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First of all, let me state that my question is not about time dilation as observed between inertial reference frames moving at different velocities, which is completely symmetrical. It's about time dilation due to gravitation/acceleration (e.g. the case of individuals againg at different rates).

Reading the following thread:

Does velocity or acceleration cause time dilation?

a couple of doubts crossed my mind.

An user answers:

you can't say the time dilation is due to velocity or due to acceleration, just that it's due to relative motion.

It's at the end of the second answer, from above. Note that the guy has a big bunch of gold badges, so I'd trust him.

Another user, below, says:

Now as a gravitational acceleration has the same effect on clocks as a rocket powered acceleration, and as a clock in a rocket powered acceleration will get slower and slower the longer the acceleration occurs, then a clock that exists say for 100 million years on the earth and is subject to gravity for the entire period and therefore an acceleration for 100 million years will tick more slowly now in 2015 than it did 100 million years ago.

So, a clock on Earth will slower steadily with respect to a clock in the intergalactic space, even if those two clocks are in no relative reciprocal motion. Is that correct? Is the first guy wrong?

A closely related issue: 1g of acceleration may seem an eerie thing, but it's not so: it has been calculated that a spaceship accelerating at a constant rate of 1g, can manage a tour along the observable universe, and get back to earth, within the lifespan of those inhabiting the ship. Obviously, they will find Earth aged billions of years. Now, if acceleration by gravity is the same as acceleration by rocket drive, we on Earth should see the rest of the universe aging at a much faster rate with respect to us, like those on the spaceship.. After all, it's the same 1g both for them and for us. Am I right? (Obviously not, but why?)

Another thought experiment: We are in a region more or less empty space, and suddenly a messive object pops up. We begin accelerating toward that object, at a conspicuous rate, or, in other words, are put in relative motion with respect to the object, and our respective relative velocities grow steadily. Note, however, that we are in free fall, and our accelerometer says "0 g". We experiment no forces, and still our velocity increases. Will we experiment any phenomena of time dilation and aging differences with respect to the people on the surface of the object?

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Time dilation is caused by gravitational fields, or equivalently, acceleration, and by relative motion. The wiki article at https://en.m.wikipedia.org/wiki/Time_dilation. A much more precise and complete derivation is given at the Living Reviews article by Ashby at http://relativity.livingreviews.org/Articles/lrr-2003-1/

Both include in the equations the effect of a gravitational field (I.e, General Relativity), approximated to first order by the Newtonian potential (Living Reviews goes further, includes the earth's rotation and asymmetry), and the velocity of the orbiting satellite (derived from the General Relativity metric -- GR, the metric and the invariance of the ds or equivalently the d$\tau$ term -- but accounting for the Special Relativity, SR, time dilation of relatively moving bodies). There is also a terms for non orbiting bodies.

The basic answer from that is that the SR effect makes satellites orbiting near the earth loose time (time dilation, time goes slower) - this happens eg for the ISS. The pure gravitational effect makes satellites further out gain time (clocks go faster than clocks on earth). The sum of the two effects has SR dilation dominating near the earth, but gravitational effects stronger (with respect to clock on the earth) the farther you are. There are plots of this in one graph, combining it all.

Of course, the speeds in those plots are for satellites in orbit around the earth, very non-relativistic. If they are traveling much faster, and the spaceship or satellite is far enough away from gravitational-causing bodies, the SR factor dominates and you can stay younger when you go fast enough. Even more so if you go into a deep gravitational well, as in the movie Interstellar.

The wiki article also has the equations for a constantly acccelerated spaceship, comes out to be that proper time (of an observer or spaceship) is a log or inverse hyperbolic function of the coordinate time (say at infinity). That what you asked for, the equation is there and you can plug in the acceleration. It's not a huge effect instantaneously at an acceleration g (the earth's gravitational well is not that deep), but it builds up over time. I'm sure there's more references on this constant acceleration case.

Interestingly, the Living Reviews extensive paper relates some of the history of the time delay discovery for GPS. Some of the engineers and others did not believe it was a real effect -- this was around 1977, and wanted to ignore it; the clocks onboard were built with a frequency that did not account for the effect, but they all decided, that just in case it was real, they would have a synthesizer for the clock oscillator that could be changed the small amount that might be needed. After some of the initial testing in orbit they adjusted the clock frequencies. Of course SR and GR, or just GR, predicted the values, but not everyone believed the physicists. There are other interesting points in that article. They wanted the clocks in orbit to be adjusted for clocks on the earth surface where the receivers would be, but of course those were also in a gravitational field (g). So they calculated the effects on those also with respect to the gravitational center of the earth where there is no field, and offset it so they would all match (to the accuracy needed) the international time standard in Paris. So, a lot of very fine timekeeping was done in that whole effort.

