I can't understand inertial reference frames

[loom91]

Hi,

Please help me, I can't make head or tail of the concept of an inertial reference frame. What is an inertial reference frame? In what fundamental way does it differ from a noninertial reference frame if all motion is relative? Thanks for your help.

Molu

 

[vanesch]

Please help me, I can't make head or tail of the concept of an inertial reference frame. What is an inertial reference frame? In what fundamental way does it differ from a noninertial reference frame if all motion is relative? Thanks for your help.

Very good question: the definition of an inertial frame is a subtle concept ; moreover it is dependent on the paradigm in which you work (Newtonian, Special relativity, general relativity...).

There is some circularity in the definition of an inertial frame: you define particles to be free of interaction if they are in uniform motion wrt the frame, and you define an inertial frame as one in which free particles have uniform motion .

Now, in practice, you can solve this by saying that you can be pretty sure that certain observable objects are essentially free of interaction, and that in certain frames their motion does indeed look like it should. Then you can say that this frame is probably close to an inertial frame for the purposes of your inquiry, and the object is a free object for the purposes of your inquiry.The best thing to do, in Newtonian physics, for instance, is to say that it is a POSTULATE of the theory that there is an inertial frame, and to postulate that there are things which are particles free of interaction.

Now, things become clearer in general relativity: there, indeed, there is no such thing as an "inertial frame" and all frames are on equal footing. But now, the concept what a coordinate system means has become more involved (for instance, it's not up to you to say what is the "time" coordinate, and what is a "space" coordinate, and sometimes, these can switch roles !).

 

[loom91]

So am I to understand that the concept of inertial reference frames in Newtonian mechanics is an imprecise concept better un--understood?

 

[rcgldr]

The classic definition for an inertial frame of reference is one without acceleration (thus the term "inertial"). Here is a link:

http://id.mind.net/~zona/mstm/physics/mechanics/framesOfReference/inertialFrame.html

I had the impression that inertial frame of references were oriented towards Newtonian physics, not general relativity.

 

[vanesch]

The classic definition for an inertial frame of reference is one without acceleration (thus the term "inertial"). Here is a link:

http://id.mind.net/~zona/mstm/physics/mechanics/framesOfReference/inertialFrame.html

I wonder how you determine that your inertial frame has "constant velocity".
Velocity with respect to what ? An inertial frame ?

 

[vanesch]

So am I to understand that the concept of inertial reference frames in Newtonian mechanics is an imprecise concept better un--understood?

No, it is well understood. But as you saw by the "attempt at definition" by the link given in Jeff Reid's post, it's difficult not to be circular if you want to stay totally general in its definition.

As I said, in *practice* there's no difficulty. For many things, a coordinate frame attached to the surface of the Earth is "good enough". In fact, we don't find any "free objects" at the surface of the Earth (apples falling from the tree do NOT describe a uniform motion), but we don't seem to mind, and say that it is due to a force working on them, the force of gravity.
If a coordinate system attached to the surface of the Earth is not good enough, we do with a coordinate system through the center of the earth, and with the axes oriented to the "fixed starry sky". That's a much better inertial frame. Things that were incomprehensible in the former one, are now understood, such as, say, the Coriolis force. We recon that it is "an effect of the rotation of the earth".
An even better inertial frame would be one that is centered on the sun, and with its axes pointing to the "fixed starry sky".
That's about as good as you can do in Newtonian mechanics. This is the frame in which many objects which can be considered somehow "free" are in uniform motion... but there aren't any, of these "free" objects ! They all are in orbits around the sun, or around a planet, or... because they ALL are subjected to the force of gravity... And that was Einstein's idea: let's include this unavoidable "force of gravity" into the definition of "inertial frame".
General relativity is the result of a deep reflexion of what it really means, an "inertial frame".

 

[rcgldr]

I wonder how you determine that your inertial frame has "constant velocity".

See if the laws of inertia apply from the inertial frame reference as mentioned by the website I linked to and many others. The point was that with an accelerating frame of refernence the laws of ineritia wouldn't apply. Again I defer to the website and several others that define inertial frame of reference in the same manner.

 

[loom91]

See if the laws of inertia apply from the inertial frame reference as mentioned by the website I linked to and many others. The point was that with an accelerating frame of refernence the laws of ineritia wouldn't apply. Again I defer to the website and several others that define inertial frame of reference in the same manner.

