Chapter 6 Curved spacetime and General Relativity 6.1 Manifolds, tangent spaces and local inertial frames A manifold is a continuous space whose points can be assigned coordinates, the number of coordinates being the dimension of the manifold [ for example a surface of a sphere is 2D, spacetime is 4D ]. A manifold is differentiable if we can define a scalar field φ at each point which can be differentiated everywhere. This is always true in Special Relativity and General Relativity. We can then define one - forms d˜φ as having components φ ∂φ and { ,α ≡ ∂xα } vectors V as linear functions which take d˜φ into the derivative of φ along a curve with tangent V: α dφ V d˜φ = Vφ = φ V = . (6.1) ∇ ,α dλ ! " Tensors can then be defined as maps from one - forms and vectors into the reals [ see chapter 3 ]. A Riemannian manifold is a differentiable manifold with a symmetric metric tensor g at each point such that g (V, V) > 0 (6.2) for any vector V, for example Euclidian 3D space. 65 CHAPTER 6. CURVED SPACETIME AND GR 66 If however g (V, V) is of indefinite sign as it is in Special and General Relativity it is called Pseudo - Riemannian. For a general spacetime with coordinates xα , the interval between two neigh- { } boring points is 2 α β ds = gαβdx dx . (6.3) In Special Relativity we can choose Minkowski coordinates such that gαβ = ηαβ everywhere. This will not be true for a general curved manifold. Since gαβ is a symmetric matrix, we can always choose a coordinate system at each point x0 in which it is transformed to the diagonal Minkowski form, i.e. there is a transformation ∂xα¯ Λα¯ (6.4) β ≡ ∂xβ such that 1 0 0 0 −0 1 0 0 g ¯(x ) = η ¯ = . (6.5) α¯β 0 α¯β 0 0 1 0 0 0 0 1 Note that the sum of the diagonal elements is conserved; this is the signature of the metric [ +2 ]. α¯ In general Λ β will not diagonalize gαβ at every point since there are ten func- α¯ β tions gαβ(x) and only four transformation functions X (x ). α¯ We can also choose Λ β so that the first derivatives of the metric vanishes at x0 i.e. ∂g αβ = 0 (6.6) ∂xγ for all α, β and γ. This implies g (xµ) = η + [(xµ)2] . (6.7) αβ αβ O That is, the metric near x0 is approximately that of Special Relativity, differences being of second order in the coordinates. This corresponds to the local inertial frame whose existence was deduced from the equivalence principle. In summary we can define a local inertial frame to be one where gαβ(x0) = ηαβ (6.8) CHAPTER 6. CURVED SPACETIME AND GR 67 for all α, β, and gαβ,γ(x0) = 0 (6.9) for all α, β, γ. However g (x ) = 0 (6.10) αβ,γµ 0 $ for at least some values of α. β, γ and µ. It reflects the fact that any curved space has a flat tangent space at every point, although these tangent spaces cannot be meshed together into a global flat space. Recall that straight lines in a flat spacetime are the worldlines of free particles; the absence of first derivative terms in the metric of a curved spacetime will mean that free particles are moving on lines that are locally straight in that coordinate system. This makes such coordinates very useful for us, since the equations of physics will be nearly as simple as they are in flat spacetime, and if they are tensor equations they will be valid in every coordinate system. 