Hail Mary Science
Spaceflight and relativity

Time dilation calculator

Pick a speed as a percentage of light speed and the calculator tells you how much time passes on Earth and how much for the crew on trips to several stars. It is Einstein's special relativity applied to an actual journey.

Key formula

γ = 1 / √(1 − v²/c²) · t_ship = t_Earth / γ

Try it

How to use the simulator

Special relativity says a clock moving very fast ticks more slowly than one at rest. The Lorentz factor, γ = 1/√(1 − v²/c²), tells you by how much. At everyday speeds γ is essentially 1, but it soars as you approach light speed. For several destinations, the table works out how long the trip takes on Earth and for the crew, assuming a constant cruising speed.

Controls

  • Speed (% of light): move the slider to change the ship's speed and the Lorentz factor.

Things to try

  1. At 97% of c, γ ≈ 4.1: each year on board equals about four on Earth.
  2. Drop to 50%: γ is only 1.15. Dilation barely matters until you're well past half the speed of light.
  3. Go to 99% and check the Andromeda row: millions of years on Earth, but "only" a few hundred thousand on board.

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What time dilation is

Special relativity starts from a fact confirmed by every experiment: the speed of light in a vacuum is the same for every observer. For that to hold, time and space cannot be absolute. A clock moving very fast relative to you ticks more slowly than yours.

The Lorentz factor (γ) measures the effect. At everyday speeds it is essentially 1: on a commercial flight the difference is a few nanoseconds per hour. But γ grows without limit as you approach light speed: 1.15 at 50%, 2.29 at 90%, 7.09 at 99% and 22.4 at 99.9%.

Why it is not an illusion

Time dilation is measured every day. Muons created in the upper atmosphere live only about 2.2 microseconds at rest. With that lifetime almost none should reach the ground, yet they arrive in large numbers because they travel close to light speed and their internal clock runs slow as seen from Earth.

GPS satellites account for it too. Their orbital speed makes their clocks lose about 7 microseconds a day, while the weaker gravity at that altitude makes them gain about 45. Without correcting both effects, positions would drift by kilometers every day.

What the calculator assumes

The table assumes the ship cruises the whole way at a constant speed. A real trip would include acceleration and braking phases in which the Lorentz factor keeps changing. Even so, the cruise-speed figure gives the right order of magnitude and shows the key effect: the trip is shorter for the crew than for those who stay home.

Worked example

Example: a trip to Tau Ceti at 90% of light speed

  1. Tau Ceti is about 11.9 light-years away. At 90% of light speed the trip takes 11.9 / 0.9 ≈ 13.2 years as measured from Earth.
  2. The Lorentz factor is γ = 1 / √(1 − 0.9²) = 1 / √0.19 ≈ 2.29.
  3. Time for the crew is 13.2 / 2.29 ≈ 5.8 years. When they arrive, they have aged 7.4 years less than the people they left on Earth.
  4. Move the slider to 99%: γ ≈ 7.1, and the same trip takes about 12 years on Earth but under 2 on board.

In Project Hail Mary

The novel's ship accelerates continuously for years and reaches a sizeable fraction of light speed. That is why time on board and time on Earth don't match, and the story takes it into account when working out how long the mission has lasted.

This tool appears in:Chapter 6The Engines Stop

Frequently asked questions

Can you travel faster than light?

Not according to known physics. Accelerating anything with mass to light speed would take infinite energy, because γ grows without limit. What relativity does allow is a trip that feels very short to the traveler, thanks to time dilation.

Does the crew notice their time running slow?

No. Inside the ship everything is normal: clocks, heartbeats and aging all run at their usual pace. The difference only shows up when comparing with Earth, for example on returning or when receiving signals.

What is the twin paradox?

If one twin travels at relativistic speed and comes back, they will be younger than the twin who stayed. It looks contradictory because each sees the other moving, but the situation isn't symmetric: only the traveler turns around and accelerates, and that breaks the symmetry.

Does gravity affect time as well?

Yes, but that is a different effect, from general relativity: time runs slower near a large mass. This calculator only covers the effect of speed.

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