I am having trouble understanding how the following statement (taken from some old notes) is true:
For a 2 dimensional space such that ds2=1u2(−du2+dv2)
the timelike geodesics are given by u2=v2+av+bwhere a,b are constants.
When I see "geodesics" I jump to the Euler-Lagrange equations. They give me ddλ(−2˙uu2)=(−˙u2+˙v2)(−2u3)⟹¨uu2−2˙u2u3=1u3(−˙u2+˙v2)⟹u¨u−˙u2−˙v2=0
Timelike implies ˙xa˙xa=−1
I can't for the life of me see how the statement results from these. Would someone mind explaining? Thanks.
Answer
I prefer to use Killing vectors and conservation laws to solve stuff like this, so let's analyze the problem using Killing vectors, and see if the results agree with your Euler-Lagrange equations.
Notice that the metric is invariant under translations of v. The associated killing vector is ∂v which in turn gives the following conserved quantity: cv=˙x⋅∂v=˙vu2
Warning; there may be simpler ways of showing what you want to show.
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