Chapter 1

General Relativity

In this chapter, we review the basic formalism of the standard theory. We do not focus on details, but on the basic facts necessary for cosmological applications to be developed in following chapters.

1.1

Before we start discussing (GR), let us review some concepts in Special Relativity (SR) and covariant equations. It all starts with problems faced with (E&M) at the end of the 19th century, which required a redefinition of our classical notions of space and .

1.1.1 Electromagnetic Equations

We start with a brief review of the electromagnetic equations.

Maxwell Equations

The Maxwell equations describe the production and propagation of the Electromagnetic (E&M) fields. They are written in differential form as

ρ E = (Gauss Law) (1.1) ∇· ǫ0 B = 0 (Nonexistence of Magnetic Monopoles) (1.2) ∇· ∂B E = (Faraday induction Law) (1.3) ∇× − ∂t ∂E B = µ j + µ ǫ (Ampere Law) (1.4) ∇× 0 0 0 ∂t where ρ is the and j is electric .

11 12 CHAPTER 1. GENERAL RELATIVITY

Charge Conservation Taking the divergence of Ampere Law: ∂( E) B = µ j + µ ǫ ∇· ∇·∇× 0∇· 0 0 ∂t ∂(ρ/ǫ ) = µ j + µ ǫ 0 0∇· 0 0 ∂t ∂ρ = µ ( j + ) 0 ∇· ∂t Therefore, charges are conserved through the continuity equation ∂ρ + j = 0 (1.5) ∂t ∇·

Electromagnetic Potentials It is convenient to define electromagnetic potentials through the source-free Maxwell Eqs. Firstly, since B = 0 B = A (1.6) ∇· → ∇× Using this expression in Faraday Law, we have ∂ A ∂A ∂A E = ∇× = E + = 0 (1.7) ∇× − ∂t ∇× − ∂t → ∇× ∂t     and the term in parenthesis must be the gradient of a field φ. The φ and the magnetic vector potential A are thus defined as ∂A E = φ (1.8) −∇ − ∂t B = A (1.9) ∇× Gauge Transformation: Invariance If φ e A are solutions of the Maxwell Eqs., then the potentials φ′ and A′ defined by

′ ∂f φ = φ (1.10) − ∂t ′ A = A + f (1.11) ∇ for a general function f(x,t), are also solutions, because ′ ′ ′ ∂A ∂f ∂A ∂( f) ∂A E = φ = φ + ∇ = φ = E (1.12) −∇ − ∂t −∇ ∇ ∂t − ∂t − ∂t −∇ − ∂t ′ ′ B = A = A + ( f)= B (1.13) ∇× ∇× ∇× ∇ Therefore, we have the freedom to choose the function f conveniently without changing the fields. The choice of f implies determining a gauge. The gauge used for the solutions to electro- magnetic waves is the Lorenz gauge ∂φ A + µ ǫ = 0 (Lorenz Gauge) (1.14) ∇· 0 0 ∂t 1.1. SPECIAL RELATIVITY 13

Electromagnetic Waves Inserting the potentials into the Maxwell Eqs, we have ∂A ∂ A ρ E = ( φ )= 2φ ∇· = (1.15) ∇· ∇· −∇ − ∂t −∇ − ∂t ǫ0 and

B = ( A)= ( A) 2A ∇× ∇× ∇× ∇ ∇· −∇ ∂E ∂ ∂A = µ j + µ ǫ = µ j + µ ǫ φ 0 0 0 ∂t 0 0 0 ∂t −∇ − ∂t   ∂φ ∂2A = µ j µ ǫ µ ǫ (1.16) 0 −∇ 0 0 ∂t − 0 0 ∂t2   These two equations imply: ∂ A ρ 2φ + ∇· = (1.17) ∇ ∂t −ǫ0 ∂2A ∂φ 2A µ ǫ = µ j + A + µ ǫ (1.18) ∇ − 0 0 ∂t2 − 0 ∇ ∇· 0 0 ∂t   Choosing the Lorenz gauge ∂φ A + µ ǫ = 0 (1.19) ∇· 0 0 ∂t these equations become

2 2 1 ∂ φ 2 ρ φ = 2 2 + φ = (1.20) −c ∂t ∇ −ǫ0 1 ∂2A 2A = + 2A = µ j (1.21) −c2 ∂t2 ∇ − 0 i.e., the potentials propagate according to the classical non-homogenous wave equation with con- stant speed equal to the speed of c2 = 1/µ ǫ . Unification: E&M Optics. 0 0 ↔ Given the E&M fields, corresponding E&M forces F act on particles of charge q and are given by:

F = q(E + v B) (1.22) × 1.1.2 Special Relativity Postulates The E&M wave equations have a constant speed of propagation. Questions:

1: With respect to what reference frame should c be measured? 2: How can we explain effects such as the disappearance of magnetic forces in a reference frame that moves with the charge? 14 CHAPTER 1. GENERAL RELATIVITY

These issues motivated the development of special relativity, which solves these problems and change our classical concepts of space and time, requiring only two postulates (in fact, only one):

Postulate 1: The laws of are the same in all inertial frames. Postulate 2: The is the same in all inertial frames.

The constancy of c follows from postulate 1, since E&M is a of physical laws in which c=const. Therefore Postulate 2 is redundant given Postulate 1.

1.1.3 Coordinates and Metric Defining contravariant coordinates

xµ =(x0,x1,x2,x3)=(ct, x, y, z) , (1.23) the ds

ds2 = dx2 + dy2 + dz2 c2dt2 = (dx0)2 +(dx1)2 +(dx2)2 +(dx3)2 µ ν − − = ηµνdx dx (1.24) defines the metric ηµν

1 0 0 0 − 0 1 0 0 ηµν =   . (1.25) 0 0 1 0  0 0 0 1      Covariant coordinates xµ are defined as

x = n xν =( ct, x, y, z) (1.26) µ µν − Similarly,

µ µν x = η xν, (1.27)

µν µ µν where ηµν is the inverse metric, i.e. η ηνα = δ α. In flat space it turns out that η = ηµν. Throughout we use the Einstein sum convention: crossed repeated indices are summed over:

3 ηµνx ηµνx (1.28) ν ≡ ν ν=0 X 1.1.4 Invariance of the Line Element: Spatial Rotations Let us first consider a simple in 3 dimensions. Consider a rod such that one of its ends is at the origin (0, 0, 0) and the other is at coordinates (x, y, z) in frame K. The rod has length l and therefore

l2 = x2 + y2 + z2 (1.29) 1.1. SPECIAL RELATIVITY 15

Now suppose we consider this same rod as seen by a frame K′ that is rotated by angle θ with respect to the z axis of K. In K′, the rod still has one end at the origin (0, 0, 0), but the other one is at new coordinates (x′,y′, z′) given by

′ ct = ct ′ x = x cos θ + y sin θ ′ y = x sin θ + y cos θ ′ − z = z as measured by system K′. Therefore:

ct′ 1 0 0 0 ct x′ 0 cos θ sin θ 0 x  ′  =     y 0 sin θ cos θ 0 y ′ −  z   0 0 0 1   z              Obviously, if we compute the length l′ measured by K′ we obtain

′ ′ ′ ′ l 2 =(x )2 +(y )2 +(z )2 = x2 + y2 + z2 = l2

Therefore the rotation transformation left the length Figure 1.1: Rotation of frame K into K′ around 2 l of the rod . A similar idea applies in 4- the z axis by angle θ in the xy plane. dimensional space-time, except that instead of rotat- ing purely in space, we introduce an operation that also involves the time coordinate: boosts.

1.1.5 Invariance of the Line Element: Lorentz Boosts Consider an inertial frame K and another K′ that moves relative to K with speed v in the x direction. For simplicity we may assume that the two frames coincide initially, i.e. at t = t′ = 0 we have x = x′ = 0. For both frames c is the same, so considering the trajectory of a light ray, we have that the line element s is null in both frames

′ ′ ′ ′ ′ s2 = x2 + y2 + z2 c2t2 =0= x 2 + y 2 + z 2 c2t 2 = s 2 (1.30) − − It can be shown (see Schutz) that this implies that the interval s2 is the same in all inertial frames, even when s2 = 0. The invariance of s2 is similar to the invariance of the length l2 of a rod 6 in 2 dimensions. But differently from l2, which is always positive, the space time s2 can be either positive (space-like events) or negative (time-like events). The Lorentz transformations relates coordinates xµ and xµ′, keeping s2 invariant (and null in the case of light). Considering e.g. a particle at rest in K′, and therefore at x = vt in K, one can show that the transformation is given by (Exercise: Prove it)

′ x0 = γ(x0 βx1) (1.31) ′ − x1 = γ(x1 βx0) (1.32) ′ ′ − x1 = x2 (1.33) ′ ′ x3 = x3 (1.34) 16 CHAPTER 1. GENERAL RELATIVITY where v β = < 1 (1.35) c 1 γ = > 1 (1.36) 1 β2 − or p µ′ ′ ∂x xµ = xν =Λµ xν (1.37) ∂xν ν with γ βγ 0 0 − βγ γ 0 0 Λµ =  −  (1.38) ν 0 0 1 0  0 0 0 1      µ 2 2 2 2 2 Notice that detΛ ν = γ β γ = (1 β )γ = 1. A similar (inverse) transformation relates the − ′− µ −1 µ coordinates in the inverse direction (x x), with β β, i.e. (Λ ν) (β)=Λ ν( β). → →− − Similarly, for covariant coordinates the transformation follows

′ ′ν ν α ν γ α ν γβ xµ = ηµνx = ηµνΛ αx = ηµνΛ γ δ α x = ηµνΛ γη xβ (1.39)

γβ η ηβα  β  Λµ |{z} or | {z }

′ ν xµ =Λµ xν (1.40) with Λ ν η Λα ηβν and also given by (as easily shown by multiplication) µ ≡ µα β   γ +βγ 0 0 − +βγ γ 0 0 Λ ν = (Λµ ) 1 =   (1.41) µ ν 0 0 1 0  0 0 0 1      Boost as space-time rotation with complex time and complex angle Finally, we may check that by identifying

γ = cosh φ (1.42) βγ = sinh φ (1.43) or equivalently β = tanh φ, we have

cosh φ sinh φ 0 0 − sinh φ cosh φ 0 0 Λµ =  −  (1.44) ν 0 0 1 0  0 0 0 1      1.1. SPECIAL RELATIVITY 17 which looks like a rotation in the ct x plane. In fact, if we use the identities: − sinh φ = i sin(iφ) (1.45) − cosh φ = cos(iφ) (1.46) we find ′ ct = cos(iθ)ct + i sin(iθ)x (1.47) ′ x = i sin(iθ)ct + cos(iθ)x (1.48) or multiplying the first equation by i, we have − ict′ cos(iφ) sin(iφ) ict − ′ = − (1.49) x sin(iφ) cos(iφ) x    −   which now can be clearly seen as a ”rotation” in the ( ict,x) plane with imaginary time, by an −− imaginary angle iφ, related to the boost factor φ = tanh 1 β.