But to use this definition one must first define the concept of force externally and then observe if in a particular frame particles have 0 acceleration when no force is being applied.

But I believe force itself is defined from Newtons second law as the time derivative of momentum. You define that force is the rate of change of momentum, but then in some particular frames say that even though the momentum-derivative is non-stationery there is no force. The definition looks like a FAPP (For All Practical Purposes) one to me rather than a theoretically sound physical one.

 

[rcgldr]

force itself is defined from Newtons second law

True, but again, deferring to the website I linked to (and many others), you can simply imagine a human in the frame of reference, where forces can be felt. You could also use an accelerometer.

 

[loom91]

True, but again, deferring to the website I linked to (and many others), you can simply imagine a human in the frame of reference, where forces can be felt. You could also use an accelerometer.

So we somehow decide what forces are acting and then see if they produce the right acceleration to check if the frame is inertial? How is this decision taken?

 

[marlon]

I wonder how you determine that your inertial frame has "constant velocity".
Velocity with respect to what ? An inertial frame ?

Well with respect to the uniform background that you need to define. I mean, without an Euclidean background, Newtonian mechanics (which is described in terms of vectors) has no meaning.

In this context, an inertial frame of reference is that frame onto which no net forces is acting. Thus, a frame with constant velocity with respect to the Euclidean background-space.


marlon

 

[loom91]

Well with respect to the uniform background that you need to define. I mean, without an Euclidean background, Newtonian mechanics (which is described in terms of vectors) has no meaning.

In this context, an inertial frame of reference is that frame onto which no net forces is acting. Thus, a frame with constant velocity with respect to the Euclidean background-space.


marlon

But without the presence of ether, you can't determine velocity against background-space! Galilean relativity forbids it.

 

[marlon]

But without the presence of ether, you can't determine velocity against background-space! Galilean relativity forbids it.

But we are talking about classical physics here, not general relativity. The concept of inertial frame does not mean anything in general relativity. That's the entire point of Einstein's work.


marlon

 

[vanesch]

Well with respect to the uniform background that you need to define. I mean, without an Euclidean background, Newtonian mechanics (which is described in terms of vectors) has no meaning.

In this context, an inertial frame of reference is that frame onto which no net forces is acting. Thus, a frame with constant velocity with respect to the Euclidean background-space.

Newtonian physics is not done on E^3 x E^1 but on a fibre bundle with base E^1 (time) and fibre (E^3). A fixed "background" Euclidean space is more "Aristotelian". In Newtonian physics, there is no natural identification between the different fibres E^3. Such an identification is exactly what is an observer frame. There needs to be enough structure on the fibre bundle to allow for the differentiation between inertial and non-inertial observers, but there should not be enough structure as to single out a preferred natural identification between the bundles (as is the case in E3 x E1).

Penrose has a great exposition about that in his book "The road to reality", in chapter 17, where he shows that Newton - who didn't know about fibre bundles - struggled a long way, and finally resolved for a "background space" in order to formulate his laws. It's a great read, and one which provides the insight of what's the difference between the Aristotelian view (an absolute space, and an absolute time), and the Newtonian (Galilean) view, with the inherent impossibility to have an absolute space.

 

[vanesch]

But without the presence of ether, you can't determine velocity against background-space! Galilean relativity forbids it.

Yes, exactly. That's the difference between an "Aristotelian" and a "Galilean" spacetime. In a galilean spacetime, there's no natural identification between the E3 spaces "at different times", while in the "Aristotelian" view, an absolute point is an absolute point.

That's what I said in my previous post.

 

[marlon]

Newtonian physics is not done on E^3 x E^1 but on a fibre bundle with base E^1 (time) and fibre (E^3). A fixed "background" Euclidean space is more "Aristotelian". In Newtonian physics, there is no natural identification between the different fibres E^3. Such an identification is exactly what is an observer frame.

This really is a bit of a cheap way out, no ?

We do not need to work with fibers when talking about Newtonian Physics. Hell, we should not be talking about fibers because of historical relevance. So, once again, one can define an ordinary Euclidean base which is used as background in the "Newtonian world". It is with respect to this base that all the necessary vectors are defined and how our intro physics books are written. So i can only repeat what i have stated before as an answer to your "question".

