Inertial & Non-Inertial Frames: Laws of Physics Explained

[Krishankant Ray]

According to the postulates of Einstein theory, laws of physics are same in all inertial frame. What about non- inertial frames? Why they can't be same in non-inertial frame?

 

[Orodruin]

You are referring to the special principle of relativity which is at the foundation of SR. The general principle of relativity extends this to arbitrary frames.

 

[bcrowell]

You are referring to the special principle of relativity which is at the foundation of SR. The general principle of relativity extends this to arbitrary frames.

This is more like Einstein's hazy original interpretation of GR, which was wrong. https://www.physicsforums.com/threads/distinction-between-special-and-general-relativity.827721/

Neither SR nor GR needs to be formulated in terms of frames of reference. Frames of reference are an optional thing that don't even need to be discussed if you don't want to.

According to the postulates of Einstein theory, laws of physics are same in all inertial frame. What about non- inertial frames? Why they can't be same in non-inertial frame?

They can, and it doesn't require GR. For example, electromagnetism can be formulated in the language of tensors, and then the laws have the same form in any coordinate system, regardless of whether the coordinate system even has an interpretation in terms of a frame of reference.

 

[Orodruin]

This is more like Einstein's hazy original interpretation of GR, which was wrong. https://www.physicsforums.com/threads/distinction-between-special-and-general-relativity.827721/

This depends on what you mean by "frame". For a discussion with a layman, I would let frame = coordinate system.

 

[loislane]

They can, and it doesn't require GR. For example, electromagnetism can be formulated in the language of tensors, and then the laws have the same form in any coordinate system, regardless of whether the coordinate system even has an interpretation in terms of a frame of reference.

Not really. All tensors are not created equal, and their invariance properties depend on the way they are defined and their domain of application as proper tensors, thus you may have cartesian tensors, pseudo-euclidean tensors, when they exceed their domain of application they become pseudotensors, meaning that they no longer follow tensor transformation laws strictly. The tensorial formulation of EM is in terms of differential forms. This type of tensors only require a smooth manifold and an affine connection, when the connection is flat as is the case in the relativistic formulation of EM in flat spacetime, such manifolds are equivalent to affine spaces with flat geometry, of which Minkowski space is an example.

This kind of tensors, a generalization of Euclidean or affine tensors that substitutes the invariance under coordinate transformations from orthogonal transformations to the also linear Lorentz transformations are not tensors anymore when going out of the domain where they are defined in, be it smooth manifolds with affine connection or the added condition of a flat connection as in electromagnetism in flat spacetime. They are restricted in their tensor condition to certain coordinate systems, like cartesian or skewed coordinates, not arbitrary ones.
This doesn't mean curvilinear coordinates cannot be used in an affine space like Minkowski, just that in the presence of curvilinear coordinates nonlinear terms will appear in the form of christoffel components not transforming as a tensor even if the geometry is flat.
Conversely there is no direct consequence from the restriction of the tensorial form to smooth manifolds with connection when going to EM in curved spacetime since the general coordinate transformations in curved spacetime refer to those up to diffeomorphisms of smootn manifolds.

Noninertial frames like for instance rotating frames involve the kind of nonlinear coordinate transformations that are not included in that set of coordinate transformations, much to the dismay of Einstein himself who thought for many years that he had managed to include absolute acceleration in his general theory of relativity

 

[martinbn]

@loislane, that doesn't make much sense.

 

[Orodruin]

@loislane, that doesn't make much sense.

It does make sense, but perhaps not at the level the OP is looking for ...

 

[loislane]

@loislane, that doesn't make much sense.

As written your argument isn't very helpful(as in specifying what parts make some sense and which not much sense to you), is it?
What specifically did you not understand?

 

[vanhees71]

I also don't understand why this shouldn't make sense. Perhaps it's a discrepancy in language?

 

[bcrowell]

Loislane's #5 reads to me as a farrago of correct statements, incorrect statements, irrelevancies, historically based misconceptions, and imprecise use of language. If we want to go into this, a thread at the "I" level is probably not the place for it.

