Yes, a clock on the surface of Mars or Pluto does indeed run at very slightly different rate relative to a clock on the Earth's surface - the gravitational potential is not as great on the surface of Mars or Pluto as on the surface of the Earth, which makes the clocks run very slightly faster than they do on the surface of the Earth. Neither clock is in an inertial frame.
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Proper length: Length of object measured at rest / object measured in the frame ... Proper time: Time interval Δt = t2 – t1 between two events measured in the ...Time dilation/length contraction - HyperPhysics
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The length of any object in a moving frame will appear foreshortened in the ... time measured in the frame in which the clock is at rest is called the "proper time
Is there a place on earth where time actually slows down?
This is a follow-up question to Is there a place on earth where time effectively stops?.
4 Answers
2.1k Views • Upvoted by
Leo C. Stein, Ph.D. from MIT, B.S. from Caltech. Specializing in gravity. •
David Simmons-Duffin, Physicist
According to relativity, if an observer looks at a clock, the rate at which time is progressing depends on the velocity of the clock relative to the observer, and the gravitational potential the clock is sitting in, relative to the gravitational potential where the observer is.
So if you travel very fast, other people would see time for you going slower. (And you would also see time for everyone else going slower.) Things get complicated if the relative velocity between the you and the observers is not constant. However, the upshot is that if you accelerate and then decelerate, you can actually end up aging less quickly then everyone else (see http://en.wikipedia.org/w
Secondly, time passes more slowly in higher gravitational potentials. Therefore, if you are at low altitude, time passes more slowly for you than for someone at the top of Mt. Everest.
Sadly, as far as humans on earth are concerned, both of these effects are quite small. The relative-motion time dilation is given by
t=t 0 1−v 2 /c 2 √
where t_0 is time elapsed in the stationary reference frame, v is the relative velocity, and c is the speed of light. The human airspeed record appears to be about 3500 km/hr (in an SR71 airplane), or about 1000 m/s. The speed of light is 3e8 m/s, so the time in the moving frame appears to be slowed by about a factor of one part in 10^11. If you fly around for a while, this is detectable by the best clocks, but certainly not a very noticeable difference.
As for the gravitational time dilation caused by a massive object with mass M, here the time difference is given by
t=t 0 1−2GM/rc 2 − − − − − − − − − − − √
where t_0 is the time measured very far from the massive object, G is the gravitational constant, and r is the distance from the center of the massive object. The observed difference between people at two distances r_1 and r_2 from the center of the earth would be
t 1 /t 2 =1−2GM/r 1 c 2 1−2GM/r 2 c 2 − − − − − − − − √
.
Since the numbers involved here strain the abilities of a normal calculator, we can do a number of expansions in the small quantities. This gives us that
t 1 /t 2 ≈1−GMc 2 hr 2 e
where h is the difference between the heights of the two clocks, and r_e is the radius of the earth. Plugging in h = 10^4 m , which is about the height of Mt. Everest, you get a time dilation of about one part in 10^12. Also very small.
I should point out that while we can't use relativity on earth to keep people young in any practical way, these effects can be used to keep smaller, inanimate objects "young". For example, a particle accelerator can accelerate subatomic particles to very very close to the speed of light. If such a particle has a fixed decay lifetime, when it is moving fast, its lifetime can be observed to be increased dramatically.
So if you travel very fast, other people would see time for you going slower. (And you would also see time for everyone else going slower.) Things get complicated if the relative velocity between the you and the observers is not constant. However, the upshot is that if you accelerate and then decelerate, you can actually end up aging less quickly then everyone else (see http://en.wikipedia.org/w
Secondly, time passes more slowly in higher gravitational potentials. Therefore, if you are at low altitude, time passes more slowly for you than for someone at the top of Mt. Everest.
Sadly, as far as humans on earth are concerned, both of these effects are quite small. The relative-motion time dilation is given by
where t_0 is time elapsed in the stationary reference frame, v is the relative velocity, and c is the speed of light. The human airspeed record appears to be about 3500 km/hr (in an SR71 airplane), or about 1000 m/s. The speed of light is 3e8 m/s, so the time in the moving frame appears to be slowed by about a factor of one part in 10^11. If you fly around for a while, this is detectable by the best clocks, but certainly not a very noticeable difference.
As for the gravitational time dilation caused by a massive object with mass M, here the time difference is given by
where t_0 is the time measured very far from the massive object, G is the gravitational constant, and r is the distance from the center of the massive object. The observed difference between people at two distances r_1 and r_2 from the center of the earth would be
Since the numbers involved here strain the abilities of a normal calculator, we can do a number of expansions in the small quantities. This gives us that
where h is the difference between the heights of the two clocks, and r_e is the radius of the earth. Plugging in h = 10^4 m , which is about the height of Mt. Everest, you get a time dilation of about one part in 10^12. Also very small.
I should point out that while we can't use relativity on earth to keep people young in any practical way, these effects can be used to keep smaller, inanimate objects "young". For example, a particle accelerator can accelerate subatomic particles to very very close to the speed of light. If such a particle has a fixed decay lifetime, when it is moving fast, its lifetime can be observed to be increased dramatically.
David Kahana,
Yes, a clock on the surface of Mars or Pluto does indeed run at very slightly different rate relative to a clock on the Earth's surface - the gravitational potential is not as great on the surface of Mars or Pluto as on the surface of the Earth, which makes the clocks run very slightly faster than they do on the surface of the Earth. Neither clock is in an inertial frame.
But there is also an effect that goes in the opposite direction since there is also relative motion between a clock fixed on the Earth's surface and one fixed on the surface of Mars or Pluto.
These effects are not large, but they do exist.
Clocks in inertial frames in another galaxy, or in equivalent gravity wells, would run at the same rates if they were measured in another galaxy, but over large distances the effect of the cosmological redshift causes clocks, or rather vibrating atoms and molecules that emit light, to appear to us to run more slowly than clocks in our galaxy. For very distant galaxies, this becomes a large effect.
Of course for very distant galaxies we have no chance whatever of actually synchronizing the clocks at all, so we can't possibly make a measurement, as we could do, in principle, for Mars and Pluto.
Comparisons of relative motion at cosmological distances are likewise very different.
But there is also an effect that goes in the opposite direction since there is also relative motion between a clock fixed on the Earth's surface and one fixed on the surface of Mars or Pluto.
These effects are not large, but they do exist.
Clocks in inertial frames in another galaxy, or in equivalent gravity wells, would run at the same rates if they were measured in another galaxy, but over large distances the effect of the cosmological redshift causes clocks, or rather vibrating atoms and molecules that emit light, to appear to us to run more slowly than clocks in our galaxy. For very distant galaxies, this becomes a large effect.
Of course for very distant galaxies we have no chance whatever of actually synchronizing the clocks at all, so we can't possibly make a measurement, as we could do, in principle, for Mars and Pluto.
Comparisons of relative motion at cosmological distances are likewise very different.
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