# Definition of an Inertial Frame

## Main Question or Discussion Point

Is there some way to define an inertial coordinate system without being cyclical (defining it with terms that require an inertial coordinate system to define)?

For example if you refer to straight lines... straight according to what coordinate system? Or if you refer to velocity... that too is a coordinate system dependant quantity. Or if you refer to a force, which is also a coordinate system dependant quantity, how would you even define a force without referring to something depending on an inertial coordinate system. And so on...

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EDIT: For clarity, I'm not worried about relativity being some kind of "trick of circular reasoning". I'm a physics student, not a crackpot. It's just that I and some other students noticed that people after start from "given an inertial frame...", and was wondering if one could define it outright.

EDIT(2): Hmmm... I of course am looking for a definition doesn't make relativity a tautology though.

EDIT(3): Oh, and the closest I've gotten so far is along the lines of: if we have a "standard clock", then we could use the second postulate (the constancy of the speed of light in an inertial frame) to define an inertial frame and the first postulate (to best match the original 'intention/wording' something like: the laws of physics written using the coordinates from an inertial coordinate system are the same regardless of the choice of inertial coordinate system) is still falsifiable. The problem is how do we define a "standard clock" without assuming we can use the second postulate? Maybe there is a better approach along different lines, but I haven't found one yet.

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JesseM
Well, inertial coordinate systems in SR are defined in terms of local readings on a grid of inertial rulers with clocks at each marking which have been synchronized by the Einstein synchronization convention. But if you're just asking how we know the rulers are moving inertially, you could put accelerometers all along the ruler, and if they all show zero G-forces the ruler is moving inertially.

Oops. I was editting my post while you were replying.

Anyway, it seems you are choosing to define standard rulers and clocks, and then also defining a synchronization convention. Is there some way to have a standard ruler or clock without using the first postulate to define an inertial frame (which basically just makes relativity a tautology)?

(Also, the notion of a physical accelerometer seems to assume the first postulate as well.)

JesseM
Oops. I was editting my post while you were replying.

Anyway, it seems you are choosing to define standard rulers and clocks, and then also defining a synchronization convention. Is there some way to have a standard ruler or clock without using the first postulate to define an inertial frame (which basically just makes relativity a tautology)?
How am I using the first postulate to define an inertial frame? I'm using accelerometers.

JesseM
(Also, the notion of a physical accelerometer seems to assume the first postulate as well.)
Can you elaborate? If I give you a physical procedure for constructing one, how am I assuming the first postulate?

Anyway, I don't see why it would be tautological even if I did define inertial frames in terms of the first postulate. Isn't it conceivable that the universe would have laws of physics such that it was impossible to find a set of coordinate systems for which it is true that 1) each system is moving at constant coordinate speed relative to every other and 2) the equations of the laws of physics are the same in each coordinate system?

How am I using the first postulate to define an inertial frame? I'm using accelerometers.
If you do not believe you are implicitly using the first postulate in your definition of an inertial frame, let's imagine for a bit that the first postulate did not hold. The physics would then look different in different inertial coordinate systems... so the rest equilibrium length of a physical object or the oscillation time of some device while at rest, would be dependent on our choice of inertial coordinate system.

So assuming we can have an standard ruler or clock or accelerometer seems to be using the first postulate in part of the definition of an inertial coordinate system.

DrGreg
Gold Member
If you do not believe you are implicitly using the first postulate in your definition of an inertial frame, let's imagine for a bit that the first postulate did not hold. The physics would then look different in different inertial coordinate systems... so the rest equilibrium length of a physical object or the oscillation time of some device while at rest, would be dependent on our choice of inertial coordinate system.

So assuming we can have an standard ruler or clock or accelerometer seems to be using the first postulate in part of the definition of an inertial coordinate system.
All the accelerometer has to do is tell the difference between "accelerated" and "unaccelerated". It doesn't need to give a numerical value to the acceleration. A very crude accelerometer would consist of two identical springs side by side. One end of each spring is attached to the observer. The other end of one spring is attached to a mass, the other end of the other spring is free. If both springs are the same length, then the observer is an inertial observer. This definition does not rely on any assumptions about transformations of lengths or times.