 

[vanhees71]

These levels get on my nerves ;-). Why is this not appropriate for an "I" level thread? It's a quite basic question, how you understand "tensors". For physicists a tensor has components which transform under a certain class of transformations from one reference frame to another. There are thus tensors with respect to rotations, other tensors with respect to Lorentz transformations and so on.

Also for physicists reference frames are very important. There may be advantages for some calculations to use the mathematicians' notion of a frame-independnet formulation of the theory, but when it comes to observations in the real world you always introduce a reference frame since you use real-world measurement apparati, which define a frame.

 

[bcrowell]

These levels get on my nerves ;-). Why is this not appropriate for an "I" level thread?

"I" is defined as undergrad level. Tensors, diffeomorphisms, etc., are not typically discussed in undergraduate courses. The OP has already received correct and incorrect answers at the level s/he indicated s/he would be able to understand. I'll start a separate thread.

 

[bcrowell]

"A"-level thread:
https://www.physicsforums.com/threads/laws-of-physics-in-noninertial-frames.834657/

 

[stevendaryl]

According to the postulates of Einstein theory, laws of physics are same in all inertial frame. What about non- inertial frames? Why they can't be same in non-inertial frame?

In the modern way of thinking about it, the "laws of physics" can be expressed in a way that doesn't depend on any particular rest frame or coordinate system. However, I think that what Einstein meant was this: (Considering only a single spatial dimension, for simplicity) If you graph the position x of a particle as a function of time t, then in an inertial frame, the graph of a "free" particle makes a straight line. The graph of a light signal also makes a straight line, and furthermore, the speed (the slope of the line) has the value c=300,000,000\ m/s (or whatever the precise number is). In a noninertial coordinate system, this will no longer be true; the "speed" of a particle need not be constant (if by "speed" you just mean the slope of the graph of x versus t). So the behavior looks different in a noninertial frame.

The modern way of thinking about these things distinguishes between physical acceleration (also called "proper acceleration") and coordinate acceleration.

 

[bcrowell]

Stevendaryl's #14 is nice -- and three cheers for an answer that's at the level requested by the OP.

However, I would say this in a different way. There is a distinction between inertial motion and noninertial motion, and that distinction is a clear one in SR and GR. However: (1) the relativistic definition of inertial motion is free-falling motion, which is different from the Newtonian definition; and (2) Einstein was simply wrong when he tried to describe this by assigning some foundational role to frames and the form of the laws of physics in various frames.

This may also be of interest: https://www.physicsforums.com/threads/distinction-between-special-and-general-relativity.827721/

 

[vanhees71]

Here I disagree. Physics only works in frames. You have to define a frame, when measuring distances and time intervals. This is done by using "clocks an rods" in a very general sense, i.e., any devices which measure space and time. Physics is about observations and measurements, and as soon as you take that into account you define a frame. To measure something you need to compare the quantity you want to measure with some normal, defining your units! If this is not I level, then you can't do physics on I level!

 

[robphy]

Here I disagree. Physics only works in frames. You have to define a frame, when measuring distances and time intervals. This is done by using "clocks an rods" in a very general sense, i.e., any devices which measure space and time. Physics is about observations and measurements, and as soon as you take that into account you define a frame. To measure something you need to compare the quantity you want to measure with some normal, defining your units! If this is not I level, then you can't do physics on I level!

I'm not sure what "only works" means.

In my opinion, there is a distinction between the "laws" of physics and "measurements by observers" that one might do to express and interpret the laws.
By analogy, a law might be the statement \vec C=\vec A+\vec B. Introducing a set of axes, I could express this in various coordinate systems... resulting in expressions that describe the "law" as well as "specifics about the observer doing the measurement". At this level, one might be interested in finding ways to compare specific-observer measurements (i.e., find transformations that map one set onto another)... but, while important in doing measurements in the lab, it clutters the essence of the law itself.