6.2 Covariant derivatives and Christoffel sym- bols In Minkowski spacetime with Minkowski coordinates (t, x, y, z) the derivative of a α vector V = V eα is just ∂V ∂V α = e , (6.11) ∂xβ ∂xβ α since the basis vectors do not vary. In a general spacetime with arbitrary coordi- nates, eα which vary from point to point so ∂V ∂V α ∂e = e + V α α . (6.12) ∂xβ ∂xβ α ∂xβ β Since ∂eα/∂x is itself a vector for a given β it can be written as a linear combi- nation of the bases vectors: ∂e α = Γµ e . (6.13) ∂xβ αβ µ The Γ’s are called Christoffel symbols [ or the metric connection ]. Thus we have: ∂V ∂V α = e + V αΓµ e , (6.14) ∂xβ ∂xβ α αβ µ CHAPTER 6. CURVED SPACETIME AND GR 68 so α ∂V ∂V µ α = eα + V Γ µβ eα . (6.15) ∂xβ ) ∂xβ * Thus we can write ∂V = V α e , (6.16) ∂xβ ;β α where α α µ α V ;β = V ,β + V Γ µβ . (6.17) α Let us now prove that V ;β are the components of a 1/1 tensor. Remember in α section 3.5 we found that V ,β was only a tensor under Poincar´e transformations α in Minkowski space with Minkowski coordinates. V ;β is the natural generalization for a general coordinate transformation. α¯ Writing Λα¯ ∂x , we have: β ≡ ∂xβ α¯ α¯ µ¯ α¯ V ;β¯ = V ,β¯ + V Γ µ¯β¯ µ ∂ α¯ ν µ¯ α¯ = Λ ¯ (Λ V ) + V Γ ¯ β ∂xµ ν µ¯β ν α¯ µ α¯ ∂V ν µ ∂Λ ν µ¯ α¯ = Λ ¯Λ + V Λ ¯ + V Γ ¯ . (6.18) β ν ∂xµ β ∂xµ µ¯β Now ∂eµ¯ λ¯ = Γ ¯e¯ , (6.19) ∂xβ¯ µ¯β λ therefore ∂eµ¯ α¯ λ¯ α¯ λ¯ α¯ α¯ w˜ = Γ ¯e¯w˜ = Γ ¯δ ¯ = Γ ¯ , (6.20) ∂xβ¯ µ¯β λ µ¯β λ µ¯β so we obtain: ν α¯ α¯ µ α¯ ∂V ν µ ∂Λ ν µ¯ ∂eµ¯ α¯ V ¯ = Λ ¯Λ + V Λ ¯ + V w˜ ;β β ν ∂xµ β ∂xµ ∂xβ¯ ν α¯ µ α¯ ∂V ν µ ∂Λ ν µ¯ γ α¯ δ ∂eµ¯ = Λ ¯Λ + V Λ ¯ + Λ V Λ w˜ β ν ∂xµ β ∂xµ γ δ ∂xβ¯ ν α¯ µ α¯ ∂V ν µ ∂Λ ν µ¯ γ α¯ δ ) ∂ ν = Λ ¯Λ + V Λ ¯ + Λ V Λ w˜ Λ ¯ (Λ e ) β ν ∂xµ β ∂xµ γ δ β ∂x) µ¯ ν ν α¯ µ α¯ ∂V ν µ ∂Λ ν µ¯ γ α¯ δ ) ν ∂eν = Λ ¯Λ + V Λ ¯ + Λ V Λ w˜ Λ ¯Λ β ν ∂xµ β ∂xµ γ δ β µ¯ ∂x) ν µ¯ γ α¯ δ ) ∂Λ µ¯ + Λ V Λ w˜ Λ ¯e (6.21) γ δ β ν ∂x) CHAPTER 6. CURVED SPACETIME AND GR 69 ν α¯ Now using Λν Λµ¯ = δν , w˜ δe = δδ and Λα¯ ∂Λ µ¯ = Λν ∂Λ ν we obtain: µ¯ γ γ ν ν ν ∂x$ − µ¯ ∂x$ ν α¯ α¯ α¯ α¯ µ ∂V ν µ ∂Λ ν α¯ ) ν δ µ ∂Λ ν ν V ¯ = Λ Λ ¯ + V Λ ¯ + Λ Λ ¯V Γ Λ ¯ V ;β ν β ∂xµ β ∂xµ δ β ν) − β ∂xµ ν α¯ µ ∂V α¯ ) ν δ = Λ Λ ¯ + Λ Λ ¯V Γ ν β ∂xµ δ β ν) ν α¯ µ ∂V δ ν = Λ νΛ β¯ + V Γ δµ , (6.22) ) ∂xµ * so α¯ α¯ µ ν V ;β¯ = Λ νΛ β¯V ;µ . (6.23) α We have shown that V ;β are indeed the components of a 1/1 tensor. We write this tensor as V = V α e w˜ β . (6.24) ∇ ;β α ⊗ It is called the covariant derivative of V. Using a Cartesian basis, the components α are just V ,β, but this is not true in general; however for a scalar φ we have: ∂φ φ φ = , φ = d˜φ , (6.25) ∇α ≡ ;α ∂xα ∇ since scalars do not depend on basis vectors. α ∂eµ α Writing Γ µβ = ∂xβ w˜ , we can find the transformation law for the components of the Christoffel symbols. α ∂eµ α α γ¯ σ¯ ∂ λ¯ Γ = w˜ = Λ w˜ Λ Λ e¯ µβ ∂xβ γ¯ β ∂xσ¯ µ λ ! " λ¯ α σ¯ λ¯ γ¯ ∂eλ¯ α σ¯ γ¯ ∂Λ µ = Λ Λ Λ w˜ + Λ Λ w˜ e¯ γ¯ β µ ∂xσ¯ γ¯ β λ ∂xσ¯ λ¯ α σ¯ λ¯ γ¯ α σ¯ ∂Λ µ = Λ Λ Λ Γ ¯ + Λ ¯Λ (6.26) γ¯ β µ λσ¯ λ β ∂xσ¯ This is just α σ¯ λ¯ α 2 λ¯ α ∂x ∂x ∂x γ¯ ∂x ∂ x Γ = Γ ¯ + . (6.27) µβ ∂xγ¯ ∂xβ ∂xµ λσ¯ ∂xλ ∂xβ ∂xµ We can calculate the covariant derivative of a one - form p˜ by using the fact that p˜(V) is a scalar for any vector V: α φ = pαV . (6.28) CHAPTER 6. CURVED SPACETIME AND GR 70 We have ∂p ∂V α φ = φ = α V α + p ∇β ,β ∂xβ α ∂xβ ∂p = α V α + p V α p V µΓα ∂xβ α ;β − α µβ ∂pα µ α α = pµΓ αβ V + pαV ;β . (6.29) )∂xβ − * Since φ and V α are tensors, the term in the parenthesis is a tensor with ∇β ;β components: p = p p Γµ . (6.30) α;β α,β − µ αβ We can extend this argument to show that T T = T T Γα Γ Γα , ∇β µν ≡ µν;β µν,β − αν µβ − µα νβ T µν T µν = T µν + T αν Γµ + T µαΓν , ∇β ≡ ;β ;β αβ αβ T µ T µ = T µ + T α Γµ T µ Γα . (6.31) ∇β ν ≡ ν;β ν,β ν αβ − α νβ α 6.3 Calculating Γ βγ from the metric α Since V ;β is a tensor we can lower the index α using the metric tensor: µ Vα;β = gαµV ;β .