In general relativity, we have a gravitational time dilation of:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{d\tau}{dt} = \frac{1}{\sqrt{1 - \frac{2GM}{rc^2}}}[/tex]

The term [itex]- \frac{2GM}{rc^2}[/itex] appears to be based on the fact that gravity is attractive. If I understand correctly, if the curvature of space-time leads to attraction, then the curvature of space-time is positive. Alternatively, some parts of space-time could be curved the other way. It could have negative curvature instead of positive curvature. Instead of only bending paths inward as in elliptical geometry, we could have variations in space-time curvature that permit bending of paths outward as in hyperbolic geometry. Why isn't this normally discussed of? I mean, in serious physics people already talk about wormholes, multiverses, parallel dimensions, and time machines, so is there some reason why we ignore the possibility that space could be curved the other way, which in many respects, sounds a lot less like science fiction than what is being entertained right now? Couldn't the curvature of space time in the hyperbolic sense result in a sign change of the effective potential? Couldn't this correspond to a [itex]\frac{d\tau}{dt} < 1[/itex], suggesting the possibility of time acceleration - time dilation's opposite?

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# Can the proper time interval be < the coordinate time interval?

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