# Inertial and non inertial frames

• bgq

#### bgq

Hi,

In the classical theory, there is an absolute rest frame, and every frame moving with constant velocity with respect to it is called inertial frame of reference. The frames that are accelerating with respect to it are called non inertial. To test whether a frame is an inertial, we test Newton's first law. Now in SR, there is no absolute rest frame, so why are there frames that are inertial while others not? In the absence of absolute rest frame, all frames should be completely equivalent.

Just to clarify the point. Consider two frames where one of them is inertial, the other frame moves towards the first with an acceleration. Now both frames are completely symmetrical to each other, for each frame, the other is accelerating towards it, so what makes one of them preferable (inertial) over the other?

## Answers and Replies

Hi,

In the classical theory, there is an absolute rest frame, and every frame moving with constant velocity with respect to it is called inertial frame of reference. The frames that are accelerating with respect to it are called non inertial. To test whether a frame is an inertial, we test Newton's first law. Now in SR, there is no absolute rest frame, so why are there frames that are inertial while others not? In the absence of absolute rest frame, all frames should be completely equivalent.

Just to clarify the point. Consider two frames where one of them is inertial, the other frame moves towards the first with an acceleration. Now both frames are completely symmetrical to each other, for each frame, the other is accelerating towards it, so what makes one of them preferable (inertial) over the other?

One of them actually IS accelerating, and could be determined to be so by a measurement IN that frame. If you are in an accelerating elevator, you are in a non-inertial frame whereas a person on the floor of the same building is in an inertial frame. BOTH frames can be accelerating relative to some other frame, but that's irrelevant.

One of them actually IS accelerating, and could be determined to be so by a measurement IN that frame. If you are in an accelerating elevator, you are in a non-inertial frame whereas a person on the floor of the same building is in an inertial frame. BOTH frames can be accelerating relative to some other frame, but that's irrelevant.
I know this, but when we say accelerating (and so non inertial), we mean accelerating with respect to a certain frame (According to both classical and special relativity). What is this frame?
According to the classical theory, the answer is very clear: It is the absolute rest frame, but in SR it seems (to me) that there is something missing. It is not the issue how to test whether a frame is inertial or not, I know the whole story of this, but the issue is what initially makes some frames inertial and the others not? If we say that non inertial frames are those that are accelerating, this has no meaning unless we specify with respect to what frame.

I know this, but when we say accelerating (and so non inertial), we mean accelerating with respect to a certain frame (According to both classical and special relativity). What is this frame?
No, we do not! Velocity is "relative" but "acceleration" is not. If you are accelerating you will feel an additional force that you do not feel when you are not accelerating.

According to the classical theory, the answer is very clear: It is the absolute rest frame, but in SR it seems (to me) that there is something missing. It is not the issue how to test whether a frame is inertial or not, I know the whole story of this, but the issue is what initially makes some frames inertial and the others not? If we say that non inertial frames are those that are accelerating, this has no meaning unless we specify with respect to what frame.

I know this, but when we say accelerating (and so non inertial), we mean accelerating with respect to a certain frame (According to both classical and special relativity). What is this frame?
According to the classical theory, the answer is very clear: It is the absolute rest frame, but in SR it seems (to me) that there is something missing. It is not the issue how to test whether a frame is inertial or not, I know the whole story of this, but the issue is what initially makes some frames inertial and the others not? If we say that non inertial frames are those that are accelerating, this has no meaning unless we specify with respect to what frame.
If you stand on a (properly aligned) scale in an accelerating frame, the reading will get bigger and bigger, but if you stand on it in an inertial frame, it won't change over time. When a jet pilot gets launched off of an aircraft carrier, he knows damn good and well that he's in an accelerating frame, but the guy on the flight deck isn't.

No, we do not! Velocity is "relative" but "acceleration" is not. If you are accelerating you will feel an additional force that you do not feel when you are not accelerating.
That seems to be exactly the point of the OP, a "relative" velocity implies treating all reference frames as equal, an absolute quantity on the contrary must define some absolute reference frame, so I'd say he is simply saying that if acceleration is considered "absolute" rather than "relative", what frame is it distinguishing as the absolute one?

In the classical theory, there is an absolute rest frame and every frame moving with constant velocity with respect to it is called inertial frame of reference

No, In galilean relativity every intertial frame is equivalent. None of them is an absolute rest frame.

To test whether a frame is an inertial, we test Newton's first law

Yes.

Now in SR, there is no absolute rest frame, so why are there frames that are inertial while others not? In the absence of absolute rest frame, all frames should be completely equivalent

In SR there are a set of intertial frames miving with constant velocity with respect to each other. All of them are equivalent, the same as in galilean relativity

As Newton laws only apply in inertial reference frames, SR laws only apply in inertial reference frames.

