Tuesday, 28 June 2016

homework and exercises - Killing vector and one-form



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.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...