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Alice and Bob, observers in deep Galilean spacetime, have identical rocket ships. Alice is at rest relative to an inertial frame. At $t=0$ she accelerates with acceleration $a_0$ in a straight line just as Bob passes in a parallel trajectory and same acceleration but with a speed $v_0$ at the time the rockets cross.

Even though they are both accelerating and their instantaneous speeds (relative to the inertial frame) are different ($v_B>v_A$), their speed relative to one another $v_{BA}$ remains constant (all in same direction)

$$v_A=at$$ $$v_B=v_0+at$$ $$v_{BA}=v_B-v_A=v_0$$

Although they are in accelerating frames within which objects are subject to a “fictitious” inertial force to the rear of the rocket and they are traveling at different instantaneous velocities, their “laws of physics” (classical mechanics) should be the same.

$$\mathbf{F}=m(\mathbf{a}-a_0\mathbf{i})$$

A ball thrown horizontally towards the front of the rocket with initial velocity $\mathbf{v}_{b0}$ and and acceleration $\mathbf{a}_b$ would follow the same trajectory in its local Cartesian coordinates

$$x=v_{b0x}t+{1\over 2}(a_{bx}-a_0)t^2$$ $$y=v_{b0y}t+{1\over 2}a_{by}t^2$$ $$x=v_{b0z}t+{1\over 2}a_{bz}t^2$$

Do Alice, Bob and all other observers in frames with the same acceleration vectors but moving at different speeds form some sort of class of noninertial frames with their own “laws of physics”? For each of them, observing themselves and each other, objects will obey these laws (at least until they run out of rocket fuel or reach relativistic speeds!). I realize this could be couched using free fall in a fixed gravitational field using Einstein’s equivalence principle but thought it would be a cleaner example with non gravitational linear acceleration.

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Your analysis is correct. Precisely as there is an equivalence class of infinite inertial frames in relative motion with a constant velocity, there are equivalence classes of non-inertial frames in relative motion with a constant velocity. In each of the frames belonging to the same equivalence class, one has to introduce the same inertial forces in addition to the real ones.

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  • $\begingroup$ Thanks. Exactly what I was looking for. Really interesting that they do form an equivalence class with the same inertial force. Look forward to learning more as I embark on a deeper dive into the formal mathematical underpinnings of reference frames (inertial and non-inertial) in Galilean spacetime. $\endgroup$
    – user175324
    Commented Jul 23, 2022 at 18:15
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One way to think about this situation is in terms of analogy between marine navigation, and spacetime navigation.

In marine navigation there is dead reckoning

For the purpose of thought demonstration I simplify: imagine a body of water that is without any current; all parts are stationary with respect to all other parts.

Two ships depart from each other, and the crews have agreed to rejoin at a future point in time.

Both ships are applying dead reckoning: they keep track of their velocity and any change of direction. That way they always know their current position relative to their point of departure, so at any point in time they can plot a course to the point where the two ships will rejoin.

A crucial factor in this ability to rejoin is the uniformity of the water. If parts of the water would move relative to other parts then the position estimate obtained with the dead reckoning would be off.

This same uniformity prevents the ships from having such a thing as an absolute position cooridinate. Every part of the body of water is indistinguishable from any other part of the water. What the dead reckoning keeps track of is relative position. For the two ships knowing relative position to point of departure is sufficient: they can rejoin at some point in the future using dead reckoning only.



For spaceships the counterpart of dead reckoning is application of an inertial navigation system.

In a spacetime with the property that Inertia is universally uniform is it is possible for two spaceships to depart from each other and rejoin at a future point in time, using inertial navigation only.

The inertial navigation system consist of acceleration sensors and change-of-orientation sensors. The system keeps track of all acceleration and of all change of orientation. To obtain position relative to the point of departure the system integrates the acceleration data twice. The first integration obtains velocity relative to the point of departure, integrating the velocity data obtains the position relative to the point of departure.

So:
Two spaceships that depart from each other can at any point in time plot a course to the point where the two spaceships will rejoin.

Crucial for this capability is that Inertia is uniform. In the absence of sources of gravitational effect Inertia is uniform.

In the case of Galilean spacetime: Inertia has the same symmetries that Euclidean geometry has. Inertia is the same at any point in spacetime, and it is the same for any orientation.

We now see that the uniformity of Inertia has two implications:
The uniformity prevents us from establishing an absolute reference of positon, and it prevents us from establishing an absolute reference of velocity.

On the other hand: the very uniformity of Inertia is the basis of obtaining an absolute reference of acceleration. For $F=ma$: the same $m$ has the same effect everywhere in spacetime. Generalizing: as long as all the factors involved in $F=ma$ are uniform throughout spacetime an absolute reference of acceleration is available.


As we know, in the presence of source of gravitational effect spacetime is not uniform, and for the purpose of navigation the gravitational effect has to be taken into account.



Historically: in theories of motion Inertia and spacetime have always been treated as one and the same entity. Treating those concepts as the same is inevitable, since there is no experiment that can distinguish between the two.

In science: if no experiment is available to measure some conceptual difference, then carrying that conceptual difference in the theory is pointless.

You can try this as an exercise: take a physics textbook and replace 'spacetime' with 'inertia'. As long as you keep in mind that there is no measurement that can distinguish between the two: after that substitution the statements will still be meaningfull statements.


Thought demonstration:
Two spaceships need to obtain their distance relative to each other, with the condition that using anything from the spectrum of electromagnetic radiation is disallowed (so using radar is out). So one ship sends out a probe that has an onboard inertial navigation system. From the acceleration of the probe the velocity of the probe is reconstructed, and from the velocity and the duration of the flight the distance between the two ships is reconstructed.

This procedure relies on uniformity: there is no distinguishing between uniformity of inertia and uniformity of spacetime.

In the context of Minkowski spacetime this uniformity property generalizes to the spectrum of electromagnetic radiation. In the context of Minkowski spacetime there is - as a matter of principle - no distinction between uniformity of propagation of electromagnetic waves and uniformity of spacetime.

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