Actually, this exactly why i like Newtonian physics. Once we need to start working with differential geometry and related tensor formalisms, things get more complicated. Eventhough the latter formalisms provide the correct physical description i think their value should not be overestimated (indeed, i am referring to general relativity now). Just look at the amount of problems we are able to solve by just using the old fashioned Newtonian physics. What problems are solved by GR ? (er, this is a rhaetorical question). All industry that we know off is based upon classical physics and QM, but where's GR ? That is also why i say that Newton's work is far more valuable than Einstein's.

regards
marlon

 

[loom91]

But we are talking about classical physics here, not general relativity. The concept of inertial frame does not mean anything in general relativity. That's the entire point of Einstein's work.


marlon

I said Galilean relativity, not general relativity.

 

[loom91]

We do not need to work with fibers when talking about Newtonian Physics. Hell, we should not be talking about fibers because of historical relevance. So, once again, one can define an ordinary Euclidean base which is used as background in the "Newtonian world". It is with respect to this base that all the necessary vectors are defined and how our intro physics books are written. So i can only repeat what i have stated before as an answer to your "question".

regards
marlon

But irrespective of whether we CAN take the background as E^3xE (which I believe we can not once Galilean relativity, which was not used by Newton but became a part of classical mechanics because Newton was wrong and Galileo was right), there is no method whatsoever for measuring a velocity against that background spacetime. It is theoretically impossible to determine a reference frame to be fixed with respect to the E^3, even if it existed. That's the whole point about the ether fiasco. There exists no absolutely stationery reference frame.

Molu

 

[vanesch]

This really is a bit of a cheap way out, no ?

We do not need to work with fibers when talking about Newtonian Physics. Hell, we should not be talking about fibers because of historical relevance. So, once again, one can define an ordinary Euclidean base which is used as background in the "Newtonian world".

Of course you can do that, just as you can do general relativity in an Euclidean space. But then you miss the essential concept. You can even do Euclidean geometry in a vector space, with an origin. The problem is that you've introduced arbitrary, unphysical concepts, and afterwards you have then to postulate that for one or other strange reason, the "detection" of these arbitrary concepts is "impossible". It is much clearer (though more sophisticated) to start directly with the right vision.

So, yes, you can say that there is "an absolute space and an absolute time" (which is essentially an Aristotelian viewpoint), and start working from there. And then you miss the entire content of the principle of Galilean relativity, which was EXACTLY the point, that there is no natural identification between "space points" at two different times, but that this is an entirely observer-dependent notion.

This is usually solved (correctly) in courses on Newtonian physics, by saying that *by postulate* there exists (at least one) inertial observer. It's this observer which can then "fix" absolute space for all times.
But then there's nothing that sets _this_ inertial observer apart from another one, and we'd be confronted with an operational difficulty: HOW do you find, operationally, this SPECIAL inertial observer for which absolute space is fixed ? You can't. When you start working from this "special" inertial observer, you arrive at a point where you find out that it is, in all respects, impossible to discern from "another" inertial observer which is in uniform motion wrt the original, "special" one. So this "absolute" observer was a totally arbitrary choice. But that means that there's no physical meaning to be attached to this absolute space in the first place, because it is totally arbitrary.
This viewpoint is EXACTLY the same as the "Lorentz ether" view in special relativity. You can specify indeed, an absolute luminoferous ether, and then postulate that things contract and times dilate wrt to this Lorentz ether. But no serious course on special relativity takes on this view, because it would make you totally MISS the MEANING of special relativity (although all calculations would be correct). In the same way as it makes you miss the meaning of special relativity by introducing an ether, an absolute euclidean space in Newtonian physics makes you miss entirely the content of the principle of Galilean relativity, of which the content is exactly, that there is NO natural way to identify "space points" at two different instances of time, and that this identification is a totally observer-dependent concept.
It is the same as doing Euclidean geometry in R^3. You miss the entire idea behind the concept of an Euclidean space when you *identify* an Euclidean space with R^3 - although of course all results will be correct.

 

[vanesch]

It is theoretically impossible to determine a reference frame to be fixed with respect to the E^3, even if it existed. That's the whole point about the ether fiasco. There exists no absolutely stationery reference frame.

