This is also why faster than light travel is impossible. You need more and more fuel the faster you go because your mass increases, which requires more fuel that causes your mass to increase, which....and on and on and so on and so forth. There isn't a fuel with that kind of exponential efficiency, because you would be getting more energy than you're putting in. Energy (like matter) cannot be created nor destroyed, only moved and stored. You would need a perpetual motion machine in order to go faster than light.

[–] quarrk@hexbear.net 9 points 1 week ago* (last edited 1 week ago)

You’re right that the behavior is similar, but the physical explanation is different, and the rate of increase for required fuel is different as a result.

The classical rocket equation is well, classical, and derived from non-relativistic Newtonian physics. Fuel requirement increases exponentially because each additional ounce of fuel itself has mass that needs to be accelerated. But importantly, according to the classical equation, it would be possible to accelerate to light speed and faster, if you could find enough fuel.

For the relativistic rocket equation, fuel requirement increases along a different curve (not exponential but hyperbolic) which results in asymptotically approaching light speed. The reason has to do with the Lorentz factor gamma (γ) which expresses the degree to which time dilates and length contracts as you approach light speed. It takes more fuel as you speed up because spacetime itself changes form so that this is true - in addition to the exponential part of the classical equation.