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I have read this question:

I think it's the opposite - when there is no gravity, the difference in time observers measure only depends on their relative velocity. It's when gravity gets very strong that you can get things like an event horizon. In the case of a black hole, the time coordinate of the faraway observer "stops" at the event horizon.

Does time exist without Gravity?

The original question was about whether time can exist without gravity, and most of the answers say yes, time can exist even without gravity. Now I am asking the opposite, whether we can think of gravity in terms of a 3D space, without the temporal dimension. Does gravity even exist in such a "space"? Can gravity be expressed without involving the temporal dimension? Or is gravity inherently temporal?

Question:

  1. Can gravity exist without time?
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  • $\begingroup$ Why the downvote? $\endgroup$
    – Árpád Szendrei
    Jun 21 at 3:18
  • $\begingroup$ I think that the question confuses the coordinate system used to model mathematically the observations/measurements of gravity with the phenomenon itself. In the Newtonian gravitation that is clear . There are answers within the framework of General Relativity which ties up space and time to model gravity, but even there, it is still part of the coordinate system used to define the functional forms of the model. $\endgroup$
    – anna v
    Jun 21 at 4:12
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    $\begingroup$ If there is no time then motion is not possible and so the whole idea of a force is mute anyway. $\endgroup$
    – rghome
    Jun 21 at 6:59
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    $\begingroup$ @rghome not mute, moot $\endgroup$
    – m4r35n357
    Jun 21 at 10:48
  • $\begingroup$ Gravitational potential is the time dilation. Gravitational acceleration is a spatial gradient of the time dilation (reflecting the change along a direction). The only meaning of “without time” I can think of is when time is stopped (the infinite time dilation). One hypothetical (an unstable) example is the inside of a hollow massive shell at its Schwarzschild radius. Time inside this shell would stand still representing the infinite gravitational potential, zero gravitational acceleration, and zero speed of light. See this for details: math.stackexchange.com/questions/3310040 $\endgroup$
    – safesphere
    Jun 21 at 16:09

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It depends what you mean by 'exist without time', and also whether we're talking about our Universe or a different hypothetical one. I've given two possible answers below, but you can clarify which you intended.

  1. We live in a universe with four spacetime dimensions: a Lorentzian manifold with signature $(-+++)$. The curvature, and hence gravity, is defined as the spacetime curvature; time is intrinsically mixed up with space. We can look at the spatial curvature on a three hypersurface (for come constant given time $t$), but this isn't really 'gravity' in the same sense, and this also doesn't seem to be what you're asking? (If it is, then the answer is an emphatic no, simply because space and time are mixed. An easy example: spatially flat FRW cosmology - the spatial curvature is zero but spacetime curvature is not).

Now I'll go presume that you're asking about a different hypothetical universe that isn't modelled by a Lorentzian manifold.

  1. If we interpret gravity simply as the curvature of spacetime (encapsulated in the Riemann tensor $R^a{}_{bcd}$), then there's no requirement that the manifold you're working with needs to have a Lorentzian signature. Gravity is still just given by the curvature of spacetime (in this setting, it's really just space). So the answer to this question would be yes, time is not needed for gravity.

To conclude, there's nothing special or necessary about the temporal part in order to have gravity, but in our universe we do have one dimension of time, so curvature (gravity) necessarily depends on it in general.

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  • $\begingroup$ Thank you so much! $\endgroup$
    – Árpád Szendrei
    Jun 21 at 3:18
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In general most notions of gravity depend on energy, and energy is the conserved current generated by invariance of translations in time. Moreover, I don't think there's any physically realistic stress-energy tensor in which all of the temporal components are 0 (in particular in order to have stress-energy you need to have energy). So practically the answer is no, it doesn't really make sense to discuss gravity without involving the temporal dimension.

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  • $\begingroup$ Thank you so much! $\endgroup$
    – Árpád Szendrei
    Jun 21 at 3:18

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