Acceleration can be measured. It is not relative.

Now, you might ask what determines this set of inertial frames. This question has not been addressed in galilean relativity and in SR. In GR, However, it turns out to be determined by the mass distribution in the universe. inertial frames are defined by free fall (geodesic motion)
which is determined by the mass distribution

Now, you might ask what determines this set of inertial frames. This question has not been addressed in galilean relativity and in SR. In GR, However, it turns out to be determined by the mass distribution in the universe. inertial frames are defined by free fall (geodesic motion)
which is determined by the mass distribution
I can't see right now the relation between the universe mass density and geodesics. Can you explain it?

Oh, ok you must be referring to the critical density notion from cosmology, but notice that this is indeed determining a reference frame in FRW cosmology.

In the classical theory, there is an absolute rest frame, and every frame moving with constant velocity with respect to it is called inertial frame of reference. The frames that are accelerating with respect to it are called non inertial. To test whether a frame is an inertial, we test Newton's first law. Now in SR, there is no absolute rest frame, so why are there frames that are inertial while others not? In the absence of absolute rest frame, all frames should be completely equivalent.

You distinguish inertial from non-inertial frames by the same test: whether Newton's first law holds.

Just to clarify the point. Consider two frames where one of them is inertial, the other frame moves towards the first with an acceleration. Now both frames are completely symmetrical to each other, for each frame, the other is accelerating towards it, so what makes one of them preferable (inertial) over the other?

Acceleration is absolute, and locally-measurable, so these two frames are not equivalent.

You distinguish inertial from non-inertial frames by the same test: whether Newton's first law holds.
The OP refers to SR, Newton laws are set in an absolute space, that is not the case in SR.

Acceleration is absolute, and locally-measurable, so these two frames are not equivalent.
See my previous post before the last two.

That seems to be exactly the point of the OP, a "relative" velocity implies treating all reference frames as equal, an absolute quantity on the contrary must define some absolute reference frame, so I'd say he is simply saying that if acceleration is considered "absolute" rather than "relative", what frame is it distinguishing as the absolute one?
Whaaat ? This logic is wrong. It's so wrong I can't think how to refute it.

Whaaat ? This logic is wrong. It's so wrong I can't think how to refute it.

That's almost a self-defeating confession for a relativity forum, come on Mentz you know better than that :tongue2:, if it is so wrong it must be easy to pinpoint what exactly is wrong with that logic.

By the way, for instance ofirg had no problem following that logic and he came to the right conclusion: this is not addressed in SR, and in GR is addressed by introducing coordinate conditions, like the FRW comoving frame.

That's almost a self-defeating confession for a relativity forum, come on Mentz you know better than that :tongue2:, if it is so wrong it must be easy to pinpoint what exactly is wrong with that logic.
What's wrong with

"Some peas are white, some peas are not- therefore there must be an absolute pea"

Translating your logic it seems to say

"Some frames are inertial, some are not. Therefore there must be an absolute frame"

an absolute quantity ... must define some absolute reference frame

What absolute frame is defined by, for example, the proper mass of an electron?

What's wrong with

"Some peas are white, some peas are not- therefore there must be an absolute pea"

Translating your logic it seems to say

"Some frames are inertial, some are not. Therefore there must be an absolute frame"
Hmmm, apparently you didn't understand what I wrote. Do you know the difference say, between saying that a quantity is relative as opposed to absolute?

What absolute frame is defined by, for example, the proper mass of an electron?
We are talking about rates of change either of position or of velocity, what's that got to do with the electron's invariant mass?

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1) I really cannot see that acceleration is not relative. It is clear that every frame measures different acceleration, so we cannot say "accelerating" without considering a frame. For example: If two cars starts move at the same position with the same acceleration in the same direction, then each one sees the other non accelerating while both are accelerating with respect to the road.

3) Consider two frames of reference approaching each other with a certain acceleration. Which one is inertial and which one is non inertial ?!
The answer is simply we cannot know! There are missing information. It is possible that both are non inertial, and it is also possible that only one of them is inertial... My point is that physics here does not only depend on the relativity between the two objects as SR suggests. GR may be able to solve this problem, but I cannot figure how.

3) Testing Newton's laws allow to identify inertial frames, but this is not my issue. I am looking for the reason that makes some frames inertial while others not. If acceleration is the reason that makes some frames non inertial, then we have to specify with respect to what.

