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The third part of the university of cambridge mathematical tripos exam held on june 3, 2003. The exam covers topics in general relativity and electromagnetism. Students were required to answer three questions out of four, each carrying equal weight. Questions on the lie derivative, killing vectors, gauge-invariant linearized vacuum perturbations of flat minkowski spacetimes, and the electromagnetic field equation. It also includes instructions for the essay question.
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Tuesday 3 June 2003 1.30 to 4.
Attempt THREE questions.
There are four questions in total. The questions carry equal weight.
Information
The signature is ( + − − − ), all connections are symmetric, and the curvature tensor conventions are defined by
Rikmn = Γikm,n − Γikn,m − ΓipmΓpkn + ΓipnΓpkm.
1 The Lie derivative with respect to a vector field X of a scalar function f and of a vector field Y are defined to be
LX f = X(f ), LX Y = [X, Y ],
where [X, Y ] is the commutator of X and Y. Write out these definitions in suffix notation and extend the definition of the Lie derivative to cover covector and
1
tensor fields. Explain briefly how you would extend the definition to
(m n
tensor fields.
Show that LX LY − LY LX = L[X,Y ].
Explain what is meant by a Killing vector. Show that a linear combination of Killing vectors with constant coefficients is a Killing vector. Show also that the commutator of two Killing vectors is a Killing vector.
The metric for an axially symmetric rotating star admits precisely two linearly independent Killing vectors T and Φ. Far from the star where fields are weak, and using a standard cylindrical polar chart (t, r, z, ϕ) we have
∂t
∂ϕ
, as (r^2 + z^2 ) → ∞.
Show that T and Φ commute.
[You may quote the Jacobi identity.]
2 Write an essay on gauge-invariant linearized vacuum perturbations of flat Minkowski spacetimes.
[You may use any information from the lecture handout included with this examination paper.]
Paper 54
