Time Dilation: Accelerating vs Inertial Frame

[Buckethead]

This is probably common knowledge to relativity aficionados but at Example 7.3 in this paper:

https://www.farmingdale.edu/faculty/peter-nolan/pdf/relativity/Ch07Rel.pdf

I was surprised to read the author showing that a ship accelerating at 1g from rest for 1 hour and reaching a speed of 1610km/s as seen by a stationary observer will show exactly the same amount of dilation as another ship flying past the same observer for 1 hour at a speed of 1610km/s. The amount of dilation in both cases is 1.0000144 hr.

The reason I find this counter-intuitive is obvious. The accelerating ship is not going at the maximum speed during the whole trip as is the inertial ship, so I would have thought it would show less dilation.

I thought about this and realized the difference must lie in the fact that the accelerating ship is experiencing gravity in addition to velocity which the inertial ship is not. And since gravity slows time, it is this gravity due to acceleration that is making the two equal.

Is my realization correct? I have read in other threads in this forum that acceleration (i.e. a ship accelerating) does not affect time dilation, only the resulting velocities at each point matter. I apologize that I cannot refer to such posts, but I remember them being in response to questions about the twin paradox. But if that were true, then the two times should be different due to the differences in the sums of the instantaneous velocities during acceleration. Can someone clarify this? Thanks.

 

[phinds]

Time dilation is an observation from a "rest frame" of an object in that rest frame that has non-zero velocity. It doesn't have an "amount", just an instantaneous value based on the velocity relative to the rest fame, so if objects at time T are going the same speed in the rest frame, it's irrelevant how they got to that speed, they will have the same value of time dilation.

Now, differential aging is different and can depend on total path-velocity.

 

[PeroK]

This is probably common knowledge to relativity aficionados but at Example 7.3 in this paper:

https://www.farmingdale.edu/faculty/peter-nolan/pdf/relativity/Ch07Rel.pdf

I was surprised to read the author showing that a ship accelerating at 1g from rest for 1 hour and reaching a speed of 1610km/s as seen by a stationary observer will show exactly the same amount of dilation as another ship flying past the same observer for 1 hour at a speed of 1610km/s. The amount of dilation in both cases is 1.0000144 hr.

The reason I find this counter-intuitive is obvious. The accelerating ship is not going at the maximum speed during the whole trip as is the inertial ship, so I would have thought it would show less dilation.

I thought about this and realized the difference must lie in the fact that the accelerating ship is experiencing gravity in addition to velocity which the inertial ship is not. And since gravity slows time, it is this gravity due to acceleration that is making the two equal.

Is my realization correct? I have read in other threads in this forum that acceleration (i.e. a ship accelerating) does not affect time dilation, only the resulting velocities at each point matter. I apologize that I cannot refer to such posts, but I remember them being in response to questions about the twin paradox. But if that were true, then the two times should be different due to the differences in the sums of the instantaneous velocities during acceleration. Can someone clarify this? Thanks.

I agree with you. The time dilation in the scenario he describes is entirely due to the relative velocity at each point in time. In particular, a clock that accelerates from rest to some speed $v$ cannot record the same proper time as a clock that travels at speed $v$ throughout, as measured in some IRF.

Also, his equation (7.38) is wrong. There is no general equivalence between "gravity" and "acceleration". The equivalence is between "gravity" and an "accelerating reference frame".

In equation (7.32) the distance $y$ refers to the distance between two clocks that are both in an accelerating reference frame. This cannot be replaced with the total distance traveled by a single acclerating clock as measured in an inertial reference frame.

That section, IMHO, is all nonsense.

PS equation 7.38 gives the rate of time dilation once a distance $y$ has been travelled. But, of course, the proper time of the clock is the integral of this factor. Not simply this final time dilation factor multiplied by the coordinate time of the motion!

In any case, the logic of this whole section is totally wrong.

 

[Ibix]

The amount of dilation in both cases is 1.0000144 hr.

Agree with @PeroK. It is true that the Lorentz gamma factor of something doing $v$ is the same however the object attained that speed. But that means that, working in an inertial frame, the elapsed time for an object during coordinate time $T$ is $$\int_0^T\frac{dt}{\gamma(t)}$$which is obviously not the same for an object where $\gamma(t)$ is a constant and where $\gamma(t)=\sqrt{1-(at/c)^2}$.

 

[SiennaTheGr8]

If a rocket with constant proper acceleration (magnitude $\alpha$) starts from rest, then after a duration $c \Delta t$ of a qualifying inertial observer's coordinate time (qualifying here means that the rocket's motion is rectilinear), the final rapidity of the rocket in that frame is:

$\phi = \sinh^{-1}\left( ( \alpha / c^2) c \Delta t \right)$,

and the elapsed proper time for the rocket is:

$c \Delta \tau_{accelerating} = \dfrac{\phi}{ \alpha / c^2}$

(see here).

If a second inertial rocket were moving with the first rocket's final rapidity $\phi$ for that same duration of observer coordinate time $c \Delta t$, its elapsed proper time would be less:

$c \Delta \tau_{inertial} = \dfrac{c \Delta t}{\cosh \phi}$

(that's just the time dilation formula).

With $\Delta t = 1$ hour and $\alpha = 1$ g:

$\Delta \tau_{accelerating} \approx 3599.99999$ seconds
$\Delta \tau_{inertial} \approx 3599.99998$ seconds.

[Edited to correct 3999 to 3599]

 

[Buckethead]

Thank you all very much for your answers. I'm glad I can take to the bank the fact that only instantaneous velocity and not acceleration have an effect on time dilation. What a surprise to see such a well written paper as being so wrong. It's hard to sort out what's true from what's not when something is presented so well such as in this paper.