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Disclaimer : I am quite a beginner in physics, so very sorry if this is too much basic of a question but I have never read anything about what I am going to ask now. Or maybe I have but did not understand ?

In Newtonian physics It takes energy (work) to move a given mass at rest From one location in space to another. I assume it is the same in general relativity. Then, as time is the fourth dimension of space-time, where does the energy to move in time comes from ? I hope my question is not too much ill-posed. Don't hesitate to edit if you feel it can help. Thanks.

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  • $\begingroup$ How much energy you have to expend depends on how fast you want to move. In the limit that energy can be made zero (in absence of a potential). As for moving in time... that's a bit more complicated. Time is that which a clock shows, so how much energy you expend on your clocks will determine how well you can measure time. If you don't expend any energy, then you can't measure time, at all. $\endgroup$ – CuriousOne Feb 28 '16 at 20:48
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When you say that:

It takes energy (work) to move a given mass at rest

I would guess you mean it takes energy to accelerate an object from rest to some non-zero velocity $v$. This is mostly true (acceleration need not necessarily involve a change in kinetic energy) and in this sense it is also true in relativity. However once you have accelerated to a constant velocity it does not take energy to move at constant velocity.

In relativity we need to consider motion in time as well as motion in space, and we replace the usual velocity by a four-velocity and acceleration by four-acceleration. Just as with normal velocity, moving at constant four-velocity doesn't use up energy in any sense so it doesn't take energy to move in time.

To change the four-velocity may require energy, though energy can be a tricky concept in general relativity. Normally we use the four-momentum instead, of which the energy is one component.

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