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

[kmarinas86]

In general relativity, we have a gravitational time dilation of:

$$\frac{d\tau}{dt} = \frac{1}{\sqrt{1 - \frac{2GM}{rc^2}}}$$

The term $- \frac{2GM}{rc^2}$ 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 $\frac{d\tau}{dt} < 1$, suggesting the possibility of time acceleration - time dilation's opposite?

 

[Bill_K]

You have it backwards, kmarinas86. Take another look at the Schwarzschild solution - it starts off ds² = (1-2M/r) dt² - ... Which means that for a particle sitting still, ds/dt = √(1 - 2M/r), which is less than 1. Elapsed proper time is less than elapsed coordinate time, not greater. The Schwarzschild field is attractive, however the terms positive/negative curvature and elliptical vs hyperbolic don't really apply.

in serious physics people already talk about wormholes, multiverses, parallel dimensions, and time machines

I'm afraid this is a false impression. Topics like these are more typically found in articles in Discover magazine. The kindest term for them is "highly speculative". Most physicists do not sit around talking about wormholes and multiverses.

 

[Matterwave]

Coordinate time is just a choice of coordinates. Of course it's possible to make it's relation with the proper time anything you'd like. Just change your coordinate system.

 

[kmarinas86]

Coordinate time is just a choice of coordinates. Of course it's possible to make it's relation with the proper time anything you'd like. Just change your coordinate system.

In general relativity (difference in the time interval due to influence of the gravitational field), just how arbitrary is it? Surely you couldn't just make the proper time interval zero, or something arbitrarily close to it, could you? There has to be some limit - right?

 

[kmarinas86]

The Schwarzschild field is attractive, however the terms positive/negative curvature and elliptical vs hyperbolic don't really apply.

So do you think scientists even know what hyperbolic curvature of spacetime would do to time dilation? Is the idea of there being the opposite of time dilation ruled out by GR, or does GR technically allow it?

 

[kmarinas86]

Can the proper time interval be > the coordinate time interval?

You have it backwards, kmarinas86. Take another look at the Schwarzschild solution - it starts off ds² = (1-2M/r) dt² - ... Which means that for a particle sitting still, ds/dt = √(1 - 2M/r), which is less than 1.

Oh ya, that's right. ds/dt is the derivative of s with respect to t. Clearly that value would be less than one when considering time dilation. So in reality:

In general relativity, we have a gravitational time dilation of:

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{rc^2}}$$

 

[Matterwave]

In general relativity (difference in the time interval due to influence of the gravitational field), just how arbitrary is it? Surely you couldn't just make the proper time interval zero, or something arbitrarily close to it, could you? There has to be some limit - right?

The proper time is the time measured by some clock and so you can't change that one arbitrarily given some process (i.e. proper time for an item to go from one place to another through some path is an invariant and not something that we can change). The coordinate time is purely arbitrary, you can make it essentially anything you want. Of course, you should choose your 4 basis vectors to at least span the 4-dimensional tangent space to the space-time you are considering, and so your time basis vector should, generally, be time-like. But it doesn't always have to be. For example, inside the event horizon of a Schwarzschild black hole, the Schwarzschild time coordinate basis vector becomes space-like.