Time dilation in SF and GPS

I'm reading Voyage to Alpha Centauri and ran into a question about relativity. The book says in one place that their ship is moving a 56.7% of the speed of light, and in another place it says that time moves about 20% slower for them relative to folks on Earth. Are those two statements consistent?

It wouldn't bother me if they weren't consistent. I ordinarily wouldn't bother to check such things. But I remember looking into time dilation before and being surprised how little effect velocity has until you get very close to the speed of light. I couldn't decide whether the relativistic effect in the novel sounded too large or too small.

If a stationary observer is watching a clock moving at velocity v, during one second of the observer's time,

seconds will have elapsed on the moving clock.

Even at 20% of the speed of light, the moving clock only appears to slow down by about 2%.

If, as in the novel, a spaceship is moving at 56.7% of the speed of light, then for every second an Earth-bound observer experiences, someone on the ship will experience (1 - 0.567^2) = 0.82 seconds. So time would run about 20% slower on the ship, as the novel says.

The author must have either done this calculation or asked someone to do it for him. I had a science fiction author ask me for something a while back, though I can't remember right now what it was.

Small velocities

You can expand the expression above in a Taylor series to get

and so velocities much smaller than the speed of light, the effect of time dilation is 0.5 v^2/c^2, a quadratic function of velocity. You can use this to confirm the comment above that when v/c = 0.2, the effect of time dilation is about 2%.

GPS satellites travel at about 14,000 km/hour, and so the effect of time dilation is on the order of 1 part in 10¹⁰. This would seem insignificant, except it amounts to milliseconds per year, and so it does make a practical difference.

For something moving 100 times slower, like a car, time dilation would be 10,000 times smaller. So time in a car driving at 90 miles per hour slows down by one part in 10¹⁴ relative to a stationary observer.

Tape measures

The math in the section above is essentially the same as the math in the post explaining why it doesn't matter much if a tape measure does run exactly straight when measuring a large distance. They both expand an expression derived from the Pythagorean theorem in a Taylor series.

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