# If the equivalence principle is true, and gravity causes time dilation, why doesn't acceleration also cause time dilation?

I'm sure I must have something very confused but I'm just reading through SE questions and articles and this seems to be a contradiction. Until recently I had no problem as I assumed that acceleration did in fact cause time dilation, then I read numerous answers from people stating clearly that when calculating time dilation, only the distance travelled matters, not the path taken. For instance it is claimed that for a body experiencing centripetal acceleration, only the tangential velocity is relevant to the time dilation effect, not the radial acceleration.

I have 2 issues with this:

1. The Equivalence Principle states that being in an accelerating reference frame is indistinguishable from standing still in a gravitational field. We know that gravity causes time dilation, so why not acceleration?
2. The Twin Paradox is supposed to be resolved by the acceleration effect. Otherwise the time dilation is completely reciprocal and no difference will be seen between the clocks. Therefore acceleration must be causing time dilation?

How can these statements be reconciled? One page tells me acceleration is irrelevant, the next tells me it's equivalent to gravity. Both cannot be true.

• What makes you think that it doesn't? If you have two clocks in an accelerating rocket, one at the tip, the other at the other end, why do you think they will be synchronized? The second is not quite right, acceleration is not what explains the twin paradox.
– MBN
Mar 3, 2021 at 10:10
• Where have you read that acceleration doesn't cause time dilation, as that most certainly isn't the case? Mar 3, 2021 at 10:11
• @Eletie Here for example: physics.stackexchange.com/questions/513012/… Mar 3, 2021 at 10:16
• @MBN See the link in my other comment. Can you elaborate on the twin paradox? Taking the usual Alice & Bob, Alice goes on the rocket, Bob stays at home. My understanding is that since time dilation due to relative velocity is reciprocal, both observers would see the other's clock running slow. In every treatment of this issue I have read, the paradox is supposedly resolved by accounting for the fact that Alice has to undergo phases of acceleration. Mar 3, 2021 at 10:19
• @Eletie Re Where have you read that acceleration doesn't cause time dilation, as that most certainly isn't the case? -- This most certainly is the case in the sense that there is no acceleration term in special relativistic time dilation formulae. What you will find is that time dilation in special relativity depends solely on velocity, but of course velocity depends on acceleration. Mar 3, 2021 at 14:15

I think the confusion here stems from trying to learn a concept by an accumulation of isolated statements in words. That may be possible for some concepts but it is not the right approach with something like time dilation. Rather, you come to understand it by building up a grasp of the wider subject of spacetime, first as it is treated by special relativity, and then by general relativity. Asking about the impact of acceleration on time dilation is like asking about the impact of acceleration on a total journey time in Newtonian physics: it is a relevant part of the physics but it does not on its own tell you very much (e.g. a large acceleration does not necessarily make the journey time smaller; it might make it longer if it is in the wrong direction). Time dilation is the name we give to the fact that the accumulation of proper time along certain worldlines is different from the accumulation of proper time along certain other worldlines, and it is connected to the way events in spacetime may or may not be said to be simultaneous depending on how systems of coordinates are set up. In the twin paradox what acceleration causes is the departure of one of the worldlines from a line of most proper time.

Here is the twin paradox in a nutshell.

Suppose there are two events $$A$$ and $$B$$ with a time-like separation. These events could be, for example, the departure and return of a rocket on planet Earth. Of all the timelike curves extending between these two events, there is a unique one along which the accumulated proper time is the most. That unique line is the one followed by a body undergoing inertial motion. For all other lines the accumulated proper time is smaller.

That's it. You can now add some story colouring by having one twin follow the unique line of most proper time, and another twin follow some other worldline. This helps to make clear that by "proper time" we mean the amount by which ordinary physical things like wrist-watches, hearts and bodies proceed through their ordinary physical evolution in their own instantaneous rest frame at each moment.

• I think I understand. It's a case of asking the wrong questions. I'm familiar with that from QFT... "how can the electron be in two places at once?" <-- wrong question, causes more confusion than answers. Been a while since I was actively studying this stuff but it's coming back to me. I'm familiar with the concept of a metric and calculating proper time, so I see what you're saying. The equations don't depend directly on acceleration, but on the path taken between the two events. There is a unique worldline between them undergoing inertial motion (geodesic?) which has the longest (1/2) Mar 4, 2021 at 13:46
• possible proper time. All other worldlines between the events will necessarily have a shorter proper time. Is it true to say that such paths always involve some form of acceleration? Being that they are not "inertial"? (2/2) Mar 4, 2021 at 13:48

The Equivalence Principle states that being in an accelerating reference frame is indistinguishable from standing still in a gravitational field. We know that gravity causes time dilation, so why not acceleration?

1: We know gravity causes time dilation

2: We know acceleration does not cause time dilation

3: We can distinguish between gravity and acceleration

If we kidnap a guy, put him in small windowless room, said guy can not tell if he is in an accelerating rocket or not.

The Twin Paradox is supposed to be resolved by the acceleration effect. Otherwise the time dilation is completely reciprocal and no difference will be seen between the clocks. Therefore acceleration must be causing time dilation?

Well, it is said really often that acceleration is important. It is said by amateurs mostly. Also acceleration not being important is said quite often, as some wiser folks try to correct the amateurs.