ − 2 = 0, cos φ = ± + 2. rg rg rg2 Only the solution with the minus sign is meaningful (cos φ < 1). Since rg ≪ R0 , we may expand this in Taylor series and find rg + O(rg3 /R03 ). cos φ ≈ − R0 23

Therefore, the angle φ is very close to π/2, φ1,2 = ±

π 2

+ε ,

This formula can be rewritten as δ=

ε≈

rg , R0

⇒

δ = 2ε =

2rg . R0

2rg R⊙ 2rg /R⊙ = . R0 /R⊙ R⊙ R0

For the Sun we have R⊙ = 6, 96 × 105 km and rg = 2, 954 km, therefore 2rg /R⊙ = 8, 489 × 10−6 = = 1, 751”

8, 489 × 10−6 × 360◦ /2π × 3600 [arc seconds]

(see R. Oloff “Geometrie der Raumzeit,” 2nd German edition, page 151).

7.2

Einstein tensor for weak field

For this problem Chapter 4 from the book Norbert Straumann “General Relativity and Relativistic Astrophysics” is useful. We have gµν = ηµν + hµν and Rµν = Γλµν,λ − Γλλµ,ν , where (...),µ denotes a derivative ∂µ (...). Here one can ask the students about the symmetry of this tensor. Furthermore 1 α 1 ,α hµ,ν + hα Γαµν = η αβ [hµβ,ν + hνβ,µ − hµν,β ] = ν,µ − hµν , 2 2

(15)

where as usual we use the convention that indices are raised or lowered with η µν ; thus e.g. hαβ ≡ η αλ hλβ . Using Eq. (15) we have 1 λ h + hλν,µλ − hµν − h,µν , Rµν = 2 µ,νλ where = η µν ∂µ ∂ν and h = hλλ = η λα hαλ . And for the Ricci scalar we obtain R = η µν Rµν = hλν,νλ − h. Thus in the linear approximation we have i 1 1h λ Gµν = Rµν − ηµν R = hµ,νλ + hλν,µλ − hµν − h,µν − ηµν hλβ,βλ + ηµν h . 2 2

Let us introduce a new variable γµν ≡ hµν − 21 ηµν h. The traces of two tensors h and γ are related by γ = −h, thus hµν ≡ γµν − 21 ηµν γ. Inserting the last expression for hµν in Gµν , we have i 1h λ λ γµ,νλ + γν,µλ − γµν − ηµν γ λβ,βλ = 2 i 1 h ,λ ,λ ,λβ = γµλ,ν + γνλ,µ − γµν − ηµν γβλ , 2

Gµν =

or finally

Gµν =

7.3

i 1 h µ,λ λ,µ β,λ γλ,ν + γν,λ − γνµ − δνµ γλ,β . 2

Gravitational perturbations I

The metric is written as gµν = ηµν + δgµν , i.e. g00 = 1 + 2Φ, g0i = B,i + Si ,

gij = −δij + 2Ψδij + 2E,ij + Fi,j + Fj,i + hij ,

(16)

where Si,i = Fi ,i = hij,i = hij η ij = 0, hij = hji . We shall use the formula for Gµν derived in Problem 7.2. All 3-dimensional indices are raised and lowered using δij , so we can write these indices in any position, as convenient: δgj0 = δg0j = −δg0j ,

δgij = −δgij .

Also note that for any quantity X we have We need to write the components of

˙ −X,j . X ,j = X, 1 ¯ γνµ = δgνµ − δνµ h, 2 24

¯h ≡ δg µ , µ

using the 3+1 decomposition: ¯ h = δgµµ = η µν δgµν = 2 (Φ − 3Ψ − ∆E) ,

γ00 = Φ + 3Ψ + ∆E,

γj0 = B,j + Sj = −γ0j = γ0j ,

γji = − (Φ − Ψ − ∆E) δij − 2E,ij − Fi,j − Fj,i − hij = γij . Now we compute 0 ˙ + 3Ψ ˙ + ∆ E˙ − B ; =Φ γλ0,λ = γ˙ 00 − γj,j

j λ γλj,λ = −γj,λ = −γ˙ 0j − γi,i = −B˙ ,j − S˙ j − (− (Φ − Ψ − ∆E) δij − 2E,ij − Fi,j − Fj,i − hij ),i

β,λ γλ,β

= −B˙ ,j − S˙ j + (Φ − Ψ + ∆E),j + ∆Fj ; i h ¨ + 3Ψ ¨ +∆ E ¨ − B˙ + −B˙ ,j − S˙ j + (Φ − Ψ + ∆E) + ∆Fj = γλ0,λ + γλj,λ =Φ ,j ,0

,j

,j

¨ + 3Ψ ¨ + ∆E ¨ − 2∆B˙ + ∆ (Φ − Ψ + ∆E) . =Φ

Then we compute each component of Gµν separately: 0,λ β,λ ¨ + 3Ψ ¨ +∆ E ¨ − B˙ =2 Φ 2G00 = 2γλ,0 − γ00 − δ00 γλ,β − ∂0 ∂0 (Φ + 3Ψ + ∆E) + ∆ (Φ + 3Ψ + ∆E) ¨ + 3Ψ ¨ + ∆E¨ + 2∆B˙ − ∆ (Φ − Ψ + ∆E) − Φ = 4∆Ψ;