It is true that, for an inertial observer to construct their own frame of reference, they need a notion of distance and time. But we needn't say precisely what these notions are to make the definition. We simply assume they exist. The postulates of special relativity (SR) then go on to make assumptions about how these coordinate systems behave.

To put it another way, Newtonian physics and Lorentz Ether Theory both postulate the existence of a single inertial frame. SR postulates the existence of an infinite number of inertial frames. And, in fact, general relativity (GR) denies the existence of inertial frames (in a global sense, although it does postulate the existence of local approximations to inertial frames, which are often described simply as "inertial frames" for convenience). So it is logically possible to define something without knowing whether such a thing actually exists. (For example, nothing stops me defining an "octocat" to be an eight-legged cat and developing a theory of octocats.)

The postulates of a theory are a set of assumptions on which the theory is based. Deriving the theory from the postulates is an exercise in logic and mathematics, and doesn't depend on whether the postulates are actually true in real universe. Indeed it may not even be possible to directly test whether the postulates are true or not. As long as there is no mutual contradiction in the postulates, then the theory "works" as a block of logical reasoning. The test of whether a logical theory is physically useful comes if the theory makes predictions that can be tested experimentally. Relativity theory passes that test, so we accept is as a "good" theory, even though some aspects of its postulates can't be proven without circular logic. (Bear in mind any theory of physics is an approximate model of reality, not reality itself. The value of a theory is how good an approximation it is, under appropriate conditions.)

For example, the concept of "speed" depends on how we define distance, time and clock-synchronisation. We postulate the speed of light is constant, but the modern definition of distance, and the way we synchronise clocks, assumes the constancy of the speed of light, so there is a circularity of logic here. It doesn't matter, because the postulate is simply an assumption on which the theory is based. For the logic of the theory we don't care precisely how distance, time and clock-synchronisation are defined, as long as, under those definitions, the speed of light is constant.

All the accelerometer has to do is tell the difference between "accelerated" and "unaccelerated". It doesn't need to give a numerical value to the acceleration. A very crude accelerometer would consist of two identical springs side by side. One end of each spring is attached to the observer. The other end of one spring is attached to a mass, the other end of the other spring is free. If both springs are the same length, then the observer is an inertial observer. This definition does not rely on any assumptions about transformations of lengths or times.
That definition DOES rely on assumptions. It assumes that the rest length of a physical object is the same in all inertial coordinate systems. It is implicitly assuming the first postulate.

If you do not believe you are implicitly using the first postulate in your definition of an inertial frame, let's imagine for a bit that the first postulate did not hold. The physics would then look different in different inertial coordinate systems... so the rest equilibrium length of a physical object would be dependent on our choice of inertial coordinate system. So your "accelerometer" may give a non-zero result even while at rest according to an inertial coordinate system.

So it is logically possible to define something without knowing whether such a thing actually exists. (For example, nothing stops me defining an "octocat" to be an eight-legged cat and developing a theory of octocats.)
Yes, I agree you can postulate something exists even if it may not. But that is not what I am discussing here. I am asking for a precise definition of what is being postulated.

Relativity theory passes that test, so we accept is as a "good" theory, even though some aspects of its postulates can't be proven without circular logic.
I'm not questioning the validity of relativity. As I mentioned, I'm a physics student. I know it works great. What I never really noticed before until discussing with some students is that physicists carry around this "intuitive" notion of an inertial frame which everyone feels is "obvious", but no one can actually define accept relative to another inertial frame. Asking some theorists they say that to avoid this mess, most theorists define special relativity purely by the symmetry it imposes on physical laws (as the historical tie to some concept of inertial frames or even electromagnetism, is just a historical tie and doesn't necessarily capture the true essence which they feel is the symmetry). He defined it as: physical laws are invarient to operations of the proper orthochronous Lorentz group. Another physicist said something similar but with the Poincare group (I don't remember the exact wording off hand).

Worded this way, you don't need to set up inertial coordinate systems to test relativity. You only need to setup ONE, using rulers, clocks, whatever. The only thing you need to assume here is translational invarience to steup this coordinate system. Measure the physical laws written in coordinates of this coordinate system and then check if they have the expected symmetry.