Let' see, there's atendency for some to get confused with the basics.
SR, first postulate, yes, the relativity principle, it prescribes that there is no preferred reference frame for the laws of physics, that is all inertial frames are considered equal, it makes no difference wrt the principle between velocities and accelerations, the principle is clear. This didn't cause any problem in the first years of SR because acceleration was not usually dealt with in the SR context at that time (the twin paradox introduced by Einstein in 1905 was a notorious exception, but gave rise to multitude of heated debates, including Einstein himself saying in the 20's that the solution required GR, which is not agreed by the majority of physicists nowadays), later it was found perfectly ok to work with acceleration in SR (Rindler coord., etc) as long as it was relative.
If one wants to declare that acceleration is absolute, that is fine, but then you are either using a weaker form of the first postulate (like it is done in GR and the use of coordinate conditions), or you are using an absolute reference frame, otherwise don't call acceleration absolute.

1) I really cannot see that acceleration is not relative. It is clear that every frame measures different acceleration, so we cannot say "accelerating" without considering a frame. For example: If two cars starts move at the same position with the same acceleration in the same direction, then each one sees the other non accelerating while both are accelerating with respect to the road.

3) Consider two frames of reference approaching each other with a certain acceleration. Which one is inertial and which one is non inertial ?!
The answer is simply we cannot know! There are missing information. It is possible that both are non inertial, and it is also possible that only one of them is inertial... My point is that physics here does not only depend on the relativity between the two objects as SR suggests. GR may be able to solve this problem, but I cannot figure how.

3) Testing Newton's laws allow to identify inertial frames, but this is not my issue. I am looking for the reason that makes some frames inertial while others not. If acceleration is the reason that makes some frames non inertial, then we have to specify with respect to what.
You have absolutely failed to understand anything you've been told about acceleration.

3) Consider two frames of reference approaching each other with a certain acceleration. Which one is inertial and which one is non inertial ?!
That is meaningless unless you specify which one or both are accelerating and by how much.
The answer is simply we cannot know! There are missing information. It is possible that both are non inertial, and it is also possible that only one of them is inertial... My point is that physics here does not only depend on the relativity between the two objects as SR suggests.
It is easy to tell if a frame is accelerating using an accelerometer. Your remark is completely wrong.

Every one of your points is wrong. I wonder why we're bothering.

1) I really cannot see that acceleration is not relative. It is clear that every frame measures different acceleration, so we cannot say "accelerating" without considering a frame. For example: If two cars starts move at the same position with the same acceleration in the same direction, then each one sees the other non accelerating while both are accelerating with respect to the road.

It may not satisfy you, but in SR, the norm of the 4-acceleration a scalar invariant. In all frames and and even non-inertial coordinates (e.g. Rindler), it comes out the same. This is called the proper acceleration. Specifically, say in, some frame in which the Minkowski metric holds, you have an accelerating object. You compute a proper acceleration for it. Transform to coordinates where the object is stationary - which implies a transform of metric diag(-1,1,1,1) to something like the Rindler metric. Now compute proper acceleration of the path (which is constant in Rindler position coordinates). You still get the same proper acceleration.

The proper acclereation computed this way is what an accelerometer would measure.

I personally don't believe there is a really good answer to 'why is some arbitrary frame inertial frame' beyond: a frame where an accelerometer measures no acceleration. GR provides a very partial answer, but far from complete. Arbitrary boundary conditions play a larger role than matter distribution. With arbitrary boundary conditions, you remain with nothing better than to measure which frames are inertial.

No, we do not! Velocity is "relative" but "acceleration" is not.

*Proper* acceleration is not relative. However, a number of people in this thread appear to be using the word "acceleration" to mean coordinate acceleration, not proper acceleration, and coordinate acceleration *is* relative. For example:

If you stand on a (properly aligned) scale in an accelerating frame, the reading will get bigger and bigger, but if you stand on it in an inertial frame, it won't change over time. When a jet pilot gets launched off of an aircraft carrier, he knows damn good and well that he's in an accelerating frame, but the guy on the flight deck isn't.

The guy on the flight deck *is* accelerating in an absolute sense; he feels nonzero proper acceleration. So in an absolute sense, he, and anyone else on the surface of the Earth, *is* in an accelerating frame. But with respect to a frame at rest on the surface of the Earth, his *coordinate* acceleration is zero.

The correct answer to the OP's question is the one Ben Niehoff gave in post #10: an "inertial" frame is a frame in which an observer at rest measures zero *proper* acceleration. A person at rest on the Earth's surface does not. Nor does Newton's First Law (the test Ben gave for an inertial frame) hold in a frame at rest on the surface of the Earth: a dropped rock falls even though there is no force on it. (In relativity, gravity is not a force.) The rest frame of the dropped rock is an inertial frame (more precisely a *local* inertial frame, if we let the rock fall far enough tidal effects will become evident and that opens a whole new can of worms).

if acceleration is considered "absolute" rather than "relative", what frame is it distinguishing as the absolute one?

