Differential geometry and Einstein’s Field Equations Otis Saint - 10584344 Centre for Mathematical Sciences, University of Plymouth

Manifolds, Vectors, and Tensors Curvature Tensors

• is founded upon the idea that space-time is a 4-dimensional , • On a curved manifold, parallel transport of a vector around a closed loop causes the which can curve in the presence of matter. vector to change direction when it is returned to its original starting point. • (Simple definition) A manifold is a type of topological space for which local regions • The change in a vector’s components after parallel transport around an infinitesimal around points, ”look like” a subset of Rn. closed loop can be approximated as I ν • (Hard definition) A manifold is a Hausdorff topological space, M, such that each point β 1 β λ αdx 0 ∆Vk = RλναV x 0 dt (5) m ∈ M is contained in some open subset Uk ⊂ M, called a coordinate patch, for which 2 dt n n there exists a homeomorphism φk : Uk → ⊂ . β Rk R • Rλνα is the , and is defined to be • The tuple (Uk, φk), is called a coordinate chart on M, and allows us to define the notion β β β β τ β τ R = ∂νΓ − ∂αΓ + Γ Γ − Γ Γ (6) of coordinates on M, which we often denote (xµ). λνα λα λν τν λα τα λν n • By contracting the upper index with one of lower indices, we obtain a new tensor of type • One of the simplest examples of a manifold is the n-sphere, S . β (0, 2), called the Ricci tensor, Rλα = Rλβα • Vectors are elements of the tangent space, TpM, and are given as differential operators • Further contraction of the Ricci tensor, by use of the metric inverse then gives the Ricci acting on smooth scalar fields defined on the manifold. The set of partial derivatives, scalar R = gλαR {∂ }, forms a basis of T M λα µ p 1 ∗ ∗ • Using both of the Ricci tensors, we can define the Einstein tensor, Gλα = Rλα − 2gλαR, • Dual vectors are elements of the dual space, Tp M, and are such that w ∈ Tp M : µ ∗ which gives the curvature of space-time a (pseudo-Riemannian manifold) TpM → R. The set of coordinate differentials {dx } forms a basis of Tp M. ∗ • Vectors are such that for V ∈ Tp we have V : Tp → R. ∗ ∗ • A tensor, Ω, of type (m, n) is a multilinear map, Ω : Tp × ... × Tp × Tp × ... × Tp → R | {z } n−times | m−{ztimes } Einstein-Hilbert action and Field Equations

• We can derive Einstein’s field equations by minimising the Einstein-Hilbert action Z √ 4 S[gµν] = R −gd x (7) V • V is is the entire space-time volume. • By varying this action it an be shown that Z Z βσ √ √ 4 βσ√ 4 δS = δg Rβσ −g + R δ −g d ξ + g −g δRβσ d ξ . (8) V V • Since for some matrix, B, we have δ ln(det(B)) = (δ ln(B)), it follows that the variation of the metric determinant is given by √ 1√ δ −g = − −g g δgβσ (9) 2 βσ Fig. 1: The tangent space of a 2-sphere • Explicitely, the Ricci tensor is defined to be α α α α ρ α ρ Rβσ = Rβασ = ∂αΓβσ − ∂σΓβα + ΓαρΓβσ − ΓσρΓβα . (10) • Varying the Ricci tensor and comparing the result with the following covariant derivatives The Metric Tensor and Levi-Civita Connection α α α ρ ρ α ρ α ∇α(δΓβσ) = ∂αδΓβσ + ΓαρδΓβσ − ΓβαδΓσρ − ΓασδΓβρ , (11) α α α ρ ρ α ρ α • The metric tensor, gµν, furnishes us with a way to measure distances through integration. ∇σ(δΓβα) = ∂σδΓβα + ΓσρδΓβα − ΓβσδΓαρ − ΓσαδΓβρ . (12) µ ν • Using the metric tensor, we write the inner product of two vectors as hv, wi = gµνv w , then leads to the Palatini identity for the variation of Rβσ µ µ where gµν = h∂µ, ∂νi, and v = v ∂µ, w = w ∂µ ∈ TpM. α α δRβσ =∇α(δΓβσ) − ∇σ(δΓβα) (13) • The line element (or metric) on a manifold endowed with a metric tensor is thus given 2 µ ν • Using (9) and (13) the variation of the Einstein-Hilbert action now becomes by ds = gµνdx dx . Z Z βσ√ 1 4 βσ√ α α 4 • Solutions to Einstein’s equations take the form of a metric. δS = = δg −g(Rβσ − gβσR) d ξ + g −g (∇α(δΓβσ) − ∇σ(δΓβα)) d ξ V 2 V • We define the of an arbitrary tensor of type (m, n) as (14) ∇ T µ1,...,µm = ∂ T µ1,...,µm + Γµ1 T α,...,µm + Γµ2 T µ1,α,...,µm + ... β ν1,...,νn β ν1,...,νn βα ν1,...,νn βα ν1,...,νn • Using a generalised version of Stoke’s theorem, it can be seen that the second integral −Γα T µ1,...,µm − Γα T µ1,...,µm − ... (1) βν1 α,...,νn βν2 ν1,α,...,νn here contributes a boundary term, which, due to space-time being asymptotically flat, µ must vanish! • The symbols Γαβ, called the connection coefficients, are given to represent the magni- tude of change in the direction of the basis vector ∂µ, of the basis vector ∂β, due to a • The principle of least action, δS = 0, then leads to Einstein’s field equations in change in the xα coordinate. vacuum 1 • In general relativity, the connection is given by the Levi-Civita connection, the cor- R − g R = 0 (15) βσ 2 βσ responding connection coefficients are often called the Christoffel symbols, defined to We can also write the Einstein tensor in terms of the functional variation as be δS 1 1 √ = R − g R (16) Γα = gαβ(∂ g + ∂ g − ∂ g ) (2) −gδgβσ βσ 2 βσ µν 2 µ νβ ν βµ β µν α α • To find the complete field equations, we need to include a new term in the action to The Christoffel symbols are such that ∇βgµν = 0 (metric compatibility) and Γµν = Γνµ (torsion free). account for mass and energy. The new action can be written as 1 S = S + S (17) 2κ EH M • Varying this action with respect to gβσ and invoking (16) then gives Parallel Transport, Geodesics and Newton’s First Law δS 1 1 1 δS √ = (R − g R) + √ M (18) δgβσ −g 2κ βσ 2 βσ −g δgβσ • Given a curve γ(t) = (xµ(λ)), defined on a manifold M, we can parallel transport a vector, V σ, along γ using the equation • Extremising this equation using δS = 0 then gives µ 1 σ dx σ Rβσ − gβσR = κTβσ . (19) DtV = ∇µV = 0 (3) 2 dλ • The energy momentum tensor Tβσ is defined to be • Parallel transport allows us to compare vectors living in different tangent spaces. n 2 δSM • In , parallel transport is path independent. Tβσ = −√ . (20) R −g δgβσ • A curve which has the property of parallel transporting its own tangent vector is called Finally, inserting κ = 8πG (which can be found using the Newtonian limit) leaves us a Geodesic. It can be shown directly from the above equation, that geodesics satisfy • c4 with the full field equations d2xρ dxσ dxν + Γρ = 0 (4) 1 8πG dλ2 σν dλ dλ R − Rg = T (21) βσ 2 βσ c4 βσ • A geodesic is a length minimising curve between 2 points on a manifold. • In curved space-time, Newton’s first law is generalised to: An object in free fall will move along geodesics in space-time.

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