1.1.6 Lorentz Transformations The Lorentz Transformations are a subset of transformations that leave the 4-dimensional line element invariant. It includes mainly Lorentz boosts and spatial rotations. It can be checked that they actually form a non-abelian , the . If we add translations to the Lorentz group, we obtain the so-called Poincar´egroup.

1.1.7 Space-like and Time-like Events Consider the interval s2 between 2 events, one of them happening at the space-time origin.

Space-like: If s2 > 0, that means x2 + y2 + z2 > c2t2 and events are not causally connected. There is not enough time for a light signal to connect the two events. This being true in one reference frame, it obviously remain true in any reference frame. These intervals are called space- like intervals. For them, we may define the proper distance σ as

σ = √s2 = c2t2 + x2 + y2 + z2 (Proper Distance) (1.50) − For space-like intervals, we mayp find a frame in which c2t2 = 0, i.e. the events are simultaneous, in which case σ = x2 + y2 + z2. But we may not find a frame in which they happened at the same , i.e. x2 + y2 + z2 = 0. For space-like events, the order in which they appear relative to p each other may change, which is fine since they are not causally connected.

Time-like: If s2 < 0, that means c2t2 > x2 + y2 + z2 and events may be causally connected. There is enough time for a light signal to connect the two events. These intervals are called time-like intervals. For them, we may define the τ as

cτ = s2 = c2t2 x2 y2 z2 (Proper Time) (1.51) − − − − For time-like intervals, wep may not findp a frame in which c2t2 = 0, i.e. the events are simultaneous. But we may find a frame in which they happened at the same position, i.e. x2 + y2 + z2 = 0, in which case τ = t. For time-like events, the order in which they appear relative to each other may not change, since they are causally connected. 18 CHAPTER 1. GENERAL RELATIVITY

1.1.8 ′ ′ ′ ′ ′ Let us consider an observer at rest in K that measures two events (x1,t1) and (x2,t2), which happen at the same location as the observer i.e. ∆x′ = x′ x′ = 0 and ∆t′ ∆τ = t′ t′ . Such 2 − 1 ≡ 2 − 1 time measured in the reference that moves with the system coincides with its proper time. Now for the K frame ∆t = t t and the two events do not happen at the same position, i.e. 2 − 1 ∆x = x x = 0. In fact, from the space Lorentz transformations we have 2 − 1 6 ′ x1 = γ(x1 βct1) (1.52) ′ − x = γ(x βct ) (1.53) 2 2 − 2 so that we have

′ ′ ′ ∆x = x x =0= γ(∆x v∆t) ∆x = v∆t (1.54) 2 − 1 − → and therefore the time Lorentz transformations

′ ct1 = γ(ct1 βx1) (1.55) ′ − ct = γ(ct βx ) (1.56) 2 2 − 2 implies

′ c∆t c∆t = γ(c∆t βv∆t)= cγ(1 β2)∆t = (1.57) − − γ so

′ ∆t = γ∆t (1.58)

Therefore, time is longer (passes slower) between events in frame K where the particles move. A more direct way to derive this result, without using the Lorentz transformations is to simply set ds2 = ds′2 in both frames. In the reference frame K, we have dx = vdt and the invariance of the line element gives us

′ ds2 = ds 2 c2dt2 + dx2 = c2dτ 2 − − c2dt2(1 β2) = c2dτ 2 − − − dt2/γ2 = dτ 2 → 1.1.9 Let us now consider two events as seen by an observer in K′, who measures a length or spatial ′ ′ ′ separation ∆x = x2 x1. Obviously, lengths must be defined from coordinates measured at the − ′ ′ same time, and therefore t1 = t2. Such spatial separation measured by a frame moving with the observer coincides with its proper distance. Now, for consistency, the length ∆x measured by K, must also be defined for simultaneous measurements, i.e. t2 = t1. From the , we then have ′ x1 = γ(x1 βct1)= γx1 (1.55) ′ − x = γ(x βct )= γx (1.56) 2 2 − 2 2 1.1. SPECIAL RELATIVITY 19 so that

′ ∆x = γ∆x (1.57) or ∆x′ ∆x = (1.58) γ

1.1.10 Scalars, 4-Vectors, As will be seen in the following sections, objects that transform in specific ways under a Lorentz transformation will be quite useful for the covariant formulation of physical theories in the context of special and general relativity. Next we define such objects, generally called tensors. A scalar S is defined as being invariant under a Lorentz transformation:

′ S = S (1.59)

A contravariant 4-vector V µ is defined by the property of transforming exactly as the contravari- ant coordinates xµ under a Lorentz transformation

′µ µ ν V =Λ νV (1.60)

A covariant 4-vector Vµ is obtained by lowering the index with the metric, as for the coordinate 4-vector:

ν Vµ = ηµνV (1.61)

Therefore from the transformation law for a contra variant 4-vector, we find

′ ′ν ν α Vµ = ηµνV = ηµνΛ αV (1.62)

defined by the property of transforming exactly as the covariant coordinates xµ under a Lorentz transformation

′ ν Vµ =Λ µVν (1.63)

A rank 2 (tensor with 2 indices, think of a matrix), T µν, is defined by the property of transforming as

′µν µ ν αβ T =Λ αΛ βT (1.64)

Higher rank tensors are defined similarly, with extra factors of Λ in their transformation law.

Examples The speed of light c, the charge q of elementary particles, the (rest) mass m and the proper time τ (for massive particles in timelike trajectories) are scalars (Lorentz invariant). As dxµ is a 4-vector and dτ is a scalar, the 4- U µ defined for massive particles as dxµ U µ = (1.65) dτ 20 CHAPTER 1. GENERAL RELATIVITY is also a 4-vector, as well as the 4- P µ = mU µ and the 4-force f µ = dP µ/dτ. The electromagnetic fields E and B form a rank 2 tensor Fµν, defined in the next sections. The product of two 4-vectors AµBν is a tensor of rank 2. µν The contraction of a rank 3 tensor, e.g. Tν , is a 4-vector. The derivative with respect to a contravariant coordinate is a covariant 4-vector

∂ ∂xβ ∂ = (1.66) ∂x′α ∂x′α ∂xβ whereas the derivative with respect to a covariant coordinate is a contravariant 4-vector

∂ ∂xβ ∂ ′ = ′ (1.67) ∂xα ∂xα ∂xβ Finally the 4-dimensional Laplacian or D’alambertian is a scalar

1 ∂2  = ∂ ∂α = + 2 (1.68) α −c2 ∂t2 ∇ In general, all quantities above are tensors of some rank. Scalars are tensors of rank 0 and vectors are tensors of rank 1.

1.1.11 and Momentum Given the 4-velocity U µ definition for massive particles, we have

dxµ dx0 dxi cdt dxi U µ = = , = ,γ =(γc,γv)= γ(c, v) (1.69) dτ dτ dτ dτ dt     We define the 4-momentum for massive particles as

P µ mU µ =(γmc, γmv) Momentum (massive particles) (1.70) ≡ The spatial component of P µ is associated with the particle relativistic momentum p, i.e. p = γmv and the zero-th component is defined as the particle relativistic energy E such that E = γmc2. Therefore, E P µ , p (1.71) ≡ c   In the limit v c we have γ = (1 β2)−1/2 1+ β2/2+ O(β4) and ≪ − ≈ v2 1 E = γmc2 = mc2 1+ mc2 + mv2 + O(β4) (1.72) 2c2 ≈ 2   p = γmv mv + O(β3) (1.73) ≈ i.e. we recover the classical expressions for momentum and kinetic energy, given we interpret mc2 as a ”rest energy”. More generally, for any particle (including massless particles, e.g. ), we may define the 4-momentum as dxµ P µ = Momentum (massive and massless particles) (1.74) dλ 1.1. SPECIAL RELATIVITY 21 where λ parametrize the particle trajectory. For massive particles this parameter can be chosen to be λ = τ/m. But for massless particles, both τ and m and zero and we must make another choice for λ. As we will see later, It turns out that we do not really need to find this parameter for massless particles, since we can always replace it in favor of the time coordinate. Notice that for massive particles U µU = U U +U U = γ2c2 +γ2v2 = γ2c2(1 β2)= c2 µ − 0 0 i i − − − − and therefore P µP = m2c2. As a result, µ − E 2 P µP = + p2 = m2c2 (1.75) µ − c −   and we obtain the relation

E2 =(pc)2 +(mc2)2 (1.76)

For a particle at rest (p = 0) we re-obtain Einstein’s relation for the rest energy E = mc2. For a massless particle (m = 0) we get E = pc.