0,λ λ,0 β,λ ˙ + 3Ψ ˙ + ∆ E˙ − B ¨,j − S¨j + Φ ˙ −Ψ ˙ + ∆E˙ 2G0j = γλ,j + γj,λ − γj0 − δj0 γλ,β = Φ −∗ −B + ∆F˙j ,j ,j ˙ ,j + ∆Sj − ∆F˙j , ¨,j + S¨j + ∆ (B,j + Sj ) = 4Ψ − B i i h h i,λ j,λ β,λ 2Gij = γλ,j + γλ,i − γji − δji γλ,β = −B˙ ,j − S˙ j + (Φ − Ψ + ∆E),j + ∆Fj + −B˙ ,i − S˙ i + (Φ − Ψ + ∆E),i + ∆Fi ,j ,i h i ¨ ¨ ¨ ˙ + ((Φ − Ψ − ∆E) δij + 2E,ij + Fi,j + Fj,i + hij ) − δij Φ + 3Ψ + ∆E − 2∆B + ∆ (Φ − Ψ + ∆E) h i ¨ + ∆ Φ − Ψ − B˙ + E ¨ . ¨ = 2 Φ − Ψ − B˙ + E + Fï,j + F¨j,i − S˙ i,j − S˙ j,i + hij − 2δij 2Ψ ,ij

λ,0 j,λ * - the origin of the minus sign here is γj,λ = −γλ,0 .

7.4

Gravitational perturbations II

Under an infinitesimal transformation xµ → xµ + ξ µ , the metric changes as γ γ gαβ → gαβ − gαγ ξ,β − gβγ ξ,α = gαβ − ξα,β − ξβ,α .

(17)

(This can be easily found from the standard formula for the change of coordinages, involving ∂ x ˜µ /∂xν .) Now let us write Eq. (17)in full, using the perturbation variables (16), the covariant components ξµ , and the decomposition ξµ = ξ 0 , ξ⊥i + ζ,i . We can write the transformation of gαβ component by component using the 3+1 decomposition, and we use the fact that the background metric is diagonal, 0 g00 → g00 − 2ξ,0 ;

g0i → g0i − ξ,i0 − ξi,0 ;

gij → gij − ξi,j − ξj,i .

To simplify calculations, we adopt the convention of raising and lowering the spatial indices i, j, ... by the Euclidean spatial metric δij rather than by ηij . This will get rid of some minus signs. We also denote ∂0 ≡ ∂t by the overdot. Thus we have g00 → g00 − 2ξ˙0 ;

g0i → g0i − ξ,i0 − ξ˙i ;

gij → gij − ξi,j − ξj,i .

Substituting the perturbation variables from Eq. (16), we get Φ → Φ − ξ˙0 ,

(18)

B,i + Si → B,i + Si −

ξ,i0

− ξ˙⊥i − ζ˙,i ,

(19)

2Ψδij + 2E,ij + Fi,j + Fj,i + hij → 2Ψδij + 2E,ij + Fi,j + Fj,i + hij − ξ⊥i,j − ξ⊥j,i − 2ζ,ij .

(20)

Now we need to separate these equations and derive the transformation laws for the individual perturbation variables. This is easy to do if we perform a Fourier transform of Eqs. (18)-(20) and pass to the Fourier space (where every variable is a function of a 3-vector k). A vector Vi is decomposed into scalar and vector components as follows, (V )

Vj = ikj V (S) + Vj

;

V (S) ≡

Vl kl , k2

(V )

Vj 25

≡ Vj − ikj

Vl kl = Vj − ikj V (S) . k2

(21)

The idea is first, to project the given vector Vi (k) onto the direction of ki , and second, to subtract the projection from Vi and to obtain the component of Vi which is transversal to ki . The imaginary unit factors are added as coefficients at kj for convenience: with these factors, the decomposition (21) translates to real space as (V )

Vj = ∂j V (S) + Vj

.

The same procedure applied to a symmetric tensor Tij leads to a decomposition into scalar, vector, and tensor components. Let us go through this procedure in more detail. First, we subtract the trace and obtain the traceless part T (1) of the tensor T , 1 (1) (1) Tij ≡ Tij − δij Tll ; Tii = 0. 3 Note the coefficient

1 3

(1)

that depends on the number of spatial dimensions (three). Now we project Tij

onto ki kj and

(2)

obtain the scalar component T (S) proportional to ki kj and the tensor Tij orthogonal to ki kj : (1)

Tij ≡

(1)

1 (2) −ki kj + δij k 2 T (S) + Tij ; 3

T (S) ≡ −

3 ki kj Tij ; 2 k4

(2)

Tij ki kj = 0.

(2)

(2)

Note that Tij is again a trace-free tensor, Tii = 0, due to the subtraction of 31 k 2 δij in the first term. Finally, we project (V )

(2) Tij

onto ki and kj separately, to obtain a “vector” part Tj

part

(T ) Tij

such that (2)

(T ) Tij kj

= 0 and

(V )

Tij = iki Tj

(T ) Tii (V )

+ ikj Ti

(V )

such that Tj

kj = 0, and a completely traceless (“tensor”)

= 0: (T )

+ Tij ;

(V )

Tj

≡i

kl (2) T , k 2 jl

(T )

Tij

(V ) (V ) (2) ≡ Tij − i ki Tj + kj Ti .

