1) Rotating System . . . [7P]

University of Regensburg SS 17

General Relativity and Cosmology

Prof. John Schliemann Mo. H34 12pm c.t. &Wed. PHY 9.2.01, 1pm c.t.

Dr. Paul Wenk Thu. 1pm c.t., PHY 9.1.10

Sheet 5

1) Rotating System . . . [7P]

Write down the equations of motion

d²x^κ

dτ² = −Γ^κ_µνdx^µ dτ

dx^ν dτ

for a rotating system as described in Ex. 2 of sheet 4. By dropping all terms of the order of O(v²/c²) and higher, identify the Coriolis and centrifugal force.

2) Covariant Derivative . . . [4P]

Calculate the covariant derivative of the field tensorA^αβ_νγδ.

3) Lie Derivative . . . [10P]

Consider a coordinate transformation x^µ7→x^0µ=x^µ−^µ. Thus, we have

A^0µ(x) =A^µ(x)−A^ν(x)∂_ν^µ+^ν∂_νA^µ(x) +O(²). (1) Now, theLie derivative (Sophus Lie 1842-1899) measures the changeA^0µ−A^µ. More general: It evaluates the change of a tensor field, along the flow of another vector field. It is defined for contravariant components as (see lecture notes)

LA^µ:=A^0µ−A^µ+O(²) (2)

=^ν∇νA^µ−A^ν∇ν^µ+O(²)≡^νA^µ;ν−A^νµ;ν+O(²), (3)

and LAµ:=A⁰µ+Aµ+O(²) (4)

=^ν∇_νA_µ+A^ν∇_µν+O(²)≡^νA_µ;ν+A^ν_ν;µ+O(²). (5) (a) If the Lie-derivative of a metric tensorg_αβ along the direction of a vector-field ξvanishes,L_ξg_αβ= 0, ξ

is called Killing vector.

(i) Show thatgαβ,µξ^µ+gνβξ^ν_,α+gανξ_,β^ν = 0 is equivalent toξα;β+ξβ;α= 0.

Hint: ξ defines a symmetry if an infinitesimal translation along ξleaves ds² invariant: δ(ds²) = 0.

Assume ξto be a tangent vector to a curve x^α(λ),ξ^α =dx^α/dλ. Use (and show) δ(dx^α) =dξ^αδλ andδg_αβ=g_αβ,µξ^µδλ.

(ii) Calculate the Killing vectors of a spherical surface with radius equal to one,ds²=dθ²+ sin²(θ)dϕ². Hint: You’ll find three independent parameters allowing for the definition of corresponding Killing vectors.

(iii) Calculate the Killing vectors of Minkowski spacetime. Show that the general solutions are given by ξ_µ=c_µ+b_µγx^γ with constants c_µ,b_µγ. Furthermore, show that this flat spacetime has ten linearly independent Killing vector fields (corresponding to the ten generators of the Poincar´e algebra).

(b) Show that if ξµ(x) is a Killing vector field andv^η(λ) is a tangent vector to a geodesic curve m(λ) then v^ηξη(x) is constant alongm.