1.1.12 Doppler Effect for Radiation The 4-momentum P µ is a 4-vector and, as such, transforms under a Lorentz transformation in the same way as the position vector xµ. Consider a particle with energy E and x momentum p when measured by frame K. We would like to know its energy E′ in the K′ frame that moves with velocity v in the x direction relative to K. For a boost in the x direction, we simply apply the transformation E′ E = γ βp (1.77) c c −   ′ E p = γ p β (1.78) − c   Now, suppose this particle is a massless coming from a source fixed at K. In this case Eγ = pγc and both equations above have the same information. Let us take the first equation then for a photon: ′ E E E γ = γ γ β γ (1.79) c c − c   and therefore ′ E = γ(1 β) E γ − γ (1 β) = − Eγ 1 β2 − p1 β = − Eγ (1.78) s1+ β

Finally, from quantum we know that Eγ = hν, where h is the Planck’s constant and ν is the photon frequency, we have the Doppler shift in the radiation frequency ν:

′ 1 β ν = − ν (1.79) s1+ β 22 CHAPTER 1. GENERAL RELATIVITY which can also be expressed in terms of the wavelength λ of the photon, since c = λν, producing

′ 1+ β λ = λ (1.80) 1 β s − So the observer at K′, who sees the source of photons moving away from him, will detect the photon with wavelength λ′ > λ. Since larger wavelengths are redder, we call this effect . The redshift z is defined as the relative difference of wavelengths

∆λ λ′ λ λ′ 1+ β z = = − = 1= 1 (1.81) λ λ λ − 1 β − s − For small β 1, we have ≪ v z (1 + β)2 1= β = (1.82) ≈ − c p So the velocity is linearly connected with the redshift of photons. If the source of photons at K′ emits a full spectrum of photons, this spectrum would be redshifted (shifted in the direction of larger wavelengths). This is a purely kinematical effect from special relativity. As we will see, there is a similar dynamical effect in general relativity due to gravitational potentials and in the cosmological context, due to the expansion of the (Hubble flow) or the fabric of space-time itself. It is this dynamical effect that is relevant for the connection between redshift of spectral lines of galaxies and the expansion of the Universe. The kinematical effect is less pronounced and relates mainly to the peculiar velocities of galaxies on top the so-called Hubble flow, being nonetheless important for effects of redshift distortions. Even though we derived this effect using the fact that light is made of quantum photons, we could have also argued at the classical level, considering light as a wave, and taking into account the space-time Lorentz transformations to compute the time between two crests from the point of view of an observer at K and an observer at K′. We obtain the same results as above.

1.1.13 Relativistic Covariance in Electromagnetism By the Special Relativity postulate, the E&M equations must have the same form (be invariant) under Lorentz transformations. Therefore, they must be written in tensorial form, since tensors transform under the specific rules described in previous sections. Tensor equations guarantee that the validity of an equation in one reference frame imply its validity in other reference frames obtained by Lorentz transformation. For instance, an equation

γ γ aAµν + bBµν = 0 (1.83)

′ µ valid in frame K is automatically valid in frame K , since we can simply apply Λ ν factors and γ γ′ turn tensors in unprimed coordinates to tensors primed coordinates, i.e. Aµν A ′ ′ . → µ ν The speed of light c, by hypothesis, is a scalar (Lorentz invariant). It is an empirical fact that the charge q of elementary particles also is, i.e. it does not change with relative movement.

If ρ is the charge density, we have that dq = ρd3x is a Lorentz invariant. But since the 4- 4 ′ µ 4 µ 4 0 3 dimensional volume element d x =detΛ νd x and detΛ ν = 1, we have that d x = dx d x is also 1.1. SPECIAL RELATIVITY 23 an invariant. Therefore, ρ must transform as the zero component of a 4-vector jµ under Lorentz transformations. Defining this 4-current as:

jµ =(cρ, j) (1.84) where j is the charge current density, we have that the scalar equation

∂jµ = 0 (1.85) ∂xµ implies charge conservation

∂ρ + j = 0 (1.86) ∂t ∇· Defining the 4-potential

Aα =(φ/c, A), (1.87) and using the fact that the D’alambertian  is a scalar, we may write the covariant 4-vector equation

Aα = µ jα (1.88) − 0 which implies the wave equations obtained before: ρ 2φ = (1.89) −ǫ0 2A = µ j (1.90) − 0 The contravariant 4-vector equation

′ ∂f A α = Aα + (1.91) ∂xα describes the gauge transformations:

′ ∂f φ = φ (1.92) − ∂t ′ A = A + f (1.93) ∇ whereas the scalar equation

∂Aα = 0 (1.94) ∂xα describes the Lorenz gauge:

1 ∂φ A + = 0 (1.95) ∇· c2 ∂t We know that the fields E and B have 6 components altogether, and from their relation to the potentials (φ, A), they must be first derivatives of Aα. Because an anti-symmetric rank 2 tensor 24 CHAPTER 1. GENERAL RELATIVITY has exactly 6 independent components (zeros in the diagonal and ij = ji), we may define the − field tensor F µν: ∂Aν ∂Aµ F µν = (1.96) ∂xµ − ∂xν

Explicity evaluation of e.g. the 01 component gives

∂A1 ∂A0 ∂A ∂(φ/c) 1 ∂A E F 01 = = x = φ = x (1.97) ∂x − ∂x ∂( ct) − ∂x − c ∂t −∇ c 0 1 −  x Proceeding similarly, we obtain all the components of F µν:

0 Ex/c Ey/c Ez/c Ex/c 0 Bz By F µν =  − −  (1.98) E /c B 0 B − y − z x  E /c B B 0   − z y − x    or alternatively

0 E /c E /c E /c − x − y − z αγ Ex/c 0 Bz By Fµν = ηµαF ηγν =  −  (1.99) E /c B 0 B y − z x  E /c B B 0   z y − x    The Maxwell’s Eqs. with source can then be written in covariant form as ∂F µν = µ jµ (1.100) ∂xν 0 from which charge conservation also follows, since F µν is anti-symmetric:

∂F µν 1 ∂F µν ∂F νµ 1 ∂ ∂jµ = + = (F µν + F νµ) = 0 = 0 ∂xµ∂xν 2 ∂xµ∂xν ∂xν∂xµ 2 ∂xµ∂xν → ∂xµ   (1.100)

Likewise, the sourceless Maxwell Eqs. can be written as ∂F ∂F ∂F µν + σµ + νσ = 0 (1.101) ∂xσ ∂xν ∂xµ With e.g. µ = 0, ν = 1, σ = 2, we have:

∂F ∂F ∂F ∂(E /c) ∂( E /c) ∂B ∂B 01 + 20 + 12 = − x + − y + z = E + = 0 ∂x2 ∂x1 ∂x0 ∂y ∂x ∂(ct) −∇ × ∂t  z (1.101) and similarly for all other components, obtaining ∂B E = (1.102) ∇× − ∂t B = 0 (1.103) ∇· 1.1. SPECIAL RELATIVITY 25

Finally, defining the 4-force dP µ f µ = (1.104) dτ for the E&M case as the covariant combination

µ µν f = qF Uν (1.105) we find that that the Lorentz force is also obtained: dp F = = q(E + v B) (1.106) dt × Note that since F µν is a rank 2 tensor, it transforms under Lorentz transformation as

′µν µ ν αβ F =Λ αΛ βF (1.107) and therefore electric and/or magnetic fields can appear in a reference frame even if they do not exist in another frame. That is why the term ”electromagnetic field” makes sense: not only the electric and magnetic fields propagate together in a wave, but they are different ”projections” of a same physical entity. This is similar to a quantum wave function ψ(x)= x ψ , which is a projection h | i of an abstract quantum state ψ in a specific representation in the x of eigenvectors of the | i | i position operator X.

1.1.14 Energy-Momentum Tensor We will typically need to consider the collective behavior of a number of particles, and will often employ the approximation of a fluid (a continuum described by macroscopic quantities such as density, pressure, entropy, viscosity, etc). Although P µ provides a complete description of the energy and momentum for a single particle, in order to describe a collection of particles or a fluid, we need to consider a symmetric tensor T µν, dubbed the energy-momentum tensor. In general, T µν is defined as

T µν = ”flux of P µ across surface of constant xν” = P µ per surface to xν. ⊥ and its components can be separately interpreted. For

ν = 0 P µ per dx1dx2dx3= ”density of P µ” → ν = 1 P µ per dx0dx2dx3 = P µ per time per area to x1 = flux of P µ in direction of x1 → ⊥ So, we may write

T µ0 = ”density of P µ ”

T µi= P µ per time per area to xi = ”flux (current density) of pµ in the xi direction” ⊥ Therefore, all components are: 26 CHAPTER 1. GENERAL RELATIVITY

T 00: density of P 0 = E : energy density

T 0i: flux of P 0 = E in the xi direction : energy flux = energy current density

T i0: density of P i : momentum density = energy flux

T ii: flux of P i in the xi direction : force f i per area to xi = momentum flux = pressure ⊥ T ij: flux of P i in the xj direction : force f i per area to xj = shear viscosity ⊥ where (flux) = (current density).

