Relativity

Tom Charnock

Contents

1 Index Notation 51.1 Units and Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Summation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Index Clashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.4 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.5 Vector Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Euclidean Space 72.1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Passive Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Worldline, Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Spatial Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Galilean Relativity 103.1 Standard Inertial Laboratories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Galilean Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Observers and Standard Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 The Galilean Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 The Newtonian Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Electromagnetism 124.1 Charge Density and Charge Density Current . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Lorentz Force Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Conservation of Electric Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Einstein’s Relativity 165.1 Speed of Light and the Aether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.1 FitzGerald Length Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Einstein’s Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.1 Elementary Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.2 Loss of Absolute Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.3 Space-Time Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Minkowski Spacetime 216.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Invariance of c2∆t2 −∆x2 −∆y2 −∆z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 Minkowski Distance Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.3.1 Imaginary Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 Metric Structure of Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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6.4.1 Lightcones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4.2 Spacetime Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.5 Minkowski Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.6 Transformations of the Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.7 Lorentz and Poincare Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.8 Index Raising and Lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Relativistic Particle Mechanics 267.1 Particle Worldline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.1.1 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.1.2 Four Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.1.3 Four-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.2 One-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.3.1 Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 Energy-Momentum Four-Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.5 Relativistic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.6 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.7 Relativistic Particle Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.7.1 Particle Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.7.2 Two-Two Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8 Relativistic Maxwell’s Equations 338.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Gauge Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 The Four Potential, Aµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.4 Maxwell-Faraday Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.5 Lorentz Transformations of Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . 358.6 Raising and Lowering Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.7 Covariant Formulation of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 36

9 General Relativity 389.1 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9.1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.1.2 General Coordinate Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

9.3 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.4 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 Tangent Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9.5.1 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.6 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 Cotangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.7.1 Coordinate Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.8 Congruences and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.9 Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.9.1 Coordinate Transformation of Tensor Field Components . . . . . . . . . . . . . . . 439.10 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.10.1 Index raising and Lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.11 Parallel Transport and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.11.1 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.11.2 Covariant Derivatives of Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 469.11.3 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.11.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.11.5 The Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.11.6 Metric Connection via Variational Calculus . . . . . . . . . . . . . . . . . . . . . . 499.11.7 Schwarzschild Connection Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 50

9.12 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.13 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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9.14 Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.14.1 Components of the Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . 529.14.2 Riemann-Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.14.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10 Relativity and Gravitation 5410.1 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.2 Tidal Acceleration in Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.3 Weak Gravitational Fields and the Newtonian Limit . . . . . . . . . . . . . . . . . . . . . 5510.4 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.5 General Relativity Matter-Free Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.6 Symmetries of the Ricci Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.7 The Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.8 Gravitational Orbit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.8.1 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.8.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

10.9 The Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.9.1 Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10.10The Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.10.1 The Stress-Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 61

10.11Affine Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.11.1 Null Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

10.12Deflection of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

11 Schwarzschild Geometry 6311.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.2 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.3 The Kretschmann Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.4 Birkhoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.5 Black Hole Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

11.5.1 Eddington-Finkelstein Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.5.2 White Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6611.5.3 Kruskal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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Chapter 1

Index Notation

1.1 Units and Dimensional Analysis

In some cases, the speed of light c can be set to unity, as can ~ in quantum mechanics. The problemwith this is that the limits where c→ 0 and c→∞ can not be explored. It makes sense to use SI unitssuch that:

mass kg Mlength m Ltime s Telectric charge C Q

The physical dimensions of a given quantity will always be given by A = MαLβT γQd, where α, β, γ, dare real numbers to be determined. Only equivalent dimensions can be added.

1.2 Index Notation

1.2.1 Summation Convention

The Einstein summation convention is a way of omitting the summation sign by using repeated indices,which is more concise.

n∑i=1

AiBi = AiBi

For a˜ = a1i+ a2j + a3k then by converting the unit vectors into i = e˜ i, j = e˜j and k = e˜k:

a˜ =∑i

aie˜ i = aie˜ iThe orthonormality relations can be summarised by e˜ i · e˜j = δij where the indices are free, becausethere is no repetition of index then there is no summation. Even if i = j then this still not a summationbecause the index labels are different. It is also interesting to note that the Kronecker Delta gives thenumber of dimension of space. For example in three dimensional space then δii = δ11 + δ22 + δ33 = 3.

1.2.2 Index Clashes

For two summations in the same expression then the index being summed over is called a dummy indexwhere any, non-repeated letter can be used:

aibicjdj = ajbjcidi

An index clash would occur when aibicidi, so that it is not known what is being summed over.

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1.2.3 Scalar Product

The scalar product of two vectors can be written as a˜ · b˜= (aie˜ i) · (bje˜j). As ai and bi are just numbers

they can be arbitrarily moved to give a˜ · b˜ = aibj(e˜ i · e˜j) = aibjδij . Because the Kronecker Delta isnon-zero only when i = j then the scalar product is:

a˜ · b˜= aibi

1.2.4 Vector Product

The cross product can be written for an orthonormal basis as:

e˜ i × e˜j = εijke˜kWhere εijk is the Levi-Civita tensor which is 1 for cyclic permutations of i, j and k, -1 for anti-cyclicpermutations of i, j and k and 0 otherwise. It can be seen that the Levi-Civita tensor and the basis vector,e˜k are summed over k. The relation of the Levi-Civita tensor to the Kronecker Delta is interesting. Fora summation over i then:

εijkεiab = δjaδkb − δjbδkaAlso a double summation over both i and j is given by:

εijkεija = 2δka

And finally when i, j and k are summed over then:

εijkεijk = 6

1.2.5 Vector Triple Product

The εijk and δij relations make the vector triple product identity much easier to find. By starting with:

a˜× ( b˜× c˜) = (aie˜ i)× (bje˜j)× (cke˜k)

Now moving the numbers ai, bj and ck to the front and using the vector product relation on the basisvectors inside the brackets gives aibjck( e˜ i) × εjkr( e˜r) where the Levi-Civita tensor can be pulled tothe front, as can e˜ i × e˜r = εirse˜s. As there is a sum present over r then the first Levi-Civita tensor

can be arranged as such εjkr = εrjk, and the second one εirs = −εris. This leaves −aibjckεrjkεrise˜s =

−aibjck(δjiδks−δjsδki)e˜s where the δij then changes the labels of a, b and c to give −aibicke˜k+aicibje˜j .This is the index notation form of the vector triple product:

a˜× (b˜× c˜) = −aibicke˜k + aicibje˜j = (a˜ · c˜)b˜− (a˜ · b˜)c˜

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Chapter 2

Euclidean Space

If there is an n-tuple coordinate x = (x1, x2, ..., xn) then the Euclidean metric on Rn is:

dE(x˜, y˜) =√

(x1 − y1)2 + (x2 − y2)2 + ...+ (xn − yn)2

This is the n-dimensional version of Pythagorus theorem, which is the distance theorem in Euclideanspace. As there are other metrics on Rn then the Euclidean metric is defined as En = (Rn, dE). Rn is notautomatically a vector space because it needs a defined concept of vector addition and multiplication bya scalar. For Rn, then vector addition is (x1, x2, ..., xn) + (y1, y2, .., yn) = (x1 + y1, x2 + y2 + ..., xn + yn)and scalar multiplication is λ(x1, x2, ..., xn) = (λx1, λx2, ..., λxn)

2.1 Transformations

2.1.1 Passive Transformation

A passive transformation is just a relabelling and as such the entire system acts exactly the same. Thismay be a change in coordinate frame in physics or a change in basis of a vector space in maths. Somepassive transformations can appear equal and opposite to active transformations, but with equal andopposite parameters. An example of this is when one person turns in front of an audience. To theaudience the person is physically turning, but the person can state that he is still with the audiencepassively rotating away in the opposite direction. Special relativity does not have any origins of inertialframes. In general relativity, global inertial frames do not exist and so all reference frames are admissible.

In a Cartesian frame F in E3 with an orthonormal basis set e˜ i : i = 1, 2, 3 and coordinates xi : i =

1, 2, 3. It is known that e˜ i · e˜j = δij . If a particle has a position P so that r˜P = xiP e˜ i then by making apassive rotation of the frame F → F ′ with the basis e˜ i : i = 1, 2, 3 where e˜ i · e˜′i = δij then the position

vector becomes r˜′P = x′iP e˜′i. The relationship between xiP and x′iP is a passive transformation andthe origin of coordinates does not move. As the origin has not moved then r˜P = r˜′P . This means thatthe new basis vectors can be represented as a linear relation of the old vectors.

e˜′i = Rije˜jRij are real. It can further be seen that e˜′i ·e˜′j = Riae˜a·Rjbe˜b. This, following normal dot productrules, can be written as RiaRja = δab and so it can indeed be written as:

RiaRjb = δij

In three dimensions then this is equivalent to RRT = I3, where RT is the matrix transpose of R whichhas the relation

[RT]ji

=[R]ij

. I3 is the 3 × 3 identity matrix. Taking the determinant of both sides

gives∣∣RRT ∣∣ = |I3|, which can be written using |I3| = 1,

∣∣RRT ∣∣ =∣∣R∣∣∣∣RT ∣∣ and

∣∣RT ∣∣ =∣∣R∣∣ as:∣∣RRT ∣∣ =

∣∣R2∣∣ = 1

This means that |R| = ±1. When |R| = 1 then the rotation is proper and is what is intuitively thoughtof as a rotation. These rotations can be characterised by three angles when in three dimensions. The set

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of all proper rotations is connected so that any two proper rotations can be transformed smoothly intoeach other. Proper rotations preserve the handedness so that a proper rotation rotates a right-handedsystem into another right-handed system.

By taking the dot product of e˜′i = Rike˜k with e˜j gives e˜j · e˜′i = e˜j · Rike˜k = Rike˜j · e˜k = Rikδjk.In other words:

e˜′i · e˜j = Rij

This means that r˜′P = r˜P can now be written as:

x′iP = RijxjP

Euclidean Distance is preserved under passive rotations because the distance between P and Q is:

S2E(P,Q) = (x˜′P − x˜Q)TGE(x˜′P − x˜′Q)

This is found by squaring the Euclidean distance equation where (x1P −x1

Q) + (x2P −x2

Q) + (x3P −x3

Q) canbe written as: (

x1P − x1

Q x2P − x2

Q x3P − x3

Q

)1 0 00 1 00 0 1

x1P − x1

Q

x2P − x2

Q

x3P − x3

Q

The identity matrtix here is the Euclidean metric, GE .

2.2 Worldline, Γ

x

y

z

P

t

Figure 2.1: Worldline

The trajectory moving through time gives the position of a particle at all times and is called the worldline.A worldline in Newtonian mechanics must be twice differentiable, except for in certain cases such as wherethere are a finite number of discontinuities as happens with impulse, or when the worldline is continuousbut not differentiable as for Brownian motion where Γ = (t, x(t), y(t), z(t)).

There are restrictions on what the worldline can do, for example: a loop in time is not allowed so that theposition is single valued at a particular time and the position can also not be discontinuous at a singletime, as it would create teleportation.

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Figure 2.2: Worldlines Which Show Loops in Time and Teleprotation

2.3 Affine Spaces

A space X with two points, P and Q, can be given by a vector in vector space V with the relationQ = P + PQ˜. This can be done with another set of points, for example S = R + RS˜ . In vector space

there must also be a zero vector were X = X + 0˜. If there is a vector from the origin in space to a pointP , then it has a vector OP˜ connected with it. If the positions of O and P are both translated by r thenthe points become O+ r and P + r, but the vector space vector remains OP˜. In Galilean relativity then

the first frame can have a basis set i, j and k. If a second reference frame has a basis i′, j′ and k′ thenby making these basis vectors the same then they both use the same vector space. It is this vector spacewhich allows direction to be defined.

2.3.1 Spatial Inversion

Under spatial inversion the original basis has a change in handedness:

e˜ i → e˜′i = −e˜ iAs the transformation is passive ri → r′i = ri. This means that the vectors r˜ = xie˜ i and r˜′ = x′ie˜ i givethe property:

x′i = −xi , x′i = xi , x′i = xi

9

Chapter 3

Galilean Relativity

The three laws of motion do not only hold in a single Absolute Frame of reference but also for anyuniformly moving frame of reference. This is called form invariance and leads to the statement thatNewton’s laws of motion are form invariant under change of inertial frame.

3.1 Standard Inertial Laboratories

There is a Cartesian frame of reference, FA, which is at rest in Newton’s Absolute Space. This meansthere are fixed unit vectors and a fixed origin, with the units of measurement being fixed to SI units. Theassociated spatial coordinates (x1, x2, x3), are called the Absolute Coordinates. An Absolute Time withcoordinate, t is also needed in the frame FA. FA can label any point, P , in four-dimensional space-time,N 4, which is distinct from spacetime because t is not associated with xi, by P (tP , x

1P , x

2P , x

3P ). In N 4

then the Euclidean distance rule applies only on the spatial coordinates. This means that between twoevents, P (tP , x

1P , x

2P , x

3P ) and Q (tQ, x

1Q, x

2Q, x

3Q), the distance SE(P,Q) must be found at tP = tQ.

This can be extended in special relativity by the fact the distance between times is also included.

3.1.1 Galilean Transformation

A pure Galilean transformationG( v˜ ) is a passive transformation from the absolute frame FA (t, x1, x2, x3)

to a new frame F ′ (t′, x′1, x′2, x′3) where:

t −→G( v˜)

t′ = t and xi −→G( v˜)

x′i = xi − vit

vi are the constant relative velocities of the frame F ′. This is a time dependent translation with norotation of coordinate axes. The origin, O′ for F ′, has the coordinates x′iO = 0 relative to F ′, whichin the Absolute Frame gives xiO = vit. O′ in FA appears to move with a constant velocity v˜ = vie˜ i.The relation can also easily be inverted so that t = t′ and xi = x′i + vit′ where O for FA moveswith a constant velocity v˜i = −vie˜ i relative to F ′. For an arbitrary worldline then any point on Γ is

xiP (t) −→ x′iP (t′) = xiP (t)− vit. The protocol for measuring velocity depends on the clock in each frame:

dxiP (t)

dt−→ dx′iP (t′)

dt′=

dt

dt′d

dt

(xiP (t)− vit

)In Galilean relativity then dt

dt′ = 1, but has different values in different types of relativity. Using the valueof one gives:

dx′iP (t′)

dt′=dxiP (t)

dt− vi

And so velocity is not form invariant in Galilean relativity, but acceleration is, as can be shown:

d2xiP (t)

dt2−→ d2x′iP (t′)

dt′2=

dt

dt′d

dt

[dt

dt′d

dt

(xiP (t)− vit

)]=d2xiP (t)

dt2

10

3.2 Observers and Standard Configurations

An observer is a localised recording device with its own clock to measure events. The frame of referenceinvolves a continuum of observers, each with a clock and recording device. The net effect can then befound by a super-observer who can analyse all of the recordings. A standard inertial frame uses Cartesiancoordinates with a standard set of units. All of the clocks of the observers are at rest in the standardinertial frame and are all synchronised. Two inertial frames are in standard configuration if the x, y andz axes are respectively parallel. If this is not true then the relative velocity of the two frames is arbitraryin direction.

3.2.1 The Galilean Group

In the same way that Euclidean transformations form a group, so do the Galilean transformations, GG.An element gG( v˜ ) ∈ GG with the pure Galilean transformation has t→ t′ = t and x˜→ x˜′ = x˜− v˜t. Thegroup axioms are satisfied for multiplication where:

gG(u˜ )gG( v˜ ) = gG(u˜+ v˜ ) = gG(v˜+ u˜) = gG( v˜ )gG(u˜ )

u˜+ v˜ is normal vector addition and as such the group commutes and is called Abelian.

3.2.2 The Newtonian Group

The Euclidean and pure Galilean transformations form a nine-parameter group called the Newtoniangroup, GN . The associated element, gN (R˜ , a˜, v˜), is combined with t→ t′ = t and x˜→ x˜′ = Rx˜ + a˜+ v˜t.The rules are then:

gN (R˜ 2, a˜ 2, v˜2)gN (R˜ 1, a˜ 1, v˜1) = gN (R˜ 1 +R˜ 2, R˜ 1a˜ 1 + a˜ 2, R˜ 1v˜1 + v˜2)

11

Chapter 4

Electromagnetism

Maxwell showed that light was an electromagnetic wave phenomenon, propagating with a fundamentalspeed c which is measured in the absolute inertial frame FA. In Galilean relativity then velocity is notinvariant but the speed of light is observed to be invariant and so goes beyond this form of relativity.This removes the ability to have an absolute frame of reference. Electromagnetism is the study oftime-dependent electric and magnetic field systems. There are generally three significant componentsinvolved; the electric field E˜ (t, r˜), the magnetic field B˜ (t, r˜) and the electric charge distribution. Thevectors representing the directions are not real space directions but instead are directions in an affinevector space.

