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- Homework Statement
- Given an AdS-Schwarzschild metric in ##(t, z, x, x_i)## coordinates, embed a surface (actually it is a null hypersurface) given by the constraint ##dV = 0## (##S = -t+x ##) using the lightcone coordinates. What is the normal vector along this surface, i.e. along the ##U##-direction?

- Relevant Equations
- AdS-Schwarzschild metric:

##ds^2 = \frac{1}{z^2} \left( -f(z) dt^2 + \frac{dz^2}{f(z)} + dx^2 +\sum_{i=1}^{d-1} (dx_i)^2 \right), \qquad f(z) = 1-\left(\frac{z}{z_h}\right)^{d+1}##

Lightcone coordinates:

##dU = dt + dx##

##dV = dt - dx##

Metric in lightcone coordinates

##ds^2 = \frac{1}{z^2} \left[ \frac{z^{d+1}}{z_h^{d+1}} \cdot \frac{dU^2 + dV^2}{4} + \left( -2 + \frac{z^{d+1}}{z_h^{d+1}} \right) \frac{dUdV}{2} + \frac{dz^2}{f(z)} + \sum_{i=1}^{d-1} (dx_i)^2 \right]##

Surface in lightcone coordinates:

##ds^2 = \frac{1}{z^2} \left[ \frac{z^{d+1}}{4 z_h^{d+1}} \cdot dU^2 + \frac{dz^2}{f(z)} + \sum_{i=1}^{d-1} (dx_i)^2 \right]##

Surface:

S = -t + x

I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##.

The normal vector is given by,

##n^\mu = g^{\mu\nu} \partial_\nu S ##

How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##?

Also, after calculating ##n^\mu## in terms of ##U##, how do I transform it back in terms of ##t## and ##x##?

The normal vector is given by,

##n^\mu = g^{\mu\nu} \partial_\nu S ##

How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##?

Also, after calculating ##n^\mu## in terms of ##U##, how do I transform it back in terms of ##t## and ##x##?