Time Dilation at High Speeds

(Updated 11 November 2012)

Bibliography

Understanding Space: An Introduction to Astronautics, Revised Edition by Jerry Jon Sellers

Special Note

I took the liberty of pre-calculating time dilation factors for various speeds over HERE for quick reference.

Equation

T' = t / SQRT(1 – (v2 / c2))

(Equation Diagram)

Equation Explained

T' = Rate multiplier that the spacecraft experiences time at.

t = Rate Multiplier of a reference point you want to calculate from. Usually 1.

v = Velocity of the spacecraft in m/sec

c = Velocity of the speed of light, or 299,792,458 m/sec.

Example:

A spacecraft is traveling at “point one light” or 10% of light speed (29,979,246 m/sec). What is the time dilation ‘slowdown’ on board compared to a fixed reference point?

1 / SQRT (1 – 29,979,2462 / 299,792,4582) = 1.005037815259 times slower than that of the fixed reference.

If 3,600 seconds have passed on a (relatively) non stationary reference point, then the amount of time that will have passed on board the ship is:

3600 / 1.005037815259 = 3,581.954 seconds

The effect is not very noticeable at lower fractions of cee, but as you approach light-speed it grows asymptotically; e.g. the faster you go, the slower time passes.

At “point nine light” (90% light speed), time dilation is 2.294157338706 times slower than a fixed point, while at “point nine nine light” (99% light speed), time dilation is 7.088812050083 times slower than a fixed point.

The Tau Factor (τ)

This time dilation can be expressed in terms of ‘tau’, which is the ratio of time passing on the craft to time on a (relatively) non stationary body.

τ = TCraft / TStationary

Where

TCraft = Amount of time that has passed on the craft.
TStationary = Amount of time that has passed on the stationary reference point.

Example I: A craft has been traveling at 90% of light speed (2.294157338706 times slower than a fixed point) for a month (730.484 hours). What’s the Tau of the craft?

(730.484 / 2.294157338706) / 730.484 = τ0.435

Example II: A craft has been traveling at 99% of light speed (7.088812050083 times slower than a fixed point) for a month (730.484 hours). What’s the Tau of the craft?

(730.484 / 7.088812050083) / 730.484 = τ0.141

Thus, you can see how one of the most famous SF stories ever written: Tau Zero by Poul Anderson got its name.