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Chapter 3: Relativity 15 2. The principle of equivalence : inertial mass gravitational mass gravitation is equivalent with a curved space-time were particles move along geodesics. 3. By a proper choice of the coordinate system it is possible to make the metric locally flat in each point x i : g αβ ( x i ) = η αβ := diag ( - 1 , 1 , 1 , 1) . The Riemann tensor is defined as: R μ ναβ T ν := α β T μ - ∇ β α T μ , where the covariant derivative is given by j a i = j a i + Γ i jk a k and j a i = j a i - Γ k ij a k . Here, Γ i jk = g il 2 g lj x k + g lk x j - g j k x l , for Euclidean spaces this reduces to: Γ i jk = 2 ¯ x l x j x k x i ¯ x l , are the Christoffel symbols . For a second-order tensor holds: [ α , β ] T μ ν = R μ σαβ T σ ν + R σ ναβ T μ σ , k a i j = k a i j - Γ l kj a i l + Γ i kl a l j , k a ij = k a ij - Γ l ki a lj - Γ l kj a jl and k a ij = k a ij + Γ i kl a lj + Γ j kl a il . The following holds: R α β μ ν =
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