In real space, the full decomposition is 1 1 (T ) (V ) (V ) Tij = Tll δij + ∂i ∂j − δij ∆ T (S) + ∂i Tj + ∂j Ti + Tij . 3 3 3 1 1 1 1 (2) (S) T ≡ ∂i ∂j Tij − Tll δij ; Tij ≡ Tij − Tll δij − ∂i ∂j − δij ∆ T (S) ; 2 ∆2 3 3 3 1 (V ) (V ) (2) (T ) (2) (V ) Tj = ∂i Tij , Tij = Tij − ∂i Tj − ∂j Ti . ∆ It may be convenient to gather the “trace” terms (the terms containing δij ) as one term, (V )

Tij = T (tr) δij + ∂i ∂j T (S) + ∂i Tj

(V )

+ ∂j Ti

(T )

+ Tij ,

T (tr) ≡

1 1 Tll − ∆T (S) . 3 3

Note that the perturbation variables Ψ, E, Fi , hij are obtained by this decomposition method, starting from the symmetric perturbation tensor δgij , with slight modifications: there are some cosmetic factors of 2 and some minus signs. Applying the decomposition method to Eqs. (18)-(20), we get the following transformation laws for the perturbation variables, Φ → Φ − ξ˙0 ,

E → E − ζ,

˙ B → B − ξ 0 − ζ,

Ψ → Ψ,

Si → Si − ξ˙⊥i ,

Fi → Fi − ξ⊥i ,

hij → hij .

Remarks: 1. It is clear that one can set Fi = 0, B = E = 0 with a coordinate transformation. Other components will then show whether the geometry is really perturbed or it’s just a coordinate transformation of a flat space. In general, there will remain 6 independent components of perturbations (Φ, Ψ, Si , hij ). 2. These considerations depend rather crucially on the silently made assumption that all the metric perturbations vanish, δgµν → 0, at spatial infinity. These boundary conditions are implicitly used when defining the Fourier transforms necessary for the tensor/vector/scalar decompositions (a Fourier transform is undefined without this boundary condition). Alternatively, one may do without Fourier transforms but then one still needs boundary conditions to solve the relevant Poisson equations for components. Without boundary conditions, there is no unique decomposition of the form Xi = A,i + Bi ,

A=

1 Xi,i , ∆

because the function A is defined up to solutions of ∆A = 0. So the tensor/vector/scalar decomposition is actually undefined without a fixed assumption about the boundary conditions. The boundary conditions δgµν → 0 at spatial infinity is a natural, physically motivated set of boundary conditions. An explicit counterexample where these boundary conditions are not satisfied: gµν = diag (A, −B, −B, −B), where A 6= 1, B 6= 1 are constants. This metric is flat but one cannot see this by using the perturbation formalism! (The component Ψ 6= 0 cannot be removed by a gauge transformation.) The reason is that this gµν is a “perturbation” of flat metric with Φ and Ψ that do not decay to zero at spatial infinity. So a coordinate transformation with ξ µ decaying to zero cannot bring this metric to ηµν . 26

8 8.1

Gravitational radiation I Gauge invariant variables

¨ and Si − F˙i are invariant Using the equations derived in Problem 7.4, it is very easy to verify that D = Φ − Ψ − B˙ + E under infinitesimal changes of coordinates (i.e. invariant under infinitesimal gauge transformations).

8.2 8.2.1

Detecting gravitational waves Using distances between particles

(This solution follows Hobson-Efstathiou-Lasenby [2006], §18.4.) Consider a plane wave moving in the z direction, (all other components of hµν are zero) hxx = −hyy = A+ e−iω(t−z) ,

hxy = hyx = A× e−iω(t−z) .

(22)

To detect the presence of this gravitational wave, let us imagine a cloud of particles initially at rest at different positions. The 4-vectors describing the particles are uµ = (1, 0, 0, 0), so one can easily see that these particles move along geodesics: uν uµ ;ν = uν uµ ,ν + Γµνα uν uα = Γµ00 , 1 Γµνα = η λµ (hλν,α + hλα,ν − hαν,λ ) , 2 1 µ Γ00 = η λµ (hλ0,0 + hλ0,0 − h00,λ ) = 0. 2 Therefore the coordinates xµ of each particle remain constant with time. However, the distance between each pair of particles is determined through the spacelike vector ∆xµ ≡ xµ(1) − xµ(2) as ∆L2 ≈ (ηµν + hµν ) ∆xµ ∆xν and will change with time because of the dependence on hµν . Since the only nonzero components of hµν are the x, y components, it is clear that only changing lengths are between particles that have some separation in the x, y directions. Therefore it is sufficient to consider a ring of particles situated in the x − y plane. The physically measured distances between the particles in the ring will change with time, i.e. the ring will experience a deformation. To visualize the deformation, it is convenient to make a local coordinate transformation (local in the neighborhood of the ring) such that the metric becomes flat, gµν xµ xν = ηµν x ˜µ x ˜ν (up to second-order terms). The trick that performs this transformation is the following, 1 1 x ˜µ = xµ + hµλ xλ ≡ xµ + hαλ xλ η αµ . 2 2 It is easy to check that gµν xµ xν ≡ (ηµν + hµν ) xµ xν = ηµν x ˜µ x ˜ν + O(h2 ).

Therefore, x˜µ can be understood as the (approximate) Cartesian coordinates where the length is given by the usual Pythagorean formula. Now if we compute the shape of the ring in these coordinates, it will be easy to interpret this shape in a straightforward way. Consider a particle with constant 3-coordinates (x, y, z). After the coordinate transformation, we have 1 (A+ x + A× y) e−iω(t−z) , 2 1 y˜ = y + (A× x − A+ y) e−iω(t−z) , 2 z˜ = z.

x ˜ =x+

To visualize the deformation, it is convenient to consider first the case A+ 6= 0, A× = 0 and then the opposite case. The deformation of the ring is squeezing in one direction and expansion in the orthogonal direction. It follows that A+ describes a deformation in the two vertical directions, while A× describes a deformation in the directions at 45◦ . Note that the deformations change the shape of the ring in the same way, except for the rotated orientation. This can be verified by performing a rotation by π4 , 1 x ˜ 1 −1 x ˜ →√ , y˜ 1 1 y˜ 2 and then it is straightforward to see that this will exchange A+ and A× .

