# Normal vector of an embedding surface

• shinobi20
In summary, to calculate the normal vector in terms of U, we first need to rewrite the surface in terms of U and then use the chain rule to transform it back into terms of t and x.
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
##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##?

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.

## 1. What is a normal vector of an embedding surface?

A normal vector of an embedding surface is a vector that is perpendicular to the surface at a specific point. It is used to describe the orientation of the surface and is an important concept in the field of differential geometry.

## 2. How is the normal vector of an embedding surface calculated?

The normal vector of an embedding surface can be calculated using the gradient of the surface's equation at a specific point. It can also be calculated using the cross product of the tangent vectors at that point.

## 3. What is the significance of the normal vector in surface embedding?

The normal vector is significant in surface embedding because it helps determine the curvature and orientation of the surface. It is also used in various mathematical and physical applications, such as in calculating surface area and determining the direction of forces acting on the surface.

## 4. Can the normal vector of an embedding surface change at different points?

Yes, the normal vector of an embedding surface can change at different points. This is because the orientation of the surface can vary, and the normal vector is dependent on the orientation of the surface at a specific point.

## 5. How does the normal vector of an embedding surface relate to the concept of surface normal?

The normal vector of an embedding surface is closely related to the concept of surface normal. The surface normal is a vector that is perpendicular to the surface at a specific point, and the normal vector of an embedding surface is essentially the same thing. However, the normal vector of an embedding surface is used more broadly in the study of surfaces in differential geometry, while surface normal is often used in computer graphics and visualization.

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