4.1 Charge Density and Charge Density Current

The charge density %(t, r˜) is the charge per unit volume:

[%] = QL−3

The charge density current j˜(t, r˜) has dimensions:

[j˜] = QLL−3T−1 = QL−2T−1

Electric charges are the ultimate sources of the electromagnetic fields and are also affected by these fields.

4.2 Lorentz Force Law

The study of mechanics of electric charges under electromagnetic forces is electrodynamics. The LorentzForce Law is:

mr˜ = qE˜ + qr˜×B˜It states that a particle in an electromagnetic field has a force equal to the effect of the electric field onthe charge and the effect of the charge moving perpendicularly to the magnetic field. The dimensions ofthe electric and magnetic field can be calculated using:

[Ei] = MLT−2Q−1 and [Bi] = MT−1Q−1

Under spatial inversion the electric field acts as expected Ei → E′i = −Ei, but the magnetic field doesn’tand so is called an axial or polar vector where Bi → B′i = Bi.

4.3 Electrostatics

The study of electric charges and electric fields which are time-independent is called electrostatics. Inthe standard inertial frame F has an electric field given by:

E˜ (t, r˜) =q

4πεo

r˜|r˜|312

Where |r˜| 6= 0 and εo is the permittivity of free space. In a volume of space with a sphere centred on thepoint charge then the flux through the sphere is:

Φ(a) =

∫a

ds˜ · E˜ =

∫a2dΩ

q

4πεo

1

a2=

q

εo

This is independent of the radius, a, of the sphere. This leads to the statement ∇ · E˜ = %εo

, which isone of Maxwell’s static equations. This can then be found by noting that for a point electric charge atthe origin of coordinates is %(t, r˜) = qδ3(r˜), where δ3(r˜) = δ(x1)d(x2)δ(x3) is the Dirac δ function. This

function is zero everywhere except at the point (x1, x2, x3), which makes it useful for placing the pointof charge. Calculating the divergence at a point not at the origin gives:

∇ ·r˜|r˜|3 = 4πδ3(r˜)

This allows the electric field outside of the point charge to be written:

E˜ (t, r˜) = −∇φ(r˜)The charge density over a region of space, V , is:

Φ(x˜) =

∫V

d3y˜%(y˜)

4πεo|x˜− y˜|This satisfies Poisson’s equation as can be seen:

∇2xφ(x˜) = ∇2

x

∫V

d3y˜%(y˜)

4πεo|x˜− y˜| =

∫V

d3y˜%(y˜)4πεo

∇2x

1

|x˜− y˜| =

∫V

d3y˜%(y˜)4πεo

(−4πδ3(x˜− y˜)) = −%(x˜)

εo

Now because E˜ (x˜) = −∇xφ(x˜) then:

E˜ (x˜) = −∇x∫V

d3y˜%(y˜)

4πεo|x˜− y˜| = −∫V

d3y˜%(y˜)4πεo

(−

x˜− y˜|x˜− y˜|3)

This shows how the scalar potential can be moved to the electrostatic potential.

4.4 Magnetostatics

Magnetostatics is the study of magnetic charges when they are stationary. As magnetic monopoles arenot seen to exist then ∇ ·B˜ = 0. In fact, Dirac described how magnetic monopoles could exist using thevector potential and B˜ = ∇ × A˜ , but where ∇ · (∇ × A˜) 6= 0. This could only work if A˜ had a line ofsingularity from the monopole to infinity. This string-like object is called a Dirac string.

4.5 Conservation of Electric Charge

The total electric charge within a region of space cannot be created or destroyed. The only way the totalelectric charge can change is by moving charges into or out of a region of space. The number of chargedparticles in the space does not have to stay the same due to opposite charges annihilating, but the overallcharge remains the same. The electric charge conservation can be written as a differential equation:

∇ · j˜+ ∂t% = 0

This is an equation of continuity, similar to that of mass in Newtonian mechanics. The difference hereis that the Newtonian mass conservation does not hold for special relativity whilst charge conservationdoes. The continuity equation leads to charge conservation as can be seen by:∫

V

d3x˜∂t%(t, x˜) = −∫V

d3x˜∇ · j˜(t, x˜)

13

Using divergence theorem provides:

d

dt

∫V

d3x˜%(t, x˜) = −∫S

ds˜ · j˜(t, x˜)

The rate of change of the charge density in a volume is equal to the charge current over the surface of thevolume. When there is no flow of current on the boundary j˜ = 0 so the total charge is instantaneously

constant. If there is a net in flow then the electric charge increases, but it decreases if j˜ points outwards

as there is a net loss.

4.6 Maxwell’s Equations

There are four of Maxwell’s equations, two homogeneous and two non-homogeneous. The first homoge-neous equation states that there are no magnetic monopoles and is:

∇ ·B˜ = 0

The second equation is Faraday’s Law of Induction and gives rise to the electromotive force:

∇× E˜ +∂B˜∂t

= 0

The first inhomogeneous equation is Gauss’ Law:

∇ · E˜ =%

εo

And the last equation is Ampere’s Circuital Law, which includes Maxwell’s correction for the displacementcurrent:

∇×B˜ − µoεo ∂E˜∂t = µoj˜µo is the permeability of free space with the dimensions [µo] = MLQ−2. As there are temporal derivativesthen it suggests that in the absence of an electric charge, % = 0 and j˜ = 0, then Maxwell’s equations are:

∇ ·B˜ = 0 ∇ · E˜ = 0

∇× E˜ +∂B˜∂t

= 0 ∇×B˜ − µoεo ∂E˜∂t = 0

These equations can be written as wave equations:(µoεo

∂2

∂t2−∇2

)E˜ = 0

(µoεo

∂2

∂t2−∇2

)B˜ = 0

The general solution here is ϕ(t, x) = f(x− vt) + g(x+ vt), where f(x− vt) is called a right-mover andg(x+ vt) is called a left-mover. The velocity here is:

v =

√1

εoµo= c

The D’Alembertian operator can be introduced which is:

=∂2

c2∂t2−∇2

The D’Alembertian operator has the dimensions of [] = T 2L−2T−2 = L−2. The wave equation canthen be written as E˜ = 0 and B˜ = 0. The plane wave solutions to the wave equations are:

E˜ = E˜ oei(k˜·x˜−iωt) + cc

14

B˜ = B˜ oei(k˜·x˜−iωt) + cc

cc is the complex conjugate and ω = |k˜|c. It is useful to note that |k˜| = 2πλ and ω = 2πν, where λ is the

wavelength and ν is the frequency. This reveals the relationship:

νλ = c

For the plane waves to be consistent with Maxwell’s equations then:

E˜ o · k˜ = 0 and B˜ o · k˜ = 0

This states that the electric and magnetic fields travel transverse to the direction of the wave propagation.Considering the other two of Maxwell’s equations gives the relationship:

k˜× E˜ o = ωB˜ oThis states that not only are the waves travelling transverse to the direction of propagation, but alsotransverse to each other.

15

Chapter 5

Einstein’s Relativity

5.1 Speed of Light and the Aether

The limits of classical mechanics had been stretched by the characteristic speed of light given by Maxwell’sequations. In classical mechanics there are three statements:

• The laws of Newtonian mechanics are form-invariant to Galilean transformations.

• Maxwell’s equations are valid equations of physics. The speed of light is an electromagnetic wave-phenomena given by c = (µoεo)

− 12 ≈ 3× 108ms−1.

• The velocity is not form invariant to Galilean transformations.

There were problems with these statements:

• The electrodynamic stability of atoms. According to Maxwell’s equations an accelerated electriccharge radiates electromagnetic energy.

• The continuous linear momentum and energy of particles. These properties must be quantum toprevent ultraviolet catastrophe.

• The orbital period of Mercury. There is a small discrepancy from the Newtonian mechanics orbitto the observed values.

These problems were addressed by quantum mechanics in the first two cases and the third solved bygeneral relativity. Another problem was the medium in which light propagated. This brought aboutthe creation of the aether, which had a frame of reference at rest. An experiment was carried out byMichelson and Morley where light was split into two transverse directions and then recombined. Thiswas tested throughout different times of the year to see how much the interference fringes moved andhence calculate the velocity with which the earth was travelling compared to the aether. The experimenthad a negative result and so an explanation of why the aether was not detected was needed.

5.1.1 FitzGerald Length Contradiction

To overcome the failed result by Michelson-Morley then FitzGerald stated that the length of materialbodies changes as they move through the aether. Light moves with constant speed c in the aether restframe. It moves at c− v in the frame of the laboratory moving with velocity v but the whole laboratoryis contracted in length so that the measurement of the speed of light is the same. The contraction in

length is dependent on f(v2

c2

).

5.2 Lorentz Transformations

The Lorentz transformation is the correct transformation between two standard inertial frames withaligned axes. If there are two standard inertial frames F and F ′ where F ′ is moving with v ≥ 0 withtrespect to F along the x-direction then the Lorentz transformation, Λ(v) is a passive transformation:

16

t −→Λ( v˜)

t′ = γ(v)(t− vx

c2

)x −→

Λ( v˜)x′ = γ(v) (x− vt)

y −→Λ( v˜)

y′ = y

z −→Λ( v˜)

z′ = z

It is standard convention to chose x as the direction of relative motion. The most notable fact here isthat time from one frame to another can be different. The Lorentz factor is introduced here:

γ(v) =1√

1− v2

c2

This is the same factor that FitzGerald referred to. It shows that nothing can travel faster than the speedof light because the square root becomes negative when v > c. The Lorentz transformation in matrixand index notation is:

x′µ = Λ(v˜)µνxν =

γ(v) −γ(v)β 0 0−γ(v)β γ(v) 0 0

0 0 1 00 0 0 1

ctxyz

Where β = v

c . The inverse transformation from F ′ to F exists as long as the determinant of the matrixΛ(v˜) is non-zero.

det Λ(v˜) = γ2(v)− γ(v)2 v2

c2= 1

As such the inverse transformation always exists. It is easy to find this inverse Lorentz transformationbecause F observes F ′ moving with velocity v˜ and by symmetry F ′ observes F moving with velocity −v˜and as such:

Λ(v˜)−1 = Λ(−v˜) =

γ(v) γ(v)β 0 0γ(v)β γ(v) 0 0

0 0 1 00 0 0 1

If the spatial axes are not aligned then rotations can be introduced using a rotation matrix:

t x y zt′ 1 0 0 0x 0y 0 Rijz 0

Table 5.1: Rotation of Axes

This then leaves only the normal Lorentz transformation. As c is such a large number then the expansionof the Lorentz factor is:

γ(v) = 1 +1

2

v2

c2+O

(v4

c4

)This means that the Lorentz transformation takes the form:

t −→Λ( v˜)

t′ = t+O(β)

x −→Λ( v˜)

x′ = x− vt+O(β2)

y −→Λ( v˜)

y′ = y

z −→Λ( v˜)

z′ = z

For ordinary speeds then β is negligible and so when β → 0 the Galilean transformations are recovered.

17

5.3 Einstein’s Principle of Relativity

Poincare showed that the Lorentz transformations form a symmetry group of Maxwell’s equations suchthat they are form invariant. Einstein used physical intuition to form the Lorentz transformations fromGalileo’s principle of relativity and Maxwell’s equations invariance in all inertial frames.

Galileo’s Principle of Relativity states that the laws of physics, excluding gravitation, take thesame form in all standard inertial reference frames. It is therefore impossible to use the laws of physicsto single out any inertial frame as special.

Maxwell’s Equations are Valid in All Inertial Frames because they take the same form in everyinertial frame. This can be seen by the fact that the speed of light is the same in every standard inertialframe.

Einstein also abolished the aether as it was not needed conceptually.

5.3.1 Elementary Derivation

It can be assumed that space and time are homogeneous so that there are no special places or timeswhere relationships change their nature. This is not unreasonable over large intervals of space and timeif gravity is excluded. In two frames set at a special event, O, called the origin of space-time coordinatesthen:

(tO, xO) = (t′O, x′O)′ = (0, 0)

Any other event which now occurs must have coordinates given by:

t′ = Mt+Nx

x′ = Rt+ Sx

M , N , R and S are independent of t and x and so the transformation from F ′ to F is linear-homogeneous.Now if F ′ is moving uniformly along the positive x-axis with speed, v, then x = vt. For the origin ofcoordinates x′ = 0 in F ′ gives the condition 0 = Rt+ Svt and as such R = −vS and the transformationis:

t′ = Mt+Nx

x′ = S(x− vt)

This begins to look familiar to the Lorentz transformations. Now the second of Einstein’s principles needsto be considered, that light travels at the same speed in all inertial frames. If a pulse of light set outfrom O = (0, 0) to an event A in the positive x-direction so that A (T, cT ) according to frame F thenaccording to frame F , A has coordinates:

t′A = MT +NcT

x′A = S(c− v)T

The velocity of light according to F ′ must satisfy:

dx′Adt′A

=ddT x

′A

ddT t′A

=S(c− v)

M +Nc= c

Assuming that M +Nc 6= 0. Now for a second event, B in the negative x-direction where B (T,−cT )then according to F ′:

t′B = MT −NcT

x′B = −S(c+ v)T

And so the speed of light is:

dx′Bdt′B

=ddT x

′B

ddT t′B

=−S(c+ v)

M −Nc= −c

18

It can be written that Mc+Nc2 = S(c+ v) and so M = S and N = −S vc2 which gives:

t′ = S(t− vx

c2

)x′ = S(x− vt)

Considering the inverse transformation from F ′ to F then there is a change in the sign of v and so:

t = S

(t+

vx′

c2

)= S2

(t− vx

c2

)+ S2 v

c2(x− vt) = S2

(1− v2

c2

)t

From this it can be seen that:

S =1√

1− v2

c2

This is the Lorentz factor.

5.3.2 Loss of Absolute Simultaneity

If there are two events, A and B in F which are simultaneous then A (T, xA, yA, zA) and B (T, xB , yB , zB). Using the Lorentz transformations for F ′ gives:

t′A = γ(v)(T − vxA

c2

)t′B = γ(v)

(T − vxB

c2

)From this it can be found that:

t′A − t′B =γ(v)v

c2(xB − xA)

Events which are simultaneous in F are not simultaneous in F ′.

5.3.3 Space-Time Diagrams

By ignoring the non-changing coordinate then x and t can be plotted to give:

x

x′

tt′

x = ct

Figure 5.1: Space-Time Diagram

19

The dark axes are in the frame F and the dashed axis are in the frame F ′. It can be seen that simultaneityis lost because times with a constant t have different values of t′ so that events which are simultaneousin F are not in F ′.

5.4 Addition of Velocities

In Newtonian mechanics then the velocities can be added easily as they are just vectors. This is not truefor special relativity. A particle, P , is moving uniformly with a velocity, u˜′ = (u1, u2, u3)′, relative to theframe F ′. The worldline of the particle is a set of events in space-time with coordinates (t′P , x

′P , y

′P , z

′P )′

so that:x′P (t′P ) = x′P (0) + u1t′P

y′P (t′P ) = y′P (0) + u2t′P

z′P (t′P ) = z′P (0) + u3t′P

In the frame F the velocities are found by the inverse Lorentz transformations so that:

tP = γ(v)(t′P +

vx′Pc2

)= γ(v)

(t′P +

vx′P (0)c2 + vu1

c2 t′P

)xP = γ(v) (x′P + vt′P ) = γ(v)(u1t′P + vt′P )

yP = y′P = y′P (0) + u2t′P

zP = z′P = z′P (0) + u3t′P

This is a linear relation and as such a uniformly moving particle in F ′ will also be moving uniformly inF . To find the velocity in F then only unprimed components can be used. This can be done by usingthe chain rule.

u1 =dxPdtP

=dxPdt′P

dt′PdtP

=u1 + v

1 + vu1

c2

u2 =dyPdtP

=dyPdt′P

dt′PdtP

=u2

γ(v)(

1 + vu1

c2

)u3 =

dzPdtP

=dzPdt′P

dt′PdtP

=u3

γ(v)(

1 + vu1

c2

)If the components of the velocity are written as u˜′ = c(sinϑ cosϕ, sinϑ sinϕ, cosϑ) then the scalar product

gives the correct value of the speed of light u˜′ · u˜′ = c2

20

Chapter 6

Minkowski Spacetime

6.1 Geometry

Minkowski spacetime rewrites the theory of special relativity in terms of geometry. If a straight line isdrawn and a locus formed by constructing perpendicular lines of length a, from the first straight linethen the lines do not intersect in Euclidean geometry.

aa

aa

aa

Figure 6.1: Euclidean Parallel Lines

This is not necessarily true for other geometries. For example in hyperbolic geometries then the two linesnever meet but it is clear that one line is not straight in Euclidean geometry.

a

a

a

a

a

Figure 6.2: Hyperbolic Parallel Lines

Non-homogeneous geometries can also exist where each point in space can have a different geometry.This is Riemannian geometry.

