Normal vector of an embedding surface

shinobi20
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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##?
 
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To calculate the normal vector in terms of U, we first need to rewrite the surface in terms of U. We can do this by writing S as a function of U:##S = S(U)##Now we can calculate the normal vector by taking the partial derivative of S with respect to U:##n^\mu = g^{\mu\nu} \partial_\nu S = g^{\mu\nu} \partial_\nu S(U)##To transform this back into terms of t and x, we can use the chain rule and the definition of U:##U = (t - f(x))##We can then rewrite the normal vector as:##n^\mu = g^{\mu\nu} \frac{\partial S}{\partial U} \frac{\partial U}{\partial t} \frac{\partial t}{\partial x} + g^{\mu\nu} \frac{\partial S}{\partial U} \frac{\partial U}{\partial x} = g^{\mu\nu} \frac{\partial S}{\partial U} (-1) \frac{\partial f}{\partial x} + g^{\mu\nu} \frac{\partial S}{\partial U} \frac{\partial U}{\partial x} ##Now that the normal vector is expressed in terms of t and x, we can use it to calculate other quantities such as the area of the surface or the mean curvature.
 
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