Normal vector of an embedding surface

[shinobi20]

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 $x$?

 

[Nate810]

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.