Construction

Given these considerations, for a system of particles labeled by n, we define the density of P α as

T α0 = P α δ3(x x ) (1.108) n − n n X and the flux (current density) of P α in the xi direction as

dxi T αi = P α n δ3(x x ) (1.109) n dt − n n X 0 Since xn = t, these two equations can be combined

dxβ T αβ = P α n δ3(x x ) (1.110) n dt − n n X

β β dxn and since Pn =(En, pn)=(mγ, mγv)= mγ(1, v)= En dt or

dxβ P β n = n (1.111) dt En and

P α P β T αβ = n n δ3(x x ) (1.112) E − n n n X From this we see that

T αβ = T βα (1.113) 1.2. 27

Perfect Fluid A perfect fluid is defined as having a local frame K′ at each point, in which the fluid is isotropic. In this (comoving) frame, the energy-momentum is spherically symmetric:

′ T 00 = ρ (1.114) ′ ′ T i0 = T 0i = 0 (1.115) ′ T ij = Pδij (1.116)

Now we go to a lab frame K, which sees the fluid moving with velocity v, such that

α α ′β x =Λ βx (1.117) and since T αβ is a tensor

αβ α β ′γδ T =Λ γΛ δT (1.118) implies

T 00 = γ2(ρ + Pv2) (1.119) i0 2 T = γ (ρ + P )vi (1.120) ij 2 ij T = γ (ρ + P )vivj + Pδ (1.121)

To check that this is a tensor, we note that these equations can be expressed in tensorial form as

T αβ =(ρ + P )U αU β + Pηαβ (1.122)

1.2 Equivalence Principle

The apparent equality of the gravitational mass mg (source of gravity) and the inertial mass mi (which responds to gravity) has profound implications. An immediate consequence is that a static and uniform gravitational field cannot be detected in a freely falling frame (e.g. observer and elevator accelerate together). Consider an observer at Laboratory frame K watching the elevator falls, for which there is a gravitational field g. For this observer

d2x m = m g (Laboratory Frame) (1.123) i dt2 g Now consider an observer at frame K′ freely falling with the elevator, whose coordinates relate to those of K as

′ 1 x = x gt2 (1.124) − 2 ′ t = t (1.125)

Inserting these in Eq. 1.123 we find

d2x′ m + m g = m g (1.126) i dt′2 i g 28 CHAPTER 1. GENERAL RELATIVITY or since mi = mg

d2x′ m = 0 (Freely Falling Frame) (1.127) i dt′2 Thus, for both K and K′, Newtons’s second law is valid. But for K there is a gravitational field, and for K′, there is not.

The Equivalence Principle says that, as in the case above, it is always possible to find a local frame where gravity disappears (a locally inertial frame or a freely falling frame). And since we are dealing with a relativistic theory now, in this locally inertial frame, the laws of special relativity should apply locally.

Locally inertial frames: freely-falling frames in small enough regions for which special rela- tivity holds locally.

Obviously, a real gravitational field is not static and uniform on large enough distances, and eventually we should be able to tell whether we are in a freely falling frame within a gravitational field or if we are in a truly inertial frame. For instance, in the case of the elevator above, the gravitational field points to the center of the Earth. As a result, the observer at K′ would eventually see things moving radially inside the elevator for no good reason and would conclude: ”Humm... I’m in a gravitational field...” On small enough regions though, we should expect an equivalence. These considerations allow us to state the equivalence principle.

Weak Equivalence Principle (WEP): ”In small enough regions of space-time, the of freely-falling particles is the same in a uniform gravitational field and in a uniformly accelerated frame, i.e. the laws of Mechanics take the same form as in an unaccelerated frame in the absence of . As a result, at every point of space-time in an arbitrary gravitational field, it is possible to choose a ”locally inertial frame” such that in small enough regions the laws of Mechanics reduce to those of special relativity.”

Obviously on large enough regions inhomogeneities and anisotropies in the gravitational field will cause tidal forces, which can be detected. The Equivalence Principle from Einstein is quite stronger, and generalizes the statement above to all laws of physics, including gravitation itself.

Strong Equivalence Principle (SEP): Replace laws of Mechanics by laws of Physics above.

Therefore, according to the SEP it is impossible to detect the existence of a gravitational field by means of local experiments of any kind. The SEP suggests that gravity is somewhat different from the other interactions in the sense that it affects and is affected by everything. The SEP motivates the description of gravity as a geometric effect, a property of space-time itself. This is even more suggestive when we contrast the Equivalence Principle and the Gauss axiom for non-Euclidean geometry:

Equivalence Principle: At any point in space-time, we can find locally inertial coordinates in which the metric is locally Minkowski, i.e. special relativity holds. 1.2. EQUIVALENCE PRINCIPLE 29

Gauss axiom: At any point in a curved surface, we can find local cartesian coordinates in which distances obey Pythagoras Theorem, i.e. Euclidean geometry holds.

This analogy in fact allows gravity to be described geometrically.

1.2.1 Geodesics Consider a massive particle in the presence of a gravitational field. From the equivalence principle, we can find a set of freely-falling coordinates ξα, similar to coordinates in the moving frame K′, but only in the local sense. In this locally inertial frame, no force is acting and we have d2ξα = 0 (1.128) dτ 2 and the proper time is

c2dτ 2 = ds2 = η dξαdξβ (1.129) − − αβ Now let us transform to another general frame (e.g. K in the laboratory) with coordinates xβ, i.e. ξα xβ. We have → dξα ∂ξα dxµ = (1.130) dτ ∂xµ dτ and therefore d2ξα d dξα = dτ 2 dτ dτ   d ∂ξα dxµ = dτ ∂xµ dτ   d ∂ξα dxµ ∂ξα d2xµ = + dτ ∂xµ dτ ∂xµ dτ 2   ∂ξα dxν dxµ ∂ξα d2xµ = + = 0 (1.128) ∂xν∂xµ dτ dτ ∂xµ dτ 2 ∂xγ Multiplying this equation by the inverse transformation ∂ξα and using ∂xγ ∂ξα = δγ (1.129) ∂ξα ∂xµ µ we have d2xγ ∂xγ ∂ξα dxν dxµ + = 0 (1.130) dτ 2 ∂ξα ∂xν∂xµ dτ dτ or d2xγ dxµ dxν + Γγ = 0 (1.131) dτ 2 µν dτ dτ γ where we defined the affine connection Γµν ∂xγ ∂2ξβ Γγ = (1.132) µν ∂ξβ ∂xµ∂xν 30 CHAPTER 1. GENERAL RELATIVITY

γ γ Note that Γµν = Γνµ. For the proper time, we have

∂ξα ∂ξβ c2dτ 2 = η dxµ dxν αβ ∂xµ ∂xν µ ν = gµνdx dx (1.132)

µ where we defined the gµν in the new coordinates x :

∂ξα ∂ξβ g = η (1.133) µν ∂xµ ∂xν αβ If we had chosen a massless particle instead (e.g. a photon), we should not use τ as the evolution variable because dτ = 0. In this case we could use any other parameter, such as the time component in the freely-falling frame, e.g. λ = ξ0. The exact same procedure would then lead us to d2xγ dxµ dxν + Γγ = 0 (1.134) dλ2 µν dλ dλ and

0= g dxµdxν (1.135) − µν Note however that we do not need to know τ or λ in order to find the motion of the particle. That is because after solving the equations we obtain either xµ(τ) or xµ(λ), and for the µ = 0 component x0(τ) = ct(τ) can be inverted to give τ(t), with which we obtain for the spatial components xi(t)= x(t), i.e. the trajectory of the particle. According to the equivalence principle, at every point x, we can find locally inertial coordinates ξα(x) that satisfy

∂ξα(x) ∂ξβ(x) ηαβ = gµν (1.136) ∂xµ ∂xν in an infinitesimal neighborhood of x, i.e. in this region the metric is locally Minkowski.

1.2.2 Metric and Connection As we will see, the metric will play an important role in General Relativity, and it is convenient to γ express the connection Γµν in terms of gµν. We start with Eq. 1.133

∂ξα ∂ξβ g = η (1.137) µν ∂xµ ∂xν αβ Since the connection is related to the second derivative of the coordinates, we differentiate this equation to make the connection appear, obtaining

∂g ∂2ξα ∂ξβ ∂ξα ∂2ξβ µν = η + η (1.138) ∂xλ ∂xλ∂xµ ∂xν αβ ∂xµ ∂xλ∂xν αβ Multiplying Eq. 1.132 by ∂ξα/∂xγ we have

∂ξα ∂ξα ∂xγ ∂2ξβ ∂2ξα Γγ = = (1.139) ∂xγ λµ ∂xγ ∂ξβ ∂xλ∂xµ ∂xλ∂xµ

α δβ | {z } 1.2. EQUIVALENCE PRINCIPLE 31 and therefore,

∂g ∂ξα ∂ξβ ∂ξα ∂ξβ µν = Γγ η + Γγ η (1.140) ∂xλ ∂xγ λµ ∂xν αβ ∂xµ ∂xγ λν αβ ∂ξα ∂ξβ ∂ξα ∂ξβ = Γγ η +Γγ η (1.141) λµ ∂xγ ∂xν αβ λν ∂xµ ∂xγ αβ

gγν gµγ | {z } | {z } and we used Eq. 1.133 yet again above, to obtain

∂g µν = g Γγ + g Γγ (1.142) ∂xλ γν λµ µγ λν

We consider this same equation 3 with interchange µ λ and ν λ ↔ ↔ ∂g µν = g Γγ + g Γγ ∂xλ γν λµ µγ λν ∂g λν = g Γγ + g Γγ ∂xµ γν µλ λγ µν ∂g µλ = g Γγ + g Γγ ∂xν γλ νµ µγ νλ

γ γ Since gµν = gνµ and Γµν = Γνµ, we add the first eq. to the second and from the result subtract the third, obtaining

