# Definition of an Inertial Frame

1. Apr 9, 2008

### JustinLevy

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.

Last edited: Apr 9, 2008
2. Apr 9, 2008

### 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.

3. Apr 9, 2008

### JustinLevy

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.)

4. Apr 9, 2008

### JesseM

How am I using the first postulate to define an inertial frame? I'm using accelerometers.

5. Apr 9, 2008

### JesseM

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?

6. Apr 9, 2008

### JustinLevy

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.

7. Apr 9, 2008

### DrGreg

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.

8. Apr 9, 2008

### JustinLevy

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.

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.

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.

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?

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.

Last edited: Apr 9, 2008
9. Apr 9, 2008

### Mentz114

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.

10. Apr 9, 2008

### paw

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.

11. Apr 10, 2008

### Chrisc

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.

12. Apr 10, 2008

### 1effect

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.

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")

DrGreg and Mentz already did. I'd like to think that I helped :-)

13. Apr 11, 2008

### JustinLevy

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.

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.

Yes, that works much better.

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.

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.

14. Apr 11, 2008

### Chrisc

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.

15. Apr 11, 2008

### D H

Staff Emeritus
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.

16. Apr 11, 2008

### Chrisc

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.

17. Apr 11, 2008

### 1effect

What is the "parity issue" that you are having"

Can you show how you can achieve such a feat?

18. Apr 11, 2008

### JustinLevy

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.

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.

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).

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.

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.

Last edited: Apr 11, 2008
19. Apr 11, 2008

### 1effect

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.

20. Apr 11, 2008

### 1effect

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.