Time Dilation Calculator
Find out how much a fast trip ages the travellers and how much the ship is shortened along its path.
What it does
Special relativity is mostly about one idea: every observer measures light at the same speed, whatever they are doing. That single fact changes what time is and what length is, and the change only becomes noticeable at speeds close to that of light. The OpenStax text on relativistic mechanics sets out the two results that matter here: time dilation and length contraction.
The Lorentz factor is the number that does the work. It is 1 divided by the square root of 1 minus the square of the speed over the speed of light, and it is 1 for a stationary object and grows without limit as the speed approaches light. Multiply the time on a moving clock by the factor and you get the time that passes on the ground. At nine tenths of light the factor is 2.2942, so a year on board is a bit over two years and three months at home.
Length contraction is the other half. An object moving relative to you is shorter along the direction of motion, by the same factor, so a 100 metre ship seen from the ground measures 43.589 metres. It is not shorter for the people on board, and nothing is compressed, which is why the two observers disagree.
The energy number is the one that makes the physics real. Kinetic energy grows as the factor minus one times the mass times the square of the speed of light, and for a large mass at a high fraction of light that is a staggering amount of energy. It is also why a spacecraft that reaches a distant destination in a reasonable time needs a power source that is almost unimaginable in human terms.
The calculator treats the trip as uniform motion, ignores gravity and takes the speed of light as the exact value NIST publishes. It also ignores the fuel, the braking and the fact that a real journey has to accelerate and turn around.
How to use
- Pick metric or imperial units.
- Enter the speed as a fraction of the speed of light, for example 0.9.
- Enter the time measured by the people on board.
- Enter the length of the ship along its direction of travel.
- Enter the mass you want the kinetic energy for.
- Read the Lorentz factor first, since every other answer comes from it.
Formula
- gamma
- The Lorentz factor, the multiplier for time and the divisor for length.
- v
- Speed relative to the observer, as a fraction of the speed of light.
- t0
- Proper time, the time measured by a clock moving with the object.
- L0
- Proper length, the length of the object in its own rest frame.
- m
- Rest mass of the object.
The speed of light is exactly 299,792,458 metres per second, which is why the numbers come out so large in the energy column. The energy shown is the increase in the mass's kinetic energy as seen from the ground frame, which is not the same as the fuel a spacecraft would have to carry, and it does not include slowing down at the end of the trip.
Worked example
Example: one year aboard at 0.9 times light
Defaults: metric units, a speed of 0.9c, one year aboard, a 100 m ship and a mass of 1,000 kg.
- Speed: 0.9 x 299,792,458 = 269,813,212 m/s.
- Lorentz factor: 1 / sqrt(1 - 0.81) = 1 / sqrt(0.19) = 2.2942.
- Time on the ground: 2.2942 x 1 year = 2.2942 years.
- Length seen from the ground: 100 / 2.2942 = 43.589 m.
- Kinetic energy: (2.2942 - 1) x 1,000 x (299,792,458 squared) = 1.1631e+20 J.
- The tool shows 2.2942, 2.2942 years, 43.589 m and 1.1631e+20 J.
Reference table
| Speed | Lorentz factor | Time on the ground | Ship length seen from Earth |
|---|---|---|---|
| 0.5c | 1.1547 | 1.1547 years | 86.603 m |
| 0.9c | 2.2942 | 2.2942 years | 43.589 m |
| 0.99c | 7.0888 | 7.0888 years | 14.107 m |
Limits
- Assumes constant speed, with no acceleration or turnaround.
- Ignores gravity, so it does not describe a trip that falls into a black hole.
- Shows the kinetic energy of the mass, not the fuel a drive system would need.
- Leaves out the deceleration at the end, which costs as much energy as the acceleration.
Common mistakes
- Dividing by the Lorentz factor for the time instead of multiplying, which gives a shorter journey instead of a longer one.
- Applying the contraction to the whole object when it only applies along the direction of motion.
- Forgetting that the light year is a distance and not a time, which matters when a trip is quoted in light years.
- Ignoring the turnaround, so a one year round trip is not two years for the travellers.
- Treating the kinetic energy as the fuel bill, which is a different quantity entirely.
Try it yourself
Quick checks. Answers sit tucked under each question.
What does one year aboard at 0.99c equal on the ground?
The Lorentz factor at 0.99c is 7.0888, so one year aboard is 7.0888 years on the ground, and a 100 m ship measures 14.107 m from here.
How much longer is a year at 0.5c than at rest?
The factor at 0.5c is 1.1547, so a year aboard is 1.1547 years on the ground, about 56 days more.
What is the kinetic energy of a 1,000 kg mass at 0.9c?
It is (2.2942 - 1) x 1,000 x (299,792,458 squared), which works out at 1.1631e+20 J.
Questions
What is time dilation?
The fact that a moving clock runs slow compared with a clock at rest, so less time passes on board a fast ship than on the ground for the same journey.
Do the travellers feel it?
No. They see their own clocks running normally and their ship the right length. The difference only shows up when the two sides compare clocks, which is why a twin paradox is about the reunion rather than the journey.
What is the Lorentz factor?
The multiplier that connects the two frames, equal to 1 divided by the square root of 1 minus the square of the speed over light. It is 1 at rest and grows without limit as the speed approaches light.
What is length contraction?
An observer who sees an object moving measures it shorter along the direction of motion, by the Lorentz factor. The people on board measure their own ship as its full length.
How much energy does it take to reach 0.9c?
For a 1,000 kg mass the kinetic energy at that speed is 1.1631e+20 J. A drive system has to supply that much, and the same again to slow down.
How close to light can a human actually go?
The energy needed grows so steeply that even a small mass needs an enormous drive. The physics has no speed limit, but the engineering does.
Sources
- OpenStax University Physics Volume 3: 5.3, time dilation
- OpenStax University Physics Volume 3: 5.4, length contraction
- NIST: the speed of light in vacuum, exactly 299,792,458 m/s