∂g ∂g ∂g µν + λν µλ = 2g Γγ (1.140) ∂xλ ∂xµ − ∂xν γν µλ

νσ νσ σ Finally, we multiply by the inverse metric g , such that g gγν = δγ and find

gνσ ∂g ∂g ∂g Γσ = µν + λν µλ (1.141) µλ 2 ∂xλ ∂xµ − ∂xν   or rearranging

gσν ∂g ∂g ∂g Γσ = µν λµ + νλ (1.142) µλ 2 ∂xλ − ∂xν ∂xµ   Suppose we parametrize the path from A to B in arbitrary coordinates x using a parameter λ that defines a family of possible trajectories, and define an action with the line element for a free particle as

B B dxµ dxν S = ds = g dλ. (1.143) µν dλ dλ ZA ZA r If we vary the path and require that it is an extremum, i.e. δS = 0, it is possible to show that we obtain the Geodesic equation. (Exercise: Show). 32 CHAPTER 1. GENERAL RELATIVITY

1.2.3 Newtonian Limit Let us consider the so-called Newtonian limit of our previous result, i.e. the case of a particle moving slowly in a weak and stationary gravitational field. For a slow (non-relativistic) particle we have dxi/dτ dt/dτ and Eq. 1.131 becomes ≪ d2xα dxµ dxν + Γα = 0 dτ 2 µν dτ dτ d2xα dx0 dx0 dx0 dxi dxi dxj + Γα + 2Γα + Γα = 0 (1.143) dτ 2 00 dτ dτ 0i dτ dτ ij dτ dτ ≈0 ≈0 so, since x0 = ct |{z} | {z } d2xα dt 2 + c2Γα = 0 (1.144) dτ 2 00 dτ   As we will see better later, the effects of gravitation are encoded in the metric gµν and its derivatives, α i.e. in gµν and Γµν. For a stationary field, time derivatives of gµν are negligible and we have

αν αν g ∂g0ν ∂g00 ∂gν0 g ∂g00 Γα =  +  = (1.145) 00 2 ∂x0 − ∂xν ∂x0 − 2 ∂xν  ≈ ≈   0 0    For weak fields, we may perturb the| {z metric} around| {z its} coordinates in (which is valid for freely-falling coordinates) as g = η + h (x), with h (x) η (1.146) αβ αβ αβ αβ ≪ αβ gαβ = ηαβ hαβ(x) (1.147) − and to first order in the metric perturbation gαν ∂g Γα = 00 (1.148) 00 − 2 ∂xν

αν αν (η h ) ∂η00 ∂h00 = −  +  (1.149) − 2 ∂xν ∂xν    =0  ηαν ∂h   00 | {z } (1.150) ≈ − 2 ∂xν (1.151) whose components become η0ν ∂h η00 ∂h Γ0 = 00 = 00 0 (1.152) 00 − 2 ∂xν − 2 ∂x0 ≈ ≈0 stat. ηiν ∂h 1 ∂h 1 ∂h Γi = 00 = ηij| {z }00 = 00 (1.153) 00 − 2 ∂xν −2 ∂xj −2 ∂xi δij |{z} (1.154) 1.2. EQUIVALENCE PRINCIPLE 33

Therefore, the components of the Geodesics equation give d2x0 dt 2 d2t + c2 Γ0 = 0 = 0 (1.155) dτ 2 00 dτ → dτ 2 =0   which says that dt/dτ =const. and |{z} d2xi dt 2 + c2Γi = 0 dτ 2 00 dτ   d2xi c2 ∂h dt 2 d2x c2 dt 2 00 = 0 = h (1.155) dτ 2 − 2 ∂xi dτ → dτ 2 2 ∇ 00 dτ     Since dt/dτ=const., we may divide the equation by (dt/dτ)2 and get d2x c2 = h (1.156) dt2 2 ∇ 00 The corresponding Newtonian result would be d2x = φ (1.157) dt2 −∇ where GM φ = (1.158) − r Therefore, we have h 00 = φ + constant (1.159) 2 − But since for large r the gravitational effects should disappear, i.e. h 0 for r , we must 00 → → ∞ have constant = 0. Therefore h = 2φ and 00 − g = (1 + 2φ) (1.160) 00 − This means that the gravitational potential is a perturbation in the time-time component of the metric relative to the Minkowski metric of flat space-time.

1.2.4 Time Dilation and Since gravitational fields are perturbations in the time-time component of the metric, we should expect the rate of clocks to change in the presence of gravitational fields. Consider a clock in an arbitrary gravitational field. By the equivalence principle, in a locally inertial frame K′ with coordinates ξα the rate is not affected and the space-time interval is given by ′ dt = dτ =( ds)1/2 =( η dξαdξβ)1/2 (1.161) − − αβ where dt′ is the time between ticks of the clock when it is at rest in the absence of gravity. Therefore, changing to coordinates xµ in an arbitrary frame K, we have

α β 1/2 ′ ∂ξ ∂ξ dt = η dxµ dxν (1.162) − αβ ∂xµ ∂xν   34 CHAPTER 1. GENERAL RELATIVITY or using ∂ξα ∂ξβ g = η (1.163) µν αβ ∂xµ ∂xν we have ′ dt =( g dxµdxν)1/2 (1.164) − µν If the clock has velocity dxµ/dt in the frame K, we have

µ ν 1/2 ′ dx dx dt = dt g (1.165) − µν dt dt   and the time interval dt between ticks in K is µ ν −1/2 ′ dx dx dt = dt g (1.166) − µν dt dt   If the clock is at rest at K, v = dx/dt = 0 and dxµ dxν g = g (1.167) µν dt dt 00 Therefore ′ − dt = dt ( g ) 1/2 (1.168) − 00 From this result we can compare the time measured at 2 different points. Consider two points 1 and 2 at rest in a stationary gravitational field. Suppose at point 2 a light pulse (wave crest) is emitted towards point 1. The time between crests dt2 measured at 2 is the same when the crests arrive at 1 and equals ′ − dt = dt ( g (x )) 1/2 (1.169) 2 − 00 2 On the other hand, the time dt1 between wave crests due to the same process happening at point 1, will be ′ − dt = dt ( g (x )) 1/2 (1.170) 1 − 00 1 Therefore, the ratio of times between 1 and 2 is − dt g (x ) 1/2 2 = 00 2 (1.171) dt g (x ) 1  00 1  i.e. the ratio of frequencies ν 1/dt will be ∝ ν g (x ) 1/2 2 = 00 2 (1.172) ν g (x ) 1  00 1  So in the weak field regime g = (1 + 2φ) and 00 − δν ν ν ν 1 + 2φ(x ) 1/2 = 2 − 1 = 2 1= 2 1 ν ν ν − 1 + 2φ(x ) − 1 1 1  1  (1 + 2φ(x ) 2φ(x ))1/2 1 ≈ 2 − 1 − φ(x ) φ(x ) (1.171) ≈ 2 − 1 1.3. TENSORS REVISITED 35

1.3 Tensors Revisited

The Principle of Equivalence (of gravity and inertia) establishes an analogy between gravity and non-Euclidean geometry. The mathematical language connecting both is that of covariant objects, i.e. tensors.

1.3.1 Principle of Equivalence Principle: Gravitational effects can be obtained by writing equations for general gravitational fields in a locally inertial frame where gravitational effects disappear (e.g. dξ2/dτ 2 = 0) and transforming to the Laboratory coordinates to find the equation in the Lab. frame.

The Principle of General Covariance is an alternative version of the Equivalence principle with the same physical content. It states that

Principle of General Covariance: A physical equation holds in general gravitational fields (i.e. in general relativity) if: a) the equation holds in the absence of gravitation; i.e. it agrees with special relativity when α gµν = ηµν and Γ µν = 0. b) the equation is generally covariant, i.e. it preserves its form under a general coordinate transformation.

Note this generalizes the covariance of electromagnetism we defined before, where we only required the equations to be valid for transformations between inertial frames. Now it’s a general transformation between any 2 frames, i.e.. general covariance. This principle follows from the equivalence principle: there is always a frame (the locally inertial frame) in which gravitational effects are absent. In this frame, condition a) holds and then condition b) guarantees that the equation holds in any other frame. The principle of general covariance is a statement only about gravitational fields, nad only holds in the local sense, i.e. on scales small compared to the scale of the gravitational fields. Therefore we expect only gµν and its first derivatives to enter the generally covariant equations, and these quantities will precisely represent the gravitational field effects.

1.3.2 Tensors Again Most of the definitions made in the context of Special Relativity still hold for scalars, vectors and tensors in the context of gravitational fields or General Relativity, provided we generalize the transformation law to be between arbitrary coordinates and not only between inertial frames. For instance, under xµ x′µ a contravariant vector V µ transforms as → ′µ ′ ∂x V µ = V ν (1.172) ∂xν and a covariant vector Vµ ν ′ ∂x V = V (1.173) µ ∂x′µ ν and a mixed tensor ′µ ρ ′λ ′ ∂x ∂x ∂x T µ λ = T κ σ (1.174) ν ∂xκ ∂x′ν ∂xσ ρ 36 CHAPTER 1. GENERAL RELATIVITY

The affine connection is not a tensor. We will see how it transforms next.

