Embedding Diagram of Weyl Metric in ##R^3##

[Onyx]

TL;DR Summary

Is it possible to make an embedding of the $\phi$=constant slice of a Weyl metric in $R^3$?

Is it possible to make an embedding of the $\phi$=constant slice of a Weyl metric in $R^3$? In particular, I'm thinking of a metric where the components are both $\rho$ and $z$ dependent.

 

[Onyx]

Has anyone seen this question?

 

[PeterDonis]

a Weyl metric

This is a very general category, so I'm not sure your question is answerable unless you can narrow things down more.

 

[Onyx]

This is a very general category, so I'm not sure your question is answerable unless you can narrow things down more.

Actually, forget about the Weyl metric for now. I am specifically trying to embed $ds^2=U^2(dx^2+dy^2)$, where $U=1+\frac{1}{\sqrt{(x-1)^2+y^2}}+\frac{1}{\sqrt{(x+1)^2+y^2}}$, into 3D Euclidean space. The only trouble is, the resulting function clearly does not have an indefinite integral, so there is the question of where to start the integration from.

 

[PeterDonis]

I am specifically trying to embed $ds^2=U^2(dx^2+dy^2)$, where $U=1+\frac{1}{\sqrt{(x-1)^2+y^2}}+\frac{1}{\sqrt{(x+1)^2+y^2}}$, into 3D Euclidean space.

Where does this line element come from?

One obvious issue is that the function $U$ is undefined at the $(x, y)$ points $(1, 0)$ and $(-1, 0)$.

 

[Onyx]

Where does this line element come from?

One obvious issue is that the function $U$ is undefined at the $(x, y)$ points $(1, 0)$ and $(-1, 0)$.

Yes, those are supposed to be the degenerate horizons of the black holes I think. I got this metric from a 2 black holed Majumdar-Papapetrou metric with black holes centered at the points you mentioned. I took the $\phi=constant$ slice and replaced what is usually $p$ and $z$ with $x$ and $y$.

 

[Onyx]

Okay, I think I figured out through pullback the form of h(x,y), the embedding function, is. It involves an indefinite integral whose answer is probably expressed with elliptic integrals in a way that I don't know. Maybe that question would be more at home in the pure math section of the website at this point.