21

6.2 Invariance of c2∆t2 −∆x2 −∆y2 −∆z2

The invariance of the speed of light under Lorentz transformation has the consequence that the distancerelation of a pulse of light sent from an event A to another event B in a frame F is:

c(tB − tA) =√

(xB − xA)2 + (yB − yA)2 + (zB − zA)2

In frame F ′ then:

c(t′B − t′A) =√

(x′B − x′A)2 + (y′B − y′A)2 + (z′B − z′A)2

Squaring both sides allows these equations to be written as:

0 = c2∆t2 + ∆x2 + ∆y2 + ∆z2

0 = c2∆t′2 + ∆x′2 + ∆y′2 + ∆z′2

This is seen to be equal and so any two events in spacetime are Lorentz invariant and form invariantunder Lorentz transformation. This therefore allows the Minkowski distance rule to be found.

6.3 Minkowski Distance Rule

The spacetime (Minkowski) distance is analogous to the distance rule in three dimensional space with ametric given by:

∆s2 = c2∆t2 − (∆x2 + ∆y2 + ∆z2)

The relative change of sign between the spatial and time intervals is arbitrary and is only chosen to bethe way written above because the time term tends to dominate and hence give imaginary ∆s.

6.3.1 Imaginary Time

The coordinate c∆t could be replaced by i∆w where w is the new fourth dimension of a Euclidean spacewhich allows the formation of:

∆s2 = ∆w2 + ∆x2 + ∆y2 + ∆z2

This is appealing because of the similarities to three dimensional geometry, but time becomes imaginaryand so this appears to have no physical meaning. This implies that w is complex which is not allowed inreal space.

6.4 Metric Structure of Minkowski Spacetime

Spacetime with ∆s2 = c2∆t2 − (∆x2 + ∆y2 + ∆z2) as the distance rule is called Minkowski spacetime,M4. A crucial feature of the Minkowski spacetime is that ∆s2 = 0 does not imply that ∆t = ∆x =∆y = ∆z = 0 as it does for lower dimensional spaces. Two events are said to be relatively timelike if∆s2(A,B) > 0 and relatively spacelike if ∆s2(A,B) < 0. Spacelike seem strange because it implies thatthe negative square root is still a real number, which can arise due to the geometry. Two separate pointswith ∆s2(A,B) = 0 are said to be relatively lightlike. A signal moving between two relatively lightlikeevents has a c velocity relation in any inertial reference frame as:

0 = c2∆t2 −∆r˜∆ · r˜In the limit that ∆t→ 0 then

∆r

∆t = u˜ so that u˜ · u˜ = c2.

22

6.4.1 Lightcones

Absolute Past

Absolute Future

Timelike

Spacelike Absolute Elsewhere

Figure 6.3: Lightcone

The set of points, B, such that ∆s2(A,B) = 0 is the lightcone of A. Any point B which are relativelytimelike are within the lightcone and any which are spacelike are outside. Points in the forward timeconeare in the absolute future and points in the backward timecone are in the absolute past. The worldlinemust always be a forward moving timelike vector. Particles moving faster than the speed of light wouldhave a spacelike worldline which exists outside of the speed of light.

Retardation effects enforce a particle and field theory. This is because at a single moment of time then acharged particle which has sent a signal to another will appear as two separate particles and a messengerin a field.

Figure 6.4: Retardation Effects When a Message is Sent

6.4.2 Spacetime Coordinates

The notation for spacetime is xµ = (ct, x, y, z), which could also be written as x = (ct, xi) or x = (ct, r˜).Coordinates are used to identify points in a manifold relative to some local coordinates patch. Somemanifolds such as spheres cannot be covered by a single coordinate patch. E3 andM4 can and Minkowskispace can also use polar coordinates as M4 is itself a manifold and so can use the coordinates patch R4.

6.5 Minkowski Metric

If two events, A and B, in spacetime have coordinates xµA and xµB then in Minkowski distance ∆s betweenthe events is:

∆s2 = ∆xµηµν∆xν

23

Where ∆xµ = xµB − xµA and η is a 4× 4 matrix:

ηµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

This is the matrix of the matrix tensor components relative to standard inertial coordinates. Spacetimeis now given a set of properties with a metric tensor attached to it. The worldlines of objects are usedto explore the physics of the spacetime metric and so this is equivalent to studying the properties ofspacetime. This means that the aether hasn’t really been removed but rather relabelled.

If A and B are said to be infinitesimally close then ∆xµ = xµB − xµA become infinitesimals dxµ and so

ds2 = dxµBηµνdxν . Expanding this gives:

ds2 = (dx0)2 − (dx1)2 − (dx2)2 − (dx3)2 = c2dt2 − dx2 − dy2 − dz2

This is called the line element and allows the metric tensor to be constructed. For example in sphericalpolar coordinates then:

ds2 = c2dt− dr2 − r2dϑ2 − r2 sin2 ϑdφ2

The metric tensor in these coordinates is therefore:

ηµν =

1 0 0 00 −1 0 00 0 −r2 00 0 0 −r2 sin2 ϑ

6.6 Transformations of the Metric Tensor

If there is an open subset U in M4 with two coordinate patches P = xµ and P ′ = x′µ covering it.Over U the Minkowski space is:

ds2 = dxµgµνdxν

Where gµν is now used as the metric rather than ηµν for when the components may not be diagonal ormay depend on the coordinates. If U is such that P can transform to P ′ as x′µ = x′µ(x) and xµ = xµ(x′).These coordinates are related in a smooth way where xµ are infinitely differentiable functions of x′ν . Thisallows the chain rule to be used to transform between the infinitesimal dxµ and dx′µ:

dx′µ = ∂νx′µdxν and dxµ = ∂′νx

µdx′ν

dxµ and dx′µ transform as the components of a type−(

10

)tensor which are vectors and inM4 can

also be called four-vectors. xµ and x′µ are not the components of a type−(

10

)tensor in principle.

Given that ds2 = dxµgµνdxν then this can be written as:

ds2 = ∂′αxµdx′αgµν∂

′βx

νdxνdx′β = dx′α(∂′αxµ∂′βx

νgµν)dx′β

By rearranging this ds2 = dx′αg′αβdx′β . Since dx′α are independent then it can be considered that:

g′αβ = ∂′αxµ∂′βx

µgµν

Knowing what the metric tensor looks like in one coordinate patch allows the metric tensor in a newcoordinate patch to be found. This is the transformation rules for the transformation of the component

of a type−(

02

)tensor. Spacetime is described by this

(02

)tensor. In general relativity there is also a(

13

)tensor which is the Riemannian curvature tensor.

24

6.7 Lorentz and Poincare Transformations

If in Minkowski spacetime, M4 there are two standard inertial frames F and F ′ then the value of suchframe is two-fold. The fist is that there is a single coordinate patch which covers M4 and the metrictensor takes on the same form. The passive transformation between standard inertial frames can involvespacetime translations (4 parameters), spacial rotations (3 parameters) and velocity boosts (3 parameters)is known as a Poincare transformation:

xµ → x′µ = Λµνxν + aµ

For the Poincare transformation then using infinitesimal gives:

dxµ → dx′µ = Λµνdxν

This can be seen because:

∂νx′µ = ∂ν [Λµνx

α + aµ] = ∂ν [Λµαxα] + ∂νa

µ

As Λµα and aµ are constant this leaves:

∂νx′µ = Λµα(∂νx

α) = Λµαδαν

There is a summation over the α and so this leaves:

∂νx′µ = Λµν

The metric tensor relative to F should be the same as those relative to F ′ and so:

ds2 = dxµηµνdxν = dx′µηµνdx

′ν

It can therefore be shown that by substituting dx′µ = Λµαdxα:

dxµηµνdxν = [Λµαdx

α]ηµν [Λνβdxβ ] = [Λαµdx

µ]ηαβ [Λβνdxν ]

From which it can be deduced that ΛαµηαβΛβν = ηµν . This is the fundamental condition that the

general Lorentz transformation matrices have to satisfy ΛαµηαβΛβν = ηµν . This can be compared withthe analogous condition obeyed by rotation matrices in E3:

RT IR = I

Taking determinants on both sides of ΛαµηαβΛβν = ηµν gives |Λ2| = 1 where |Λ| = 1 are proper Lorentztransformation. For example when µ = ν = 0 then:

Λα0ηαβΛβ0 = η00 = 1

Whenever |Λ| = −1 then the Lorentz transformation are improper and are related to time, space orspacetime inversion.

6.8 Index Raising and Lowering

If there are a set of numbers or objects Aµ : µ = 0, 1, 2, 3, a lower index set Aµ : µ = 0, 1, 2, 3 can bedefined via:

Aµ = gµνAν

For Cartesian coordinates where gµν = ηµν then A0 = A0 and Ai = −Ai. Aµ does not have to be thecomponents of a four-vector for example xµ are not the components of a four-vector but it is still truethat xµ = ηµνx

ν . The Lorentz transformation can also be lowered using a similar form:

Λµν = ηµαΛαν

Raising the indices is done using the inverse metric tensor so that an object with a set of componentsgµν = gαµgµν = δαν . Given a set Bµ then raising the indices shows:

Bµ = gµαBα

Raising and lowering indices allows shortening of rules such as the Minkowski distance rule:

ds2 = dxµdxµ

25

Chapter 7

Relativistic Particle Mechanics

7.1 Particle Worldline

An important property of a particle worldline are that a cross-sectional cut through a single (simultaneityof) time in an inertial reference frame contains at most only one point of a given particle worldline. Thislinks into the theory that electrons are moving forward in time and positrons move backwards in time.It can be also predicted that there could be only one electron/positron which travels forwards andbackwards in time and moves around in space to create all the particles. Another important property isthat a worldline does not disappear spontaneously and then reappear at a different time or place. Thismeans that particle worldlines are continuous in the non-mathematical sense.

7.1.1 Proper Time

A worldline W inM2 can be parameterised by some real path parameter λ, running from λi to λf > λi.It can be noted that proper time is not a quality of spacetime but rather dependent on the worldline andas such it is not integrable. Relative to the inertial frame F then the events of the worldline are given bycoordinates xµ(λ), µ = 0, 1, 2, 3, 4. W is assumed to be at least twice differentiable with respect to λ atevery point. This is the same as Newtonian mechanics, but the path parameter is now no longer absolutetime. The Minkowski distance square between A and B is:

∆s2(A,B) = ∆xµηµν∆xν

The particle can be restricted to timelike worldlines by instating ∆s2(A,B) ≥ 0. For any pair of eventsA and B in W then ∆s2(A,B) > 0 ∀A,B ∈ W . Now taking the limit where xµ → 0 then the distancerule becomes ds2 = dxµηµνdx

ν . Differentiating this with respect to λ then:(ds

dλ

)2

=dxµ

dληµν

dxν

dλ> 0

Taking the square root then this can in fact be integrated along the worldline to give:

sf − si =

∫ λf

λi

dλds

dλ=

∫ λf

λi

dλ

√dxµ

dληµν

dxν

dλ

It is clear to see that sf − si is path-dependent and has the physical dimensions of length. The pathparameter can be chosen, where it increases monotonically along W . It is most useful to close λ to beproportional to s along W so that:

ds

dλ= c

λ is the proper time along the worldline and it is often denoted by τ . This is useful for timelike worldlinesbut not for null or spacelike ones.

7.1.2 Four Velocity

Using proper time as the path parameter for the timelike worldline then the coordinates relative to anyinertial reference frame satisfies:

c2 = xµηµν xν

26

Where x ≡ dxµ

dτ is the four-velocity. It can also be denoted by uµ and holds the condition uµuµ = c2.

From time dilation, T ′ =√

1−v˜·v˜c2 T then it can be seen that T ′ is equivalent to the proper time τ so

that when T → 0 then:

dτ

dt=

√√√√1−

dx

dt·dx

dtc2

dx

dt is the instantaneous three-velocity of the clock at an event in the inertial frame. This allows thedefinition of:

t = γ(v˜)This means that the four velocity can be written as:

uµ ≡ (ct, x˜) = (cγ, γv˜)7.1.3 Four-Tensors

In three-vector algebra then a vector v˜ can be described in terms of an orthonormal basis sete˜,ı, i = 1, 2, 3

these vectors can be written as linear summation:

v˜ = vie˜ iIn passive rotations then e˜ i → e˜′i = Rije˜j and vi → v′i = Rijv

j and so v˜ → v˜′ = v˜. Much the

same thing happens in special relativity. M4 becomes an affine space with the Minkowski pseudometric,M4 = (R4, η). For any two events P and Q then:

Q = P + a˜If a basis is chosen so that

e˜µ, µ = 0, 1, 2, 3

then the four-vector a˜ can be written as:

a˜ = aµe˜µAny event, P in M4 will have coordinates xµP relative to a frame F , which has O as the origin ofcoordinates, is given by:

P = O + xνP e˜νPoincare transformation can be made from the old origin O to O′ with Lorentz transformations to give:

P = O′ + x′µP e˜′µNow taking x′µP = Λµνx

νP + aµ, where ΛµαηµνΛνβ = ηαβ . The shift in the origin in coordinates is found

by substituting this into the above equation to give:

O = O′ + aµe˜′µIt is clear that e˜ν = Λµνe˜′µ and using the rules to raise and lower indices Λ α

µ Λµν = δαν and ΛαµΛ µβ = δαβ

and from which e˜′µ = Λ νµ e˜ν . A four-vector is invariant to Lorentz transformations but can be written

according to the frame as:v˜ = vµe˜µ and v˜ = v′µe˜′µ

It can be seen from this that v′µ = Λµνv′ν and e˜′µ = Λ ν

µ e˜ν .

7.2 One-Forms

Given a vector space then a mapping into R or C by a linear functional where T (αv˜+βu˜) = αT (v˜)+βT (u˜).If addition and multiplication by scalars of linear functional is possible then the set of the linear functionalsis a vector space called the dual space, V ∗. The dimension of the dual space is the same as the originalspace. The elements of the dual space are called one-forms, ω. Acting on a one-form on a vector gives ascalar denoted ω(v˜). If there is a basis e˜µ for V then a basis eµ for V ∗ can always be found whichhas the condition:

eµ(e˜ν) = dµν

27

An arbitrary one-form is a combination of the basis elements and components with the form ω = ωµeµ.

The index is the opposite way to that of vectors:

ω(v˜) = ωµeµ(vνe˜ν) = ωµv

ν eµ(e˜ν) = ωµvνdµν = ωµv

µ

Under Lorentz transformation then µ˜ → e˜ ′µ = Λ νµ e˜ν for basis vectors and for basis one-forms then

eµ → e′µ = Λµν eν . Under the Lorentz transformation then it is clear that ω(v˜) is a Lorentz scalar and

so is invariant to Lorentz transformations.

7.3 Metric Tensor

One-forms can be used to elegantly describe the metric tensor. The metric tensor η is a type-

(02

)tensor

given by:ηµν e

µ ⊗ e ν

Where ⊗ is the tensor product. The use of the tensor product can be seen by taking two copies of thedual space and taking an element from each. The tensor product of the elements ω1 and ω2 is a memberof the vector space V ∗⊗V ∗, but there are more members than just these created from linear combinations

of elements which are not separable. ηµν eµ ⊗ e ν acts on a type-

(20

)tensor such as:

Tαβe˜α ⊗ e˜βThis happens because that in the same way that one-forms act on a vector to create a scalar, then the

type-

(02

)tensor acts on the type-

(20

)tensor, which maps into R.

