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 – 0.567²) = 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 v²/c², 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.
The biggest relativistic adjustment to GPS clocks is one based on general relativity. Roughly put, clocks that are higher in a gravity field run faster.
Here’s a short summary from https://physicscentral.com/explore/writers/will.cfm
But in a relativistic world, things are not simple. The satellite clocks are moving at 14,000 km/hr in orbits that circle the Earth twice per day, much faster than clocks on the surface of the Earth, and Einstein’s theory of special relativity says that rapidly moving clocks tick more slowly, by about seven microseconds (millionths of a second) per day.
Also, the orbiting clocks are 20,000 km above the Earth, and experience gravity that is four times weaker than that on the ground. Einstein’s general relativity theory says that gravity curves space and time, resulting in a tendency for the orbiting clocks to tick slightly faster, by about 45 microseconds per day. The net result is that time on a GPS satellite clock advances faster than a clock on the ground by about 38 microseconds per day.