There may be problems with that line of reasoning, but it makes sense to me at the moment. And besides, I am getting off topic now. The point I wanted to make was to stress that, as I already stated, I am NOT arguing relativity is wrong due to some kind of "trick of circular reasoning". So I don't want to waste any more time on that. What I am trying to see is if we can, using our current understanding, go back to the historical definitions and make them precise yet not make relativity a tautology or our definition of inertial frames ad hoc to match experiment.

For the logic of the theory we don't care precisely how distance, time and clock-synchronisation are defined, as long as, under those definitions, the speed of light is constant.
This seems to be arguing to use the second postulate to define an inertial coordinate system. Paraphrasing to: As long as the second postulate holds, regardless of precisely how distance, time and clock-synchronisation are defined, it is an inertial coordinate system.

Is that what you are saying?

It is true that, for an inertial observer to construct their own frame of reference, they need a notion of distance and time. But we needn't say precisely what these notions are to make the definition. We simply assume they exist. The postulates of special relativity (SR) then go on to make assumptions about how these coordinate systems behave.
A student showed me a very interesting point today which demonstrates the importance in how we define an inertial frame. Above I said if I was given something I could use as a "standard clock", then combined with the second postulate I could construct an inertial frame. You seem to be saying the same thing here: "we needn't say precisely what these notions [of length or time] are to make the definition. We simply assume they exist." I was amazed when the other student asked me to define an inertial coordinate system given a "standard clock" and the second postulate, he then proceeded to show my definition made relativity not match experiment. I could only fix this by adding what felt like an adhoc rule into my definition of inertial frame in order to make the predictions of relativity match experiment.

I hadn't considered that before. It complicates even more the act of defining an inertial frame precisely so as to fit with relativity but not make relativity a tautology. I was left feeling like I shoved in an adhoc requirement into the definition of inertial frame just to make the predictions of relativity match experiment. A sour taste indeed.

I'm starting to agree with the theorists... we shouldn't be so intuitively tied to the historical pedagogy of special relativity... for there is no way to nicely make it precise. At the same time though, I am glad the teachers introduced relativity using the historical approach, for I definitely didn't need to know about that "preciseness" at the time and jumping straight to some symmetry group arguement of the theorists would have confused me.

Oh well, if "inertial frame" needs to remain a loosely defined intuitive concept except for defining it in retrospect, I guess I will learn to live with it.

If anyone has other ideas though, please let me know.

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Mentz114
Gold Member
I always understood that the first postulate is the definition of an inertial frame. Paraphrasing

"There exists a set of frames in which the laws of physics are the same as the ones we know."

This excludes places (?) where f=/= ma, for instance. So, everytime you see the term "intertial frame" substitute "a frame where the laws of physics ( as we know them) hold".

No definition required.

paw
That definition DOES rely on assumptions. It assumes that the rest length of a physical object is the same in all inertial coordinate systems. It is implicitly assuming the first postulate.
DrGregs accelerometer does not rely on assumptions if you look carefully. His accelerometer only requires the springs be the same length to the observer. There's no other frame required for this test.

Still, Mentz114 gives a much simpler solution above.

JustinLevy, the way I understand it, the problem you're having is with the idea that defining an inertial frame of reference requires the use of the first postulate, the principle of relativity. But you don't want to do that because it is using an assumption that leads to a conclusion that proves the assumption (the principle of relativity) and this results in a tautology.
If that is your concern, consider it this way:
Make an assumption that leads to a conclusion that "when tested" proves the assumption.
Hopefully that helps. It is not the assumption giving weight to a conclusion that proves the assumption.
It is the successful tests of the conclusion's predictions (falsifiability) that proves the conclusion and therefore the assumption.
In theory development it is sometimes very useful to make intuitive leaps like this. Einstein was the kind of long jumps. What counts as theory and distinguishes it from speculation, is whether that leap finds you landing on your feet or your face.

That definition DOES rely on assumptions. It assumes that the rest length of a physical object is the same in all inertial coordinate systems. It is implicitly assuming the first postulate.
No, it doesn't. The definition relies on the construction of the accelerometer in order to detect the presence of acceleration. If acceleration is present (non-inertial frame) the springs look different. If acceleration absent (inertial frame), the springs have equal lengths.