Yes, indeed. The mathematical structure should not introduce fundamental concepts which are - in principle - operationally undefinable. This is indeed, what goes conceptually wrong with things such as "absolute space" or "ether".

 

[rcgldr]

During the Apollo space missions to the moon and back, and also in the case of satellites that have traveled to the edges of our solar system, how much GR math was involved? I have the impression that Newtonian physics, combined with minor mid-course corrections were more than accurate enough to send objects at reasonably fast speeds and distances.

My point here is that for many Earth bound and space bound modes of travel, Newtonian physics is close enough to be useful.

It's in these Newtonian base applications where an inertial frame of reference would make sense. Again, if a web search is done for inertial frame of reference, most of the hits will include a simple description, which is good enough for most pratical puposes.

 

[vanesch]

It's in these Newtonian base applications where an inertial frame of reference would make sense. Again, if a web search is done for inertial frame of reference, most of the hits will include a simple description, which is good enough for most pratical puposes.

That's what I said, too: *practically* there is no problem ; and a good *practical approximation* of an inertial frame is taking a coordinate system centered on the sun, and with axes pointing to the remote stars. This is the system in which most if not all "space mission" calculations are performed, and even calibrated (using small telescopes on satellites which keep a certain far away star in the collimator).
And why is it a practically "good" reference system ? Well, because we can apply Newton's laws in it and it works quite well !

 

[marlon]

But without the presence of ether, you can't determine velocity against background-space! Galilean relativity forbids it.

Yes you can. The concept of absolute velocity is dealt with (meaning that one does not need absolute v anymore) in special relativity (this is NOT Galilean Relativity).

marlon

 

[marlon]

But irrespective of whether we CAN take the background as E^3xE (which I believe we can not once Galilean relativity, which was not used by Newton but became a part of classical mechanics because Newton was wrong and Galileo was right), there is no method whatsoever for measuring a velocity against that background spacetime.
Molu

Well, this is all very true bit really irrelevant to this discussion. I never stated that classical physics was perfect or that there are no problems concerning relativity. But relativity is NOT the point. The point is that we can write down the entire Newtonian physics formalism by just chosing one Euclidean base that we look at as background. All the results of point particle mechanics, rigid rotators, statica, etc etc will be found very nicely. So again, all the usefull mechanics just requires one basic background. That's the beauty of it all

marlon

 

[marlon]

So, yes, you can say that there is "an absolute space and an absolute time" (which is essentially an Aristotelian viewpoint), and start working from there. And then you miss the entire content of the principle of Galilean relativity, which was EXACTLY the point, that there is no natural identification between "space points" at two different times, but that this is an entirely observer-dependent notion.

But my point is that all these notions of relativity are not the most important aspect. You say they are the key concept but i disagree. This is what i am trying to say. All the usefull results coming from classical physics can just be derived using a simple Euclidean background space. I mean, can't you see the difference in relevance to us when you compare classical physics with respect to what Lorentz transformations have taught us (Please, look at this question as being a rhaetorical one ?

Just to be clear : i never stated that classical physics is perfect and that the formalism has no flaws.

marlon

 

[vanesch]

Yes you can. The concept of absolute velocity is dealt with (meaning that one does not need absolute v anymore) in special relativity (this is NOT Galilean Relativity).

Eh, no, the idea that there is not such a thing as absolute velocity IS (Galilean) relativity.
The only difference between Galilean relativity and Newtonian gravity on one side, and Special and General relativity on the other side, is the extra postulate that the velocity of light should be independent of the observer's frame (and as such, there's no other solution than to make time relative and not absolute anymore).

Galilean/Newtonian relativity and Einstein relativity swap the following postulates:
G/N: time is absolute, the same for all observers
E: velocity of light is the same for all observers

But the "no absolute velocity" is NOT the thing that was new in special relativity, its original idea was found by Galileo.

Both G/N and E share the postulate that there is no such thing as absolute velocity. If you want to drop that, you go back to Aristotle's natural philosophy, where "rest" is the "natural state of motion" without a "mover".

So when, in an intro course on Newtonian physics, one starts to work in R^3, then it should be clear that this is NOT absolute space, but *the frame of an inertial observer, choosen at random*.