27

8.2.2

Using geodesic deviation equation

PLEASE NOTE: The commonly found arguments that use the geodesic deviation equation are suspect because the geodesic deviation equation uses coordinates ξ µ rather than gauge-invariant quantities. A cloud of particles at rest in the gravitational field hµν described by Eq. (22) will stay indefinitely at rest in the coordinate system (ξ µ = const) even though the distances between particles will change with time. See arxiv:gr-qc/0605033 for nice explanations. The solution given above is simple and straightforward. The argument using the geodesic deviation (see Carroll, Chapter 6, p. 152-154) goes like this: The geodesic deviation equation can be simplified for a deviation vector S σ corresponding to nonrelativistic (almost stationary) particles moving with 4-velocity approximately equal to (1, 0, 0, 0), d2 S σ = Rσ00λ S λ . dt2 The Riemann tensor to first order in h can be expressed as ¨σ Rσ00λ = h λ (note that hµ0 = 0). Therefore, the geodesic deviation equation becomes ¨σSλ, S¨σ = h λ x ¨ S = ω 2 (hxx S x + hxy S y ) = ω 2 (A+ S x + A× S y ) e−iω(t−z) , Sÿ = ω 2 (hyx S x + hyy S y ) = ω 2 (A× S x − A+ S y ) e−iω(t−z) , and there is no change in the z direction.

8.3

Poisson equation

The general solution of the Poisson equation, ∆φ = 4πρ, with boundary conditions φ → 0 at infinity, is easy to find using the Fourier transform:

−k 2 φ(k) = 4πρ(k), Z Z Z Z d3 k ik·x 4πρ(k) d3 k ik·x 4π 3 −ik·y φ(x) = − e e = − d ye ρ(y) = d3 y ρ(y)G(x − y), 3 3 k2 k2 (2π) (2π)

where G(x) is the Green’s function, Z Z π Z ∞ Z 1 ∞ 1 2 dk d3 k ik·x 4π ik|x| cos θ =− sin kx = − . dk dθ sin θ e =− G(x) = − 3e 2 k π 0 π |x| 0 k |x| (2π) 0 Here we used the known integral

Therefore

Z

∞ 0

sin z 1 dz = z 2 φ(x) = −

One can denote this integral more concisely,

Z

Z

+∞

−∞

1 sin z dz = π. z 2

d3 y ρ(y). |x − y|

(23)

1 ρ, ∆ 1 where the operator ∆ is just a shorthand notation for the integral in Eq. (23). Note that the function ρ must fall off sufficiently rapidly as |x| → ∞ or else the integral (23) will not converge. It is sufficient that |ρ(x)| ∼ |x|−2−ε at large |x| (where ε > 0). φ = 4π

8.4

Metric perturbations 1

An arbitrary 3-vector Xi (such as T 0 i ) is decomposed into scalar and vector parts as follows, Xi = a,i + bi ,

bi,i = 0.

To determine an explicit expression for a, let us compute the divergence of Xi , Xi,i = a,ii = ∆a. Therefore a(x) = − One can write more concisely

1 4π

Z

a=

d3 y Xi,i (y). |x − y| 1 Xi,i . ∆ 28

8.5

Metric perturbations 2

The energy-momentum tensor Tµν is decomposed as Ti0 = α,i + βi ,

α≡

1 0 T , ∆ k,k

βi ≡ Ti0 − (T )i

Tki = µδik + λ,ik + σi,k + σk,i + Tk βi,i = σi,i = 0,

(T )i Ti

(T )i Tk,i

= 0,

1 0 T ∆ k,k

;

,i

,

= 0.

We need to verify that the equation

1 (S˙ i − Fï ) = σi , (24) 16πG which represents the vector part of the spatial Einstein equation (here σi is the vector part of the spatial Tij ), also follows from the conservation of Tµν and from the other Einstein equations. To calculate the components λ, µ of the EMT, we compute −

Tii = ∆λ + 3µ, i Tk,i = ∆λ,k + µ,k + ∆σk , i Tk,ik = ∆∆λ + ∆µ.

Now we solve this system of equations and find 3 1 i 1 i 1 1 , λ= Tii − Tk,ik µ= T − T i, 2 ∆ 2 ∆ k,ik 2 i 1 1 i 1 i − Tk,ik , (T ) Tki = Tki − µδki − λ,ik − σi,k − σk,i . σj = Tj,i ∆ ∆ ∆ ,j Note that the operator ∆12 applied to a function f (x) is defined only if the function f has a sufficiently fast decay at −3−ε |x| → ∞. It is sufficient that |f (x)| ∼ |x| with ε > 0 at large |x|. This is a faster decay than that required by the 1 operator ∆ . The Einstein equations are

h ¨ − 1 S˜˙ i,j D,ij − δij ∆D + 2Ψ 2

where we have denoted for brevity

2∆Ψ = 8πGT00 , ˙ ,i + 1 ∆S˜i = 8πGTi0 = 8πG (α,i + βi ) , 2Ψ 2 i 1 (T )i ˙ + S˜j,i + hij = 8πGTji = 8πG µδik + λ,ik + σi,k + σk,i + Tk , 2 S˜i ≡ Si − F˙i ,

˙ D ≡ Φ − Ψ + B − E,

which are gauge-invariant variables. In the 3+1 decomposition, the Einstein equations become ∆Ψ = 4πGT00 , ˙ = 4πGα, Ψ

(25) (26)

∆S˜i = 16πGβi , D = 8πGλ, ¨ = −8πGµ, ∆D + 2Ψ

(27) (28) (29)

S˜˙ i = −16πGσi ,

hij =

(T )i 16πGTj .