I'm not questioning the validity of relativity. As I mentioned, I'm a physics student. I know it works great. What I never really noticed before until discussing with some students is that physicists carry around this "intuitive" notion of an inertial frame which everyone feels is "obvious", but no one can actually define accept relative to another inertial frame.
You must mean "except", right?
You are mistaken here as well. I think DrGreg gave you the "class representative". One inertial frame is the frame that has the accelerometer indicating zero acceleration. All frames in relative uniform motion wrt the "class representative" are also inertial frames. This is what the physicists mean when thy define inertial frames "relative to another inertial frame". It is not wrt another inertial frame, it is wrt the inertial frame that containts the accelerometer that shows zero acceleration (the "class representative")

So I don't want to waste any more time on that. What I am trying to see is if we can, using our current understanding, go back to the historical definitions and make them precise yet not make relativity a tautology or our definition of inertial frames ad hoc to match experiment.
DrGreg and Mentz already did. I'd like to think that I helped :-)

Thank you everyone, this gave me much to think about and has helped alot.
I love this forum for we can have (while it may seem boring basic stuff to you guys) very informative discussions.

DrGregs accelerometer does not rely on assumptions if you look carefully.
You are right, his accelerometer does not depend on the assumption I listed. Rereading it, I see I must have misread it the first time. Makes sense now. Thanks for pointing that out.

But that still leaves the issue of defining a standard ruler (or clock) if we try to build up an inertial coordinate system while avoiding using the first postulate to define one.

Still, Mentz114 gives a much simpler solution above.
Yes, that works much better.

I always understood that the first postulate is the definition of an inertial frame. Paraphrasing

"There exists a set of frames in which the laws of physics are the same as the ones we know."
This works great.
The problem is I was paraphrasing it something like "The laws of physics are the same in all inertial frames". And so using that as a definition could lead (with no contradictions from the postulates) that there is only one inertial frame or something.

I like your wording a lot, and that does seem the simplest solution.
There is no need to worry about standard length or time to setup coordinate systems, because it is basically absorbed into applying the physics self-consistently for the definition. Even the "parity problem" I had above from trying to build up a coordinate system with clocks and light goes away.

Thank you.
That is simple... the first postulate is a definition of inertial frames and declaring a set of them exist, the second postulate is then still falsifiable. Perfect.

You are mistaken here as well. I think DrGreg gave you the "class representative". One inertial frame is the frame that has the accelerometer indicating zero acceleration. All frames in relative uniform motion wrt the "class representative" are also inertial frames.
That clearly won't work because of the parity issue, and also because one can even define coordinate systems with "uniform motion wrt [the given inertial frame]" in which the speed of light is not constant (basically, Newton's first law can be preserved even though the synchronization convention is changed).

The only thing I'm left with now, is I wish I could have a better understanding for WHY relativity wouldn't predict that empty space doesn't care whether you use a right handed coordinate system or a left handed coordinate system. Experiment shows that the vaccuum does not have this symmetry, so we can just include this into "known physics" and thus subsume it into our definition of an inertial frame as Mentz suggested. That clearly works, but feels quite dissatisfying.

I looked it up and ignoring Parity transformations is what the theorist meant with the adjective "proper" infront of "orthochronous Lorentz group".

Throwing away a whole subclass of frames that could equally have been chosen as "inertial frames" is bothering me. We're forced to throw out one or the other, but the fact that we could choose, and even choose arbitrarily, whether we keep right or left handed coordinate systems bothers me.

Is there some way of looking at this that would be more inciteful and not seem so adhoc?
I'm clearly still missing something, for this seems to destroy part of the beauty of SR.

JustinLevy, before you focus on parity transformations I would suggest you go back and think through your conclusions above.
The first postulate is NOT a definition of inertial frame.
An inertial frame is one in which Newton's first two laws hold, which is to say all observations of test bodies show no evidence of acceleration or gravitation, which is why its called an "inertial" frame.
The first postulate does not define this in any way.
"the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."
It states the principle of relativity.
You are taking the term "frame of reference" to mean "inertial frame" it does not. It can be inferred when those equations being tested are Newton's first two laws, but the first postulate purposely omits "inertial" for that very reason, because it is a "principle" not a definition.
Which can be paraphrased:
upon the validation of the equations of mechanics from place to place or from time to time, it follows that the laws of physics will not change from place to place or from time to time.
If you attempt to read into it anything more than the principle of relativity you will find yourself thinking in circles.

