Ludwig Maximilian University of Munich (LMU) General Relativity, TC1, Prof. Dr. V. Mukhanov, WS 2014/15 Instructors: • Dr. Alex Vikman (Tuesday, 2-4 pm, Theresienstr. 37, A 449, Wednesday 2-4 pm, Theresienstr. 37, A 450), • Dr. Ted Erler (Thursday 2-4 pm, Theresienstr. 37, A 450), • Dr. Paul Hunt (Thursday 4-6 pm, Theresienstr. 37, A 450) https://www.physik.uni-muenchen.de/lehre/vorlesungen/wise_14_15/TC1_-General-Relativity/index.html Problem Sheet 4 (discussion 10.11-14.11.2014) 4.1 Locally Inertial Reference Frame α has the value (0) Γα and is symmetric (0) Γα = Suppose the Christoffel symbol1 at a point x(0) µν µν α can be constructed by defining new coordinates ξ α such that locally inertial system at point x(0) α ξα (x) = xα − x(0) + (0) Γα νµ .A 1 (0) α µ µ ν . Γµν x − x(0) xν − x(0) 2 Prove explicitly that the Christoffel symbol for the new coordinates vanishes at the origin ξα = 0 correα . sponding to the point x(0) 4.2 Geodesics a) Show that the geodesic equation can be written in the following form d uα 1 ∂ gβγ β γ − u u = 0. ds 2 ∂xα b) What happens, if the metric does not depend on a coordinate, say on x1 ? c) Show that gαβ uα uβ is a constant along the geodesic. 4.3 Minimal Length and Maximal Proper Time Derive the geodesic equation from the requirement that a spacelike geodesic is a curve of extremal (usually minimal) spacial distance between two points, while a timelike geodesic corresponds to the extremal (usually maximal) proper time between events. 1 We use the term ‘Christoffel symbol’ for the components of any connection, not only for the components of the Levi-Civita connection as it is done in some textbooks. 4.4 Commutator of Covariant Derivatives Show that uα;β;γ − uα;γ;β = Rαδγβ uδ , where the Riemann tensor is defined by α R δγβ = ∂ Γαδβ ∂xγ − ∂ Γαδγ ∂xβ + Γασγ Γσδβ − Γασβ Γσδγ . 4.5 Parallel Transport and Riemann Tensor Consider a vector Aα parallel-transported along a small closed curve xµ (s) . Show that the change in Aα after the parallel transport can be approximately expressed as 1 β xβ dxγ , δAα ≡ Γαγ (x)Aβ dxγ ≈ Rδαβγ Aδ 2 where it is assumed that the area within the closed curve is very small. Hint: Use a locally inertial coordinate system where Γαβγ = 0 at one point. Also, show that α β x dx = − xβ dxα . 4.6 Riemann Tensor a) Using the symmetry properties of the Riemann tensor Rαβγδ , compute the number of independent components of Rαβγδ in an n -dimensional space (n ≥ 2 ). b) Prove the Bianchi identity Rαβγδ;σ + Rαβσγ;δ + Rαβδσ;γ = 0. c) Compute the Einstein tensor Gαβ in an arbitrary two-dimensional space. Hint: First determine the independent components of Rαβγδ . 4.7 A Useful Operator Consider an operator which depends on two vectors a and b Oˆ (a, b) = [∇ a , ∇ b ] − ∇[a,b] , where ∇ is the covariant derivative / Levi-Civita connection, so that ∇ a = aµ ∇µ , [∇ a , ∇ b ] = ∇ a ∇ b − ∇ b ∇ a and [a, b] = ab − ba. Show that this operator defines a tensor. Find components of this tensor. 2