Metric and Kronecker We defined the metric for a coordinates xµ in a general frame as

∂ξα ∂ξβ g = η (1.175) µν αβ ∂xµ ∂xν In a different x′µ the metric becomes

α β ′ ∂ξ ∂ξ g = η (1.176) µν αβ ∂x′µ ∂x′ν ∂ξα ∂xρ ∂ξβ ∂xσ = η (1.177) αβ ∂xρ ∂x′µ ∂xσ ∂x′ν ∂ξα ∂ξβ ∂xρ ∂xσ = η (1.178) αβ ∂xρ ∂xσ ∂x′µ ∂x′ν

gρσ ∂xρ ∂xσ = |g {z } (1.179) ρσ ∂x′µ ∂x′ν (1.180)

So it is really a covariant tensor. The inverse metric gλµ is a contravariant tensor and satisfies

λµ λ g gµν = δν (1.181)

The Kronecker is a mixed tensor, because

∂x′ρ ∂xν ∂x′ρ ∂xµ δµ = = δρ (1.182) ν ∂xµ ∂x′σ ∂xµ ∂x′σ σ Operations like linear combinations, direct products and contractions can be performed on tensors to generate other tensors as usual. µν And the metric gµν and its inverse g can be used to lower/raise indices:

µρ ρ gµνT σ = Tν σ (1.183) and

µν ρ µρ g Tν σ = T σ (1.184)

Finally,

λµ λ g gµν = δν (1.185) and

λµ κν λµ κ λκ g g gµν = g δµ = g (1.186)

κ δµ | {z } 1.3. TENSORS REVISITED 37

Volume Element Let us consider the of the metric (seen as a matrix):

g = Det gµν (1.187) From the metric transformation rule we have ρ σ ρ σ ′ ∂x ∂x ∂x ∂x g = g = g (1.188) µν ρσ ∂x′µ ∂x′ν ∂x′µ ρσ ∂x′ν Since the determinant of a product of matrices is the product of the , we have

ρ 2 ′ 2 ′ ∂x ∂x g = Det g = g (1.189) ∂x′µ ∂x  

where the determinant above is the Jacobian of the transformation between coordinates. On the other hand, the 4-dimensional volume element transforms as ′ ′ ∂x d4x = d4x ∂x

g g = d4x = − d4x (1.189) g′ g′ r r− and we put the minus sign, because e.g. Det η = 1. Therefore µν − ′ g′ d4x = √ g d4x (1.190) − − i.e. √ g d4x is an invariant (scalar)p volume element. This generalizes the case in Minkowski space − where √ g = √ η = 1. − − 1.3.3 Transformation of the Affine Connection The affine connection was defined as ∂xλ ∂2ξα Γλ = (1.191) µν ∂ξα ∂xµ∂xν where ξα(xµ) are locally inertial coordinates. Passing to coordinates x′µ we have

′λ 2 α ′ ∂x ∂ ξ Γ λ = µν ∂ξα ∂x′µ∂x′ν ′λ α ∂x ∂ ∂ξ ′ = (now change x x) ∂ξα ∂x′µ ∂x′ν →   ∂x′λ ∂xρ ∂ ∂xσ ∂ξα = ∂xρ ∂ξα ∂x′µ ∂x′ν ∂xσ   ∂x′λ ∂xρ ∂xσ ∂xτ ∂2ξα ∂2xσ ∂ξα = + (1.189) ∂xρ ∂ξα ∂x′ν ∂x′µ ∂τ∂xσ ∂x′µ∂′ν ∂xσ   ρ ρ The combination of the first two terms in blue gives Γτσ and the last two give δσ, therefore

′λ τ σ ′λ 2 ρ ′ ∂x ∂x ∂x ∂x ∂ x Γ λ = Γρ + (1.190) µν ∂xρ ∂x′µ ∂x′ν τσ ∂xρ ∂x′µ∂x′ν 38 CHAPTER 1. GENERAL RELATIVITY

λ The first term on the right is what we would expect if Γµν were a tensor. We may re-express the second inhomogenous term, by differentiating the identity

∂x′λ ∂xρ = δλ (1.191) ∂xρ ∂x′ν ν with respect to x′µ, obtaining

∂x′λ ∂2xρ ∂2x′λ ∂xρ + = 0 ∂xρ ∂x′µ∂x′ν ∂x′µ∂xρ ∂x′ν ∂x′λ ∂2xρ ∂xσ ∂2x′λ ∂xρ + = 0 ∂xρ ∂x′µ∂x′ν ∂x′µ ∂xσ∂xρ ∂x′ν or ∂x′λ ∂2xρ ∂xρ ∂xσ ∂2x′λ = (1.190) ∂xρ ∂x′µ∂x′ν −∂x′ν ∂x′µ ∂xσ∂xρ Therefore the transformation of the affine connection becomes

′λ τ σ ρ σ 2 ′λ ′ ∂x ∂x ∂x ∂x ∂x ∂ x Γ λ = Γρ (1.191) µν ∂xρ ∂x′µ ∂x′ν τσ − ∂x′ν ∂x′µ ∂xρ∂xσ

1.3.4 Covariant Differentiation Differently from Minkowski space, in general coordinates the derivative of tensors is not a tensor. For instance, for a contravariante tensor:

′µ ′ ∂x V µ = V ν (1.192) ∂xν and its derivative is ∂V ′µ ∂x′µ ∂V ν ∂2x′µ = + V ν ∂x′λ ∂xν ∂x′λ ∂xν∂x′λ or ∂V ′µ ∂x′µ ∂xρ ∂V ν ∂2x′µ ∂xρ = + V ν (1.192) ∂x′λ ∂xν ∂x′λ ∂xρ ∂xν∂xρ ∂x′λ The first term on the right is what we would expect if ∂V µ/∂xλ were a tensor.

λ ν Now, combining the transformations for Γµν and V we have

′µ ρ σ 2 ′µ ρ σ ′κ ′ ′ ∂x ∂x ∂x ∂ x ∂x ∂x ∂x Γ µ V κ = Γν V η λκ ∂xν ∂x′λ ∂x′κ ρσ − ∂xρ∂xσ ∂x′λ ∂x′κ ∂xη   σ The terms in blue both give δη and we get

′µ ρ 2 ′µ ρ ′ ′ ∂x ∂x ∂ x ∂x Γ µ V κ = Γν V σ V σ (1.192) λκ ∂xν ∂x′λ ρσ − ∂xρ∂xσ ∂x′λ 2 ′ ∂ x µ ∂xρ ν ρ ν ′ V ∂x ∂x ∂x λ | {z } 1.3. TENSORS REVISITED 39

Therefore, adding the two equations above, the inhomogeneous terms cancel out and we get

′µ ′µ ρ ν ∂V ′ ′ ∂x ∂x ∂V + Γ µ V κ = + Γν V σ (1.193) ∂x′λ λκ ∂xν ∂x′λ ∂xρ ρσ   Thus, the combination in brackets transform as a tensor. This is defined as the covariant derivative ∂V V µ = V µ = + Γµ V κ (1.194) ∇λ ;λ ∂xλ λκ

Similarly we may define a covariant derivative for a covariant vector Vµ ∂V V = V = µ Γκ V (1.195) ∇λ µ µ;λ ∂xλ − µλ κ which is also a tensor, as can be checked. This can be extended to a general tensor, with a +Γ for each contravariant index and a Γ for each covariant index. For instance − ∂T µσ T µσ = λ + Γµ T νσ + Γσ T µν Γκ T µσ (1.196) λ;ρ ∂xρ ρν λ ρν λ − λρ κ The covariant derivative of the metric is zero, as can be checked, using Eq. 1.142: ∂g g = µν Γρ g Γρ g = 0 (1.197) µν;λ ∂xλ − λµ ρν − λν µρ Importance of covariant derivatives for forming covariant equations: µν µν 1) They transform tensors into tensors, i.e. if A is a tensor, so is λA . ∇ λ 2) They reduce to ordinary derivatives in the absence of gravity (when gµν = ηµν and Γµν = 0).

Therefore, the principle of general covariance allows us to apply the following algorithm to obtain equations that are generally covariant and true in the presence of gravity: a) Write the equation in special relativity (which holds in the absence of gravitation) b) Replace η g µν → µν c) Replace ∂/∂xµ . →∇µ Divergence and Curl The covariant derivative of a scalar is simply the usual gradient: ∂S S = (1.198) ;µ ∂xµ Now since ∂V V = µ Γλ V µ;ν ∂xν − µν λ ∂V V = ν Γλ V ν;µ ∂xµ − νµ λ and the afine connection is symmetric on the lower indices, we define the covariant curl ∂V ∂V V V = µ ν (1.197) µ;ν − ν;µ ∂xν − ∂xµ 40 CHAPTER 1. GENERAL RELATIVITY which is just the ordinary curl. We also define the covariant divergent ∂V µ V µ = + Γµ V λ (1.198) ;µ ∂xµ µλ and since 1 ∂g ∂g ∂g Γµ = gµρ ρµ + ρλ µλ µλ 2 ∂xλ ∂xµ − ∂xρ   µ ρ 1 ∂gρµ ∂g ∂g = gµρ + λ λ  (1.198) 2 ∂xλ ∂xµ − ∂xρ    0    and | {z } 1 ∂g Γµ = gµρ ρµ (1.199) µλ 2 ∂xλ Now for a general matrix M, we have

Tr (ln M) = lnDet M (1.200)

Notice this is obvious for a diagonal matrix and can be shown to be true more generally by diago- nalization with a matrix S such that M ′ = S−1MS is diagonal. Differentiating this equation with respect to xλ, we have

∂ ∂ ∂ Tr ln M = lnDet M = ln( Det M) ∂xλ ∂xλ ∂xλ −   − ∂ 1 ∂ Tr M 1 M = Det M (1.200) ∂xλ Det M ∂xλ   we may apply this for M = gρµ, so that Det M = g and we get