(30) (31)

The conservation law of the EMT in 3+1 decomposition looks like this, j 0 T0,0 + T0,j = 0,

This gives therefore

j 0 Ti,0 + Ti,j = 0.

T˙00 = ∆α,

α˙ ,i + β˙ i + ∆λ,i + µ,i + ∆σi = 0,

T˙00 = ∆α,

α˙ + ∆λ + µ = 0,

β˙ i + ∆σi = 0.

(32)

(33)

Then it is easy to see that Eqs. (26), (29), and (30) are consequences of Eqs. (25), (27), (28), and the conservation laws (33). In particular, 1 1 S˜˙ i = ∂t 16πGβi = − 16πG∆σi = −16πGσi . ∆ ∆ 29

9

Gravitational radiation II

9.1

Projection of the matter tensor

a) First note that Pab is a projector, Pab Pbc = Pac , and its image has dimension 2, that is, the trace of Pab is 2, Pii = 3 − ni ni = 2. Therefore, for any Xab we have (T )

1 1 Xii = Pia Xab Pbi − Pii Pab Xab = Pab Xab − 2Pab Xab = 0. 2 2

b) We compute (T )

Xik,i

1 1 1 = Pia Xab Pbk − Pik Pab Xab = Pia Pbk − Pik Pab Xab + Pia Pbk − Pik Pab Xab,i . 2 2 2 ,i ,i

(34)

Note that the projection kills any component proportional to Ri because Pia Ri = 0. At the same time, Xab,i is proportional to Ri because h i Ri ~ Xab,i = X(t − |R|) = − X ′. R ,i Therefore the second term in Eq. (34) vanishes: 1 Pia Pbk − Pik Pab Ri = 0. 2

So only the first term remains, (T )

Xik,i

1 = Pia Pbk − Pik Pab Xab . 2 ,i

However, this term contains derivatives of Pab , which are also sometimes proportional to Ri . We compute Ri,a δia 1 Ri Ri Ra Ri = = Pia , − 2 |R|,a = − Pik,a = −ni,a nk − ni nk,a ; ni,a = 3 R ,a R R R R R 1 2 Pik,a = − (Pai nk + Pak ni ) , Pik,i = − nk , (note that Pak nk = 0) R R 1 1 1 = Pia,i Pbk − Pik,i Pab + Pia Pbk,i − Pik Pab,i Pia Pbk − Pik Pab 2 2 2 ,i 1 1 1 2 Pik (Pai nb + Pbi na ) = − na Pbk + nk Pab − Pia (Pbi nk + Pik nb ) + R R R 2R 1 1 1 −2na Pbk + nk Pab − Pab nk − Pak nb + Pak nb + Pbk na = R 2 2 1 3 1 =− Pbk na + Pak nb . R 2 2 ~ than Pab , as required. This is higher-order in 1/|R|

9.2

Matter sources

The question is to verify the following property, (T )

Xik = (T ) Qik ,

where Qik = Xik −

Z

1 δik r2 T 0 0 d3 r. 3

It is easy to see that Xik differs from Qik only by a term of the form A(t, R)δij . The transverse-traceless part of δij is zero, 1 Pia Pbk − Pik Pab δab = 0. 2 Therefore the transverse-traceless parts of Xik and Qik are the same.

30

9.3

Energy-momentum tensor of gravitational waves

See Hobson-Efstathiou-Lasenby [2006], §17.11. We need to compute the second-order terms in the Einstein tensor. The idea is to separate the second-order terms already in the Ricci tensor. We will also try to simplify things by using the fact that hµν is purely transverse-traceless; h0α = 0, hii = 0, hik,i = 0. It follows that η µν hµν = 0, hµν ,µ = 0. Also, it is given that the EMT of matter vanishes, Tµν = 0, which we will use below. First we decompose the metric, gµν = ηµν + hµν , g µν = η µν − hµν ;

note that now indices are always raised and lowered using ηµν . We need to compute the Ricci tensor to second order. The Christoffel symbol up to second order is 1 λµ (1)λ (2)λ η − hλµ (hµα,β + hµβ,α − hαβ,µ ) = Γαβ + Γαβ , 2 1 = η λµ (hαµ,β + hβµ,α − hαβ,µ ) , 2 1 = − hλµ (hαµ,β + hβµ,α − hαβ,µ ) . 2

Γλαβ = (1)λ

Γαβ

(2)λ

Γαβ The Ricci tensor is

(1)

(2)

Rαβ = Γλαβ,λ − Γλλα,β + Γλλν Γναβ − Γλβν Γναλ = Rαβ + Rαβ , (1)

(1)λ

(1)λ

(2)

(2)λ

(2)λ

Rαβ = Γαβ,λ − Γλα,β ,

(1)λ (1)ν

(1)λ (1)ν

Rαβ = Γαβ,λ − Γλα,β + Γλν Γαβ − Γβν Γαλ . Let us now evaluate these expressions and simplify as much as possible, as early as possible: 1 1 λµ (1)λ Γαλ = η λµ hλµ,α = η hλµ ,α = 0, 2 2 1 1 (1) (1)λ (1)λ Rαβ = Γαβ,λ − Γλα,β = η λµ (hαµ,βλ + hβµ,αλ − hαβ,µλ ) = − hαβ , 2 2 (1)

because of the transverse traceless property of hµν . Now, since Rαβ is found from the first-order Einstein equation 1 (1) Rαβ − ηαβ R(1) = 8πGTαβ , 2 and it is given that Tαβ = 0. Hence, we have hαβ = 0. Let us now evaluate derivatives of the second-order terms in the Christoffel symbols: 1 1 (2)λ Γαλ = − hλµ (hαµ,λ + hλµ,α − hαλ,µ ) = − hλµ hλµ,α , 2 2 1 λµ 1 λµ (2)λ −Γαλ,β = h hλµ,αβ + h,α hλµ,β , 2 2 1 λµ (2)λ Γαβ,λ = − h (hαµ,βλ + hβµ,αλ − hαβ,µλ ) 2 (1)λ (1)ν

