Relativistic Rocket

A rocket departs Earth for another star. We divide the trip into three phases:
  • During the boost phase, it accelerates at a fixed rate. This phase is powered because the rocket is firing.
  • During the cruise phase, it coasts at a fixed velocity. This phase is unpowered.
  • During the deceleration phase, it decelerates at the same rate as it had originally accelerated. That is, deceleration phase is a mirror of the boost phase. This phase is also powered.
The rocket thus arrives at the target star with zero velocity relative to the start point--ignoring issues like gravity, resistance of the interstellar medium, relative motions of the stars, etc.
This page computes a number of interesting properties of such a trip, given just a few bits of information. Fill in what you know and it computes the rest.

Powered Phase Questions.

What is the acceleration?

Answer any one of the next five questions.

How far does it travel under power?
How far does it travel during boost alone?
What velocity does it reach?
How long does it accelerate? (ship time)
How long does it accelerate (earth time)?

Cruise Phase Questions.
Answer any one of the following six questions. "Total" in this section includes boost, cruise, and deceleration, so "total distance" is the entire distance from Earth to the star.

What is the total distance travelled?
How long is the whole trip? (ship time)
How long is the whole trip? (earth time)
How far does it travel during cruise?
How long does it cruise? (ship time)
How long does it cruise? (earth time)

Other Statistics
At max velocity, time slows down and lengths contract by a factor, tau, while and mass increases by its reciprocal, gamma.

How much "time slip" is there between the crew and the people on Earth?
Earth minus Ship Time

How much reaction mass is needed, assuming we use anti-matter fuel and our exhaust is coherent gamma rays.
Mass per kilogram to cruise velocity
Energy per kilogram to cruise velocity
Mass per kilogram to destination
Energy per kilogram to destination

Suppose there were no such thing as relativity. What would the times look like? (For cruise segments, this keeps the distances constant.)
Newtonian Boost Time
Newtonian Cruise Velocity
Newtonian Cruise Time
Newtonian Total Time

Other relativity numbers. You should read the Relativistic Rocket (below) for explanations.

Link for sharing these calculations:

All calculations taken from the Relativistic Rocket (Don Koks and Phillip Gibbs, 2006)


  1. Hi

    How would you compute length contraction and energy increase (relativistic) for a constant acceleration rocket?

  2. PS: my email address is

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  13. Hi Greg. I am wondering if maybe you could guide me through the use of your calculator to help me with a story. I need to get my crew to the inner layer of the Oort Cloud; 2000 AU from Earth, give or take. According to one link I found, the best I could do for "hard sci-fi" tech was the fusion rocket design (,) which, based on estimated distances, would take about 3.7 years to reach that distance, provided I had the calculations right. This isn't quite the speed of light, obviously, so I am having trouble figuring out your calculator. However, I imagine that the same principle of continuous acceleration could work. I guess I'm trying to reverse engineer the calculator; I want to know how long the ship would have to accelerate and decelerate for in order to be safe but reach the requisite speed. Also, I want to know if that 3.7 years at speeds that are pushing, but not achieving, light speeds, would have relativistic effects on the crew. Does this make sense, or are the two technologies (fusion rockets and continuous acceleration) completely incompatible? Thanks for your help!

    1. I should pay more attention to this page. If you're still interested, it's just 167 days--not 3.7 years. But that assumes 1g acceleration to the half-way point and then 1g deceleration from there.

  14. I am sorry for the lenguage but JUST FUCKING EPIC blog entry. I love you, sir or madam. I absolutely love you! This is exactly what I was looking for. ¡Thank you!

  15. Nice but it could use one improvement--remembering which values were entered. I was looking at a constant end velocity with various accelerations but when I changed the acceleration it holds the first number constant rather than the one I entered.

    1. I've thought about it, but that turns out to be really hard to do in a systematic way. The trouble is that some entries force other ones, and even figuring out which ones can't be changed is a challenge.

  16. Hello, I don't understand one point, when there is acceleration = 1g on the top of the form, it means that ship and crew suffer the same "gravity feel" as those people, that did stay on Earth. In this case, ship time and earth time should be the same even after return, no matter final speed the ship reaches. In your calculator they differs. How is it possible? Thank you very much

    1. @siloton You are mixing up the fact that gravity slows time with the fact that speed slows time. Furthermore, what counts is the gravitational field, not what you actually feel. Being in orbit doesn't change the slowing due to gravity (but adds slowing due to speed.)

      Both factors are minute at the scale that we experience here on Earth. The only time the average person will encounter the effects is with GPS (and even then you don't know it unless you dig into the equations that run your system.)

      To get a noticeable slowing from speed requires velocities that are an appreciable fraction of lightspeed. On Earth that means the realm of the particle accelerator.

      To get a noticeable effect from gravity requires getting very close to a neutron star or black hole--exceedingly dangerous places to be.