D H
Staff Emeritus
The first postulate is NOT a definition of inertial frame.
Newton's first law of motion is not true for all observers. The modern interpretation of Newton's first law is that it defines the concept of an inertial reference frame. In particular, an inertial reference frame is one in which the Galilean principle of inertia is valid. Without this viewpoint, the first law is merely an unnecessary specialization of the second law. With this viewpoint, the first law is fundamental and establishes the framework for the second law.

D_H, I may be confused, but I understood the reference to the "first postulate" as used in JustinLevy's posts, to refer to the first postulate of Einstein's paper "On the Electrodynamics of Moving Bodies" not the Newton's first law of motion.

That clearly won't work because of the parity issue,
What is the "parity issue" that you are having"

and also because one can even define coordinate systems with "uniform motion wrt [the given inertial frame]" in which the speed of light is not constant (basically, Newton's first law can be preserved even though the synchronization convention is changed).
Can you show how you can achieve such a feat?

JustinLevy, before you focus on parity transformations I would suggest you go back and think through your conclusions above.
The first postulate is NOT a definition of inertial frame.
I like Mentz and paw's viewpoint on this. Defining an inertial frame that way works fine and does not make SR a tautology.

If you want to offer an alternative definition, I am interested to here it.

An inertial frame is one in which Newton's first two laws hold, which is to say all observations of test bodies show no evidence of acceleration or gravitation, which is why its called an "inertial" frame.
Using that definition, since Newton's laws are invariant under parity transformation, we would get both right AND left handed inertial coordinate systems being labelled inertial coordinate systems... and relativity would make a prediction that has been proved wrong.

So that is not an acceptable definition.

The modern interpretation of Newton's first law is that it defines the concept of an inertial reference frame.
It defines a property of inertial frames, but that cannot be used to define an inertial frame. For there are coordinate systems in which Newton's first law holds, but the speed of light is not isotropic let alone the same constant as in an inertial frame (this is done by changing the simultaneity convention).

What is the "parity issue" that you are having"
If we are not careful, our definition of inertial frames would contain both right and left handed inertial coordinate systems. For them to be equivalent, the laws of physics would have to have parity symmetry. It turns out parity is violated. So our definition of inertial frames must pick right OR left handed inertial coordinate systems.

and also because one can even define coordinate systems with "uniform motion wrt [the given inertial frame]" in which the speed of light is not constant (basically, Newton's first law can be preserved even though the synchronization convention is changed).
Can you show how you can achieve such a feat?
As a simple example, Galilean transformations give a coordinate system in which Newton's first law still holds, but the speed of light is not the same constant.

For a more involved sample, the following transformation from an inertial coordinate system yields another coordinate system that labels spatial distances according to physical rulers that would agree with an inertial frame, and labels time coordinates as what a clock would measure, and Newton's law still holds, yet the speed of light is not constant.

x' = gamma (x - beta ct)
y' = y
z' = z
ct' = ct / gamma

where gamma = 1/sqrt(1 - beta^2), and beta = v/c, where v is the velocity of the new coordinate system's spatial origin according to the inertial frame.

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As a simple example, Galilean transformations give a coordinate system in which Newton's first law still holds, but the speed of light is not the same constant.

For a more involved sample, the following transformation from an inertial coordinate system yields another coordinate system that labels spatial distances according to physical rulers that would agree with an inertial frame, and labels time coordinates as what a clock would measure, and Newton's law still holds, yet the speed of light is not constant.

x' = gamma (x - beta ct)
y' = y
z' = z
ct' = ct / gamma

where gamma = 1/sqrt(1 - beta^2), and beta = v/c, where v is the velocity of the new coordinate system's spatial origin according to the inertial frame.
But this is not what I have given you, I have given you an operational (based on experiment) way to define inertial frames.
The two transformations that you are talking about above (Galilei and respectivelly Selleri) come later , after you choose your clock synchronization scheme. The clock synchronization scheme and the transforms that you derive from it doesn't make the frame any less inertial because we know from experiment that indeed , frames in uniform motion wrt an inertial frame are inertial themselves.