1 − ∂ 1 1 ∂ 1 1 ∂ 1 ∂ Γµ = (g ) 1 g = g = ( g)= ln( g) µλ 2 ρµ ∂xλ ρµ 2 g ∂xλ 2 ( g) ∂xλ − 2 ∂xλ − − ∂ 1 ∂ = ln √ g = √ g (1.200) ∂xλ − √ g ∂xλ − − Therefore ∂V µ 1 ∂ V µ = + √ g V λ (1.201) ;µ ∂xµ √ g ∂xλ −  −  or 1 ∂ V µ = √ gV µ (1.202) ;µ √ g ∂xµ − −  One consequence is the covariant Gauss theorem ∂ d4x√ g V µ = d4x √ gV µ = 0 (1.203) − ;µ ∂xµ − Z Z  1.3. TENSORS REVISITED 41 provided V µ vanishes at infinity. This relation also allows us to write 1 ∂ T µν = √ gT µν + Γν T µλ (1.204) ;µ √ g ∂xµ − µλ −  If T µν = T µν, the last term is zero and − 1 ∂ Aµν = √ gAµν (1.205) ;µ √ g ∂xµ − −  Another property of anti-symmetric tensors is that

Aµν;λ + Aλµ;ν + Aνλ;µ = Aµν,λ + Aλµ,ν + Aνλ,µ (1.206)

1.3.5 Covariant Differentiation Along a Curve So far we have discussed covariant derivatives of tensors defined over all space-time, in which case the derivatives are with respect to a general space-time point xµ. Now we consider tensors that are defined only over a parametrized curve xµ(τ), e.g. the particle momentum P µ(τ) over the particle trajectory xµ(τ). In this case, we define a covariant derivative with respect to τ. For a contravariant vector Aµ(τ), we have

′µ ′µ ∂x ν A (τ)= ν A (τ) (1.207) ∂x ν  x (τ) Differentiating with respect to τ, we have

dA′µ(τ) ∂x′µ dA′µ(τ) ∂2x′µ dxλ = + Aν(τ) (1.208) dτ ∂xν dτ ∂xν∂xλ dτ The second term prevents this from being a tensor and is similar to the term that prevented the affine connection from being a tensor. Therefore we define the combination

DAµ dAµ dxλ + Γµ Aν (1.209) Dτ ≡ dτ νλ dτ and it can easily be shown that this combination is a vector, i.e. DA′µ ∂x′µ DAµ = (1.210) Dτ ∂xν Dτ

Similarly, for a covariant vector Bµ DB dB dxν µ µ Γλ B (1.211) Dτ ≡ dτ − µν dτ λ µ and for a mixed tensor Tν we define DT µ dT µ dxλ dxλ ν ν + Γµ T ρ Γσ T µ (1.212) Dτ ≡ dτ λρ dτ ν − λν dτ σ If the tensor defined on a curve is actually a tensor field defined over all space-time, we have that ∂T µ T µ = ν + Γµ T ρ Γσ T µ (1.213) ν;λ ∂xλ λρ ν − λν σ 42 CHAPTER 1. GENERAL RELATIVITY and therefore, clearly, the derivative along the curve can be obtained from the covariant derivative as DT µ dxλ ν = T µ (1.214) Dτ ν;λ dτ Suppose that a vector Aµ carried along a curve does not change as a function of τ when viewed from a locally inertial frame ξµ(xν(τ)). In this frame, then

dAµ = 0 (1.215) dτ µ Γ νλ = 0 (1.216)

Therefore, in this frame DAµ = 0 (1.217) Dτ From the Principle of General Covariance, if this tensor equation is true on a locally inertial frame, it must be true on all frames, and therefore:

DAµ dAµ dxλ + Γµ Aν = 0 Vector defined by Parallel Transport (1.218) Dτ ≡ dτ νλ dτ A vector defined such that the above equation is true, is said to be defined by parallel transport along the curve. This is in fact the case when Aµ = U µ = dxµ/dτ, for which case

dU µ dxλ d2xµ dxλ dxν + Γµ U ν = + Γµ = 0 (1.219) dτ νλ dτ dτ 2 νλ dτ dτ Therefore, a free particle moves on a geodesics and under parallel transport along its trajectory.

1.4 Curvature

We would like to infer the gravitational field equations by applying the Principle of General Co- variance to gravitation itself. in order to do that we must write equations in covariant form that reduce to the Newtonian expectation for weak fields. In order to write such tensor equation we must find out what tensors can be constructed from the metric tensor and its derivatives.

1.4.1 Curvature Tensors Let us first consider tensors that can be formed from the metric and its first derivative. Since we can go to a locally inertial frame where the derivatives vanish, any covariant equation made out of the metric and its derivatives will in fact be a function only of the metric in the locally inertial frame. From the PGC the covariant equation will only depend on the metric. Therefore we must consider tensors that depend on the metric, and its first and second derivative. Since the affine connection is related to the first derivatives of the metric, we may look for tensors that depend on the first derivative of the affine connection. Since

∂xλ ∂x′ρ ∂x′σ ∂xλ ∂2x′τ Γλ = Γτ + (1.220) µν ∂x′τ ∂xµ ∂xν ρσ ∂x′τ ∂xµ∂xν 1.4. CURVATURE 43

Isolating the second derivative, we have

2 ′τ ′τ ′ρ ′σ ∂ x ∂x ∂x ∂x ′ = Γλ Γ τ (1.221) ∂xµ∂xν ∂xλ µν − ∂xµ ∂xν ρσ

Differentiating this equation with respect to xκ

3 ′τ 2 ′τ 2 ′ρ ′σ ′ρ 2 ′σ ∂ x ∂ x ∂ x ∂x ′ ∂x ∂ x ′ = Γλ Γ τ Γ τ ∂xκ∂xµ∂xν ∂xκ∂xλ µν − ∂xκ∂xµ ∂xν ρσ − ∂xµ ∂xκ∂xν ρσ ′ ∂x′τ ∂Γλ ∂x′ρ ∂x′σ ∂x′η ∂Γ τ + µν ρσ (1.221) ∂xλ ∂xκ − ∂xµ ∂xν ∂xκ ∂x′η

Now we replace the second derivaties above by Eq. 1.221, and obtain

3 ′τ ′τ ′ρ ′σ ∂ x ∂x ∂x ∂x ′ = Γλ Γη Γ τ ∂xκ∂xµ∂xν µν ∂xη κλ − ∂xκ ∂xλ ρσ   ′σ ′ρ ′η ′ξ ∂x ′ ∂x ∂x ∂x ′ Γ τ Γη Γ ρ − ∂xν ρσ ∂xη κµ − ∂xκ ∂xµ ηξ   ′ρ ′σ ′η ′ξ ∂x ′ ∂x ∂x ∂x ′ Γ τ Γη Γ σ − ∂xµ ρσ ∂xη κν − ∂xκ ∂xν ηξ  ′  ∂x′τ ∂Γλ ∂x′ρ ∂x′σ ∂x′η ∂Γ τ + µν ρσ (1.219) ∂xλ ∂xκ − ∂xµ ∂xν ∂xκ ∂x′η

Collecting terms, interchanging indices ν κ in this equation, and subtracting from the original ↔ equation, the third derivatives drop out and we the following equation

′τ µ ν κ ′ ∂x ∂x ∂x ∂x R τ = Rλ (1.220) ρση ∂xλ ∂x′ρ ∂x′σ ∂x′η µνκ where we define the Riemann curvature tensor

∂Γλ ∂Γλ Rλ = µν µκ + Γη Γλ Γη Γλ Riemman Tensor (1.221) µνκ ∂xκ − ∂xν µν κη − µκ νη

It can be shown that the Riemann tensor is the unique tensor that can be formed from the metric and its first and second derivatives. We can also form tensors of lower rank by contracting the Riemann Tensor. Contracting the first and third indices, we obtain the Ricci tensor

λκ κ Rµν = g Rλµκν = R µκν Ricci Tensor (1.222)

Contracting the Ricci tensor on its indices we obtain the Ricci scalar

µν µ R = g Rµν = R µ Ricci Scalar (1.223)

It can also be shown that these are the only tensor and scalar that can be formed from the Riemann tensor and the metric. 44 CHAPTER 1. GENERAL RELATIVITY

1.4.2 Parallel Transport on Closed Paths

Consider a vector Sµ which is carried along a curve C by parallell transport, therefore satisfying dS dxν µ = Γλ S . (1.224) dτ µν dτ λ

One can solve this equation to describe the ”trajectory” of the vector Sµ from point 1 to point 2 along the curve. If we set point 1 and point 2 equal, the vector will have returned to its original location. The question is: will it be still the same vector after then round-trip? A detailed calculation shows that the vector will change by 1 ∆S = Rσ S xρdxν (1.225) µ 2 µνρ σ IC σ Therefore, in a flat space-time, in which Rµνρ = 0, the vector would be exactly the same. However, on a space-time with intrinsic curvature, the vector will be different.

1.4.3 Commutation of Covariant Derivatives ν κ If we apply the covariant derivative to a covariant vector Vµ twice (e.g. with respect to x and x in this order) and then reverse the order (i.e. xκ and xν, we find that subtracting these two cases we get

V V = Rσ V (1.226) µ;ν;κ − µ;κ;ν − µνκ σ Therefore, if the Riemann tensor vanishes, covariant derivatives commute (as they should in flat space). That means that we are in fact in a flat space (maybe written in weird coordinates). On the other hand, for a space-time with curvature, the covariant derivatives do not communte.