η(Tαβe˜α ⊗ e˜β) = ηµν eµ ⊗ e ν(Tαβe˜α ⊗ e˜β) = ηµνT

αβ eµ( e˜α)e ν( e˜β) = ηµνTαβdµαd

νβ = ηµνT

νµ = T νµ

The metric tensors can also be regarded as taking two four-vectors as arguments. For u˜ = uµe˜µ andv˜ = vµe˜µ then:

η(u˜, v˜) = ηµν eµ ⊗ e ν(uαvβe˜α ⊗ e˜β) = ηµνu

µvν = uνvν = uνuν

This defines the relativistic inner product for these two four-vectors then u˜ · v˜ ≡ η(u˜, v˜) where, fortimelike vectors η(u˜, u˜) > 0, lightlike/null four-vectors satisfy η(u˜, u˜) = 0 and spacelike four-vectorssatisfies η(u˜, u˜) < 0. Instantaneous four-velocity u˜ ≡ xµe˜µ along a timelike worldline satisfies the rule

u˜ · u˜ = c2.

7.3.1 Four-Acceleration

An acceleration can be defined by aµ ≡ xµ ≡ uµ. This means that the Lorentz factor must also bedifferentiated:

γ =dt

dτ

d

dt

(1−

v˜ · v˜c2

)− 12

= γ

(−1

2

)(1−

v˜ · v˜c2

)− 32 −2v˜ · a˜

c2= γ

4v˜ · a˜c2

As v˜ = dtdτ

ddtv˜ = γa˜ then:

aµ =(γ4v˜ · a˜c, γ4

v˜ · a˜c2

v˜+ γ2a˜)

As the four-velocity along the worldline has the relation uµuµ = c2 then differentiating on both sidesthen it can be seen that uµaµ = 0.

7.4 Energy-Momentum Four-Vector

The four-momentum pµ of a particle along a timelike worldline has:

pµ ≡ moxµ

mo is the rest mass, which is constant. All electrons and positrons have the same rest mass, even thoughtheir total mass is different. Protons and antiprotons have the same rest mass also, although this is

28

different to that of the electrons. There are also zero rest mass particles, such as the photon. Undera standard Lorentz transformation then the coordinates transform as xµ → x′µ ≡ Λµνx

ν for an inertialframe F → F ′ and the four-velocity transforms as a four-vector xµ → x′µ ≡ Λµν x

ν so the four-momentumtransforms according to the same rule:

pµ → p′µ ≡ Λµνpν

The Lorentz invariant term is called the mass-shell constraint is shown by:

pµηµνpν = m2

oc2

It is called the mass-shell constraint because the inner product of the momentum is p˜ · p˜ = (p0)2 + p˜ · p˜ =

moc2 which confines the particles to the surface of this spacetime curve. Virtual particles can exist,

which do not obey this particular constraint. This can be seen in quantum electrodynamics wherescattering particles creates virtual particles for the interaction and destroys mass conservation. Thezeroth component of the four-momentum, p0, is related to the relativistic energy of the particle p0 ≡ E

c .The other components give the relativistic linear momentum, p˜ = γmov˜. This means that the mass-shell

constraint is equivalent to:E2 = c2p˜ · p˜+m2

oc4

For a particle of zero velocity then the linear momentum is zero and so Eo = moc2. The Hamiltonian of

a free relativistic point particle as seen from a non-relativistic frame is:

H( p˜, x˜, t) = c√p˜ · p˜+m2

oc2

Writing v ≡ |v˜| then the total energy is E = moc2(

1− v2

c2

). When v > 0 then a Taylor expansion can

be made:

E = moc2

(1− v2

c2

)− 12

= moc2

(1 +

1

2

v2

c2+O

(v4

c4

))= moc

2 +1

2mov

2 +O(c−2)

The second term is just the Newtonian kinetic energy.

7.5 Relativistic Forces

For a particle on a timelike worldline with a rest mass then the instantaneous rate of change of momentumis:

d

dτp˜ = F˜

This is the relativistic force four-vector F˜ = Fµe˜µ. This can be described as the relativistic force or

the equation of motion which determines a worldline given a relativistic force (γf0, γf˜). By changingddτ = γ d

dt then γ ddt (moγc,moγv˜) = γ(f0, f˜) and so f0 = d

dt (moγc) and ddtp˜ = f˜. This makes the force

look like a three-vector form p˜ ≡ moγv˜ where v˜ is the spatial part of the four-momentum. Taking the

mass-shell constraint p˜ · p˜ ≡ η( p˜, p˜ ) = pµpµ = moc

2 then it can be seen that f0 is the power, or the rate

of change of energy as can be seen by:d

dt(mγc2) = v˜ · f˜

7.6 Photons

Photons are particles which travel at the speed of light. It should be noted the particle velocity has amaximum speed of c but the wave velocity can be faster. This can be thought of as a subliminal speed, vaccompanied by a pilot wave of speed w, which holds the relation vw = c2. This means that for a photontravelling at v = c then the associated de Broglie wave also travels at w = c. For other real particlesthen v < c and so w > c. It is this wave which can give rise to quantum correlations, but no physicalsignal can travel faster than c. A photon must have a null four-vector u˜ · u˜ = η(u˜, u˜) = 0. Photons can

be described to have a non-zero four-momentum p = pµe˜µ with components pµ = (Ec , p˜) which satisfies

the condition pµpµ which is equivalent to E2 = c2p˜ · p˜.

29

7.7 Relativistic Particle Collisions

Unlike Newtonian mass, the rest mass is not conserved in special relativity. When relativistic particlesinteract in an isolated region of spacetime then looking at the four-momentum before and after theinteraction has the rule:

N∑n=1

pµn =

M∑m=1

qµm

pµn are the components of the individual four-momenta of the N incoming particles and qµm are thecomponents of the four-momenta of the M outgoing particles. The number of particles is not necessarilyconserved and it is also unusual for N 6= 2 because it is hard to set up a three particle collision.

qµ1qµ2 qµm

pµ2pµ1

I

Figure 7.1: Conservation of Four-Momentum During a Collision

Conserved quantities such as the total energy-momentum and the total angular momentum are linkedto spacetime and other symmetries. As Minkowski spacetime is homogeneous and flat and so there isno curvature and as such no gravity. There are also no regions where the particles undergo unexpectedmomentum changes.

7.7.1 Particle Decays

Some so-called elementary particles, such as neutrons and muons have a finite lifetime with characteristichalf-lives. An unstable particle of rest mass M can decay into two particles of rest mass m1 and m2.The momentum four-vector of the unstable particle is pµ and for the decay components qµ1 and qµ2 . Themass-shell constraints are pµpµ = M2c2, qµ1 q1µ = m2

1c2 and qµ2 q2µ = m2

2c2. The obvious frame to use is

that of the rest frame of the unstable particle. This means:

pµ = (mc, 0˜)qµ1 =

(√q˜1 · q˜1 +m2

1c2, q˜1

)qµ1 =

(√q˜2 · q˜2 +m2

2c2, q˜2

)

Applying the conservation gives pµ = qµ1 + qµ2 and substituting in the components means that:

Mc =√q˜1 · q˜2 +m2

1c2 +

√q˜2 · q˜2 +m2

2c2

30

0˜ = q˜1 + q˜2 so that q˜1 = −q˜2 = q˜Finding the value of the scalar product of q˜ gives:

q˜ · q˜ =c2(M2 +m2

1 −m22)2 − 4M2m2

1c2

4M2

The rest mass cannot be conserved because m1 + m2 = M only if q˜1 = −q˜2 = 0 which is fictitious

because the two masses do not move apart and so it is in fact just the same single particle.

7.7.2 Two-Two Scattering

If two particles collide and two particles come out then the mass shell constraints are:

pµ1p1µ = M21 c

2 pµ2p2µ = M22 c

2

qµ1 q1µ = m21c

2 qµ2 q2µ = m22c

2

The conservation of energy-momentum can be stated as pµ1 + pµ2 = qµ1 + qµ2 . This scattering can bedescribed by the Feynman diagram:

e−

e−

kµ

e−

e−

Figure 7.2: Two-Two (Compton) Scattering

But this is not the only option as it could occur that:

e−

e−

kµ

e−

e−

Figure 7.3: Different Interaction of Two-Two (Compton) Scattering

This process must be taken into account as well as higher order interactions and the sum of all theprocesses give the probability of the scattering occurring. The wavy line indicates a virtual photon whichin quantum field theory is a carrier of exchange forces. The four-momentum must be conserved betweeneach vertex so that pµ1 = qµ1 + kµ and pµ2 + kµ = qµ2 where:

kµ = pµ1 − qµ1

The virtual particles are not able to be observed and so it is not a physical particle as they are not onthe mass-shell. The four-momentum of a virtual photon is:

kµkµ = (p1 − q1)µ(p1 − q1)µ = t

Photons which have travelled over natural scale lengths, such as the distances to galaxies or the distancefrom a wall, will be on the mass-shell. It is only below these realistic scales do virtual photons or particles

31

come into existence.

There are three Lorentz invariants apart from the rest mass for the four-momentum. These are:

s = (p1 + p2) · (p1 + p2)

t = (p1 − q1) · (p1 − q1)

u = (p1 − q2) · (p1 − q2)

s describes the energy scale of collisions and is what is maximised when designing equipment and t isthe momentum transfer from one particle to the other and describes the deflection. u is redundant intwo-two scattering. The invariant quantity can be written as:

s+ t+ u = (M21 +M2

2 +m21 +m2

2)c2

If one particle is stationary in the laboratory with pµ1 = (Mpc, 0˜) and another is accelerated to a large

energy E = c√p˜ · p˜+M2

p c2 then pµ2 =

(√p˜ · p˜+M2

p c2, p˜)

and so:

s = (p1 + p2) · (p1 + p2) = 2Mp(Mpc2 + E) ∼ 2MpE

If both particles can be accelerated with the opposite momentum then s goes as the square of the energy

rather than the linearly, where pµ1 =

(√p˜ · p˜+Mpc2,−p˜

)so that:

s =4E2

c2

32

Chapter 8

Relativistic Maxwell’s Equations

8.1 Maxwell’s Equations

Maxwell’s equations do not look relativistic as they are tied to a frame of reference. Despite appearancesthey are indeed Lorentz invariant. The magnetic and electric field are written as a second rank tensornamed the Maxwell-Faraday tensor. To find this then firstly the fields are written as B˜ = ∇ × A˜ andE˜ = −∇φ− ∂tA˜ . As any vector field F˜ , which is twice differentiable is F˜ = −∇ϕ +∇× S˜. The vectorand scalar potentials are differentiable everywhere except at the point charge, so avoiding this allows anunlimited amount of differentiation. Maxwell’s equations are identities because of this so that:

∇ ·B˜ = ∇ · (∇×A˜)= 0

∇× E˜ + ∂tB˜ = ∇× (−∇φ− ∂tA˜) + ∂t(∇×A˜)= −∇×∇φ−∇× ∂tA˜ + ∂t(∇×A˜)= 0

The six degrees of freedom of the electric and magnetic field are kept within only four components. Theinhomogeneous equations can be written:

−∇2φ− ∂t∇ ·A˜ =%

ε

∇× (∇×A˜) + µoεo∂t(∇φ+ ∂tA˜) = µoj˜These do not look elegant but they can be made much more so.

8.2 Gauge Potentials

As φ and A˜ are arbitrary then gauge transformations can be made so that φ → φ′ = φ + ∂tχ andA˜ → A˜ ′ = A˜ − ∇χ. This replacement of the potentials is a gauge transformation of the second kind.A gauge transformation of the first kind is created by the addition of a constant, φ → φ′ = φ + k andA˜ → A˜ ′ = A˜ − k˜. The second kind gauge transformation components can be spacetime dependent fields.The electric and magnetic fields are invariant under a transformation of the second kind.

33

E˜ ′ = −∇φ′ − ∂tA˜ ′= −∇(φ+ ∂tχ)− ∂t(A˜ −∇χ)= −∇φ− ∂tA˜ −∇∂tχ+ ∂t∇χ= −∇φ− ∂tA˜= E˜

B˜ ′ = ∇×A˜ ′= ∇× (A˜ −∇χ)= ∇×A˜ −∇×∇χ= ∇×A˜= B˜

The electric and magnetic fields are the physical measurable quantities, but the potentials are not. Thegauge degrees of freedom can be described by starting with a Lagrangian:

L =m

2(x− y)

2

Having a gauge degree of freedom form x → x′ = x + f(x, y, z) and y → y′ = y + f(x, y, z) indicatesarbitrary redundancies in the mathematics:

L → L′ =m

2(x′ − y′)2

=m

2

(x+

d

dtf − y − d

dtf

)2

=m

2(x− y)

2= L

To see the redundancies then if the coordinate transformation u = x− y and v = x+ y is made then:

L → L =m

2u2

The physical dynamics only depend on one degree of freedom. Having two coordinates, x and y, istherefore not necessary.

8.3 The Four Potential, Aµ

The four components of the electromagnetic four-potential are:

Aµ =

(φ

c,Ai)

Where φ is divided by c to give it the correct dimensions from dimensional analysis. Now the inhomo-geneous equations can be expressed by the four-potential where it can be noted that ∇ × (∇ × A˜) =

∇(∇ ·A˜)−∇2A˜ to give:

−∂i∂iA0c− c∂0∂iAi =

%

εo

This can be simplified further by dividing by c:

∂i∂iA0 − ∂0∂iA

i = µo%c

The second inhomogeneous equation is:

−∂j∂iAi + ∂i∂iAj + ∂0(−∂jA0 + ∂0A

j) = µojj

The charge current j˜ satisfies the equation of continuity:

∇ · j˜+ ∂t% = ∂0(c%) + ∂iji = 0

It can be seen that % and j˜ are in fact both components of a four-current density:

jµ ≡ (c%, ji)

34

The equation of continuity is greatly simplified to ∂µjµ = 0. This is form invariant to Poincare trans-

formations xµ −→P

x′µ = Λµνxν + aν . The components transform according to the rules of a four-vector

field:jµ(x) −→

Pj′µ(x′) = Λµνj

ν(x)

The electric charge density appears to increase with velocity as the charge remains constant but thevolume decreases due to FitzGerald contraction. Electric currents are also taken into account in theLorentz transformation. The two inhomogeneous equations are now given as:

∂i(∂iA0 − ∂0Ai) = µoj

0

−∂j∂iAi + ∂i∂iAj + ∂0(−∂jA0 + ∂0A

j) = µojj

The Maxwell-Faraday (Field Strength) tensor can now be defined:

Fµν = ∂µAν − ∂νAµ

This tensor is antisymmetric, Fµν = −F νµ and Fµµ = 0, so there are six components available, exactlythe amount needed to contain the electric and magnetic fields It can be seen that ∂µF

µ0 = µoj0 and

∂µFµj = µoj

j so that the elegant solution is found:

∂µFµν = µoj

ν

This equation transforms Lorentz covariantly when Fµν transforms as the components of a type- 20 tensor

field:Fµν(x) −→

PF ′µν(x′) = ΛµαΛνβF

αβ(x)

And as such the four-potential transforms as:

Aµ(x) −→P

A′µ(x′) = ΛµνAν(x)

The gauge potential is now equivalent to Aµ −→χ

A′µ = Aµ + ∂χ which is consistent with Aµ being the

components of a Lorentz four-vector.

8.4 Maxwell-Faraday Tensor

The Maxwell-Faraday tensor is gauge invariant as:

F ′µν = ∂µA′ν − ∂νA′µ= ∂µ(Aν + ∂νχ)− ∂ν(Aµ + ∂µχ)= ∂µAν + ∂µ∂νχ− ∂νAµ − ∂ν∂µχ= ∂µAν − ∂νAµ= Fµν

This tensor can be filled using F 0i = ∂0Ai − ∂iA0 = 1c∂tA

i + ∂iφc = −E

i

c and F ij = ∂iAj − ∂jAi =−∂iAj + ∂jA

i = εijkBk so that:

Fµν =

0 −E

1

c −E2

c −E3

cE1

c 0 −B3 B2

E2

c B3 0 −B1

E3

c −B2 B1 0

8.5 Lorentz Transformations of Electric and Magnetic Fields

For two standard inertial frames, F and F ′ with x′µ = Λµνxν + aµ then:

F ′µν = ΛµαΛνβFαβ

35

Writing this in terms of matrix multiplication necessitates the indices to be next to each other as [F ′]µν =[Λ]µα[F ]αβ [Λ]Tνβ .