If we are not careful, our definition of inertial frames would contain both right and left handed inertial coordinate systems. For them to be equivalent, the laws of physics would have to have parity symmetry. It turns out parity is violated. So our definition of inertial frames must pick right OR left handed inertial coordinate systems.
Thank you for your clarification. It is clear that all right handed frames will form a class and all the left handed frames will form a different class. No reason to mix the two.

JustinLevy, I cannot in good conscience, let this thread drop thinking you, as a physics student, will be marching on to new ideas without the foundation necessary to discern principle.
You clearly have an ability to think beyond the obvious, a quality that is rare but necessary in a physicist.

Start with the paraphrased version you said "works much better" from Mentz144.
"There exists a set of frames in which the laws of physics are the same as the ones we know"
This conjecture (hypothesis-approaching speculation) requires proving the existence of a set of frames and the sameness of the laws tested in them.
All Einstein did was conjecture from empirical evidence and on the axiomatic foundations of the equations, that "IF" we find the equations upheld in "any" frame, we can take, on the strength of their axioms, they must uphold the laws in these frames. This principle of relativity is just that, a very sound "principle". Unless we are wiling to bring into question, the validity of the axioms of mathematics as they may or may not hold from "one place to another" or "at one time or another" this principle of relativity becomes foundational to all of physics. It is a framework within which all of physics must operate. It is a statement as much, if not more, about the validity of mathematics than the physical mechanics math describes.
With this simple, first principle reasoning, we can test Newton's laws in "any" frame. If Newton's laws are upheld, the frame is qualified from "any" to an "inertial" frame. If they fail the frame is qualified from "any" to an accelerating frame. (with the additional qualification of gravitational acceleration to come later by GR)
So, it is the test of Newton's laws that define an inertial frame, NOT the first postulate.
All this affords us is a means of qualifying frames wrt the laws, it does not define how we reason, or construct a frame.

You are concerned that the first postulate must a-priori hold, in order for more than one frame (spatially and/or temporally separated) to be defined. This is the circular thinking I warned you about earlier. By omitting the necessary qualification of the equations of mechanics being upheld as the criteria for validating the laws in any frame, you make the assumption that the first postulate states the laws "are" all the same in any frame. It does not say this and because it does not say this, it is not defining an inertial frame.
What is says is that if the equations are upheld, the laws are upheld. The "theory" then goes on to point out that if the laws are upheld, then by virtue of the empirical evidence of the constancy of the speed of light, we must accept as the only reasonable explanation, the Lorentz translation of our measures and thus the Lorentz invariance of the laws.

I hope this helps.

I feel that my original question has been answered.
There appears to be some disagreement of my understanding of this topic however. To me it seems these lingering disagreements are just semantic issues (or possibly preferred pedagogy issues). In the off chance that there is some actual physics difference in our understanding, I'll respond. If any poster or bystander sees that this is indeed just a semantic or preferred pedagogy issue, please clearly point out where this is arising so we can conclude this without belaboring the point.

But this is not what I have given you, I have given you an operational (based on experiment) way to define inertial frames.
All you gave was that an inertial frame is moving at constant velocity with respect to an inertial frame.

I consider the term "inertial frame" to be a set of inertial coordinate systems which are related by translation of the origin and spatial rotations. If you mean something else, please explain your definition to prevent confusion.

The point I made above was that yes, an inertial frame is moving at constant velocity with respect to an inertial frame. HOWEVER, this is a necessary but not sufficient property to define an inertial frame given one, for there are many coordinate systems which have that property and are not inertial coordinate systems.

The two transformations that you are talking about above (Galilei and respectivelly Selleri) come later , after you choose your clock synchronization scheme. The clock synchronization scheme and the transforms that you derive from it doesn't make the frame any less inertial
You really seem to be using a different relation between the terms inertial coordinate system and inertial frame that what I listed above (I believe that is the conventional definition). So I really do hope this is just a semantic issue. If not, I would appreciate some further clarification here.

JustinLevy, I cannot in good conscience, let this thread drop thinking you, as a physics student, will be marching on to new ideas without the foundation necessary to discern principle.
I am somewhat shocked to hear you feel so strongly about this, for what you wrote after this doesn't seem very enlightenning. There seems to be some communication problem.

