A Signs in the definition of anti de Sitter spacetimes

1. Apr 25, 2017

spaghetti3451

Consider the definition (https://en.wikipedia.org/wiki/Hyperboloid) of the hyperboloid in $\mathbb{R}^3$ with the metric

$$ds^{2}=dx^{2}+dy^{2}+dz^{2}.$$

The one-sheet (hyperbolic) hyperboloid is a connected surface with a negative Gaussian curvature at every point. The equation is

$$x^{2}+y^{2}-z^{2} = R^{2}.$$

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Now consider the definition (https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Definition_and_properties) of the anti-de-Sitter space AdS$_{2}$ in $\mathbb{E}^{2,1}$ with the metric

$$ds^{2} = - dt^{2} + dx^{2} - dy^{2}.$$

This is a connected surface with a negative Riemann curvature. The equation

$$- t^{2} + x^{2} - y^{2} = - R^{2}.$$

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Why can we not obtain the anti-de-Sitter space AdS$_{2}$ by

1. putting a negative sign in front of $dx^{2}$ in the metric of $\mathbb{R}^3$, and

2. putting a negative sign in front of $x^2$ in the equation of the hyperboloid in $\mathbb{R}^3$?

2. Apr 30, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. May 1, 2017

martinbn

I am not sure I understand the question, but of course you can consider a surface with equation $-t^2-x^2+y^2=-R^2$. To be consistent with the notations you should write it as $-t_1^2-t_2^2+x_1^2=-R^2$.