# Is there a correct way to describe delta-v for a relativistic spacecraft?

So if I have a spaceship that has an extremely powerful e.g. antimatter rocket (not FTL, relativity applies) and can accelerate up to say 60% of light-speed, then coast to Alpha Centauri, then decelerate back to zero... well, it can't be right to say the spaceship has a delta-v capacity of 120% of light-speed, can it?

Because that would imply it could accelerate to 120% of light-speed if it didn't care about slowing down again, which would violate relativity.

So what's the correct way to describe this capability? Is there such a thing as relativistic delta-v, or would some other term need to be used?

• I've added a tag, feel free to revert if you see fit. Feb 26, 2021 at 21:31
• Are you interested in the delta-v for the sake of calculating the ratio of initial mass with fuel to final payload mass, or for some other calculation? Feb 26, 2021 at 22:20
• Perhaps en.wikipedia.org/wiki/… is relevant? Given that there is a way of calculating delta-v for relativistic situations and the relevant professionals can be expected to know whether non-relativistic (ordinary) delta-v or relativistic delta-v is being used in any given situation, it doesn't seem like an issue to have a non-relativistic delta-v greater than the speed of light. Feb 26, 2021 at 22:20
• @GrumpyYoungMan - If you use delta-v > c in the relativistic version of the Tsiolkovsky rocket equation on that page (which is also discussed in this section of the Tsiolkovsky rocket equation page), you get the physically meaningless conclusion that the ratio of m0 (initial mass including fuel) to m1 (final payload mass with fuel expended) would be an imaginary number. Feb 26, 2021 at 22:57
• (cont.) For a problem where a rocket does two linear accelerations in different directions, the first one giving it a velocity v relative to its rest frame at the start, the second a velocity u relative to its rest frame at the beginning of the second acceleration, I think for the purpose of the Tsiolkovsky eq. you'd want to imagine what its delta-v relative to the starting frame would be if both accelerations had been in the same direction, using the relativistic velocity addition formula. Feb 26, 2021 at 23:00

Short answer: Yes, as long as each burn doesn't get too far into relativistic speeds, you can go back and forth to a sub light speed enough times to add up to more than 1c.

Long answer: Depends on how you look at it! I like to use Rapidity

If the non-relativistic Tsiolkovsky equation gives you a delta-V of 2c, that's fine, though you still cannot ever go faster than the speed of light by using Newtonian means! You can only achieve faster than c velocities by warping space or using wormholes. Delta-V is a measure of how much velocity change you have, not total final speed, which is limited by relativity.

Explaining further can get very technical, but to do my best, here goes: what you do is you add coefficients of the speed of light (v/c) and find the hyperbolic tangent of that value.

tanh(x) only gets to 1 at infinity, so, any sum inside of it will only ever approach 1.

Examples:

If your total dv is 2c, your v is 2c and c is still c, so 2/1 = 2. How fast can you go if you made a single burn in one direction? Since tanh(2) = 0.96402758007... c, you can get to almost 97% the speed of light! Quite fast, but good luck slowing down.

Now, let's take the other approach. "How many times can I go to .1c and back?" .1c/1c = .1/1 = .1, simply enough, since we're already using a coefficient of c to look at speed. [2 / (tanh^(-1)(.1))] = 19.9332... so you can do it almost 20 times.

Why almost 20 and not actually 20? Because going to 10% the speed of light 19 times has some additive relativistic effect.

If you want me to go even deeper into detail with reference frames for a stationary observer relative to the ship speeding up, slowing down, and coming back, that goes far beyond my abilities.

• I have no Idea what you are saying or how this might answer the question. The numbers seem to make no sense, not to mention the reverse-capitalization. Feb 27, 2021 at 0:42
• Have you heard of rapidity? Feb 27, 2021 at 0:43
• Please see my edits to the answer. Feb 27, 2021 at 0:49
• That's nice, it works as an answer as you've added the term " rapidity " with a link. Terms are what the OP was asking for after all. +1 Feb 27, 2021 at 0:54
• When you say "If a normal rocket equation gives you a dV of 2c", does "normal rocket equation" refer to the non-relativistic Tsiolkovsky equation, non-relativistic velocity addition, or something else? Feb 27, 2021 at 1:25