Why does the principle of relativity hold between two inertial observers when one of them had to accelerate?

Question

I've been studying Special Relativity, and am in need of some clarification on the Principle of Special Relativity. I understand that if there are two inertial observers with a constant relative velocity between them, the laws of Physics are the same in both frames of reference, and thus there is no way one can say that one person is at rest and the other is moving and vice-versa. While I understand that this principle applies only when both observers are in inertial frames of reference, it is also true that one of those observers had to accelerate to get to a constant relative velocity (during which time one frame is non-inertial, and SR doesn't apply). Connecting these two facts, is there a deeper principle at work? In other words - does this imply that after acceleration, when there is a constant relative velocity between the two observers, information about which observer accelerated has been lost?

Answer 1

The principle means that it does not matter which observer accelerated beforehand, or indeed, whether they both did- once they are in inertial motion relative to each other, either can be taken to be at rest.

More generally, you should not imagine that the rules of SR require observers and clocks. Instead you should consider it a more general and abstract idea which means that you can pick any inertial reference frame you like and the laws of physics will take the same form it it. You should be able to see from this that since frames of reference are entirely abstract constructs, they can be taken to be moving at arbitrary speeds relative to each other without the requirement for acceleration to have taken place.

Answer 2

Your question implies an absolute rest frame in which the two observers start, and then one goes to his new inertial frame. This whole idea violates the ethos of relativity, and when you assume an absolute reframe, either explicitly or implicitly, you're going to find things that seems to say "there is an absolute rest frame".

The idea that SR blows up under linear acceleration is also premature. It certainly failed when considering gravitation, and starts to break when objects spin near the speed of light...as curvature in the metric is present. But linear acceleration (Rindler Coordinates) aren't too much of a deviation.

When an inertial frame under goes acceleration, relative to another frame, you have to keep track of the Lorentz factor $\gamma(t)$, as it is not constant. When comparing clocks, you also need to keep track of the every changing hyperplane of simultaneity, as the clock bias back at the stationary frame is constantly changing in the accelerated frame, per:

$$ t_{\rm stationary} = \gamma(t')\Big(t'+\frac{v(t')x(t')}{c^2}\Big) $$

So if why have two frames that being coincident, and then on accelerates to a new velocity, all that really matters is the total proper time of the accelerated path:

$$\Delta\tau' =\int_Pd\tau'=\int\frac{ds'}c$$

$$\Delta\tau' = \int_P\frac 1 c \sqrt{\eta_{\mu\nu}dx'^{\mu}dx'^{\nu}}$$

$$\Delta\tau' = \int\sqrt{1-\frac{v(t')^2}{c^2}}=\int\frac{dt}{\gamma(t')}$$

Of course both observers agree on that integral. Note that nowhere in the calculation does acceleration appear.

Answer 3

Sorry for my poor english. My native language is french.

Interestingly, the subject is still object of discussion in recent works. I quickly try to summarize the point of view of J Rafaleski in the book "relativity matters" (Springer, 2017)

• The principle of relativity does not apply to accelerated frames. If a train is longer than a tunnel in the mountain frame of reference, it is indeed the train that undergoes the Lorentz contraction.

• This can be verified from a photo taken in the frame linked to the mountain. The train is shorter. It is indeed the train which has undergone the acceleration which is contracted.

• Is it possible for an observer of the train to remember the history of this contraction? the answer is yes, using the device based on the ideas of J Bell. The instantaneous length of the train can be measured from the train. The same "Bell's device" device would not show any contraction of the mountain.

You can look at Bell's spaceship paradox

To conclude, I quote the text (p 36):

Simplicius : But if only the train and the tunnel are present in the universe, how can you tell who is subject to acceleration ?

Professor : Both Einstein and Poincaré appreciated Mach's ideas about "who is accelerated" and I believe that one of the reasons Einstein reintroduced relativistically invariant aether was to be able to recognize who is accelerated, who is subjecte to a force.