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Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity after differentiation? Can the derivative provide information about the stability of reality?

If differentiation does not provide information about how reality changes with respect to time, then what is the fundamental force in the universe that drives every action and reaction?

I'm not quite familiar with metric tensors and tensor differentiation, so I would really appreciate a simplified explanation.

Thank you!

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    $\begingroup$ Maybe you could say more about what you mean by "differentiate spacetime." Differentiating X with respect to time tells us how X changes, but spacetime diagrams do not ever change. A spacetime diagram is a representation of change, but the diagram itself is a static, unmoving thing. $\endgroup$
    – Solomon Slow
    Commented Jul 8 at 18:32
  • $\begingroup$ This is a little like asking "what is the derivative of velocity with respect to the x-component of velocity". But as Solomon points out, we usually mean spacetime to mean the set of all events, i.e. the set of all time-space coordinates. Remember that time is not a parameter in relativity, it is a coordinate. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 8 at 18:35
  • $\begingroup$ Do you mean the derivative of the position of some object in spacetime, or of some function of spacetime? $\endgroup$
    – Pato Galmarini
    Commented Jul 8 at 18:36
  • $\begingroup$ Also, "stability of reality" is ill-defined. Given that we've directly observed black holes, we know that there are regions of spacetime that are so torn up by an object's gravity that it stops behaving normally; maybe that's what you mean by "unstable reality". What you might mean by the "derivative of spacetime" is the derivative of the metric tensor, which tells you how to measure gravitational acceleration/distance. $\endgroup$
    – controlgroup
    Commented Jul 8 at 20:46

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I think what you're reaching for is the four vector velocity of an object. That is, differentiating it's position with respect to time. You started with a four dimensional vector (position) and ended with a four dimensional vector which is not it's velocity vector in space, but it's velocity vector in spacetime.

This is normally done with respect to the proper time, not the local coordinate of time in that frame.

There is one very interesting property, which is that the magnitude of the four-velocity is always the speed of light (there's a more precise statement of this on the Wikipedia page I linked to).

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