All Einstein did was conjecture from empirical evidence and on the axiomatic foundations of the equations, that "IF" we find the equations upheld in "any" frame, we can take, on the strength of their axioms, they must uphold the laws in these frames.
I am not understanding your semantics here that you seem to put great weight and importance on. What is the difference between the physics equations giving correct predictions in a frame and the "laws upholding" in this frame?

This principle of relativity is just that, a very sound "principle". Unless we are wiling to bring into question, the validity of the axioms of mathematics as they may or may not hold from "one place to another" or "at one time or another" this principle of relativity becomes foundational to all of physics.
What do you mean by "axioms of mathematics" holding at different places or times? This seems a completely different (and highly bizarre) question as opposed to whether physics is invarient to translations in time or space (which to me seems to be a question of momentum or energy conservation as opposed to lorentz invarience).

And you also seem to be making an (important) distinction between a "principle" and a "postulate" which I am not seeing.

It is a framework within which all of physics must operate. It is a statement as much, if not more, about the validity of mathematics than the physical mechanics math describes.
More about the validity of mathematics than physics? I don't think energy or momentum conservation or lorentz invariance have any hold on the validity of mathematics. You've completely lost me on this train of reasoning.

With this simple, first principle reasoning, we can test Newton's laws in "any" frame. If Newton's laws are upheld, the frame is qualified from "any" to an "inertial" frame. If they fail the frame is qualified from "any" to an accelerating frame. (with the additional qualification of gravitational acceleration to come later by GR)
So, it is the test of Newton's laws that define an inertial frame, NOT the first postulate.
You object to me using the first postulate to define inertial frames. And then you go and define inertial frames using a subset of the physics!? Furthermore, you chose the subset (Newton's laws in classical mechanics) which IS parity invariant (and therefore clearly can't be used to define an inertial frame on its own).

I do not agree here.
We can argue forever what physicists in the past have precisely considered inertial frames, but we should be able to at least agree that the definition in some way must allow us to distinguish right handed from left handed coordinate systems. The answer to my question which Mentz suggested does this. Using only Newton's laws does not.

You are concerned that the first postulate must a-priori hold, in order for more than one frame (spatially and/or temporally separated) to be defined. This is the circular thinking I warned you about earlier.
This is not circular at all. As mentioned, the theory of special relativity is still falsifiable using this definition. So experimental tests of it do not involve circular logic. The experiments really do support the theory. I don't understand your criticism of this definition.

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All you gave was that an inertial frame is moving at constant velocity with respect to an inertial frame.
Wrong. I gave you the definition of the inertial frame, i.e. the "class representative" followed by the fact that any frame in uniform motion wrt the "class representative" is also an inertial frame.

I consider the term "inertial frame" to be a set of inertial coordinate systems which are related by translation of the origin and spatial rotations. If you mean something else, please explain your definition to prevent confusion.
This is exactly what I explained to you. It is good to see that you understood that.

The point I made above was that yes, an inertial frame is moving at constant velocity with respect to an inertial frame.
Good, the best way to check to see if someone understood is to play back his/her words.:-)

HOWEVER, this is a necessary but not sufficient property to define an inertial frame given one, for there are many coordinate systems which have that property and are not inertial coordinate systems.
What gives you this belief? Our experimental knowledge points out that you are mistaken, the frames in relative uniform motion wrt a known inertial frame are also inertial. The two examples that you gave are still inertial frames, as explained to you in post #19. If you don't believe that, just equip every such frame with an accelerometer, what would the accelerometer indicate?

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Wrong. I gave you the definition of the inertial frame, i.e. the "class representative" followed by the fact that any frame in uniform motion wrt the "class representative" is also an inertial frame.
You did not define an inertial frame. You just told me a property of them. As I explained that property is a necessary but not sufficient property to define an inertial frame.

I consider the term "inertial frame" to be a set of inertial coordinate systems which are related by translation of the origin and spatial rotations. If you mean something else, please explain your definition to prevent confusion.
This is exactly what I explained to you. It is good to see that you understood that.
This is not what you explained to me, thus the confusion. (for reference, see your post https://www.physicsforums.com/showpost.php?p=1684470&postcount=12 )

And despite what you say, you still seem to be using a different definition. Your definition seems to be (to get it to match your statements) something like:
An "inertial frame" is a set of inertial coordinate systems which are related by translation of the origin, spatial rotations, and clock synchronization changes.

