Obtain Normal Vector of Bubble Wall in Spacetime

[John Greger]

TL;DR Summary

Want to compute the normal vector of a bubble wall embedded in a spacetime with metric.

So say I have a bubble embedded in a spacetime with metric:
$$ds^2 = -dt^2 + a(t) ( dr^2 + r^2 d\Omega^2_2) $$

how do I compute the normal vector if I assume the wall of the bubble the metric represents follows a time-like trajectory, for any $a(t)$?

Since we are interested in dynamical solutions, we let the positions on the brane be $(t,r, \theta, \phi) \rightarrow (T(\tau), R(\tau), \theta, \phi)$.

Here:
https://arxiv.org/pdf/hep-ph/0003173.pdf
in equation 3,4, they have computed the norm for the brane in a similar fashion but for a different metric. They had $ds^2 = -u(r) dt^2 + \frac{1}{u(r)}dr^2 + r^2 d\Omega_3^2$ and got $N_a = (-\dot{R}, \dot{T},0,0,0)$ so I also expect to have a #\dot{T}# in my normal.

 

[martinbn]

Can you give more information, so that we don't have to read the paper. For instance, this

Since we are interested in dynamical solutions, we let the positions on the brane be $(t,r, \theta, \phi) \rightarrow (T(\tau), R(\tau), \theta, \phi)$.

looks like a curve not a surface. Or is this a one parameter family of surfaces, $\tau$ the parameter of the family and $\theta, \phi$ the surface parameters?

 

[John Greger]

Can you give more information, so that we don't have to read the paper. For instance, this

looks like a curve not a surface. Or is this a one parameter family of surfaces, $\tau$ the parameter of the family and $\theta, \phi$ the surface parameters?

Hi!The dot is $\partial_\tau$. The parameters R and T are coordinates on the bubble wall as functions of proper time. Sorry for not giving more context from the paper, I believe it is not very useful, they more or less just state the norm of a bubble wall in a different spacetime background, but I do not understand how they obtained it either. But I believe it could give a hint on the form of the normal vector I am looking for but I am not sure.