this post was submitted on 27 Jul 2023
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The energy necessary to accelerate a bit more increases as the speed that object is travelling gets closer to the speed of light and indeed mathematically as the speed of the object gets closer to the actual speed of light the energy necessary to accelerate it gets closer to infinity.
If I remember it correctly it's because in the General Relativity Theory in the acceleration equation (not sure anymore if it was the one that relates Force to Acceleration or the Acceleration and Velocity one) the mass of the object isn't actually a constant amount but depends on its current velocity (it's as if the mass became larger with velocity). Just like for the whole time dilation stuff, this effect comes from the main equations and only really becomes noticeable closer to lightspeed (i.e. at relativistic speeds, called that because that's when you notice the effects of the Theory of Relativity, such as time dilation).
So the energy necessary to get to speeds below lightspeed yet close enough to have relatistic effects is not at all infinite, hence it is possible to reach relativistic speeds with the whole time dilation, redshift of light and other such effects, maybe even with current technology (I think we already have the tech to accelerate a ship to near lightspeed by having ground-based lasers pointed at a mirror on the back of it (so the light gets reflected - though that stuff has incredibly low acceleration as you're literally pushing that ship with photons plus ground-based lasers inside the Earth's athmosphere would was tons of energy due to the athmosphere).
In fact the very same experiments with subatomic particles that showed time dilation effects in their decay (which I mentioned in another comment) also showed it is possible to accelerate something to the point of having time dilation effects without infinite energy (if it happenned then it was possible to make it happen :))
Those effects only occur from the perspective of, for example, Earth. If you could exert force upon a space ship traveling away from you, it would require more energy to continue to accelerate it as its speed increases relative to you. But from the reference frame of the ship, this does not hold. Now, of course it will still require an insane amount of energy to maintain a constant acceleration, but from the spaceship's perspective energy expenditure is constant (assuming the mass of the ship isn't changing due to fuel loss).
If I remember it correctly from the point of view of the ship it looks like the propulsion becomes less effective, tough I confess my recolection of what I learned about the General Theory Of Relativity in my Physics degree almost 3 decades ago (which I never completed, by the way, so I'm an EE not a Physicist) breaks at around this point so I might be completelly off on this.