    2. Ahh thank you Sir. I appreciate your effort! I thought speed itself has no effect on time speed. In example of ship and Earth consider this, how the "Time" knows if the Ship recedes from Earth, or Earth recedes away from Ship and so on what "side" be "slower"? Now, if I take acceleration as an only cause for time dilatation now it makes perfect sense for me coz now time knows exactly how fast on each side he exactly runs :) Of course I would love to inspect another explanations. Thank you ones more...

    3. Both the ship and Earth see each other's clocks as going slow. However, that only applies so long as nobody accelerates. Once you introduce an accelerated reference frame that fixes the paradox, the observer that went to the stars and came back will always be younger than his twin that stayed home. Since you can't change velocity without an accelerated reference frame there are no paradoxes.

    4. Oh you have patience with me man, If you explain to me, I will send you a donation I think :). OK exactly as you write, those clocks "are same" until one of those subjects go through acceleration / deceleration so they can be compared. You say you cannot take velicity without acceleration, yes, but if we make an mental experiment, considering one can reach speed of light without acceleration and back also, there should be zero time shift I suppose? So if yes then acceleration is the only cause or if no, can you please direct me to some equations on Wiki or somewhere that sums how much time you would skip during the same travel "without acceleration"? Thank you ones more.

    5. And maybe I can make myself even more clear, you wrote: "Since you can't change velocity without an accelerated reference frame". Well, in a case of 1g on an Ship same as on a Earth, how the "accelerated reference frame" knows i am the Ship and not the Earth? Those observers cannot tell a difference.

    6. Sir, I will pay you for an explanation. Is it acceptable? Thank you

    7. Hi Siloton, you're correct that it's not specifically due to acceleration that there's a time difference in the Twin Paradox. Rather, it's because the twin that leaves earth and comes back, even if he accelerates instantly, cannot remain in the SAME reference frame on his whole trip because he has to change direction to come back. Changing direction, even if done instantaneously, is a change to a DIFFERENT inertial reference frame.

    8. Hello Mike. Thank you very much for trying to help me understand. Since my previous posts, I did more research. The instant direction change was not a good scenario. Better stick to standard 1g acceleration example. In the past when I did read basic relativity book, Google did not exist. I made in my mind fixed idea that gravity and acceleration due to principle of equivalence has exactly same influence on speed coz the "gravity feel" and it made perfect sense in my brain. Im not psychist, just dumb curious person. Since original post I did realize current mainstream science consider Earth and Rocket not equivalent reference frame, by principles I do not understand yet and maybe I will never do. But the main problem that broke my theory was finding that there is no experiment on the time dillatation using centrifuge. This experiment would support my idea, but i did not find any. Do you know of such experiment? I will tell you why this idea still cannot leave my mind easily. PPL usually imagine that scenario in real universe. Lot of stars and galaxies. From point of time dilatation they should not need to exists. So if I imagine the Universe containing JUST an Earth and a Rocket and nothing else, then how the reference frame knows Rocket did change direction from receding to approaching and not Earth? Gravity feel always 1g. There are no other objects to tell so. Maybe principle of equivalence do not work 100% due to... presence of gravitons? I don't know but I did submit that I will not find satisfying answer in current life period, maybe never. Still thank you for your effort! :)

    9. Worrying about gravity (General Relativity) is, I think, messing you up, since the entire Twin Paradox can be explained using only Special Relativity.

      Earth's gravity has nothing to do with it -- the difference in age of the two twins would be the same if the starting point was a zero-g space station. (There might be a very tiny factor caused by Earth's gravity, but it's negligible for the purpose of this discussion, I believe.)

      So what is the difference between the two twins?

      Twin #1, who stays on the Zero-G space station, or on the Earth, never changes inertial reference frame, because he never experiences any acceleration. He stays in a single frame.

      Twin #2 (the one on the rocket) can measure his acceleration using an accelerometer, or even just by feeling it. Especially when he changes direction. And if he had a telescope pointed at his brother's house, where there was an accelerometer setup with a giant display, he would see the whole time that his brother is not undergoing acceleration.

      Therefore, this situation is not symmetrical.

      If you use instant acceleration, it is symmetrical up until the turn-around point, at which point there is a kind of "jump".

      I found this discussion helpful, both the accepted answer and additional answer my Muphrid:

  17. I didn't answer your previous question because you're getting out of my depth. I was hoping someone who knows more would answer.

    1. Sorry to bother you. I would like to pay only for a chance to discuss a problem with you. If there is a discrepancy we should find it quickly. But if you're sure your calculations are right even if principles cannot be formulated... well its up to you Sir, thank you for your time anyway...

  18. What a fantastic resource! Thank you so much for creating and posting this calculator.

    1. Glad you liked it! I keep meaning to do a few more for things like planets, orbits, eclipses, etc. but I've never got around to it.

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