(in the last line we used hλµ ,λ = 0). Finally, we tackle the term Γβν Γαλ . In this term, it helps to write 1 λ (1)λ hα,β + hλβ,α − h,λ Γαβ = αβ , 2 µν where again the indices are raised via η since we only need this term to first order. Then we can simplify this expression by grouping together terms where α, β appear in similar positions: (1)λ (1)ν 4Γβν Γαλ = hλβ,ν + hλν,β − h,λ hνα,λ + hνλ,α − h,ν βν αλ ,ν ,λ ν ,λ ν ,λ ,ν λ ν λ ν λ (expand brackets) = hλβ,ν hνα,λ + hλβ,ν hνλ,α − hλβ,ν h,ν αλ + hν,β hα,λ + hν,β hλ,α − hν,β hαλ − hβν hα,λ − hβν hλ,α + hβν hαλ

,ν ,ν ,ν ,λ ,λ λν λ λν λν λν λ λν (move, rename λ, ν) = h,ν βλ hαν + hβλ,ν h,α − hβ,ν hαλ + h,β hαν,λ + h,β hλν,α − h,β hαλ,ν − hβ,ν hλα − hβν,λ h,α + hβλ hαν ,ν ,λ λν λ λν λν (gather terms) = 2h,ν βλ hαν + (hβλ,ν − hβν,λ ) h,α − 2hβ,ν hαλ + h,β (hαν,λ − hαλ,ν ) + h,β hλν,α

,ν ,λ ,ν λ (symmetry of hµν ) = hλν,α hλν ,β + 2hβλ hαν − 2hβ,ν hαλ . (2)

Finally, we put together the expression for Rαβ : (2)

1 λµ h (−hαµ,βλ − hβµ,αλ + hαβ,µλ + hλµ,αβ ) + 2 1 = hλµ (−hαµ,βλ − hβµ,αλ + hαβ,µλ + hλµ,αβ ) + 2

Rαβ =

31

1 λµ 1 ,ν ,λ ,ν λ hλν,α hλν h,α hλµ,β − ,β + 2hβλ hαν − 2hβ,ν hαλ 2 4 1 λµ 1 ,ν λ ,λ hαλ hβ,ν − h,ν h,α hλµ,β + βλ hαν . 4 2

(35)

The Ricci scalar is (2)

R(2) = η αβ Rαβ =

3 1 λµ 1 1 ,ν αλ 1 ,α hαλ h,ν − hαλ,ν hαν,λ = hλµ,α hλµ,α − hαλ,ν hαν,λ , h hλµ + hλµ ,α hλµ + 2 4 2 4 2

(36)

where we again used the transverse traceless property of hµν and also hαβ = 0. Note that the first-order Ricci scalar is zero, R(1) = 0, since Tαβ = 0. For this reason we may use η αβ in Eq. (36), otherwise we would have to write (1) (2) (η + h) R + R and pick up a second-order term hR(1) . Again, since R(1) = 0, we may use ηµν rather than gµν to compute the Einstein tensor: 1 (2) (2) Gαβ = Rαβ − ηαβ R(2) . 2 We do not write the answer explicitly since it is a combination of Eqs. (35) and (36). (2) (2) Now let us perform an averaging of the quantity Gαβ over both space and time. In other words, we integrate Gαβ over a 4-dimensional region such that hµν = 0 and hµν,α = 0 on the boundary of that region. Then h∂µ (...)i = 0 and so we may integrate by parts, for example hAµ Bν,α i = − hAµ,α Bν i , as long as Aµ Bν contains first powers of hαβ or hαβ,γ , so that boundary terms vanish. Then, for example,

λµ

h hλµ,αβ = − hλµ ,α hλµ,β , E D

λµ = 0, h h hαν,βλ = − hλµ αν,β ,λ

(37)

hh,ν αλ hβµ,ν i = − hhαλ hβµ i = 0,

by hαβ = 0 and by the transverse traceless property of hµν . Many terms cancel in this way; for instance, R(2) = 0. Finally, we get D E 1 1 1 ,ν λ (2) ,ν ,λ h h − h h hλµ (−hαµ,βλ − hβµ,αλ + hαβ,µλ + hλµ,αβ ) + hλµ h + Gαβ = λµ,β βλ αν 2 4 ,α 2 αλ β,ν 1 λµ 1 λµ 1

= h hλµ,αβ + h,α hλµ,β = − hλµ ,α hλµ,β 2 4 4 using Eq. (37). Finally, we obtain the required equation, (GW)

9.4

Tµν = −

1 D (2) E 1 ij 1 λµ h hλµ,β = h hij,β . Gαβ = 8πG 32πG ,α 32πG ,α

Power of emitted radiation

To derive the relations 1 dΩ = δ lm , 4π 3 Z dΩ 1 lm kr nl nm nk nr = (δ δ + δ lk δ mr + δ lk δ mr ), 4π 15 Z

nl nm

let us consider the generating function gΩ (ql ) ≡

Z

dΩ exp −inl ql , 4π

which is a function of a vector argument ql . After computing gΩ (ql ) it will be easy to obtain integrals such as the above: Z ∂ ∂ l m dΩ i gΩ (qj ) , etc. =i nn 4π ∂ql qm qj =0 The computation is easy if we introduce spherical coordinates with the z axis parallel to the vector ql , then nl ql = |q| cos θ, √ where |q| ≡ ql ql , and then we have Z 2π Z π −2i sin |q| 1 1 2π dφ dθ sin θ exp [−i |q| cos θ] = 4π 0 4π −i |q| 0 1 1 1 sin |q| 2 3 = 1 − ql ql + (ql ql ) − (ql ql ) + ... = |q| 3! 5! 7!

