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When we solve the twin paradox we say something like the traveling twin has a Rindler Metric while the stationary twin has a Minkowski metric, or more plainly, the traveling twin experiences non-zero proper acceleration, while the stationary twin experiences zero proper acceleration.

We define proper acceleration to be acceleration with respect to a MCF (momentarily co-moving frame of reference) which is inertial.

But that means the MCF is traveling at constant speed with respect to another inertial frame of reference.

My question is, the MCF is inertial with respect to what? Is there an absolute frame of reference, from which all frames accelerate? Why have no experiments preferred a specific set of frames of reference as truly inertial? Do we need absolute space to define acceleration?

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  • $\begingroup$ The MCF's speed can be 0. $\endgroup$ – JEB Jun 11 at 16:00
  • $\begingroup$ Can you give more background as to why you think the property of being inertial is a relative property? In other words, why do you think frames are inertial "with respect to something else"? $\endgroup$ – BioPhysicist Jun 11 at 16:01
  • $\begingroup$ An inertial frame is by definition moving with constant velocity with respect to all other possible inertial frames. $\endgroup$ – Joeseph123 Jun 11 at 16:03
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    $\begingroup$ Acceleration is not a necessary factor in the Twin Paradox. It only serves to confuse. $\endgroup$ – m4r35n357 Jun 11 at 17:25
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    $\begingroup$ Agreed and upvoted ;) $\endgroup$ – m4r35n357 Jun 11 at 17:58
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I think this question has been answered in bits and pieces elsewhere but either in a slightly different context or with different emphasis, so I'm adding a separate answer.

Everything you say except in the last paragraph is correct, so I won't iterate the physics expressed therein. Coming directly to the four related questions you ask in your last paragraph:

  • A momentarily comoving inertial frame is comoving with respect to the twin but it is inertial on its own. Or, in other words, the property of being inertial is a property of a frame of reference itself, not a relation defined between a given frame of reference and another frame of reference.

  • This raises the question of how to determine if a frame is inertial. It's very simple: you throw a lot of (free) particles in different directions and if all of them travel with a constant velocity with respect to your frame then your frame is an inertial frame. The fact that such frames exist is not a mathematical fact, but we do the experiments and find out that they do exist. This is the first law of Newton. And it is easy to see that all inertial frames would be moving at a constant velocity w.r.t. each other. This means that the standard of whether something is truly accelerated or not is defined with respect to these inertial frames, but not w.r.t. some absolute space (we cannot pindown such absolute space because all inertial frames are moving w.r.t. each other and are completely equivalent, and also, we don't need an absolute space). Accelerometers read the acceleration of the object that it is attached to with respect to this class of inertial frames.

  • Experiments indeed have found out inertial frames of reference, precisely by doing the kind of experiments I describe in my second bullet point. There is an important point here, that we learn from general relativity, that it's impossible to find global inertial frames in a universe with gravity, but, we can always find local inertial frames (i.e. frames that act as an inertial frame in a small enough region of space and time). Thus, when we say things about inertial frames in Newtonian mechanics and special relativity, they're actually supposed to be statements about such local inertial frames. So, for example, what we actually mean is that an accelerometer reads the acceleration of the object that it's attached to with respect to the class of inertial frames in its local vicinity.

  • Your fourth question has been answered in my second bullet point.


Addendum

I have addressed the question of how to determine if a given frame is an inertial frame or not in my second bullet point. However, as the OP mentions in the comment, it is an important question to ask as to what in nature determines that a particular frame of reference is an inertial frame or not. This question is an unanswered question in both Newtonian mechanics and special relativity (and Einstein emphasizes this point, for example, in his book The Special and the General Theory of Relativity). However, this question does get answered in general relativity. A local inertial frame is the one that is attached to a freely falling object. In other words, all the frames that are moving at a constant velocity with respect to a freely falling object (in its local vicinity) constitute the class of local inertial frames. You might find my answer to a related question of interest: https://physics.stackexchange.com/a/553692/20427.

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  • $\begingroup$ My question is what determines the class of inertial frames in the local vicinity of the object. We need a physical reference medium to determine acceleration. $\endgroup$ – Joeseph123 Jun 11 at 17:14
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    $\begingroup$ @Joseph123 Good question, I'll add a few lines in my answer to clarify this point. $\endgroup$ – Dvij D.C. Jun 11 at 17:16
  • $\begingroup$ A simple idea worth considering: In the frame of reference of a car accelerating with respect to a stationary observer, the observer should feel a "pseudo force." Of course, that doesn't happen in real life; I don't feel a force pushing me back because the car passing by decided to accelerate away. $\endgroup$ – Joeseph123 Jun 11 at 17:23
  • $\begingroup$ Generally speaking, there is an ambiguity of choice of accelerating reference frames, and thus the need for an absolute frame of reference. $\endgroup$ – Joeseph123 Jun 11 at 17:25
  • $\begingroup$ @Joeseph123 I have edited my answer. Regarding your comments, no, a car accelerating w.r.t. an inertial frame should feel a pseudo force (not w.r.t. a stationary frame, what is a stationary frame anyway? stationary w.r.t. whom? ;)). And you absolutely do feel a pseudo force when your car is accelerating with respect to ground (which is a fair enough approximation of an inertial frame). That's why you should wear seat belts and have air cushions in your car ;) $\endgroup$ – Dvij D.C. Jun 11 at 17:31
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I wonder if maybe the following is on your mind:

Let's say we read a news report that says: "The growth of the swarm of locusts is accelerating." (When locusts swarm they shorten their reproductive cycle, thus accelerating the rate at which the swarm grows.)

Let's say the size of the swarm is represented in terms of the combined biomass of the swarm. Given a defined size of the swarm the rate-of-growth of the swarm is defined and then the time derivative of rate-of-growth is acceleration of the rate-of-growth.

Obviously, in order for rate-of-growth to exist, and for rate-of-rate-of-growth to exist, the size of the thing must be a definable state. (It would be absurd to suggest: there is no such thing as 'the size of the swarm', but we can meaningfully say that growth of the swarm is accelerating.)

So, does similar logic apply in theory of motion?
That is, we have that velocity is the time derivative of position, and that acceleration is the time derivative of velocity. If it is claimed that acceleration is absolute does that logically imply that whatever it is a derivative of must be absolute too?


My understanding is that in theory of motion this is dealt with as follows:
The set of all coordinate systems with a uniform velocity with respect to each other is defined as an equivalence class of inertial coordinate systems. Mathematically, acceleration with respect to that equivalence class is uniquely defined because with respect to every member of the equivalence class of inertial coordinate systems acceleration is the same.

So this is an example where a mathematical property (a property of the operation of taking a derivative) is applied as a physics theory.

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