8
$\begingroup$

If, you hypothetically started 2 clocks at exactly the same time, and placed one on Earth, and one on a rocket to Mars, and setup a video feed of both clocks, how would they compare as the 2nd clock approaches Mars?

If on the shortest possible distance between Earth and Mars light takes 3 minutes to travel from one planet to the other, then I would assume the clock would be 3 minutes out of sync by the time it reaches Mars? Does this mean that from Earth’s perspective, the clock heading to Mars would appear ever-so-slightly in slow motion?

$\endgroup$
11
  • 4
    $\begingroup$ You may want to look up time dilation. $\endgroup$ May 1, 2023 at 8:12
  • 10
    $\begingroup$ @fyrepenguin Is this time dilation, or just the Doppler effect? Critically, if you brought the Mars clock back to Earth, it surely wouldn't be off by 6 minutes when it arrived. $\endgroup$
    – Cadence
    May 1, 2023 at 11:09
  • 4
    $\begingroup$ On a video, the only clock effects you'd be able to see are due to the time it takes the signal to travel & the Doppler effect. Time dilation is what happens after you compensate for that stuff. We have a few questions on Physics.SE about the time dilation of Earth vs Mars due to the combined effects of the orbital speeds and the gravitational potentials of the Sun, Earth & Mars. Eg, physics.stackexchange.com/q/213106/123208 But those effects are too small to see on a video. $\endgroup$
    – PM 2Ring
    May 1, 2023 at 16:05
  • 1
    $\begingroup$ FWIW, here's a graph of Earth time dilation, relative to the clock of an observer at rest relative to the Sun but unaffected by the Solar System's gravity well. i.sstatic.net/sJioV.png The calculations use the correct SR & GR equations, but assume that the Earth is moving in a perfect Kepler ellipse around the Sun, ignoring all the other bodies in the Solar System. $\endgroup$
    – PM 2Ring
    May 1, 2023 at 16:08
  • 2
    $\begingroup$ @jwenting Depends on how good your clocks are. Time dilation is important when you have high precision clocks. $\endgroup$
    – John Doty
    May 2, 2023 at 16:40

2 Answers 2

21
$\begingroup$

The main idea here is just that it takes time for light to travel, which is the same thing as the Doppler shift (the longitudinal version which applies at all speeds, not the relativistic transverse one). Relativistic effects like time dilation (the moving clock ticks slower) and gravitational redshift (the clock at the bottom of Earth's gravity well ticks slower than the one in flight) cause small differences, but the travel time Doppler is still obvious even when the others are ignorable.

The clocks stay synchronized as they move apart. The only reason they appear to differ is that when one of them is far away, you're not seeing it in the present. You're seeing it as it was some time ago — and the farther away it is, the longer ago you're seeing it. When the clock traveling to Mars is one light-minute away, you're seeing it as it was one minute ago, because it took one minute for the image taken a minute ago to travel to you at the speed of light. When the Mars clock is two light-minutes away, you're seeing it as it was two minutes ago, and so on.

One light-minute is just under 18 million km. At a speed of 11 km/s, it takes about 19 days to get that far away. The moving clock appears to be ticking slower, at 99.9963% of the rate of the stationary clock, so it loses one minute every 19 days. The fractional change in frequency (1 part in 27,360) is just the ratio of velocities, because in time $t$ it moves $vt$ farther away, causing an additional signal travel time delay of $vt/c$. That's the Doppler shift.

As in Cadence's comment, this only happens on the way out. If the mission includes a return trip, the sign of the shift reverses. The traveling clock appears to be ticking slower when moving away, but when the relative velocity goes to zero, the two clocks tick at the same apparent speed, maintaining a constant offset. It doesn't stay zero for more than an instant, though. As in Paŭlo Ebermann's comment, since the planets have different velocities, the distance between them is constantly changing, with direction depending on what part of the orbital cycle we're in, so the moment of zero range rate is not in Mars orbit itself unless the timing gets extremely lucky. When the traveling clock turns around and heads back, it appears to be ticking faster, again by exactly the Doppler factor of $v/c$, so that it comes gradually back into synch when it arrives home. As in PM 2Ring's comment, any difference between the two clocks when they are brought back to the same position constitutes a measurement of relativistic effects due to the differing acceleration histories of the two clocks.

$\endgroup$
3
  • 2
    $\begingroup$ While the clock is at Mars, the distance also continuously is changing. You might want to note that too. $\endgroup$ May 1, 2023 at 23:35
  • 1
    $\begingroup$ Don't forget that the video would never be perfectly in sync anyway, as the coding/decoding takes a finite time too, usually in the region of one to three seconds, though this could be assumed to be a constant offset. $\endgroup$
    – MikeB
    May 3, 2023 at 6:35
  • 1
    $\begingroup$ @MikeB: The question supposes you "set up a video feed of both clocks", so you'd only care about the difference in delay between the two respective encoders. $\endgroup$
    – Ben Voigt
    May 3, 2023 at 14:51
11
$\begingroup$

Since any practical means of interplanetary communication will rely on electromagnetic waves (probably radio, but laser beams would also qualify), we can assume that the video signal from the Mars clock travels at the speed of light (299792458 m/s).

The distance between Earth and Mars varies between 33.9 million miles (54.6 Gm, 3.04 light-minutes) and 250 million miles (401 Gm, 22.3 light-minutes), depending on where each planet is in its orbit.

As the spacecraft gets farther from Earth and closer to Mars, its clock will appear to gradually slow down, as its signal takes longer to reach Earth. Once it lands, the clock skew will vary between 3 and 22 minutes based on the planets' relative position.

Relativistic time dilation would have some effect on the clock skew, but not much. AFAIK, the fastest spacecraft ever built is the Parker Solar Probe, with a top speed of 0.00064 c. Using the time dilation formula $\frac{t}{t'} = \sqrt{1 - (\frac{v}{c})^2}$, the clock on the spacecraft would run at 0.9999998 times the speed of the clock on Earth. Mars itself moves at a maximum speed of only 0.0000884 c relative to Earth, giving a time dilation factor of 0.999999996. The difference is just too small for anyone to notice.

$\endgroup$
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.