0 −E′1

c −E′2

c −′E3

cE′1

c 0 −B′3 B′2

E′2

c B′3 0 −B′1E′3

c −B′2 B′1 0

=

0 −E

1

c γβB3 − γE2

c −γβB2 − γE3

cE1

c 0 −γB3 + γβE2

c γB2 + γβE3

c

−γβB3 + γE2

c γB3 − γβE2

c 0 −B1

γβB2 + γE3

c −γB2 − γβE3

c B1 0

This is easier to read in the form (E′1, E′2, E′3) = (E′, γE2 − γvB3, γE3 + γvB2) and (B′1, B′2, B′3) =(B1, γB2 + γvE

3

c2 , γB3 − γvE

2

c2

)or in three-vector form then:

E˜ ′‖ = E˜‖ and E˜ ′⊥ = γ(v)(E˜⊥ + v˜×B˜⊥)

B˜ ′‖ = B˜ ‖ and B˜ ′⊥ = γ(v)

(B˜⊥ − v˜× E˜ ⊥

c

)

8.6 Raising and Lowering Indices

Lowering the indices of Fµν creates a type- 02 tensor field:

F = Fµν eµ ⊗ e ν

The components of this is:

Fµν =

0 E1

cE2

cE3

c

−E1

c 0 −B3 B2

−E2

c B3 0 −B1

−E3

c −B2 B1 0

This can be considered a mapping of the Maxwell-Faraday tensor into R.

F (F ) = Fµν u˜µ ⊗ e ν(Fαβe˜ α ⊗ e˜ β)= FµνF

αβ eµ(e˜ α)e ν(e˜ β)= FµνF

αβdµαdνβ

= FµνFµν

This is a Lorentz Scalar Field. It can be calculated in terms of electromagnetic fields in a chosen frameof reference gives:

FµνFµν = −

2E˜ · E˜c2

+ 2B˜ ·B˜And so E˜ · E˜ − c2B˜ ·B˜ is a Lorentz scalar field.

8.7 Covariant Formulation of Maxwell’s Equations

By applying ∂ν to both sides of the inhomogeneous Maxwell’s equations gives:

∂ν∂µFµν = µo∂νj

ν

As Fµν is antisymmetric and ∂ν∂µ is symmetric then it states that ∂ν∂µFµν = 0 and so the continuity

equation is shown to be valid:∂νj

ν = 0

Maxwell’s homogeneous equations can be obtained by creating the dual Maxwell-Faraday tensor, F ∗ ≡F ∗µν e

µ ⊗ e ν , where a four-dimensional Levi-Civita tensor needs to be defined, εαβµν which has the sameproperties as the three-dimensional tensor and ε0ijk = εijk. The components of the dual tensor are:

F ∗µν ≡1

2εµναβF

αβ

36

This has the very odd property of swapping the electric and magnetic fields in the Maxwell-Faradaytensor which reveals the symmetry property between the vector space and the dual space.

F ∗µν =

0 −B1 −B2 −B3

B1 0 −E3

cE2

c

B2 E3

c 0 −E1

c

B3 −E2

cE1

c 0

It can be seen that F ∗µν ≡ 1

2εµναβ(∂αAβ−∂βAα) = εµναβ∂αAβ . Taking the partial derivative to this must

be zero due to the antisymmetric properties of εµναβ and the symmetry of ∂µ∂α, This reveals ∂µF ∗µν = 0and hence the homogeneous equations:

∂µF ∗µ0 = ∂0F ∗00 + ∂iF ∗i0 = ∇ ·B˜ = 0

And:

∂µF ∗µi = ∂0F ∗0i + ∂iF ∗ji =1

c∂t(−Bi)− ∂

(−εjik

Ek

c

)= ∇× E˜ + ∂tB˜ = 0˜

37

Chapter 9

General Relativity

9.1 Differential Geometry

9.1.1 Manifolds

Manifolds are topological spaces which locally look like Euclidean spaces. An n-dimensional manifold,M is a topological space where each point P ∈ M has an open neighbourhood UP ⊆ M. This has acontinuous one-to-one map, F , onto an open subset F(UP ) of Rn for some n. F is called the framefunction and n is the dimension.

P

UP

F

G

M

Rn

y˜Px˜P

F(UP )

G(UP )

R⊗ R⊗ R⊗ ...⊗ R(1) (2) (3) (n)

(x1, x2, x3, ..., xn) = x˜

Figure 9.1: Mapping of a Point P ∈ M into xP ∈ Rn

R is more than just the abstract manifold, M, as it has coordinate and can be used to describe M. Anopen subset of R describes the open neighbourhood UP . If there is a second mapping G into Rn it couldbe in a completely different area of Rn. As the mapping is one-to-one then each point which is unique in

38

Rn is also unique in M.

There is not necessarily a map for all of M into Rn which is why the manifold looks like Rn locally butnot globally. Given P ∈M, UP and F then local coordinates are given by the coordinates of the image:

P −→F

xP ≡ (x1P , x

2P , ...x

nP ) = xaP

An open subset U ofM with a frame function F : U → Rn is called a coordinate patch. General relativityuses real manifolds based on a map into Rn but complex manifolds mapped onto C can be conceived.

9.1.2 General Coordinate Transforms

The coordinates in one image can be found by making a coordinate transformation, generally denoted T ,but which really exists as moving back along F to P and then along F ′.

P

U

U ′

F

M

Rn

x˜P

x˜′PF ′(U ′)

F(U)

F ′ T

Figure 9.2: General Coordinate Transformation from Two Different Open Subsets of M

If the partial derivatives of order k of x′a with respect to xb exists and are continuous functions thenthe frames F and F ′ are Ck-related. A Ck atlas is the charts which cover a given manifold. When twocharts overlap (intersection is non-zero) then they are Ck-related. A differentiable manifold is a manifoldwhere there is a C1-related atlas. All coordinate transforms are non-singular over U ∩ U ′ so that theinverse transformation exists.

J ≡ det(∂ax′B) 6= 0

9.2 Scalar Fields

A real scalar field, f , is a real valued function over the manifold so that for a point P ∈ M, f assigns avalue fP ∈ R. The value of fP of the field at P is independent of coordinate patch and so is intrinsic.

A complex scalar field, g, assigns a complex number gP ∈ C for each P ∈M. A scalar field is independentof any coordinate patch or frame. If (U,F) is a coordinate patch with P in it then the scalar field canbe written as a function of n real coordinates.

f(x1P , x

2P , .., x

nP ) = fP

A scalar field, f , is continuous at P if f(x) is continuous at xP . It is also differentiable if ∂if(x) exists atxP . The set of scalar fields on manifold M is S(M), where if f is a real scalar field then f ∈ S(M,R).

39

9.2.1 Coordinate Transformations

Given two coordinate patches, (U,F) and (U ′,F ′), containing P then:

fP = f(x1P , x

2P , ..., x

nP ) = f ′(x′1P , x

′2P , ..., x

′nP )

If the scalar field is differentiable then the chain rule can be used to obtain:

∂

∂xaf(x˜) =

∂x′b

∂xa∂

∂x′bf ′(x˜′)

This can be written more elegantly as:∂af = ∂ax

′b∂′bf′

9.3 Curves

A curve, Cλ, is a differential mapping of an open interval, I, of R intoM, where λ is the path parameter.For a patch (U,F) containing Pλ the coordinates xaP are differentiable functions of the curve parameter,λ, xaP = xa(λP ).

P2

P1

M

Rn

x(λ)

I

λ λR1

Figure 9.3: The Curve of I into Two Points in the Manifold Mapped onto the Real Numbers

The set of curves on a manifold M is C(M), where a point P ∈ M is CP . A scalar field, f , is over Mwhich contains two points P and Q with parameter values λP and λQ. The function of the points P andQ are:

fP ≡ f(xP ) ≡ f(x(λP )) ≡ f(λP )

fQ ≡ f(xQ) ≡ f(x(λQ)) ≡ f(λQ)

A real analytic function has derivatives of all orders which agrees with the Taylor Series in the neigh-bourhood of every point. If all scalar fields are real analytic functions of λ along the curve then:

fQ = f(λQ)

= f(λP + λQ − λP )

= f(λP ) + (λQ − λP ) df(λ)dλ

∣∣∣λ=λP

+ 12! (λQ − λP )2 d2f(λ)

dλ2

∣∣∣λ=λP

+ ...

=∑∞n=0

(λQ−λP )2

n!dn(λ)dλn

∣∣∣λ=λP

= e((λQ−λP ) ddλ )f(λ)

∣∣∣∣λ=λp

= UQP fP

Where UQP is the displacement operator. A useful property of the displacement is that UPQUQP = URP .

40

9.4 Directional Derivatives

For a n-dimensional real manifold M with a scalar field f , then the curve Cλ passing though P ∈ Mwhere λ = λP has f(λ) which can be evaluated by the directional derivative:

d

dλf(λP ) = lim

ε→0

f(λP + ε)− f(λP )

ε

This directional derivative is independent of any coordinate patch and so is an intrinsic property of themanifold, the scalar filed and the curve. For a coordinate patch (U,F) containing P then introducingthe coordinates xa for all points in U allows the directional derivative to be evaluated.:

d

dλf(λP ) =

d

dλf(x˜(λ))

∣∣∣∣λ=λP

=dxa(λ)

dλ

∂f(x˜)

∂xa

∣∣∣∣λ=λP

Since f is arbitrary then:

d

dλ= vaP

∂

∂xawhere vaP =

dxa(λP )

dλ≡ dxa(λ)

dλ

∣∣∣∣λ=λP

This therefore depends on the curve Cλ in the coordinate patch (U,F).

9.5 Tangent Vector Space

Another curve C ′µ also passing through P with path parameter µ ∈ I ′ ∈ R evaluated by C ′µ has:

d

dµ≡ waP

∂

∂xµwhere waP =

dxa(µP )

dµ≡ dxa(µ)

dµ

∣∣∣∣µ=µP

CλCµ

M

R

λP

P

µP

R

Figure 9.4: Two Curves in the Manifold M Both Passing Through P

Introducing arbitrary real constants α and β then:

αd

dλ+ β

d

dµ= αvaP

∂

∂xa+ βwaP

∂

∂xa= (αvaP + βwaP )

∂

∂xa

This is also a directional derivative operator at P . The coefficients can be written as a linear combination:

waP ≡ αvaP + βwaP

The directional derivative operators of P form a real vector space known as the tangent vector space,TP , at P . This vector space has the dimensionality as the manifold. The elements of this vector space,TP are operators which act on scalar functions and take in values in R. For a coordinate patch (U,F)containing P , a convenient basis for TP is B = ∂a, a = 1, 2, ...n where ∂a ≡ ∂

∂xa known as a coordinatebasis. This coordinate basis always commutes, ∂a∂b = ∂b∂a. It is possible to find bases for TP which arenet coordinates bases such as B′ = E˜ a where e˜ae˜b 6= e˜be˜a. This is called a non-coordinate basis.

41

9.5.1 Tangent Bundles

A tangent bundle TM is the set of all tangent vector spaces over the manifold and is an example of afibre bundle.

9.6 Vector Fields

A vector field selects one vector from the tangent space TP for each point P ∈ M. It is a cross-sectionof the tangent bundle.

M

Q

P

TP

TQ

v˜Pv˜Q

Figure 9.5: Vector Field Defined for Two Points P and Q

The vector field is denoted V(M which, for coordinate patch (U,F), then a vector field v˜ ∈ V(M) canbe expressed by v˜ = va(x)∂a, where va(x) are real-valued functions of the coordinates xa induced by

F . A differential vector field v˜ has coordinates va of v˜ which exist and are continuos in F(U) when ∂avb

exist.

9.7 Cotangent Space

For every tangent space, TP , there is a dual space T ∗P called the cotangent space at P with elements wP .

9.7.1 Coordinate Bases

For P ∈ M then (U,F) contains P so that P ∈ U and F : U → Rn induces a set of coordinates xacalled induced coordinates. A natural coordinate basis for TP is ∂a, a = 1, 2, .., n. A natural basis for

T ∗P is the conjugate basis dxa, a = 1, 2, .., d where dxa(∂b) = δab .

9.8 Congruences and Flows

For a vector field v˜ and a coordinate patch (U,F) then v˜ = vµ(x)∂µ in that patch. Flow lines are given

by v˜ = ddλ = dxµ

dλ ∂µ. These can be equated to give:

vµ(x)∂µ =dxµ

∂λ∂µ

This means the set of equations to the number of dimensions is found

dxµ

dλ= vµ(x) , µ = 1, 2, ..., n

These equations define flow lines or congruence, C(v˜) associated with the vector field. Two flow lines maydiverge or converge but can never cross.

42

9.9 Tensor Field

A tensor field of type-

(pq

)of a point P ∈M is a linear functional of p one-form fields and q vector fields,

ω1 ⊗ ... ⊗ ωpv˜1 ⊗ ... ⊗ v˜q. These correspond to p T ∗p ⊗ ... ⊗ T ∗p with coordinate basis dxµ1 , ..., dxνp and q

T⊗q ...⊗ Tq with coordinate basis ∂1α, ..., ∂

qβ . This tensor can be written as a scalar field:

T = Tµ1·µp...

ν1···νq(x)dxµq ⊗ ...⊗ dxµp ⊗ ∂1

ν1 ⊗ ...⊗ ∂qνq

A one-form field ω = ων(x)dxν is a type-

(01

)tensor and a vector field vv˜µ(x)∂µ is a type-

(01

). This

means that ω(v˜) = v˜(ω) is a scalar field, which can be seen to be true as:

nµ∂µ(ωαdxα) = vµωα∂µdx

α = vµωαδαµ = vµωµ

Here dxα and ∂µ are the basis vectors of the cotangent and tangent vector spaces, not the differentialoperators. The difference between the tangent vector space and the cotangent space is that the tangentvector space directly comes from the manifold. The set of component field of T relative to the coordinatepatch is T a1a2···ap b1b2···bq (x). The scalar field is T (S) = Tµ1...µp

ν1···νqSµ1···µpν1···νq , where T is type-(

pq

)and S is type-

(qp

)and so overall the entire filed is just a set of real numbers

(00

).

9.9.1 Coordinate Transformation of Tensor Field Components

Suppose v˜ ∈ V(M) and ω ∈ V∗(M) then these are independent of a coordinate patch. It is useful todiscuss the fields using coordinate patches. Given (U,F) and (U ′,F ′) which are intersecting, U ∩U ′ 6= 0

then over U v˜ = va∂a and ω = ωadxa and over U ′ then v˜ = v′a∂′a and ω = ω′adx

′a. Now over U ∩U ′ then

there is a coordinate transformation T from F to F so over the subset of M then va∂a = va(∂ax′b)∂′b =

v′b∂′b from which it can be seen that:

vb −→T

v′b = (∂ax′b)va

Because the coordinate transformation is invertible over U ∩ U ′ then:

v′a −→T−1

va = (∂′bxa)v′b

The case for the one-form basis is:dxa −→

Tdx′a = (∂bx

′a)dxb

And:ωa −→

Tω′a = (∂′ax

b)ωb

Under a coordinate patch transformation the type-

(pq

)tensor field shows:

Aa1a2···ap b1b2···bq = (∂r1x′a1)(∂r2x

′a2)...(∂rpx′ap)(∂′b1x

s1)(∂′b2xs2)...(∂′bqx

sq )Ar1r2···rp s1s2···sq

9.10 The Metric Tensor

Given a manifold M of dimension n, a metric tensor g is a symmetric type-

(02

)tensor field. It defines

a symmetric bilinear form so that for two vector fields, u˜ and v˜ over the manifold:

g(u˜, v˜) = g(v˜, u˜)

This is a scalar field. g acts on tensor products of the form u˜⊗v˜ and linear combinations of these tensors:

g(u˜, v˜) = g(u˜⊗ v˜)43

Conceptually these are not the same because u˜⊗ v˜ 6= v˜⊗ u˜ as the first u˜ comes from V1 and the second

comes from V2. Over a given coordinate patch (U,F) then g = gab(x)dxa ⊗ dxb where gab(x) are thedifferentiable components of a real, symmetric n × n matrix, gab(x) = gba(x). With two vector fieldsu˜ = ua∂a and v˜ = va∂a it can be shown:

g(u˜, v˜) = gab(x)dxa ⊗ dxbur∂r ⊗ vs∂s= gab(x)urvsdxa(∂r)dx

b(∂s)= gab(x)urvsdard

bs

= gab(x)uavb

The line element ds is generally ds2 = gab(x)dxadxb as the metric tensor gives the distance relationshipbetween points in a manifold separated by infinitesimal coordinate intervals dxa. Much of the timethe components gab instead of g are referred to as the metric tensor. Under a coordinate transform ofxa −→

Tx′a = x′a(x) then the components of the metric tensor transform according to:

gab(x) −→T

g′ab(x′) = (∂′ax

r)(∂′bxs)grs(x)

9.10.1 Index raising and Lowering

If only one vector field u˜ is selected and contracted with g then a type-

(01

)tensor field is found:

g(∗, u˜)

This is a one-form field because: (02

)(10

)∗ →

(01

)This is a very intuitive reason for raising and lovering indices. This can be shown by u˜ −→g ω ≡ g(∗, u˜).