To me, it seems more like:
from the myriad of measurements that observers make, we abstract ("tease out") a common feature that expresses the "physical law" (without regard to observers). Hopefully, this can be done cleanly so that, if we wish, we could re-insert an observer in order to provide a way to predict what that observer would measure.

 

[bcrowell]

Here I disagree. Physics only works in frames. You have to define a frame, when measuring distances and time intervals. This is done by using "clocks an rods" in a very general sense, i.e., any devices which measure space and time. Physics is about observations and measurements, and as soon as you take that into account you define a frame. To measure something you need to compare the quantity you want to measure with some normal, defining your units! If this is not I level, then you can't do physics on I level!

This is not really true. In principle, all measurements in relativity can be reduced to what a geometer would call incidence relations, meaning basically intersections. For example, if two particles collide, their world-lines intersect, and this is an observable fact. You can have, e.g., clocks without having a frame of reference. A frame of reference is a very high-tech, elaborate thing that is the result of a program of surveying by techniques such as Einstein synchronization. It's an optional convenience.

 

[vanhees71]

A clock defines a reference frame, namely its restframe.

 

[Dale]

A clock defines a reference frame, namely its restframe.

This is not correct. We have already discussed this previously. The GPS system implements a frame called the Earth centered inertial frame. In this frame none of the components of the GPS system are at rest.

 

[bcrowell]

To be fair to vanhees71, I should admit that the view I'm expressing in #18 shows me changing my mind on something where I'd expressed the opposite view recently on this site.

 

[bcrowell]

A clock defines a reference frame, namely its restframe.

This is not correct. We have already discussed this previously. The GPS system implements a frame called the Earth centered inertial frame. In this frame none of the components of the GPS system are at rest.

I don't see any contradiction between these two statements. Vanhees71 says that for any clock, a frame exists which is the clock's rest frame. This is true. DaleSpam says it is not true that for any frame, a clock exists such that the frame is its rest frame. This is also true.

 

[Dale]

Vanhees71 says that for any clock, a frame exists which is the clock's rest frame. This is true.

It could be that is what he means, but based on previous conversations I think that he means that the clock IS the rest frame, not merely that there exists a frame where the clock is at rest. He seems to not distinguish between measurement devices and reference frames, which is why I bring the GPS ECI frame in since the distinction between the frame and the measuring devices is so apparent.

@vanhees71 if I am misunderstanding your current position on this issue then I do apologize. I may be dragging in remnants of a previous discussion on a position you no longer hold.

 

[vanhees71]

This is not correct. We have already discussed this previously. The GPS system implements a frame called the Earth centered inertial frame. In this frame none of the components of the GPS system are at rest.

It doesn't matter which frame the device defines. It is important that it defines a frame and that you understand to which frame the "pointer readings" refer to. In the GPS this is obviously very well understood, because otherwise it wouldnt' function with the desired precision we all love when using a navigation system!

Perhaps it was misleading, how I formulated this statement. I had in mind an idealized localized clock. The GPS consists of several satellites in orbits around the Earth. There it might not even be possible to find a common restframe of this "clock" as a whole. Nevertheless you can handle the GPS only, if you have a well-defined frame of reference, and obviously this is the case, because you quote a frame called "earth centered inertial frame". I'm not familiar with the details of the workings of the GPS, but even without this knowledge it is clear that you cannot make sense of the signals of the GPS satellites without knowing about this (or any other appropriate) reference frame.

 

[vanhees71]

To be fair to vanhees71, I should admit that the view I'm expressing in #18 shows me changing my mind on something where I'd expressed the opposite view recently on this site.

If we wouldn't change our minds on things debated here, this whole forum would be useless, because then we wouldn't learn anything from it, and I learn a lot. Particularly one learns that one has not understood things, which one has thought to have understood them before (like in my case with the spaceship paradox this summer :-)). So it's a good thing!