I disagree with this strongly, because you are now including coordinate systems in which empty space is not described isotropically (nor is the speed of light a constant).

HOWEVER, this is a necessary but not sufficient property to define an inertial frame given one, for there are many coordinate systems which have that property and are not inertial coordinate systems.
What gives you this belief? Our experimental knowledge points out that you are mistaken, the frames in relative uniform motion wrt a known inertial frame are also inertial. The two examples that you gave are still inertial frames, as explained to you in post #19. If you don't believe that, just equip every such frame with an accelerometer, what would the accelerometer indicate?
What gives me that belief? I gave you two examples. If that doesn't show you I'm not sure what more to say.

Basically, if you want to use that as your definition for inertial frames, this includes coordinate systems which experimentally disagree with the postulates of relativity. So using your definition, relativity has been experimentally proven wrong. I would hope that this would cause you to pause and consider that your definition, while containing a necessary property, is not specific enough ... it is too inclusive and is wrong.

You did not define an inertial frame. You just told me a property of them. As I explained that property is a necessary but not sufficient property to define an inertial frame.
I gave you an operational definition. It is not my problem that you still don't understand it.

This is not what you explained to me, thus the confusion. (for reference, see your post https://www.physicsforums.com/showpost.php?p=1684470&postcount=12 )
Which is a superset of what mainstream physics uses as a defintion. See here, for example.

"With this in mind, we can define inertial frames collectively as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set."

I don't particularly like citing wiki but this time they are right on the money.

And despite what you say, you still seem to be using a different definition. Your definition seems to be (to get it to match your statements) something like:
An "inertial frame" is a set of inertial coordinate systems which are related by translation of the origin, spatial rotations, and clock synchronization changes.
I don't know how you managed to get the "spatial rotations" in? You don't seem to cite anything correctly. Here is what I said exactly:

"The two transformations that you are talking about above (Galilei and respectivelly Selleri) come later , after you choose your clock synchronization scheme. The clock synchronization scheme and the transforms that you derive from it doesn't make the frame any less inertial because we know from experiment that indeed , frames in uniform motion wrt an inertial frame are inertial themselves."

I disagree with this strongly, because you are now including coordinate systems in which empty space is not described isotropically (nor is the speed of light a constant).
Ah, I see your misconception. This misconception comes through clearer in your next paragraph :-)

What gives me that belief? I gave you two examples. If that doesn't show you I'm not sure what more to say.

Basically, if you want to use that as your definition for inertial frames, this includes coordinate systems which experimentally disagree with the postulates of relativity. So using your definition, relativity has been experimentally proven wrong. I would hope that this would cause you to pause and consider that your definition, while containing a necessary property, is not specific enough ... it is too inclusive and is wrong.
Both your examples are wrong, you are mixing the fact that Galilei and Selleri transforms predict anisotropic light speed for uniform relative motion of the frames with the fact that inertial frames (see wiki above) can indeed be defined as the frames being in relative uniform motion wrt a known inertial frame (what I call "class representative", i.e. the frames that contains the accelerometer showing zero acceleration).
The former is a mathematical model of reality, the latter is the experimental reality itself. You should know better, especially since you seem so interested in this fudamenatl issue. Let's take the Selleri case, since it is less known and more interesting.
While the Selleri transforms predict[/] anisotropic light speed for the class of inertial reference frames related by their equations:
-no such anisotropy has ever been observed
-(here comes the kicker) it is well known that the Selleri theory is one of the many theories of relativity that is experimentally indistinguishable from SR.

So, the Selleri reference frames are as inertial as any of the frames used by SR (the only thing that is different is the convention for clock synchronization) . All the experiments will return the same exact results as SR, courtesy of the uniform relative motion. To conclude, the definition that you have been given , is correct. I suggest that you study the whole wiki article, if you have more misunderstandings about what the Selleri relativity really says wrt inertial reference frames feel free to ask, I'll try to help :-)
I can also recommend a very good book on the subject if you are interested :-)

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