1.4.4 Properties of Rλµνκ σ From the definition of R µνκ and using

σ Rλµνκ = gλσR µνκ (1.227) one can easily check the following properties of the Riemann tensor: 1) in the block indices 12 and 34:

Rλµνκ = Rνκλµ (1.228)

2) Antisymmetry in the 12 and 34 indices:

R = R = R =+R (1.229) λµνκ − µλνκ − λµκν µλκν 3) Cyclicity in indices 234:

Rλµνκ + Rλκµν + Rλνκµ = 0 (1.230)

λν Since Rµκ = g Rλµνκ, property 1) implies

Rµκ = Rκµ (1.231) 1.5. EINSTEIN EQUATIONS 45

1.4.5 Bianchi Identities Using the definition of the Riemann tensor in terms of the affine connection, and the definition of the affine connection in terms of the metric, one can show that, in a locally inertial frame (in which α Γµν = 0, but not its derivatives) the covariant derivative of the Riemann Tensor is

1 ∂ ∂2g ∂2g ∂2g ∂2g R = λν µν λκ + µκ (1.232) λµνκ;η 2 ∂xη ∂xκ∂xµ − ∂xκ∂xλ − ∂xµ∂xν ∂xν∂xλ   If we permute indices ν, κ and η cyclically, we obtain the Bianchi identities:

Rλµνκ;η + Rλµην;κ + Rλµκη;ν = 0 (1.233)

We can contract indices λ and ν in the above equation to obtain

R R + Rν = 0 (1.234) µκ;η − µη;κ µκη;ν and contract indices once more to finally obtain 1 (Rµ δµR) = 0 (1.235) ν − 2 ν ;µ or, raising the ν index 1 (Rµν gµνR) = 0 (1.236) − 2 ;µ 1.5 Einstein Equations

1.5.1 Derivation (or Informed Guessing) We may now proceed to find the gravitational field equations applying the Principle of General Covariance for the gravity itself. Obviously we can only guess the equations and hope they are correct. The final veredict comes from experiments and observations. We start with the weak static field limit, where we have found that

g = (1 + 2φ) (1.237) 00 − and the Newtonian potential φ is determined from the Poisson equation:

2φ = 4πGρ (1.238) ∇

Since T00 = ρ, we may write this equation as

2g = 8πGT (1.239) ∇ 00 − 00 Because we want a covariant equation written in terms of tensors, we may guess that the correct equation valid for general energy-momentum tensor would be

G = 8πGT (1.240) µν − µν

In order to find the tensor Gµν we impose the following conditions: 46 CHAPTER 1. GENERAL RELATIVITY

1) Gµν is a tensor 2) Gµν consists of only terms containing 2 derivatives of the metric, i.e. either terms linear in the second derivative of gµν or quadratic in the first derivative of gµν. 3) Tµν is symmetric therefore Gµν must be symemtric. µ 4) Tµν is covariantly conserved, i.e. T ν;µ = 0 therefore Gµν must satisfy the same condition, µ i.e. G ν;µ = 0. 5) For weak stationary fields the 00 component must reduce to

G 2φ (1.241) 00 ≈∇

In order to satisfy 1) and 2), we saw that Gµν must be constructed out of contractions of the Riemann tensor. We saw that the only two tensors that can be generated by contractions are the Ricci tensor Rµν and the curvature scalar R. Therefore, we must have

Gµν = C1Rµν + C2gµνR (1.242)

From this we have

α αµ αµ αµ G ν = g Gµν = C1g Rµν + C2g gµνR α α = C1R ν + C2δν R (1.242) and conservation property 4) requires

α α α G ν;α = C1R ν;α + C2δν R;α (1.243)

α α and the Bianchi identity tells us that R ν;α = δν R;α/2, so C Gα = 1 + C R = 0 (1.244) ν;α 2 2 ;ν   This implies C = C /2 and 2 − 1 1 G = C R g R (1.245) µν 1 µν − 2 µν   One can then impose condition 5) to show that C1 = 1 in order for the Newtonian limit to be satisfied. Therefore 1 G = R g R = 8πGT (1.246) µν µν − 2 µν − µν There is a sign convention in the Riemann Tensor that differs from Dodelson, so be careful.

1.5.2 Action Principle The Einstein-Hilbert action in vacuum is given by

S = d4x√ gR. (1.247) EH,vac − Z Varying the action with respect to the metric gµν we have

δS = d4x√ gδR + Rδ√ g. (1.248) EH,vac − − Z 1.5. EINSTEIN EQUATIONS 47

Using Eq. (1.275) we can write

µν µν δR = g δRµν + Rµνδg . (1.249)

From the heuristic expression in Eq. (1.273), we have

δRρ = (δΓρ ) (δΓρ ) , (1.250) µλν ∇λ νµ −∇ν λµ and, therefore, using metric compatibility

gµνδR = gµν (δΓλ ) (δΓλ ) µν ∇λ νµ −∇ν λµ h i = gµν(δΓσ ) gµσ(δΓλ ) . (1.250) ∇σ νµ − λµ h i Since this is a total derivative, it gives no contribution to the variation of the action. Finally, for a general matrix A we have

Tr(ln A) = lndet A. (1.251)

Notice this is obvious for a diagonal matrix and can be shown to be true more generally by diago- nalization with a matrix S such that A′ = S−1AS is diagonal. Variation of Eq. (1.251) yields

− 1 Tr(A 1δA)= δ det A. (1.252) det A µν −1 −1 µλ Applying these results to A = g , so that det A = g and (A δA)νλ = gµνδg we have

µν −1 gµνδg = gδg . (1.253)

Therefore we obtain the useful relation

1 − − δ√ g = ( g 3/2)δg 1 − −2 − 1 = √ gg δgµν . (1.253) −2 − µν Combining these results in the variation of the action we obtain 1 δS = dx4√ g R Rg δgµν , (1.254) EH,vac − µν − 2 µν Z   and for arbitrary variations δgµν, we have 1 δS 1 EH,vac = R Rg = 0 , (1.255) √ g δgµν µν − 2 µν − which we recognize as the Einstein equations in vacuum. Adding to SEH,vac the matter action Sm

S d4x√ g , (1.256) m ≡ − Lm Z and defining the matter energy-momentum tensor as 2 δS T − m , (1.257) µν ≡ √ g δgµν − 48 CHAPTER 1. GENERAL RELATIVITY the variation of the total Einstein-Hilbert action SEH = SEH,vac/(16πG)+ Sm then leads to the standard Einstein equations 1 R Rg = 8πGT . (1.258) µν − 2 µν µν

Defining the Einstein tensor Gµν as 1 G R Rg , (1.259) µν ≡ µν − 2 µν the Einstein equations become

Gµν = 8πGTµν , (1.260) Taking the trace of this equation, we have 1 G = gµνG = gµνR Rgµνg = R 2R = 8πG T (1.261) µν µν − 2 µν − δµν =4 which implies | {z } R = 8πGT (1.262) − Using this result, we may express the Einstein tensor as 1 G R Rg = R + 4πGTg , (1.263) µν ≡ µν − 2 µν µν µν and the Einstein equations can be written as

Rµν + 4πGTgµν = 8πGTµν (1.264) or R = 4πG (2T Tg ) (1.265) µν µν − µν 1.6 Summary

We summarize our main results in General Relativity below, for easy reference later.

In 4-dimensional space, a covariant nth-rank tensor Aαβ...κ is an object which has n indices (α,β,...,γ) and 4n components. A tensor of rank zero is a scalar, a tensor of rank 1 is a vector and a tensor of rank 2 is a matrix. Covariant tensors are defined by their transformation properties under a change of coordinates (x x′ ), where A A′ as α → λ αβ...γ → λµ...ν 4 ′ ∂xα ∂xβ ∂xγ Aλµ...ν = ′ ′ ... ′ Aαβ...γ . (1.266) ∂xλ ∂xµ ∂xν α,β,...,γX =1 Tensors with upper indices Bαβ...γ are denoted contravariant tensors and transform similarly ′ but with inverse derivatives (∂xλ/∂xα, etc.). The metric gµν is a tensor of rank 2 defined by the relation between the coordinate elements dxµ and the line element ds

2 µ ν ds = gµνdx dx . (1.267) 1.6. SUMMARY 49

The metric is symmetric gµν = gνµ and can be used to lower/raise indices in tensors as e.g.

ρ νρ Aµ = gµνA . (1.268) The invariant volume element is a scalar defined as

dV = √ g dx4 , (1.269) − σ where g = det(gµν). We define the Christoffel symbol Γµν or affine connection as 1 Γσ = gσρ(∂ g + ∂ g ∂ g ) . (1.270) µν 2 µ νρ ν ρµ − ρ µν Covariant derivative of tensors are then defined as

Aνγ = ∂ Aνγ + Γν Aλγ + Γγ Aνλ . (1.271) ∇µ µ µλ µλ The connection defined in this way is metric compatible

gµν = g = 0 , (1.271) ∇ρ ∇ρ µν and torsion-free γ γ Γµν = Γνµ . (1.271) The change in vector V ρ under parallel transport is measured by the of two covariant derivatives

[ , ]V ρ = V ρ V ρ . (1.272) ∇µ ∇ν ∇µ∇ν −∇ν∇µ Carrying out the expansion of terms and requiring that it is of the form

[ , ]V ρ Rρ V σ , (1.273) ∇µ ∇ν ∼ σµν motivates one to define the Riemann curvature tensor as

Rρ = ∂ Γρ + Γρ Γλ (µ ν) σµν µ σν µλ σν − ↔ = Γρ Γρ (heuristic) , (1.273) ∇µ σν −∇ν σµ where the last line is a heuristic expression since the connection is not a tensor. The Ricci tensor is obtained by contraction of the first and third indices in the Riemann tensor

ρσ ρ Rµν = g Rρµσν = R µρν , (1.274) and the Ricci scalar is defined by contracting the Ricci tensor

µν µ R = g Rµν = R µ . (1.275) Finally, the Einstein Equations are 1 G R Rg = 8πGT , (1.276) µν ≡ µν − 2 µν µν or equivalently

R = 4πG (2T Tg ) (1.277) µν µν − µν 50 CHAPTER 1. GENERAL RELATIVITY