Say I’m in my spaceship with full thrusters. I turn off the thrust and coast. I’ll continue coasting until some force, like gravity, slows me down or I bump into something. But how come just moving through space doesn’t take energy?
It’s common to show space being warped by gravity similar to a ball on a sheet. There is an indentation where the ball is. If the ball moves around, the indentation moves accordingly. But in a sheet, it takes energy to deform the sheet as the ball moves around. So not only does the rolling ball have to deal with rolling resistance slowing it down, it also loses energy from deforming the sheet.
My spaceship, although small, will warp the space around it a little bit. Why does that warping of space as I move through it not take away energy?
In the rubber-sheet analogy, there’s no loss in energy to the sheet because the sheet is assumed to be perfectly elastic. So any energy lost in “stretching out” the rubber sheet in front of your spaceship is regained when the sheet contracts back to its unstretched shape. It’s very similar to how a spring works: it takes energy to stretch out a spring, but that energy can be regained when you let the spring move back to its normal length.
Now, your spaceship will lose a little bit of energy to the “rubber sheet” if it accelerates, in the form of gravitational waves. Basically, any object that accelerates (i.e., doesn’t move at a constant velocity) will send out ripples in the “rubber sheet”. These ripples have a certain energy associated with them, and so your spaceship will lose energy to these ripples. Unless your spaceship’s mass or its acceleration are stupidly large, though, the amount of energy lost to this effect will be miniscule.
Traveling through space at a constant velocity doesn’t require any more energy because it’s indistinguishable from standing still while the universe travels by you.
General relativity (where the whole notion of distortion of space comes from) is an extension to and therefore inherently consistent with special relativity. And the key tenet of special relativity is that there is no privileged reference frame. As your spaceship is traveling through space at constant speed, it’s exactly like it’s sitting at rest in space, and therefore there’s no motion for “spatial friction” to oppose.
I think there IS a privileged reference frame, in the sense relevant to the OP. Gravity waves carry energy. The reference frame that is stationary with respect to some inverse distance weighted average of all the surrounding masses is special. Objects moving with respect to this reference frame feel a drag relative to it.
It isn’t too far off to draw an analogy with viscous drag in fluids. There is a mechanism that passes energy from a moving body into the fluid, causing the fluid to have little internal relative motions. If we call the reference frame that moves with the average of the fluid a rest frame, then constant motion requires power. Of course if you define a different rest frame you can show the fluid doing work to the object.
I think the same thing applies to charge, given a moving charged particle relative to a background collection of objects with various charges. The moving charged particle has to disturb them all a little, on which effort it spends some of its kinetic energy.
This isn’t about relativity, really - it’s about interactions between populations that have various statistical descriptions of their motion.