And then the question (which is usually not dealt with) is: how do you *operationally* define such an inertial frame ?
And the "answer" would be: look at *free* particles, if they describe a uniform motion then you have an inertial frame.
But clearly not all particles are free particles. So what are free particles ?
"free particles are particles on which no force acts".
Hmmm, and what's a force ? How do I know what is the force acting on a particle ? Well write out Newton's second equation *in an inertial frame*

For instance, at the surface of the earth, there are no free particles: they all suffer gravity at least. The thing that comes closest are light rays, and there, we're not going to apply Newton's theory.

But in practice we don't care: we put a frame through the center of the earth, with axis pointing to the stars, and that's already quite good. It is even acceptable to have the axes turn with the Earth for many situations, and have the reference frame fixed to the surface of the earth. But this is because of the difference in orders of magnitude in the accelerations in the problem we study in textbooks and those of the approximate reference frames. For an ant on a dust particle in a turbulent airflow, the "practical approach" suddenly becomes much less evident. There you encounter really the circularity of the operational definition of "inertial frame".

 

[marlon]

Eh, no, the idea that there is not such a thing as absolute velocity IS (Galilean) relativity.

Eh, yes, time in Galilean relativity is absolute. Given this, the velocity transformation from one frame to another is NOT correct in Galilean relativity. Only Special relativity deals with these issues in the correct manner. Special relativity deals with absolute velocity and general relativity deals with absolute acceleration. It is a simple as that.

But the "no absolute velocity" is NOT the thing that was new in special relativity, its original idea was found by Galileo.

I never stated that this was a basic postulate of special relativity. I just said that SR solves the issue with absolute velocity.

But in practice we don't care: we put a frame through the center of the earth, with axis pointing to the stars, and that's already quite good.

You know, it is funny how you first start out with obecting against issues that are irrelevant and in the end you say this. Your above quote is my entire point. One can whine about relativity all he/she wants, but what's the use ? I mean, again, i do realize the flaws (with respect to a clear definition of inertial frame) in classical physics but why can't you see that most of the useful things we get out of this formalism don't even need all of this. Working with a simple Euclidean base will do just fine.

Doesn't this last remark account for anything ? So many people are busy with whinning about how classical physics is wrong and how "relativity" solves the problems. They really should ask themselves this : Does the practical relevance of classical physics not exceed the theoretical correctness of SR/GR ?



marlon

 

[vanesch]

Eh, yes, time in Galilean relativity is absolute. Given this, the velocity transformation from one frame to another is NOT correct in Galilean relativity. Only Special relativity deals with these issues in the correct manner. Special relativity deals with absolute velocity and general relativity deals with absolute acceleration. It is a simple as that.

Let's put things in perspective: you started by saying that in Newtonian physics, with Galilean relativity, SPACE is absolute. This is not correct. What is correct, is that in this view, TIME is absolute, but SPACE isn't.
It is the entire difference, mathematically speaking, between the product space E^3 x E^1 and an E^3 fibre bundle over E^1: in the former, there is ONE, absolute, space E^3. In the latter, there's an independent E^3 for each element of E^1. The IDENTIFICATION between the different E^3 is exactly what a reference frame is all about, and THIS is what Galileo stated (but not in the language of fibre bundles of course).
If there is an absolute space (as was the pre-galilean concept), then there' s no such issue, of course: a point in absolute space keeps its identity all over time, so there is a preferred reference frame, which assigns the SAME coordinates to this absolute point for all times.
In both approaches, time is absolute: the concept of simultaneity is a physical concept, independent of the observer.


Special relativity doesn't address this issue. It addresses the issue of the discovery that LIGHT SPEED IS AN INVARIANT. This screws up one of the two concepts:
1) the idea of galilean relativity that space is NOT absolute
2) the idea that time is absolute

The invariance of the speed of light came about experimentally, and also through the Maxwell equations. It didn't need to be this way, but nature turned out to be such that the speed of light was an absolute concept.

After this discovery, one first REJECTED 1). One PUT GALILEAN RELATIVITY aside, to go back to the Aristotelian idea of absolute space. This is the Lorentz ether theory, and the absolute space (and absolute time) are defined by the frame of the luminoferous ether.

What Einstein did, was to RESTORE Galilean relativity (1), and to REJECT 2), the physicality of absolute time.