 

[Dale]

It doesn't matter which frame the device defines. It is important that it defines a frame and that you understand to which frame the "pointer readings" refer to. In the GPS this is obviously very well understood, because otherwise it wouldnt' function with the desired precision we all love when using a navigation system!

Perhaps it was misleading, how I formulated this statement. I had in mind an idealized localized clock. The GPS consists of several satellites in orbits around the Earth. There it might not even be possible to find a common restframe of this "clock" as a whole. Nevertheless you can handle the GPS only, if you have a well-defined frame of reference, and obviously this is the case, because you quote a frame called "earth centered inertial frame". I'm not familiar with the details of the workings of the GPS, but even without this knowledge it is clear that you cannot make sense of the signals of the GPS satellites without knowing about this (or any other appropriate) reference frame.

I don't think that your position on this is tenable. Either the device defines the reference frame or the analysis does. If the device defines the reference frame, and not the analysis, then it does matter which frame it is. If the device defines the frame then there can be no freedom in the analysis to choose a different frame than the one defined by the device.

That is clearly false. Any frame may be defined in the analysis. Therefore the frame cannot be defined by the device.

That is what the GPS does. It takes a whole bunch of measurements from a bunch of different clocks and transmitters and it analyzes those measurements in order to get coordinates in the ECI frame. The ECI frame is not intrinsic to any of the measurements involved, it is a result of the analysis of the measurements.

I understand that you are reacting to a perceived overemphasis of the geometry. You want to emphasize the importance of measurements and reference frames. That is fine, but you are going too far in the process.

 

[vanhees71]

Of course, I can use any frame I like in my analysis, but the device produces pointer readings which refer to a frame. As I said, I'm not familiar with the inner workings of the GPS, but it's clear that it is a prerequesite to know, how the measurements "from a bunch of different clocks and transmitters" define this ECI frame. As with any measurement you must be able to define a frame of reference from the physics of the appartus used to measure anything. Without a clear understanding about the frame of the apparatus (or in this case the various frames the different satellites of the GPS define) you'd not have any basis for your analysis finally defining this frame. You are contradicting yourself, because on the one hand you say the ECI frame is defined by the analysis of the various signals from the clocks and transmitters and on the other hand you say the "ECI frame is not intrinsic to any of the measurements involved".Of course it is, and that's why the GPS works with high precision! As far as I understand it, as a sideproduct it's another great successful test for GR.

 

[Dale]

the device produces pointer readings which refer to a frame.

A device produces pointer readings which are frame invariant.

 

[stevendaryl]

A clock defines a reference frame, namely its restframe.

Could somebody repeat the definition of "rest frame"? Some people seem to use it just to mean a standard of rest (which the name "rest frame" suggests). A traveling clock defines a standard of rest for itself (namely, it considers itself to be at rest), but to say whether other objects are at rest, it seems to me that the clock must also carry along something like a triad of spatial vectors in order to say that the other object is maintaining the same spatial relationship to the clock (that is, it's not moving relative to the clock). So I sort of think that you need a tetrad to define (even a local) a frame, not just a clock.

 

[vanhees71]

Ok, let me try it more theoretically now:

In SR by definition the rest frame is given by a sequence of instantaneous rest frames of the particle along the timelike world line of this particle, formally defined by a comoving Vierbein (tetrad). It is also convenient to use non-rotating Vierbeins, i.e., you define the sequence of instantaneous rest frames by Fermi-Walker transport of the initial arbitrarily chosen Vierbein along the particle's trajectory.

Of course, you do not need to do that but you can as well conveniently stay in a fixed inertial frame, which is defined by a family of synchronized clocks all at rest to each other. The moving ideal clock will then show its proper time (by definition), which is given by
$$\tau(t)=\int_0^t \mathrm{d} t' \sqrt{1-\vec{\beta}^2(t')},$$
where $\vec{\beta}=\vec{v}/c$, $\vec{v}=\mathrm{d} \vec{x}/\mathrm{d} t$, and $t$ the coordinate time of your inertial frame.