I've read countless answers and other sources on the question of whether time dilation is caused both by velocity and acceleration or only by velocity, but they all look at things only from an inertial frame and/or use the popular version of the clock postulate, so they don't seem to answer the questions I pose here. Many of them also talk about it in terms of relative lengths of different paths through spacetime, but I think that sidesteps the question of whether acceleration itself can cause those lengths to change in accelerating frames (e.g., by causing a different metric to apply).
It's stated in numerous sources (e.g., the article "Does a clock's acceleration affect its timing rate?" by Don Koks, posted by John Baez, and this section in Wikipedia) and in answers by reputable users of this site (e.g., here and here) that the clock postulate of special relativity says that time dilation (among other relativistic effects) is caused only by relative velocity and that acceleration itself has no direct effect on it.
But this seems problematic to me because it only ever seems to be true in inertial reference frames. For example, in the (non-inertial) frame of the traveling twin in the twin paradox, the earthbound twin's time slows down during each inertial leg of the trip, but during the turnaround, it speeds up to an extent that indicates that the traveler's acceleration towards the earth is equivalent to a gravitational field that he resists while it pulls the earth towards him thus causing him to experience something equivalent to gravitational time dilation relative to the earth. How can it be said that this kind of time dilation is caused by velocity rather than acceleration?
Edit: To be more explicit about what the clock postulate claims according to some: In a comment, Dale says that "In a non-inertial frame, time dilation depends on velocity and position, not acceleration." The answers given here so far, however, don't seem to support this statement. How can it be supported? Edit: Dale has addressed this in his answer.
Now, there seems to be an alternative formulation of the postulate. According to this paper:
The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric.
Perhaps someone here will show me that this formulation is logically equivalent to the other one (in all frames, including non-inertial ones), but from what I can tell, it actually allows acceleration to cause time dilation. For example, in the traveling twin's frame where you have to use a metric like the Rindler metric during the turnaround (per the paper, "the restriction to Minkowski spacetime and inertial motion has been dropped"), you find that his curve is shorter than the earthbound twin's during that acceleration and thus that his own time dilates relative to the earth's, apparently due to his acceleration towards it. Therefore, unlike the other version of the postulate, this one works in non-inertial frames and seems to be consistent with acceleration directly causing time dilation. Is this true? If so, is this version of the postulate more correct?
Best Answer
This is indeed the correct general formulation of the clock hypothesis. This formulation applies for all reference frames, inertial or not, and for all spacetimes, flat or curved. The only restriction is that the coordinate basis must have one timelike basis, $dt$.
Because of this, time dilation can be written as $$\frac{1}{\gamma}=\frac{d\tau}{dt}$$ where $\gamma$ is the time dilation factor and $d\tau$ is the proper time, which is related to the metric by $$ds^2=-c^2 d\tau^2=g_{\mu\nu}dx^\mu dx^\nu$$
So, for an inertial frame we have $$ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$$ $$ \frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$ $$\frac{1}{\gamma}=\sqrt{1-\frac{v^2}{c^2}}$$ which is the usual familiar time dilation formula.
Note that it has the property that it depends only on the velocity and not the acceleration. When calculating $d\tau/dt$ we get terms like $dx^2/dt^2$, which is the square of a component of velocity, not an acceleration like $d^2x/dt^2$ would be.
Although the general formula does work in non-inertial frames, it still does not give a situation where acceleration directly causes time dilation. Let’s work it out for the Rindler metric you mentioned. For convenience I will use units where $c=1$: $$ds^2= -(gx)^2 dt^2+ dx^2+dy^2+dz^2$$ $$\frac{d\tau^2}{dt^2}=(gx)^2-\frac{dx^2}{dt^2}-\frac{dy^2}{dt^2}-\frac{dz^2}{dt^2}$$ $$\frac{1}{\gamma}=\sqrt{(gx)^2-v^2}$$
Now, you might be tempted to say “look, it has $g$ which is a pseudo gravitational acceleration, so the Rindler time dilation is directly related to acceleration”. However, closer inspection shows that it is not just $g$, but $gx$ which is the form of a gravitational potential, not gravitational acceleration.
Furthermore although $g$ has units of acceleration, it is a property of the particular coordinates chosen, not the acceleration of any worldline. The time dilation of a specific object does not depend on that object’s acceleration, only its position and velocity with respect to the chosen coordinates. If an object is momentarily at rest ($v=0$) at some $x=x_0$ then it’s time dilation is $\gamma=1/(gx_0)$, regardless of what that object’s acceleration is. So this time dilation term is a function of position, not acceleration.
It turns out that this is a general fact. In the metric the terms $g_{\mu\nu}$ are functions of position only, not velocity nor acceleration. And so when you divide by $dt^2$ you only get terms that are functions of position times velocities, like $$g_{xy}\frac{dx}{dt}\frac{dy}{dt}$$ You simply cannot get anything else because of the form of the metric.