# Q & A: relativity of time

Q:
I want to apply time dilation to a practical (not really) example. Can you please show how I would calculate this? (I know the answer will be pitifully insignificant, but this is for my own interest and instructional purposes.) Given: two identical people exist on Earth and Mars and are born at the same instant, and will both die precicely at 80 years local time. Neglecting all other effects beyond the relative speed of the planets, how much longer will the person on Earth experience his life as being? (assuming he could somehow be aware of the instant the Martian guy died.) You can assume Earth velocity = 29.8 km/s and Mars velocity = 24.1 km/s. Please make all reasonable simplifying assumptions. I am interested in understanding what formulas would be used, how, and if things like the motion of the solar system / Milky Way through spacetime would be relevant to the problem. I'm not interested in an ironclad answer (although I am interested in the order of magnitude.) Thanks so much! (I know this is probably more like work than some questions.)
- Scott
Denver, CO
A:
One aspect of your question is easy: "how much longer will the person on Earth experience his life as being?" Each person experiences a 100% self-consistent local time. That means all their local clocks (heartbeat, aging, wristwatch,....) stay synchronized. So each experiences an 80 year life.

To figure out whether the lifespan of the Martian seems lengthened or shortened to the Earthling, and by how much, requires a little calculation. Because the distance between these planets doesn't, on average, change much over time, we will ignore any changes in the signal-transmission time from the start to the end of these lives. Thus we end up with the two observers agreeing on which one lives longer or shorter, unlike in cases where the signal transmission times change and both can believe that the other one lives longer.

We are happy to follow your excellent advice to make all appropriate approximations (calculating things only to lowest order) but it turns out that that is not consistent with only considering effects of their relative speed. The reason is that the same force holding things in orbit (gravity) shows up directly in the General Relativistic time shifts. It turns out that for these orbital problems, its effect is twice as big as the Special Relativistic velocity effects.

Let's look at the whole problem from the point of view of someone at rest with respect to the sun but well away from it gravitationally. (Directly using the E or M frame is a nuisance, because these have important acceleration effects.) He sees both E and M slowed down by SR effects, by a factor sqrt(1-v2/c2) ~= 1-v2/2c2. (The last approximation is plenty good here.) v is the orbital speed, c is the speed of light. Of course E and M have different v's, as you noted.

Meanwhile, each clock rate is changed by the GR gravitational effect on time by a factor of  1+Φ/c2, where Φ is the gravitational potential. The classical virial theorem says Φ=-v2, on the average over an orbit.

So our observer sees each as being slowed by by a factor 1-3v2/2c2. (Again to lowest order.) Since E's v is almost exactly 10-4 c, our clocks seem slow by a factor 1-1.5*10-8. M's clocks are only slowed by (24.1/29.8)2 as much, about 65% as much change.  So that's roughly 0.5*10-8 difference between the two. Out of 80 years (roughly 25*108 seconds), that's a roughly 12 second difference. We Earthlings think the Martian's life is tragically shortened by that amount.

Mike W.

(published on 11/03/2010)