But this price to pay (to make time also relative) came about in order to reconcile the NEW IDEA of the constancy of the speed of light with the former idea of Galilean relativity. So Einstein REVIVED the concept of Galileo that there was not an absolute space.

I never stated that this was a basic postulate of special relativity. I just said that SR solves the issue with absolute velocity.

No, it doesn't. It was ALREADY solved by Galileo. What special relativity did was to address the EXTRA ISSUE of absolute light velocity.

You know, it is funny how you first start out with obecting against issues that are irrelevant and in the end you say this. Your above quote is my entire point. One can whine about relativity all he/she wants, but what's the use ? I mean, again, i do realize the flaws (with respect to a clear definition of inertial frame) in classical physics but why can't you see that most of the useful things we get out of this formalism don't even need all of this. Working with a simple Euclidean base will do just fine.

There are no FLAWS in classical physics concerning the relativity of velocity. It turns out that our universe is not like that, but it is logically perfectly all right. Working in a simple Euclidean base is OF COURSE ALL RIGHT once you have choosen your inertial frame! That's exactly what it is all about. But the problem is with the operational definition of that inertial frame. How you connect the laboratory procedure with the points in your Euclidean space (or better, R^3). This problem remains the same, btw, in special relativity. It is only in the approach of general relativity that this is solved through general covariance, and one can in fact do the same in classical physics. Cartan did it (but it is a more complicated theory than Einstein general relativity).

You didn't say how the ant in her lab based on her dust particle, in the turbulent air flow, is going to define the mapping of its measurements on its Euclidean basis operationally.

 

[marlon]

Let's put things in perspective: you started by saying that in Newtonian physics, with Galilean relativity, SPACE is absolute.

No, i never said that. What i said was that one can derive all results from classical physics by using just a simple Euclidean background that you need to define the necessary vectors. I have stated this several times.

It is the entire difference, mathematically speaking, between the product space E^3 x E^1 and an E^3 fibre bundle over E^1: in the former, there is ONE, absolute, space E^3. In the latter, there's an independent E^3 for each element of E^1. The IDENTIFICATION between the different E^3 is exactly what a reference frame is all about, and THIS is what Galileo stated (but not in the language of fibre bundles of course).

Again, you don't need to bring in the concept of fibre bundles into this discussion because it is irrelevant. Again, i never stated that space is absolute.

What Einstein did, was to RESTORE Galilean relativity (1), and to REJECT 2), the physicality of absolute time.

This is a poor formulation i my opinion since if he would have restored Galilean relativity, the physicality of absolute time cannot be rejected.
I would say that after special relativity, the issue of absolute velocity was solved but general relativity solves the issue of absolute acceleration.

No, it doesn't. It was ALREADY solved by Galileo. What special relativity did was to address the EXTRA ISSUE of absolute light velocity.

No, you are wrong. It was NOT solved by Galileo because the transformation of velocity under Galileo is incorrect because c is not an universal constant here. Just look at when the velocity transformation under Galileo and Lorentz are equal (only very small velocities). The Galileo Transformation is wrong and therefore does not solve the absolute v issue. Only special relativity does.

There are no FLAWS in classical physics concerning the relativity of velocity. It turns out that our universe is not like that, but it is logically perfectly all right.

Well, err, the formalism may be logically all right but if one starts from incorrect postulates, the model is just plain wrong. That's all there is to it. Aso, like you say yourself, our universe does not behave like that...Err, i think that says it all.

Working in a simple Euclidean base is OF COURSE ALL RIGHT once you have choosen your inertial frame! That's exactly what it is all about.

Well, that's what i have been saying all along. The only difference between you and me is that i say : just pick the reference frame with it's origin in the Earth's center and all is ok. No need for "defining" an inertial frame properly and no need for relativity all along.

But the problem is with the operational definition of that inertial frame.
How you connect the laboratory procedure with the points in your Euclidean space (or better, R^3).

Why would i want to do that ? For the 1000th time, classical physics works just fine starting from one simple Euclidean base. I can build devices with that, calculate the stability of walls, calculate equilibria, etc etc ...

What can i calculate with relativity that would make our lives better ?

Do you see my point ?

marlon

 

[leright]

I've always thought the absolute reference frame was considered to be the ether (not that there is such a thing, however). A stationary reference frame has zero velocity wrt the ether, and an inertial reference frame has a constant velocity wrt the ether.