If you travel closer to the speed of light and/or in a stronger gravitational field (I.e, a greater spacetime curvature), the effects are bigger, measurement is easier, but you somehow have to get the energy needed to get there.

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  • $\begingroup$ "It's not a huge effect instantaneously at an acceleration g (the earth's gravitational well is not that deep), but it builds up over time." Exactly. So, I wonder why our clocks, on Earth, don't lag behind the average cosmic clock: after all, it's like we are on a starship constantly accelerating at 1g.. $\endgroup$ – MadHatter Nov 19 '16 at 15:11
  • $\begingroup$ Well, it's different. For cosmological effects, you need to compare things using the metric for cosmology, the FLRW solution. The universe is expanding, and also slowly accelerating. The expansion causes redshifts,a factor of 1000 for the cosmic background. But there's no time at infinity, the universe has no flat limit at infinity, it also expands there/then. Use cosmological comoving time as a reference. We don't increase v with the g the earth firmness holds us, not free floating, account for that also. Time is practically measured at the inernational time keeping ref in Paris. $\endgroup$ – Bob Bee Nov 19 '16 at 21:19
  • $\begingroup$ So bottom line, we take ourselves as one reference, and calculate cosmological effects using that FLRW metric. Eg we adjust the GPS satellite clocks so it measures time in that Paris reference (plus or minus some really small effects which are also adjusted). And if we have to change the reference in some calculations for the earth's center, with no g, we do that also. $\endgroup$ – Bob Bee Nov 19 '16 at 21:24
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It's my answer that you're referring to when you cite the statement:

you can't say the time dilation is due to velocity or due to acceleration, just that it's due to relative motion

so let me clarify that my statement assumes we are talking abut special relativity so spacetime is flat and there are no effects due to gravity. The other answer is quite correct that in a curved spacetime there will also be gravitational time dilation. In a curved spacetime the time dilation will have a contribution due to relative motion and a contribution due to gravitational effects.

Once you need to include gravitational time dilation the calculations can get quite complicated, but the underlying principle is actually straightforward. I go into this in some detail in What is time dilation really? so I'll direct you to that question rather than repeat it here.

Re your closely related issue: the gravitational acceleration at the Earth's surface is 1g but it falls as you mive away from the Earth. We don't see the rest of the universe aging at a much faster rate with respect to us because of the fall off of gravity with distance. In a hypothetical situation where the gravitational acceleration remained constant with height the rate of aging of the rest of the universe would increase without limit.

Understanding the difference between gravitational acceleration and regular acceleration can be a bit tricky. If you're interested I discuss this in How can you accelerate without moving?

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  • $\begingroup$ Thanks for your answer. I'll read the answers you pointed out, and ask here, if you don't mind, should something be still unclear. $\endgroup$ – MadHatter Nov 16 '16 at 15:40
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I would like to dispute the fact that gravity exists. When I accelerate in a car I feel myself pushed into the back rest of my seat (Newton's 3 rd law for for every action there an equal and opposite reaction). However when I jump out of a plane I feel nothing but the wind blowing in my face yet I am accelerating toward earth. In both cases I am gaining energy. Now, there is the matter of Einstein's curved space time. He said that the area surrounding earth experienced a slowing down of time this has been confirmed by experiments. The closer you get to earth the slower time gets. Now if you placed time coordinates throughout space the coordinates would be all the same except when you got close to bodies of matter where they would get larger the closer you got to a body. Now Newtons 1st law says a body in motion stays in motion, a body at rest stays at rest. Now if a body was traveling through space on our equal coordinates it would remain at a constant speed until I got close to the earth where the coordinates were bigger then it would appear to accelerate. Notice I said appear to speed up even though it really hadn't as the coordinates were simple bigger. It simply moving at a constant speed according to Newtons first law. That can be shown to be true through a ratio. d/t We all know that if you increase the distant in the numerator you increasing the speed hence the hence you are pushed into the back rest of a car, you are accelerating but if decrease denominator and make the coordinates larger and fewer than in order to keep going the same speed the distance has to increase to keep the ratio the same, thus it appears to accelerate. Ie. If you are going 30 km/hour when you reach earth you are really going 30 km/half hour which is 60km/hour.

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