Compute Gradient in GR: Step-by-Step Guide

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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.
 
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##N_{\mu;\nu}## is the covariant derivative of ##N_\mu##.
 
  • #3
martinbn said:
##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..
 
  • #4
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.
 
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