gΩ (ql ) =

32

We have used the Taylor expansion for convenience of evaluating derivatives at |q| = 0. These derivatives can be found as follows, ∂gΩ 2 4 1 1 2 2 = − ql + ql |q| − ... = − + |q| ql , ∂ql 3! 5! 3 30 ∂ 2 gΩ 1 1 1 2 = − + |q| δkl + ql qk , ∂qk ∂ql 3 30 15 ∂ 3 gΩ 1 = (qj δkl + qk δjl + ql δjk ) , ∂qj ∂qk ∂ql 15 ∂ 4 gΩ 1 = (δjm δkl + δkm δjl + δlm δjk ) . ∂qj ∂qk ∂ql ∂qm 15 Now we compute the intensity of radiation. The flux of radiation in the direction nk is integrate this flux through a sphere of radius R: Z Z E D R2 dE nk . = R2 d2 Ω (GW ) T0k nk = d2 Ω hij h ij,k ,0 dt 32πG

(GW )

T0k nk , and we need to

The perturbation hij is found from the Einstein equation. It was derived in the lecture that, in the leading order in 1/R, we have ¨ ij (t − |R|) ~ Q , ~ |R| 1 (T T ) Qij ≡ Pai Pbj − Pab Pij Qab . 2 (T T )

hij = 2G

The projection tensor Pij is defined in Problem 9.1. The tensor Qij is defined by Z 1 2 Qij (t) ≡ d3 x xi xj − |x| δij T00 (x, t) 3 and is by definition trace-free, Qii = 0. Thus we have 16πG (T T ) ¨ Rk Qij (t − |R|) , RZ R D ... ... E G dE = d2 Ω (T T ) Qij (T T ) Qij . dt 8π

hij,k =

It remains to compute the average over the sphere of

... (T T ) ... Qij Qij . ... Consider any symmetric, trace-free tensor Aij instead of Qij ; the transverse-traceless part of Aij is defined by 1 (T T ) Aij ≡ Pai Pbj − Pab Pij Aab . 2 (T T )

Since Aii = 0, we have Aab Pab = −Aab na nb and so 1 1 (T T ) (T T ) Aij Aij = Pai Pbj − Pab Pij Pci Pdj − Pcd Pij Aab Acd 2 2 1 1 = Pac Pbd − Pab Pcd Aab Acd = Pac Pbd − na nb nc nd Aab Acd 2 2 1 = Aab Aab − 2Aac Abc na nb + Aab Acd na nb nc nd . 2 After integration over the sphere, according to formulas derived above, we have (again note that δab Aab = 0 and Aab = Aba ) Z 1 1 1 d2 Ω Aac Abc na nb = Aac Abc δab = Aab Aab , 4π 3 3 Z 2 1 1 Aab Acd (δab δcd + δac δbd + δad δbc ) = Aab Aab , d2 Ω Aab Acd na nb nc nd = 4π 15 15 and thus

1 4π

Z

2

d Ω

(T T )

Aij

(T T )

2 2 1 2 = Aab Aab . Aij = Aab Aab 1 − + 3 2 15 5 33

... Finally, substituting Qij instead of Aij , we find Z D ... ... E G ... ... dE G 1 Qij Qij . = d2 Ω (T T ) Qij (T T ) Qij = dt 2 4π 5

(38)

The angular brackets hi indicate that we must perform an averaging over spacetime domains. This means, for us, that we need to average over time (since Qij is a function only of time). Averaging is performed over timescales larger than the typical timescale of change in the source. For instance, if the source is a rotating body, then averaging must be performed over several periods of rotation.

10

Derivation: gravitational waves in flat spacetime

The metric is assumed to be of the form gµν = ηµν + hµν , where ηµν = diag(1, −1, −1, −1) is the Minkowski metric for flat space and hµν is a small perturbation which is assumed to fall off to zero quickly at infinity. We start with a 3+1 decomposition of the metric perturbation hµν and compute the Einstein tensor (see Problems 7.2, 7.3, 7.4) in terms of the perturbation variables Φ, Ψ, etc. We also decompose the matter energy-momentum tensor Tµν and obtain the Einstein equations separately for each component (8.5). The result is that (a) the variables E, B, Fi can be set to zero by choosing a coordinate system; (b) if there is no matter (vacuum) the scalar and vector components of the metric perturbation are equal to zero; (c) the tensor component hij satisfies the wave equation (31). Solutions of the wave equation in four dimensions with retarded boundary condition can be written using the known Green’s function. For instance, if Z 1 A(t − |r − R| , r) f (t, r) = A(t, r) ⇒ f (t, R) = − . d3 r 4π |r − R| We will use this formula for f ≡ (T ) hij and A ≡ 16πG(T ) Tji . Now, we are interested in describing the radiation sent far away by a matter distribution, so we take the limit |R| ≫ |r|, and then we can approximately set Z 4G (T ) d3 r (T ) Tji (t − |r − R| , r). hij ≈ − |R| Now we use a trick (See Hobson-Efstathiou-Lasenby, §17.9) to express the components Tij through T00 ; it is much easier to compute with T00 because this is just the energy density of matter. Consider first the tensor Tji rather than its transverse-traceless part (T ) Tji . The trick is to write the integral (out of sheer luck) Z Z d3 r ∂a ∂b ri rj T ab = 2 d3 r T ij . Then we integrate by parts and use the conservation laws (32),

2

Z

i 0 i 0 0 Tj,i = −Tj,0 , Tj,ij = −Tj,0j = T0,00 ≡ T¨00 ; Z Z d3 r T ij = d3 r T ab ,ab ri rj = − d3 r ri rj T¨00 .

