Compute Gradient in GR: Step-by-Step Guide

[John Greger]

TL;DR Summary

I'm trying to compute the extrinsic curvature. I have the formula and everything I need to plug into the formula. But I get confused when executing this calculation..

I'm trying to compute the extrinsic curvature. I have the formula and everything I need to plug into the formula. But I get confused when executing this calculation..

I have that $ds^2_{interior} = -u(r)dt^2 + (u(r))^{-1} dr^2 + r^2 d\Omega_3^2$. This is a metric describing the interior and exterior of a bubble. The extrinsic curvacture is given by
\begin{equation}
K_{ab} = N_{\mu; \nu} (\frac{\partial x^\mu}{\partial y^a})(\frac{\partial x^\nu}{\partial y^b}) = ( \frac{\partial N_\mu}{\partial x^\nu } - \Gamma^{k}_{\mu \nu} N_{k}) )(\frac{\partial x^\mu}{\partial y^a})(\frac{\partial x^\nu}{\partial y^b}).
\end{equation}

$N_{\mu; \nu}$ is the norm of the bubble which is given by $N_a = (-\dot{R}, \dot{T},0,0,0)$. Since we are looking for dyamical on the brane, we don't care about the angular part of the metric: $(t,r, \Omega) \rightarrow (T(\tau), R(\tau), \Omega)$.

$x^\mu$ labels bulk metric (I think$(\tau, r, \Omega)$) and $y^a$ labels coordinate on the brane (I think $(T(\tau), R(\tau), \Omega)$.

The nonzero christoffel sumbols of the Metric are $\Gamma^r_{rr} = \frac{\partial u / \partial r}{u(r)} = \Gamma^t_{rt}$.

I don't know howto substitute all of this into (1). Should I sum all possible combinations of indices or should I sum the following two combinations $(\mu, \nu) = (\tau, r) ; (a,b)= (T(\tau), R(\tau))$ and $(\mu, \nu) = (r, \tau) ; (a,b)= (R(\tau), T(\tau))$?

If I get some initial help here it will be straight forward to take it from there I think.

P.S the answer should be $$K_{a,b} = -\frac{1}{u \dot{T}}[\ddot{R} + (1/2) \frac{\partial u}{\partial R}] + u(R) \dot{T} R$$ but I cannot really arrive at this. Any help to get this expression is much apprichiated.

 

[martinbn]

$N_{\mu;\nu}$ is the covariant derivative of $N_\mu$.

 

[John Greger]

$N_{\mu;\nu}$ is the covariant derivative of $N_\mu$.

Hi! Many thanks for your answer. Yes indeed, might have abused language. I believe I expanded it in equation 1 the right way. But I feel that I am confused about how to sum the indices accordingly..

 

[pervect]

You sum the indices with the Einstein summation convention, which basically says you sum over all possible values of repeated indices.

a and b appear on the left hand side, so you are computing 16 quantities, as a and b both vary from 0 to 3 (or possibly 1 to 4, depending on your notation).

For each specific value of a and b, you have one component of $K_{ab}$. To get the value of that component, you sum over all values of k, $\mu$, and $\nu$. Thus if you wrote it out longhand, in general each of the 16 components of $K_{ab}$ would be the sum of 64 terms. Hopefully, though, your metric is simple enough that many of the terms are zero.