A coordinate patch can then be selected so:

g = gµν(x)dxµ ⊗ dxν

So u˜ = uµ∂µ is operated on as:

g(∗, u˜) = gµν(x)dxµ ⊗ dxν(uα∂α)

= gµν(x)uαdxµ[dxν(∂α)]

= gµν(x)uαdxµδαν= gµν(x)uν dxµ

= uµdxµ

It can be defined that uµ ≡ gµν(x)uν . This process, of multiplying gµν by uν and summing over ν, iscalled contraction. In general if Aµ are the components of a vector then Aµ are:

Aµ(x) = gµν(x)Aν(x)

Under coordinate transformation then:

gµν(x) −→T

g′µν(x′) = ∂′µxα∂′νx

βgαβ(x)

Aµ(x) −→T

A′µ(x′) = ∂′αxµAα(x)

So that:Aµ(x) −→

TA′µ(x′) = ∂′µx

αAα(x)

The inverse metric tensor g−1 is a type-

(20

)tensor defined by:

g−1 ≡ gµν(x)∂µ ⊗ ∂ν

44

It can be seen that gµνgνα = δµα. Given a one-form filed ω(x) then the inverse metric can be used tocontract and give a vector field.

ω = gµν(x)∂µ ⊗ ∂ν(ωαdxα)

= gµν(x)ωα∂µ[∂ν(dxα)]= gµν(x)ωαd

αν ∂µ

= gµν(x)ων∂µ

So gµν can be seen as an index raising operator, ωµ(x) ≡ gµν(x)ων(x). For a tensor Tαβ... µν... ≡gααgββ ...g

µµgνν...Tαβ... µν....

9.11 Parallel Transport and Curvature

In a manifold, direction is contextual. If there is a vector field v˜ over M of n dimensions then thevalues of v˜P and v˜Q of points P,Q ∈ M lie in different tangent spaces, TP and TQ. These do not havecorresponding components in these vector spaces. Vectors can only be compared when they are in thesame vector space and this means that two separate tangent spaces cannot be used.

P P

Q Q

S2 S2

A B

v˜P

v˜‖

v˜P

v˜‖Figure 9.6: The Path Travelled Does Not Necessarily Give The Same Answer At The End

9.11.1 Covariant Derivatives

A manifold M with two vector fields u˜ and v˜ have a similar point P ∈ M. u˜ has a flow line Cu(P )through P . The vector v˜ at P can be parallel transported along the flow line and at each point Q can

be compared to the transported version of v˜P at Q called v˜‖QP with the actual vector field at Q, v˜Q. Todo this a covariant derivative is needed as two vectors cannot be compared in different tangent vectorspaces.

The parameterised representation of a vector field, Cu(P ) with parameter λ is:

u˜P =d

dλ

∣∣∣∣λ=λP

P corresponds to λ = λP . If Q 6= P is another point on Cu(P ) where λ 6= P then v˜(λQ) ≡ v˜Q and

v˜‖(λQ) ≡ v˜‖QP . The covariant derivative ∇uv˜|P is defined as:

∇uv˜P = limλQ→λP

v˜‖(λQ)− v˜(λQ)

λQ − λP

This cannot be done if v˜‖(λQ) is not the same vector space as v˜(λQ). Covariant differentiation does notchange the character of the object being differentiated so that a covariant derivative of a vector is a vector

and the covariant derivative of a scalar is a scalar. It can be seen that ∇uv˜‖P = 0 because v˜‖(λQ) = v˜‖QP .Since parallel transport of scalars does not involve any direction other than along the path then:

∇uf ≡ u˜(f) ≡ df

dλ

45

The general properties of derivatives hold for the covariant derivative, such as the Leibniz property:

d

dx(f(x)g(x)) =

(d

dλf(x)

)g(x) + f(x)

d

dλg(x)

If f is a scalar field then:

∇u(fv˜) = f∇uv˜+ (∇uf)v˜ = f∇uv˜+∇u(f)v˜∇u(v˜⊗ w˜) = ∇uv˜⊗ w˜ + v˜⊗∇uw˜∇u(ω(v˜)) = (∇uω)(v˜) + ω(∇uv˜)

For any scalar field f then ∇fu = f∇u and for any two vector ∇u+v = ∇u+∇v and so it can be seen that∇fu+gv = f∇u + g∇v. The covariant derivative operator is not a tensor field because ∇u(fv˜) 6= f∇u(v˜),which is not the same rule as ω(fv˜) = fω(v˜).9.11.2 Covariant Derivatives of Basis Vectors

For a basis field e˜i for the manifold then covariant derivatives can define the elements. If the covariantderivative of the basis field is denoted ∇i ≡ ∇e˜i along the flow line of the basis vector then because the

covariant derivative of a vector is another vector then:

∇ie˜j = Γkjie˜kWhere Γkji are the affine connection coefficients. The full set of coefficients determine the affine con-

nection. ∇ie˜ i is a vector in the tangent space represented by the components Γkjie˜k. ji are the metricconnections and are symmetric although Einstein tried to violate this symmetry to create a unified theory,which did not work. It is very common to write fi ≡ ∇if even if e˜ i is not a coordinate basis. If it is a

coordinate basis then ∂,i = ∂if . A vector field v˜ = vie˜ i has a covariant derivative:

∇iv˜ = ∇i(vje˜j) = (∇ivj)e˜j + vj∇ie˜jThis can be written in terms of the affine connection coefficients:

∇iv˜ = vj,ie˜j + vjΓkjie˜kAnother simplification is that vk,i + vjΓkji ≡k;i and as such:

∇iv˜ = vk;ie˜kFor one forms then ∇i

e j(e˜k)

= ∇i(djk) = 0 hence (∇ie j)(e˜k) + e j(∇ie˜k) = 0. As the covariant

derivative does not change the type of object then a one-form must remain a one-form, ∇ie j = Γjkiek. It

can be seen that this set of coefficients are Γjik = −Γjki and so ∇ie j = −Γjkiek. For an arbitrary one-form

field ω = ωiei then:

∇iω = ωj,iωj − ωjΓjkie

k = ωj;iej

Where ωi;j = ωi,j − Γkijωk. The general rule for tensors is:

∇mT = T i···j k···l;me˜i ⊗ · · · ⊗ e˜j ⊗ e k ⊗ · · · elWhere:

T i···j k···l;m = T i···j k···l,m + ΓinmTn···j

k···l + · · ·+ ΓjnmTi···n

k···l − ΓnkmTi···j

n···l − · · · − ΓnlmTi···j

k···n

46

9.11.3 Exponentials

A real analytic f of a single real variable can be expanded using Taylor expansion which is written as:

f(b) = f(a+ b− a) = f(a) + (b− a)df(a)

dx+ · · · =

( ∞∑n=0

(b− a)n

n!

dn

dxn

)f

∣∣∣∣x=a

And so this is in the form of an exponential:

f(b) = e(b−a)Dfa

Where D ≡ ddx . This defines the parallel transport because the same process can be made:

v˜‖QP = v˜Q + (λQ − λP )∇uv˜Q +1

2(λQ − λP )2∇u∇uv˜Q + · · ·

This again can be made into an exponential:

v˜‖QP = e(λQ−λP )∇uv˜QHere the limit where limλQ→λP v˜‖QP = v˜P is necessary and so:

∇uv˜P ≡ limλQ−λP

v˜‖QP − v˜QλQ − λP

= limλQ−λP

e(λQ−λP )∇uv˜Q − v˜QλQ − λP

= limλQ−λP

∇uv˜Q +O(λQ − λP )

= ∇uv˜PThis can be extended to multiple displacements. Parallel transporting v˜ from P → Q along vector fieldu˜ with a parameter change, a, then from Q→ R along v˜ by parameter change, b, it can be seen that:

w˜‖RQP = eb∇vea∇uw˜Reb∇v may not commute with ea∇u and so the order needs to be taken into account. This effect is calledcurvature.

9.11.4 Geodesics

A geodesic is a curve that parallel transports its own tangent vector. A given set of connections Γαµνfor a coordinate patch then the equations for any vector field u˜ for the geodesic equation is ∇uu˜ = 0. If

the coordinate patch is (U,F) then u˜ = ui(x)∂i and then along the curve it can be written:

u˜ =dxi

dλ∂i

∇uu˜ = ∇ dxj

dλ ∂j

(dxi

dλ∂i

)

=dxj

dλ∇j(dxi

dλ∂i

)

=dxj

dλ

(dxi

dλ

);j

∂i

=dxj

dλ

[(dxi

dλ

),j

+ Γijkdxk

dλ

]∂i

=

[dxj

dλ∂j

(dxi

dλ

)+ Γijk

dxj

dλ

dxk

dλ

]∂i = 0

This means that the equation of a geodesic is:

xk + Γkjixj xi = 0

47

9.11.5 The Metric Connection

The affine connection allows different parallel transports to be considered. The parallel transport maychange but the scalar product must be the same and not dependent on the path:

v˜‖QP · w˜‖QP = v˜P · w˜P

P

Q

ΓQP v˜Pw˜P

ϑ

TangentPlane at P

TangentPlane at Q

w˜‖P

v˜‖Pϑ

Figure 9.7: Inner Product From Parallel Transport

The connections are such that the parallel transported pair have the same inner product as the originalvectors. The metric tensor of the inner product in terms of the metric is:

gQ(v˜‖QP , w˜‖QP ) = gP (v˜P , w˜P )

Supposing that v˜ and w˜ are parallel transported along a flow line u˜ taking covariant derivatives withrespect to u˜ is:

∇ug(v˜‖, w˜‖)

= ∇ug (v˜‖, w˜) + g(∇uv˜‖, w˜‖) + g(v˜‖,∇uw˜‖) = 0

So by construction ∇uv˜‖ = 0 and ∇uw˜‖ = 0 and so:

∇ug (v˜‖, w˜‖) = 0

This is true for all value of v˜ and w˜ so ∇ug = 0 for all u˜. The covariant derivative of the metric mustbe zero for this connection. For a coordinate patch then ∇αg = 0 so the components gµν;α = 0 and assuch the connection coefficients Γαβλ can be determined. These can be found as:

gαβ;µ = gαβ,µ − Γλαµgλβ − Γλβµgαλ

The components can be cyclically permuted and hence:

gαβ,µ = Γλαµgλβ + Γλβµgαλ

gβµ,α = Γλβαgλµ + Γλµαgβλ

gµα,β = Γλµβgλα + Γλαβgµλ

Another condition must now be enforced. This is the commutability of the covariant derivatives withrespect to the vector fields u˜ and v˜: ∇u∇v = ∇v∇uThis type of connection is called torsion-free. Applying this to a scalar field shows:

∇µ∇ν = ∇µ∂νf = ∂µ∂νf + Γανµ∂αf

48

∇ν∇µ = ∇ν∂µf = ∂ν∂µf + Γαµν∂αf

This means that Γανµ = Γαµν . Now going back to the metric components then:

gαβ,µ + gβµ,α − gµα,β = Γλαµgλβ + Γλβµgαλ + Γλβαgλµ + Γλµαgβλ − Γλµβgλα − Γλαβgµλ = 2Γλαµgλβ

This means that:

Γαµν =1

2gαβ (gβν,µ + gµβ,ν − gµν,β)

These are the Christoffel symbols of the second kind and can be written asααµ

. Lowering the index

gives the Christoffel symbols of the first kind, [µν, α] ≡ gαβΓβµν .

9.11.6 Metric Connection via Variational Calculus

Spherical Connection Coefficients On a sphere of radius a the distance rule is ds2 = a2dϑ2 +a2 sin2 ϑdϕ2 and so the metric is:

ds2 = gµνdxµdxν = gϑϑdϑdϑ+ gϑϕdϑdϕ+ gϕϑdϕdϑ+ gϕϕ

In matrix form this is:

gµν =

ϑ ϕ

ϑ(a2 0

)ϕ 0 a2 sin2 ϑ

There are eight components, two of which are symmetric, Γϑϑϑ, Γϑϑϕ = Γϑϕϑ, Γϑϕϕ, Γϕϑϑ, Γϕϑϕ = Γϕϕϑ and Γϕϕϕ.The easiest way to find the values of these components is to construct a Lagrangian and calculate theEuler-Lagrange equations for the path parameter. For the case of the spherical space then the Lagrangianis:

L =1

2a2ϑ2 +

1

2a2 sin2 ϑϕ2

The Euler-Lagrange equations are:d

ds

(∂L∂x

)=∂L∂x

For the ϑ component the:

∂L∂ϑ

= a2ϑ =⇒ d

ds

(∂L∂ϑ

)= a2ϑ and

∂L∂ϑ

= a2 sinϑ cosϑϕ2

The equation of motion is therefore ϑ − sinϑ cosϑϕϕ = 0, which is of the form needed to construct themetric connection:

Γϑϕϕ = − sinϑ cosϑ

For the ϕ component then:

∂L∂ϕ

= a2 sin2 ϑϕ =⇒ d

ds

(∂L∂ϕ

)= a22 sinϑ cosϑϑϕ+ a2 sin2 ϑϕ and

∂L∂ϕ

= 0

The equation can be written down in the form of the metric connection equation as:

ϕ+sinϑ cosϑϑϕ

sin2 ϑ+

sinϑ cosϑϕϑ

sin2 ϑ= 0

And as such the metric connection is:

Γϕϑϕ = Γϕϕϑ = cotϑ

In spherical space there are two non-zero connection coefficients Γϑϕϕ and Γϕϑϕ = Γϕϕϑ. The other four arezero.

49

9.11.7 Schwarzschild Connection Coefficients

Introducing the Schwarzschild metric:

ds2 =

(1− 2µ

r

)c2dt2 − dr2(

1− 2µr

) − r2dΩ2

Where µ = Gmc2 and dΩ2 = r2(dϑ2 + sin2 ϑdϕ2) which describes the surface area of the spacetime. The

connection coefficients can again be calculated from the Euler-Lagrange equations, where there are nocross terms.

gµν =

t r ϑ ϕ

t

(1− 2µ

r

)c2 0 0 0

r 0 − 11− 2µ

r

0 0

ϑ 0 0 −r2 0ϕ 0 0 0 −r2 sin2 ϑ

The metric connections can now be found in the same way as with the sphere, but with greater complexity.

L =1

2

(1− 2µ

r

)c2t2 − 1

2

r2

1− 2µr

− 1

2r2ϑ2 − 1

2r2 sin2 ϑϕ2

For the time coefficient then:

∂L∂t

=

(1− 2µ

r

)c2t =⇒ d

ds

(∂L∂ϕ

)=

(1− 2µ

r

)c2t+

2µ

r2c2rt and

∂L∂ϕ

= 0

And so the equation of motion is:

t+2µrt

r2(1− 2µ

r

) = 0

So the connection coefficients:

Γtrt = Γttr =µ

r2(1− 2µ

r

)9.12 Torsion

For two vector fields u˜ and v˜ then [u˜, v˜] and ∇uv˜ − ∇vu˜ are both vector fields and are antisymmetric.When ∇uv˜−∇vu˜ = [u˜, v˜] then the connection is symmetric. In a coordinate bases then:

∇µ∂ν −∇ν∂µ = [∂µ, ∂ν ]

Where ∇µ∂ν = Γανµ∂α and ∇ν∂µ = Γαµν∂α and [∂µ, ∂ν ] = 0 then for a symmetric connection Γανµ∂α −Γαµν∂α = 0 and finally Γανµ = Γαµν .