You don't need one. Proper acceleration is absolute because it's a direct, local observable; you can measure it locally with an accelerometer.

You don't need one. Proper acceleration is absolute because it's a direct, local observable; you can measure it locally with an accelerometer.

Sure, that is what I'm trying to explain but it seems when people hears "absolute frame" they get in the defensive mode.

But note that the local observable explanation is not fully satisfactory in this context, because it just changes the name from absolute to local observable without entering into what makes it a local observable as opposed to velocity for instance.

But note that the local observable explanation is not fully satisfactory in this context, because it just changes the name from absolute to local observable without entering into what makes it a local observable as opposed to velocity for instance.

Well, perhaps I can add to the confusion.

Here's what I would say: *relative* velocity between two observers passing through the same event *is* a local observable. More precisely: the relative velocity between two local inertial frames at a given event is a local observable.

Does that help any?

I have a tendency to drop the word "proper" in discussions about relativity. "Proper" in the sense of "proper acceleration" is being used according to its original Latin meaning, "one's own". That is, "proper acceleration" means "my own acceleration, according to measurements I can do, myself." This is what is meant by "locally measurable".

What the OP is neglecting is that acceleration is, in fact, locally measurable. One measures the acceleration of one's own frame by releasing objects at rest and measuring their deviation from Newton's first law. That is, by dropping things.

By contrast, there is no such thing as "proper velocity", because there is no physical way to measure such a quantity.

Why do I prefer to drop the word "proper"? Because I think the old terminology of "proper" quantities versus "coordinate" quantities is confusing things. This is physics, and we are interested in physical quantities. Physical quantities are those that are measurable: that is, the "proper" ones. Really I think phrases such as "coordinate acceleration" ought to go the way of "relativistic mass": an outdated, misrepresented idea from an earlier time of less sophisticated understanding.

It is also important to distinguish "coordinates" and "frames". A system of coordinates is a set of labels attached to various spacetime points. All it tells you is that, say, (x,y,z,t) corresponds to that place over there at that time.

A frame, on the other hand, is a local system of measuring rods and clocks. There is no apriori reason this should have anything to do with the underlying coordinate system. A frame is just a collection of vectors at a point, of known lengths and angles, against which you can make standard measurements of other vectors at that point.

We are talking about rates of change either of position or of velocity, what's that got to do with the electron's invariant mass?

It is a counter-example to the claim that "an absolute quantity ... must define some absolute reference frame".

Acceleration and Velocity are not the same thing, which seems to be what you think.

So take two space ships moving relative to each other, it doesn't mean one is accelerating.

It is a counter-example to the claim that "an absolute quantity ... must define some absolute reference frame".

Ok, maybe I should have specified I was referring to vectorial quantities.

Well, perhaps I can add to the confusion.

Here's what I would say: *relative* velocity between two observers passing through the same event *is* a local observable. More precisely: the relative velocity between two local inertial frames at a given event is a local observable.

Does that help any?
Yes, it helps to stress what I was saying about the "local observable explanation" not really solving the relative vs absolute thing.

An inertial frame is harder to push than a non-inertial frame

Yes, it helps to stress what I was saying about the "local observable explanation" not really solving the relative vs absolute thing.
What do you mean by 'really solving' ?

I think the OPs original question

Just to clarify the point. Consider two frames where one of them is inertial, the other frame moves towards the first with an acceleration. Now both frames are completely symmetrical to each other, for each frame, the other is accelerating towards it, so what makes one of them preferable (inertial) over the other?

has been answered. Where does this absolute thing come in ?

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An inertial frame is harder to push than a non-inertial frame

That is a mood dependent variable.

3) Testing Newton's laws allow to identify inertial frames, but this is not my issue. I am looking for the reason that makes some frames inertial while others not.
So you're looking for the reason that makes Newton's laws true in some frames while not true in others?

What are the alternatives?

Could Newton's laws be true in all frames? Surely not; if a freely moving object moves at constant velocity in one frame, there are other (non-inertial) frames in which its velocity is not constant.

Could Newton's laws be true in no frames? In the context of special relativity and Newtonian mechanics (i.e. ignoring general relativity), we have many centuries of experiments verifying Newton's laws.

Your remark is completely wrong.

Every one of your points is wrong. I wonder why we're bothering.

If so, can you please give me simple clear consistent undebatable answers to the following two simple clear questions?

1) Does the acceleration depend on the frame of reference?

2) a) If (1) is Yes, then when you say that non inertial frames are accelerating and inertial frames not. With respect to what frame you mean?
b) If (1) is NO, then how can you explain the very clear fact that different frames may disagree about whether an object is accelerating or not like the example on post #19.