MATH0025
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MATH0025
Answer all questions .
1. The setting of this question is an n-dimensional manifold M equipped with metric tensor gij and covariant derivative 5a . We work in a local coordinate system
Xi . Let C be a curve written as Xi (λ) and let its tangent vector be Ti .
(a) Consider a vector Va evaluated along the curve C and define
∆Va := Va (Xi (λ + δλ)) * Va (Xi (λ))
where δλ / 1. Write ∆Va in terms of the curve’s tangent vector, δλ and the derivatives of Va with respect to the local coordinates.
(b) Introduce the object
Va := Ti ∂i Va .
Write this quantity in terms of ∆Va and δλ .
(c) When discussing the geodesic deviation equation we introduced the nota- tion
Va := T 5ii Va .
Write the equation for parallel transport using this notation and also write the geodesic (or autoparallel) equation in this notation.
(d) Motivate the introduction of the connection Γj(i)k using the object ∆Va .
(e) Let Ui be a vector with length e = Us Us , (e 0) which is parallelly transported along the curve C . Find a condition on gij such that e is preserved under parallel transport. (25 marks)
2. The setting of this question is the 2-dimensional manifold M with line element ds2 = gij dXi dXj = cosh(y)╱dx2 + x2 dy2 、+ x sinh(y) dx dy ,
where Xi = {x, y} and i = 1, 2.
(a) Using any suitable method, find the non-vanishing Christoffel symbol com- ponents.
(b) Show that the geodesic equations can be brought into the form
+ xy˙2 = 0 ,
y¨ + x˙ y˙ = 0 .
(c) Show that x2 y˙ is a conserved quantity along the geodesics.
(d) By solving the geodesic equations, or otherwise, find an improved coordi- nate system to show that this is a flat 2-dimensional manifold. (25 marks)
3. The setting of this question is a 4-dimensional Lorentzian manifold M as it is used in General Relativity. Consider the perfect fluid energy-momentum tensor given by
Tij = (ρ + p)ui uj + pgij ,
where ρ is the energy density, p is the isotropic pressure and ui is the 4-velocity which satisfies ui ui = *1.
(a) Define the so-called projection hij := gij + ui uj . Show that
hj(i)hk(j) = hk(i) and hj(i)uj = 0 ,
and also
Tij = ρui uj + phij .
Justify why hj(i) is called a projection.
(b) Show that ui 5j ui = 0.
(c) Show that uj 5i Tij = 0 is equivalent to
u 5ii ρ + (ρ + p)5i ui = 0 .
(d) Next, show that hj(k)5i Tij = 0 is equivalent to
(*)
hik 5ip + (ρ + p)(hj(k)ui 5i uj ) = 0 .
(e) Assuming weak gravitational fields and small velocities (compared to the speed of light) rewrite equation (*) and interpret your result. (25 marks)
4. The line element of a Schwarzschild black hole in an external magnetic field is given by
ds2 = *A2 (1 * 2m/r)dt2 + A2 (1 * 2m/r) − 1 dr2 + A2 r2 dθ 2 + A −2 r2 sin2θdφ2 , where the function A = A(r, θ) is given by
A = 1 + B0(2)r2 sin2θ .
The magnetic field is denoted by B0 . When discussing geodesics θ = π/2 may be assumed without further justifications .
(a) Determine the possible location(s) of singularities, coordinate singularities and horizons and compare the results with the usual Schwarzschild solution. (b) Show that the geodesic equations give rise to two conserved quantities.
State the relevant equations and give a physical interpretation.
(c) Show that the geodesic equation for radial null geodesics can be written as
dr E
Explain the meaning of the two signs and determine the function a(r).
(d) Integrate the geodesic equation for radial null geodesics and show that r = 2m can be reached for a finite parameter value λ* . (An explicit expression for λ* does not have to be derived.)
(e) Show that the geodesic equations can be written in the form of an ‘energy’ equation
r˙2 + ╱ 1 * 、f (r) = .
Find the explicit form of the function f (r). (25 marks)
2023-04-24