For non-symmetric connections then the Torsion tensor, T , is a type-

(12

)tensor:

∇uv˜−∇vu˜− [u˜, v˜] = T (∗;u˜, v˜)∗ accommodates a one-form field and hence it acts like a vector field. For vector fields e˜µ which neednot be a coordinate basis then:

∇eµe˜ν −∇eνe˜µ − [e˜µ, e˜ν ] = Tανµe˜α50

v˜ v˜Figure 9.8: Vector in a Torsion-Free and a Non-Torsion-Free Space

A parallel transported vector rotates or twists as it is moved parallel to a symmetric set of connections.

9.13 Curvature

For a manifold M with a scalar field, f , and vector field, u˜, then a congruence Cu(λ) can be used toconstruct a displacement operator. A point P inM with a flow line CP where at P the parameter valueis λP . If Q is another point on CP with parameter λQ then fQ = UQP (u˜)fP where:

UQP (u˜)fP ≡ e(λQ−λP ) ddλ f∣∣∣λ=λP

= e(λQ−λP )u˜f ∣∣∣P

It can also be noted that fP = UPQ(u˜)fQ where:

UPQ(u˜)fQ ≡ e(λP−λQ) ddλ f∣∣∣λ=λQ

= e(λP−λQ)u˜f ∣∣∣P

This means that UPQ(u˜)UQP (u˜) = 1. By introducing another vector field v˜ with a new congruence Cv(µ)then a displacement can be made from P to Q along Cu(λ) with a parameter change λQ − λP = a thena second displacement from Q to R along Cv(µ) with a parameter change µR − µQ = b. The scalar fieldis therefore:

fR = URQ(v˜)UQP (u˜)fP = ebv˜eau˜f ∣∣∣P

Now if the order is reversed so that from P to S along Cv(µ) with a change of a and then from S to Talong Cu(λ) with a change b. This leaves:

fT = UTS(u˜USP (v˜)fP = eau˜ebv˜f ∣∣∣P

This can leave an extra displacement from T to R.

P

Q

R

T

S

bv˜au˜

au˜

bv˜ab[v˜, u˜]

Figure 9.9: Two Paths from P to R and P to T

This extra displacement is denoted by:

URT = URQ(v˜)UQP (u˜)UPS(v˜)UST (u˜)

51

Expanding this into the exponentials and then making a Taylor expansion to second order it can be seenthat:

URT = 1 + ab[v˜, u˜] + · · · ≈ eab[v˜,u˜]

And so R 6= T when [v˜, u˜] 6= 0.

9.14 Riemann Curvature Tensor

The parallel transport of a vector field A˜ around a closed path on the manifold:

P −→bv˜ Q −→

au˜ T −→ab[v˜,u˜]

R −→−bv˜ Q −→−au˜ P

Comparing the final direction of the parallel transported vector with the initial vector gives:

A˜‖PQRTSP = e−a∇ue−b∇veab∇[v,u]ea∇ueb∇vA˜P

Expanding the exponentials up to quadratic order finally leaves:

A˜‖PQRSTP ≡(1 + ab[∇u,∇v] + ab∇[v,u] + · · ·

)A˜P

This means that the difference between the parallel transported vector and the initial vector to the lowestorder is:

∆A˜‖PQRSTP = abR(∗;A˜ , u˜, v˜)P

This is the Riemann Curvature Tensor and is a type-

(13

), which acts like a vector field:

R(∗;A˜ , u˜, v˜) ≡ [∇u,∇v]−∇[u,v]

A˜

This acts on a one-form, such as the basis, to map it into the real numbers.

9.14.1 Components of the Riemann Curvature Tensor

If e˜µ is a basis for vector fields and eµ is the conjugate basis then the components Tα1···αpβ1···βq of

a type-

(pq

)tensor:

T (∗, ∗, · · · , ∗︸ ︷︷ ︸p one-forms

; ∗, ∗, · · · , ∗︸ ︷︷ ︸q vectors

) ≡ Tα1···αpβ1···βqe˜α1

⊗ · · · ⊗ e˜αp ⊗ eβ1 ⊗ · · · ⊗ eβq

Are:

Tα1···αpβ1···βq = T (eα1 , eα2 , · · · , eαp ;u˜β1

, e˜β2, · · · , e˜βq )

In the way the Riemann Curvature tensor can be defined by:

R(∗; ∗, ∗, ∗) = Rαβµνe˜α ⊗ e β ⊗ eµ ⊗ e νWhere Rαβµν are the components of the Riemann Curvature tensor, which by expansion can be seen as:

Rαβµν = Γαβν,µ − Γαβµ,ν + ΓλλµΓλβν − ΓαλνΓλβµ − cλµνΓαβλ

This has non-linear terms and so it is particularly difficult to solve. It is this non-linearity which is whygeneral relativity is non-renormalisable. Here cλµνe˜λ = [e˜µ, e˜ν ] for a non-coordinate basis, which showshow much the bases do net commute with each other. As such in a coordinate basis [∂µ, ∂ν ] = 0 and sothe last term is equal to zero and so:

Rαβµν = Γαβν,µ − Γαβµ,ν + ΓλλµΓλβν − ΓαλνΓλβµ

52

9.14.2 Riemann-Normal Coordinates

In general, the derivatives of the metric tensor components, gµν,λ, in an arbitrary coordinate frame arenon-zero. For a flat manifold then the Riemann tensor is zero even when gµν,λ is non-zero, such as inspherical polar coordinates in E3. It can be set up that the derivatives of gµν,λ go to zero at a point Pwhich is called the base point no that:

gµν,α∣∣P

= 0 and Γαµν∣∣P

= 0 ∀α, µ, ν

For a Lorentz manifold then the coordinates can be arranged so that the metric tensor components at Pare given by gµν

∣∣P

= ηµν . A standard local inertial frame can be set up as a base point where particletravelling on geodesics are freely moving so that:

xµ + Γµαβ xαxβ = 0

These components are called Riemann-normal coordinates which are useful because tensor componentssimplify the base point. The Riemann tensor in these coordinates are:

Rαβµν∣∣P

= Γαβν,µ − Γαβµ,ν∣∣P

9.14.3 Symmetries

In general a type-

(13

)tensor will have n4 independent components at each point in an n dimensional

manifold. For n = 4 then there are 256 independent component fields. The symmetries can reduce this.It is clear to see that Rαβµν = −Rαβνµ which is evident by inspection. More conditions can be found and

so in total there are n2(n2−1)12 independent components, 20 for n = 4. This is still not simple in anyway,

but better than 256.

53

Chapter 10

Relativity and Gravitation

10.1 Principle of Equivalence

As inertial mass is equivalent to the gravitational mass then Galileo’s Law can be stated: an objects fallat the same rate if air resistance is neglected. Over limited scales of time and space then the effects ofgravitation in a freely falling local laboratory are indistinguishable to the effect of force-free physics. Thiscan be written as Einstein’s principle of equivalence:

In a freely falling laboratory occupying a small region of spacetime,the laws of physics are those of special relativity

In this case gravitation is no longer a force but a manifestation of the curvature of spacetime. Thiscurvature is caused by the presence of matter. A particle has a natural tendency to follow a geodesic inspacetime and the observer accelerates along a non-geodesic path.

10.2 Tidal Acceleration in Newtonian Gravity

The gravitational potential is a scalar field, φ, which satisfies Poisson’s equation:

∇2φ = 4πG%m

The density of a single particle is %m(x˜) = mδ3(x˜). When there are no particles then the Newtonianvacuum field equation is found:

∇2φ = 0

For two free falling particles, P and Q, the standard coordinates are assigned as xP (t) and xQ(t). If P andQ lie on a curve with a path parameter λ then the trajectories are xP (t) = x(t, λP ) and xQ(t) = x(t, λQ),and thus the equations of motion are:

∂2x(t, λP )

∂t2= −∇φ(x(t, λP )) and

∂2x(t, λQ)

∂t2= −∇φ(x(t, λQ))

Subtracting these two equations of motion from each other, dividing by the difference between λQ andλP and taking the limit where λQ → λP then:

∂2

∂t2

(∂xi

∂λ

)= −∂i∂jφ

∂xj

∂λ

By defining ξi ≡ ∂xi

∂λ then the tidal equation is:

d2ξi

dt2= −Hijξ

j

Where Hij ≡ ∂i∂jφ. The eigenvalues of Hij describe the action of the tidal force, whether it is attractiveor repulsive depends on the sign. If the sum of the eigenvalues equals the matrix then:

Tr[H] = ∂i∂iφ = 0

54

10.3 Weak Gravitational Fields and the Newtonian Limit

If there is no gravitation then spacetime has the form of Minkowski spacetime. If a weak gravitationalfield is added then there is a slight curvature imposed:

gµν = ηµν + hµν

The metric is static such that it does net vary in time, ∂0gµν = 0. A particle falling under gravity followsthe path of a geodesic:

d2xµ

dλ2+ Γµαβ

dxα

dλ

dxβ

dλ

The conventional parameter is proper time where:

c2 = gµν(x)dxµ

dτ

dxµ

dτ

Now to recover the Newtonian limit then small speeds are considered:

dxi

dt

dxi

dt c2

It is deduced that∣∣∣dxidτ ∣∣∣ dx0

dτ . This means that:

d2xµ

dτ2+ Γµ00

dx0

dτ

dx0

dτ= 0

From this Γµ00 can be expanded to give:

Γµ00 = −1

2

(ηµj + hµj

)∂jh00

This means Γ000 ≈ 0, Γi00 ≈ − 1

2 (−dij)∂jh00 = 12∂ih00 and so:

d2x0

dτ2≈ 0 and

d2xi

dτ2≈ −1

2∂ih00

dx0

dτ

dx0

dτ= −c

2

2∂ih00

This can be compared to the Newtonian gravitational equation:

d2xi

dt2= −∂iφ

This means that the weak gravitational field is h00 = 2φc2 so the metric is:

g00 = 1 +2φ

c2

The value of h00

2 at the surface of the Earth is −10−9, at the surface of the sum is −10−6 and atthe surface of a white dwarf is −10−4. This shows that even under relatively extreme conditions weakcurvature effects are able to explain the gravitational forces. Exceptions to this occur in black holephysics. Looking at the change in properties, c2dτ2 = gµνdx

µdxν ≈ g00c2dt2, which shows that:

dτ =

√1 +

2φ

c2dt

This is the time dilation on a weak gravitational field. In Newtonian gravitation then only the differencebetween gravitational potentials in measurable but g00 cannot be changed arbitrarily as it could changephysical objects, such as the time via dilation.

55

10.4 Geodesic Deviation

From the tidal acceleration d2ξi

dt2 = −Hijξj then in general relativity then the particles become geodesics

passing through P and Q along integral curve u˜ where ∇uu˜ = 0. For a single coordinate patch xµcontaining both points then:

d2xµPds2

+ Γµαβ(P )dxαPds

dxβPds

= 0

d2xµQds2

+ Γµαβ(Q)dxαQds

dxβQds

= 0

Where s is the affine parameter for the geodesic. If these are Riemann-normal coordinate with base pointP so that Γµαβ(P ) = 0 then:

d2xµPds2

= 0

It cannot be assumed that Γµαβ(Q) = 0 as Q is not the base point. for a curve CP (λ) from P to Q then:

xµQ = xµP + λdxµPdλ

+O(λ2)

Then:

Γµαβ(Q) = Γµαβ(P ) + λdxµPdλ

∂νΓµαβ(P ) +O(λ2) = λdxνPdλ

Γµαβ(P ),ν +O(λ2)

The geodesic through Q satisfies the equation:

∂2

∂s2

(xµP + λ

∂xµP∂λ

+O(λ2)

)=

−(λ∂xνP∂λ

Γµαβ(P ),ν +O(λ2)

)∂

∂s

(xαP + λ

∂xαP∂λ

+O(λ2)

)× ∂

∂s

(xβP + λ

∂xβP∂λ

+O(λ2)

)To the order λ then the equation:

d2

ds2

dxµPdλ

= −dxνP

dλΓµαβ(P ),ν

dxαPds

dxβPds

If ξµ ≡ dxµ

dλ then:

ξµP = −Γµαβ(P ),νξν xαP x

βP

The dot denotes differentiation with respect to s. The covariant derivatives of ξµ are:

D

Dsξµ and

D2

Ds2ξµ

These are defined because moving along the curve changes the tangent vector space. These derivativesare components of vectors:

D

Dsξµ ≡ xαξµ;α = xα

(ξµ,α + Γµαβξ

β)

= ξµ + Γµαβ xαξβ

And:D2

Ds2ξµP =

D

Ds

(ξµP + Γµαβ xi

αξβP

)= ξµP + Γµαβ,ν x

νξβP

At the base point then:D2

Ds2ξµP = −Γµαβ(P ),νξ

ν xαP xβP + Γµαβ,νP x

νP x

αP ξ

βP

This can be rearranged as:D2

Ds2ξµP +

(Γµαβ,νP − Γµαν,βP

)ξν xαP x

βP = 0

It can be seen from this that the Riemann curvature tensor is present:

D2

Ds2ξµP +Rµανβξ

ν xαP xβP = 0

This is the equation of geodesic deviation.

56

10.5 General Relativity Matter-Free Equations

The tidal equation in Newtonian gravitation is:

d2ξi

dt2= −∂i∂jφGξi

The equation of geodesic deviation in a manifold is:

D2

dτ2ξµ = −Rµανβξ

ν xαxβ

In the matter-free situation then the Newtonian equation becomes:

∂i∂iφG = 0

From this it can be suggested that in general relativity then the matter-free equations should be:

Rµαµβ = xαxβ = 0

As xα and xβ are arbitrary then this equation can be written as:

Rµαµβ = 0

Noting the symmetries of the Riemann curvature tensor, Rµαµβ = −Rµαβµ then the Ricci tensor can be

defined as Rαβ ≡ Rµαβµ so in matter-free space:

Rαβ = 0

This is a set of coupled second order differential equations. Any distortions in the spacetime which satisfyRαβ = 0 indicate an oscillating metric, which represents gravitational waves.

10.6 Symmetries of the Ricci Tensor

The Ricci tensor is symmetric as can be seen from:

Rαβ ≡ Rµαβµ = gµλRλαβµ = −gµλRλαµβ = −gµλRµβλα = gµλRµβαλ = gλµRµβαλ = Rλβαλ = Rβα

The Ricci tensor has ten independent components.

10.7 The Schwarzschild Metric

Einstein’s matter-free field equations are:

Rαβ = 0

The obvious solutions to these are:

gµν = ηµν

There are more non-trivial solutions, such as the Schwarzschild metric. By choosing time, t, and sphericalpolar coordinates, r, ϑ and ϕ then r is defined by the surface area on a sphere:

r2

∫ π

0

∫ 2π

0

sinϑdϑdϕ = 4πr2

The distance r has a radius which is defined by Einstein’s vacuum field equations. In the stationary,radially symmetric situation the line element is:

ds2 = eA(r)c2dt2 − eB(r)dr2 − r2dϑ2 − r2 sin2 ϑdϑ2

57

A(r) and B(r) can be found from the field equations. There is no distortions in the angular coordinatesand so only time and distance can be distorted. The non-zero connections can be calculated and thenfrom these the Riemann curvature tensor can be found. Then the Ricci tensor components are found:

Rtt =eAc2

4reB(−rA′b′ + r(A′)2 + 2rA′′ + 4A′

)Rrr =

1

4r

(−rA′B′ + r(A′)2 + 2rA′′ − 4B′

)Rϑϑ = − 1

2eB(−rA′ + rB′ + 2eB − 2

)Rϕϕ = Rϑϑ sin2 ϑ

Using the field equations then it can be seen that A′ = −B′ so A(r) = −B(r) + k. By making t→ e−k2 t

then A(r) = −B(r) is eliminated and −2rA′ + 2e−A − 2 = 0 gives:

d

dr

(reA

)= 1

And so:

eA(r) = 1− 2µ

r

This gives the Schwarzschild metric:

ds2 =

(1− 2µ

r

)c2dt2 − 1

1− 2µr

dr2 − r2dϑ2 − r2 sin2 ϑdϕ2

If r < 2µ then the space and time coordinates change sign and so inside the critical radius then thephysics changes. This the effective event horizon of a black hole. Using g00 ≈ 1+ 2φ

r then a point particleof mass M has a Schwarzschild metric:

ds2 =

(1− 2GM

rc2

)c2dt2 − 1

1− 2GMrc2

dr2 − r2dϑ2 − r2 sin2 ϑdϕ2

In the limit that r →∞ then the metric becomes that of Minkowski.