T ij ,ij = −T¨00 .

Now, we need to obtain the transverse-traceless part of the tensor. In principle, we have the formulas for this (see Problem 8.5). But they are very complicated. A shortcut is to notice that the projection operator Pab does the job (Problems 9.1 and 9.2), at least in the leading order in 1/|R|. (We are only interested in everything to leading order in 1/ |R| since all smaller terms will not give any flux of radiated energy.) The result is (T )

2G (T T ) ¨ Qik (t − |R|), |R| Z 1 2 3 0 Qik (t) ≡ d r T0 (r, t) ri rk − r δik . 3

hik (R, t) =

(39)

The tensor Qik is the quadrupole moment of energy distribution; it is a traceless and symmetric tensor. In principle, we could just use the integral Z d3 r T00 (r, t)ri rk , (40)

because the transverse-traceless parts of (40) and of Qik are the same, but it is more convenient to use Qik . Since we found the tensor perturbation (T ) hij , now we would like to compute the energy radiated in the gravitational waves. For this we need the energy-momentum tensor of gravitational waves. This is a rather nontrivial object, since in general the gravitational field does not have any energy-momentum tensor. In the case of gravitational waves in flat background spacetime, one can define some quantity (GW ) Tµν which looks like the energy-momentum tensor of 34

gravitational waves (but actually is not even a generally covariant tensor). We will compute this quantity below. This quantity is useful because it gives the correct value of the energy after one integrates over a large region of spacetime. The real justification for using this procedure is complicated and is beyond the scope of this introductory course of General Relativity. We will only show a heuristic justification, which is the following. Gravitation is sensitive to every kind of energy, because the energy-momentum tensor acts as a “source” for gravity (it is on the right-hand side of the Einstein equation). So gravitation should be also sensitive to the energy in gravitational waves. One expects that the energy-momentum tensor for gravitational waves, (GW ) Tµν (if we know how to compute it), will act as an additional source for gravity, like every other energy-momentum tensor for other kinds of matter. We will guess the formula for (GW ) Tµν as follows. We can write the Einstein equation and expand it in powers of the perturbation hµν : (1)α

Gα β [ηµν + hµν ] = Gβ

(2)α

[h] + Gβ

[h] + ... = 8πGTβα .

(41)

Here G(1) is the first-order Einstein tensor, G(2) is the second-order etc. First we solve only to first-order in h (this is (1) what we have been doing so far) and then we will get an approximate solution hµν : (1)α

Gβ

[h(1) ] = 8πGTβα .

(42)

This solution disregards the effect of gravitational waves and only takes into account the effect of matter Tβα . We can try to get a more precise solution by using the second-order terms in Eq. (41). Then we will get a correction h(2) to the solution; the solution g = η + h(1) + h(2) will be more precise. From Eq. (41) we find (1)α

Gβ

(2)α

[h(1) + h(2) ] + Gβ

[h(1) ] = 8πGTβα .

Now this is similar to Eq. (42), but it looks as if there is an additional term in the energy-momentum tensor, which we may rewrite as o n (1)α Gβ [h(1) + h(2) ] = 8πG Tβα + (GW) Tβα , (GW)

Tβα ≡ −

1 (2)α (1) G [h ]. 8πG β

This motivates us to say that the EMT for gravitational waves is given by this formula. But of course this is not a real derivation because this does not show why the quantity (GW) Tµν has anything to do with the energy carried by waves. (2)α The second-order terms Gβ are computed in Problem 9.3. The result is used to compute the power radiated in (2)α

gravitational waves (Problem 9.4). Note that the calculation of Gβ uses averaging over spacetime in an essential way. Thus, the result is an averaged power radiated during a long time—much longer than the typical time scale of change in the sources—and averaged over large distances, much larger than the typical length scale of the sources. This kind of averaging is assumed in Eq. (38). It remains unclear exactly how one performs averaging over space and time. One can use the formula (38) to compute the gravitational radiation emitted by nonrelativistic matter far away from those places where the matter is contained. The distribution of the energy density, T00 (r, t), should be given. Then one computes the tensor Qij according to Eq. (39), by integrating over space where the matter is contained. Finally, one ... computes the third derivative Qij , the trace, and averages over long time, as indicated in Eq. (38). If we want to insert factors of c, we replace G by Gc−9 .

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A.3

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You may copy and distribute a Modified Version of the Document under the conditions of sections A.1 and A.2 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. C. State on the Title page the name of the publisher of the Modified Version, as the publisher. D. Preserve all the copyright notices of the Document. E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document’s license notice. H. Include an unaltered copy of this License. I. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled “History” in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. K. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant Section. O. Preserve any Warranty Disclaimers. If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version’s license notice. These titles must be distinct from any other section titles. You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

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The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

Combining documents You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”

Collections of documents You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

Aggregation with independent works A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation’s users beyond what the individual works permit. When the Document is included an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section A.2 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

Translation Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section A.3. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warrany Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled “Acknowledgements”,“Dedications”, or “History”, the requirement (section A.3) to Preserve its Title (section A.0) will typically require changing the actual title.

Termination You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

Future revisions of this license The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and 38

conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

ADDENDUM: How to use this License for your documents To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (c) . Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.” line with this: with the Invariant Sections being , with the Front-Cover Texts being , and with the Back-Cover Texts being . If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

Copyright Copyright (c) 2000, 2001, 2002 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

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General Relativity Solutions to Exercises by Sergei Winitzki

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