p. 21 in this paper (http://arxiv.org/abs/0704.0247)
V is Killing vector, where V2=−4bˉb, which means it is timelike Killing vector.
The authors say:
From V2=−4|b|2 and V=∂t as a vector we get Vt=−4|b|2,
My question here is how did the authors set Vt equal to this value?
They add,
From V2=−4|b|2 and V=∂t as a vector we get Vt=−4|b|2, so that V = −4|b|^2(dt+σ) as a one-form, with σ_t = 0.
Why did they assume that?
Answer
My question here is how did the authors set V_t equal to this value?
On page 21 the authors say: "Let us choose coordinates (t, z, x_i) such that V = \partial_t and i = 1, 2."
So they chose the coordinates such that V=\partial_t, which means V^t=1. Note that the other components of V^\mu are zeros. Next we have V^2=V_\mu V^\mu=-4|b|^2=V_t V^t+V_{x_1}V^{x_1}+V_{x_2}V^{x_2}+V_{z}V^{z}=V_t*1, from which you find V_t=-4|b|^2. Here we used V^{x_i}=V^z=0.
Though what I don't get is their requirement that σt be equal to zero and why did they place a dt next to it. Why did they assume that?
\sigma is a general one form on coordinates x^i, z, which means \sigma=\sigma_{1}dx^1+\sigma_2 dx^2+\sigma_3 dz, note that later they use the gauge freedom to set \sigma_z=0. They chose the coordinates to fix V_t and the rest it is the most general one form on x^i. For example, the most general one form on coordinates t,x^1, x^2,z is \alpha=\alpha_0 dt+\alpha_1 dx^1+\alpha_2 dx^2+\alpha_3 dz. Compare it to their expression for V (after (4.7)) and you will see that they chose only the first components, the rest is arbitrary.
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