10.8 Gravitational Orbit Theory

10.8.1 Newtonian

The Euler-Lagrange equations can be solved for the Lagrangian:

L =m

2

(r2 + r2ϑ2 + r2 sin2 ϑϑ2

)+GMm

r

By following this through then it can be seen that orbits form with radii:

1

r=GM

h2(1 + e cosϕ)

Where e is the eccentricity and h = r2ϕ is the orbital velocity. Defining the semi-major axis as a =h2

GM(1−e2) means that:

h2 = aGM(1− e2)

It is clear that all orbits are closed and so no precession of the perihelion is allowed.

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10.8.2 General Relativity

Mercury’s perihelion was observed to precess at about 1.4o per century. This could mostly be accountedfor by the effects of other planets but there was still 43 seconds of arc per century which could not beexplained. It can be rectified using general relativity. Assuming that the sun is the source of gravitationalcurvature then the Schwarzschild metric can be used:

ds2 =

(1− 2GM

rc2

)c2dt2 − 1

1− 2GMrc2

dr2 − r2dϑ2 − r2 sin2 ϑdϕ2

This gives the connection coefficients which can be used in the geodesic equations where 2GMrc2 ≡ µ:

t+2µ

r2(1− 2µ

r

) rt = 0

r +

(1− 2µ

r

)µc2

r2t− µ

r2(1− 2µ

r

) r2 − r(

1− 2µ

r

)ϑ2 − r

(1− 2µ

r

)sin2 ϑϕ2 = 0

ϑ+2

rrϑ− sinϑ cosϑϕ2 = 0

ϕ+2

rrϕ+ 2 cotϑϑϕ = 0

It can be seen that ϑ(0) = π2 and ϑ(0) = 0. In fact as there is no asymmetry to alter ϑ from the plane

then ϑ(t) = π2 . This simplifies equations to:

t+2µ

r2(1− 2µ

r

) rt = 0

r +

(1− 2µ

r

)µc2

r2t2 − µ

r2(1− 2µ

r

) r2 − r(

1− 2µ

r

)ϕ2 = 0

ϕ+2

rrϕ = 0

This last equations reveals r2ϕ = h. This means that ϕ increases with τ and so ϕ can be used as theproper time parameter. The first geodesic equation therefore shows:

d

dτ

((1− 2µ

r

)t

)= 0

This means that:

t =b

1− 2µr

This can be substituted into the r along with the ϕ then reduced to give:

r +µc2

r2+ 3µϕ2 − rϕ2 = 0

Defining 1r ≡ u and u′ = du(ϕ)

dϕ then:

u′′ + u =µc2

h2+ 3µu2

This has an extra term when compared with the Newtonian form. Writing the Newtonian equation as:

u′′o + uo =µc2

h2

Then the solution is:

uo(ϕ) =µc2

h2(1 + e cosϕ)

The general relativity solution is then:u = uo + ∆u

59

Where:

∆u′′ + ∆u = 3µ

(µc2

h2(1 + e cosϕ) + ∆u

)2

By introducing α ≡ 3µ2c2

h2 then the solution is:

u(ϕ) ≈ µc2

h2(1 + e cos((1− α)ϕ))

The non oscillating ϕ term causes a constant shift which allows Mercury’s perihelion to precess. Thechange in ϕ is:

∆ϕ =6πGMa(1− e2)c2

If the correct numbers are entered then the precession of Mercury’s perihelion is 42.8 seconds of arc.

10.9 The Einstein Tensor

The Einstein tensor is a type-

(02

)tensor defined by:

Gµν ≡ Rµν −1

2gµνR

R is the Ricci scalar defined by R ≡ Rµµ = gµνRνµ.

10.9.1 Bianchi Identity

The Einstein tensor can be found from the Bianchi identity:

∇λRαβµν +∇µRαβνλ +∇νRαβλµ = 0

Now raising the first index and contracting α with ν shows:

∇λRνβµν +∇µRνβνλ +∇νRνβλµ = 0

As Rνβµν = Rβµ and Rνβνλ = −Rνβλν = −Rβλ then:

∇λRβµ −∇µRβλ +∇νRνβλµ = 0

Nom raising β and contracting with λ indicates:

∇λRλµ −∇µR+∇νRνλλµ = 0

Using Rνλλµ = Rλνµλ = Rνµ then:

∇λRλµ −∇µR+∇νRνµ = 0

This is equivalent to:

∇λ(Rλµ − 1

2gλµR

)= 0

This is the matter-free Einstein Field equation:

∇λGλµ = 0

This has the same form as a conservation law and so it suggests that something is conserved in thestructure of spacetime.

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10.10 The Einstein Field Equations

10.10.1 The Stress-Energy-Momentum Tensor

As mass and the number of particles transform as %′ = %γ2 then it is seen that it must be containedwithin a high rank tensor than a vector. This is the stress-energy-momentum tensor:

T = Tµνe˜µ ⊗ e˜νThis means that TµνP = ΛµαΛνβT

αβ = ΓµαTαβΛνβ . In the instantaneous rest frame of a dust or non-

interacting collection of particles then:

[TµνP ] =

%c2 0 0 00 0 0 00 0 0 00 0 0 0

Now for a perfect fluid there is an associated pressure which in the instantaneous rest frame is:

[Tµν ] =

%c2 0 0 00 p 0 00 0 p 00 0 0 p

This can be rewritten Tµν =

(%+ p

c2

)uµuν − pgµν . % and p are both Lorentz scalars. The stress-energy-

momentum tensor is symmetric due to pressure being isotropic. The conservation of momentum andmass takes the form:

∂µTµν = 0 in Special Relativity and ∇µTµν = 0 in General Relativity

This means that there is no global meaning to the conservation of energy, there is only the four localexpessions taken over the values of ν. As ∇µGµν = 0 and ∇µTµν = 0 then a relationship between thetwo is found as:

Rµν −1

2gµν = −κTµν

Where κ = 8πGc4 . This equation is very similar to the conservation laws of electromagnetism, Aµ = jµ.

A cosmological constant can be added to the field equations, as Einstein did to create a static universeor as is now done to create an expanding spacetime, which has the form:

Rµν −1

2gµνR+ Λgµν = −κTµν

In the weak field limit then for a spherical mass:

g˜ = −∇φG = −GMr3

r˜+c2Λ

3r˜

It can be seen that there is gravitational repulsion at large scales due to the cosmological constant.

10.11 Affine Parameters

Paralel transporting a vector field u˜ along its congruence gives ∇uu˜ = 0 with the geodesic coordinateequation:

d2xµ

dλ2+ Γµαβ

dxα

dλ

dxβ

dλ= 0

Now this can be reparameterised along onother path parameter, say lambda→ λ(η) so that:

dxµ(η)

dη=dλ

dη

dxµ(λ)

dλ

Placing this into the geodesic equation reveals:

d2xµ

dη2+ Γµαβ

dxα

dη

dxβ

dη=d2λ

dη2

dxµ

dλ

Now as long as d2λdη2 = 0 then the geodesic equation has the same form as for λ and so λ = aη + b must

be the transformation. This is an affine (linear inhomogeneous) transformation.

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10.11.1 Null Curves

When light travels along a geodesic then there is no concept of proper time and so:

gµν(x)dxµ(λ)dx(λ) = 0

Any parametrisation consistent with the conditions gµν xµxν = 0 and xµ + Γµαβ x

αxβ = 0 can thereforebe used to describe the null curve.

10.12 Deflection of Light

Using a path parameter λ which describes light along a null worldline then:

0 =

(1− 2µ

r

)c2t2 − 1(

1− 2µr

) r2 − r2ϑ2 − r2 sin2 ϑϕ2

For a plane then it is useful to use ϑ = π2 so that using the impact parameter

(1− 2µ

r

)t = b and the

angular momentum r2ϕ = h then the equations reduce to:

r + 3µϕ2 − rϕ2 = 0

The path parameter can be written as ϕ so that r(λ) = u−1(ϕ) where ϕ(λ). This means that r = −hu′and r = −h2u2u′′ which gives:

u′′ + u = 3µu2

b r

ϕ

∆ϕ2

Figure 10.1: The Deflection of Light due to the Schwarzchild Geometry

As this is equation is non-linear then perturbationary theroy must be used such that when 3µu2 = 0 thenu = sinϕ

b is a solution where:

u =sinϕ

b+ ∆u

The perturbation satisfies the approximate equation:

d2∆u

dϕ2+ ∆ϕ = 3µ

sin2 ϕ

b2

This can be solved by the integral ∆u = 3µ2b2

(1 + 1

3 cos2 ϕ)

so that in taking the limit of r → ∞ thensinϕ→ ϕ and cosϕ→ 1 such that:

∆ϕ ≈ 4GM

c2b

For light grazing the sun’s surface then ∆ϕ ≈ 1.75′′.

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Chapter 11

Schwarzschild Geometry

11.1 Coordinates

The line element is ds2 = gµνdxµdxν . Now if dxµ = 0 for all coordinates except one at xα then

ds2α = gαα(dxα)2 where there is no sum over the α. If xα is timelike then ds2 is positive, ii is null if

ds2 = 0 and is spacelike if ds2 is negative.

For a two-dimensional Minkowski spacetime then ds2 = c2dt2 − dx2 and a displacement dxµ = (dt, 0)then ds2

t = c2dt2 > 0 and t is timelike. Now for a displacement dxµ = (0, dx) then ds2x = −dx2 and as

such x is spacelike. The general line element can be written as:

ds2 = (cdt− dx)(cdt+ dx)

The coordinates can be changed from t, x to u, y where u ≡ ct− x and y ≡ x then:

ds2 = du2 + 2dudy

It can be seen that du is a timelike coordinate whilst dy is a null coordinate. If another coordinate patchv, z is used where v ≡ ct+ x and z ≡ t. This has a Jacobian of −1 so is useful over the whole manifoldso that:

ds2 = −dv2 + 2cdvdz

v is a spacelike coordinate and z is a null coordinate. If, finally, a transformation is made to u, v whereu ≡ ct− x and v ≡ ct+ x then the Jacobian is 2c and is alright for the whole manifold so that:

ds2 = dudv

Both v and u are null coordinates.

11.2 Vector Fields

A vector can be said to be timelike if g(v˜, v˜)|P > 0, null if g(v˜, v˜)|P = 0 and spacelike if g(v˜, v˜)P < 0.The sign of g(v˜, v˜ is called the signature of v˜ relative to g. It is independent of the choice of coordinatepatch.

11.3 The Kretschmann Invariant

By imposing Gµν = 0 near a gravitating point mass then it implies that the Ricci tensor vanishes,Rµν = 0. This means that the Ricci scalar is zero, Rµµ = R = 0. This suggests that the curvature is zero,but it can be seen that this is not true. By looking at all of the information of the Riemann curvaturetensor it is seen that:

RK ≡ RαβµνRαβµν

This is the Kretschmann invariant. For the Schwarzschild metric then:

RK =48µ2

r6r 6= 0

63

This shows that there is no singularity at r = 2µ which is suggested by the metric. The Schwarzschildcoordinates are not useful to describe the metric. There is also no coordinate patch at r = 0. This abreak down of the model.

11.4 Birkhoff’s Theorem

The spacetime outside a general spherically symmetric matter distribution is the Schwarzschild geometry:

ds2 = A(t, r)dt2 −B(t, r)dr2 − r2dΩ2

The spacetime need not necessarily be spherically symmetric.

11.5 Black Hole Geometry

The associated Schwarzschild geometry of a proton is about 10−50cm which is far below the Planck length`P ≈ 1.62× 10−33cm. For the sun then the Schwarzschild radius is about 2.95km.

Photos moving radially have dϑ = dϕ = 0 and so the given Schwarzschild coordinates are:

cdt

dr= ± r

r − 2µ

Integrating this gives ct = −r−2µ ln |r−2µ|+constant for inward flowing photons and ct = r+2µ ln |r−2µ|+ constant for outflowing photons.

ct ct

r r2µ 2µ

Figure 11.1: Null Geodesics Inside and Outside the Schwarzschild Radius

For light then g00dt2 = grr(x)dr2 which allows lightcones to be defined all over the spacetime. Bringing

together both above diagrams shows that the lightcones change direction so that the timelike coordinatehas become the spacelike coordinate.

64

ct

r2µ

Figure 11.2: Lightcones on Either Side of the Event Horizon

11.5.1 Eddington-Finkelstein Coordinates

To get a better description of r = 2µ then a photon going inwards uses the constant found from theintegration as the advanced time parameter:

dp = cdt+r

r − 2µdr

Substituting this into the Schwarzschild metric gives:

ds2 =

(1− 2µ

r

)dp2 − 2dpdr − r2dΩ2

This no longer has a singularity at r = 2µ and r has become a null coordinate. For radially movingphotons then:

(1− 2µ

r

)(dp

dr

)2

= 2dp

dr

This has two solutions, p = 0 and p = 2(1− 2µ

r

)−1. The first equation shows that p is constant and the

second shows:

p = 2r + 4µ ln |r − 2µ|+ constant

Making another coordinate transformation with this new constant gives ct = p − r = ct + 2µ ln |r − 2µ|which reveals the metric:

ds2 =

(1− 2µ

r

)c2dt 2 − 4µc

rdtdr −

(1 +

2µ

r

)dr2 − r2dΩ2

These are the advanced Eddington-Finkelstein coordinates, (t, r, ϑ, ϕ). The incoming and outgoing photonworldlines are ct = −r+constant and ct = r+4µ|r−2µ|+constant respectively. Using these coordinatesit can be seen what happens to a particle falling into a black hole.

65

ct

r2µ

Figure 11.3: The Timelike Coordinate Curves into What Can Be Perceived a Spacelike Coor-dinate

It is possible to cross the point r = 2µ in a finite proper time. The Schwarzschild radius defines the eventhorizon so that although signals can be sent in, none can return.

11.5.2 White Holes

Retarded Eddington-Finkelstein coordinates can be defined using the retarded time parameter:

q ≡ ct− r − 2µ ln |r − 2µ|

The Schwarzschild metric is:

ds2 =

(1− 2µ

r

)dq2 + 2dqdr − r2dΩ2

This means that the radial null geodesics satisfy(dqdr

)2

= − 2rr−2µ

dqdr so that the solutions are q = r +

constant and q = −2r − 4µ ln |r − 2µ|+ constant where the time parameter is now:

ct∗ = ct− 2µ ln |r − 2µ|

Outgoing photons cross the event horizon at a finite time t∗ bit ingoing photons take an infinite time.The particles are expelled from the singularity.

11.5.3 Kruskal Coordinates

A system of coordinates can be set up such that both the ingoing and the outgoing radial photon geodesics

are continuous straight lines. By defining ct = 12 (p + q) and r = 1

2 (p − q) = r + 2µ ln∣∣∣ r2µ − 1

∣∣∣ then the

metric is:

ds2 =

(1− 2µ

r

)((c2dt2 − dr

)− r2dΩ2

When ϑ = ϕ = constant then the two dimensional Minkowski spacetime is recovered apart from theconformal scaling factor

(1− 2µ

r

). There is a problem still at r = 2µ so by defining p ≡ e

p4µ and

q ≡ −e−q4µ which leads to v ≡ 1

2 (p+ q) and u ≡ 12 (p− q) so the Schwarzschild geometry is:

ds2 =32µ2

re−

r2µ (dv2 − du2)− r2dΩ2

It can be seen that radial null geodesics have be found as 0 = dv2 − du2.

66