Signs in the definition of anti de Sitter spacetimes

[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##?

 

[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##.