# Inertial reference frame

I have heard the following oppinion:
Whether a reference frame is inertial is completely determined by whether Newton's laws are applicable for particles moving at low (that is, nonrelativistic) speeds in that reference frame.
Do you agree with it?

Meir Achuz
Homework Helper
Gold Member
But there could be non-inertial effects that become negligible at NR speeds.

That is a good starting point. Good enough for introductory classes.

But no, I would not agree with it.
As has already come up on this board a few times, the first law provides a necessary condition, but is not sufficient to define an inertial frame. As for the other two laws...

The third law is not really true except at zero spatial separation (even if you want to consider v << c, it is still observable in electrodynamics, the forces between two spatially separated particles need not be equal and opposite).

The second law can be considered as merely defining force. Force = dp/dt. Therefore, to "test" this you would need to specify force laws (newton's gravitational law, lorentz force law, etc.). If you go down this road, you are basically defining inertial frames using the first postulate of relativity (it is an inertial frame if the physical laws have the same form as in all other inertial frames).

Other definitions I have heard:
An inertial coordinate system is one which describes space homogenously and isotropically, and time homogenously. In other words, the metric is (-1,1,1,1). This alone is sufficient to derive the form of the relation between inertial frames.

Unfortunately that definition implies time and spatial inversion symmetry, which we know the fundemental laws of physics do not have. So you'd have to add some qualifiers on that.

Another way I've seen inertial frames defined:
The laws of physics can be written in a coordinate system independent form. Special relativity postulates the laws of physics have poincare symmetry. Therefore there is a preferred "set" of coordinate systems under this symmetry. The inertial frames are such a set (and this set is not necessarily unique, for either right or left handed coordinate systems would work, but they are not related by poincare symmetry).

Dale
Mentor
2020 Award
The third law is not really true except at zero spatial separation (even if you want to consider v << c, it is still observable in electrodynamics, the forces between two spatially separated particles need not be equal and opposite).
I wouldn't say it this way. I would instead say that the third law is correct and all interactions are local. The third law is essentially the conservation of momentum, which always holds, so to me it seems confusing to phrase it the way you did: "not true except X" where X is a condition that always holds.

Can't we all just get along?

An inertial coordinate system is one which describes space homogenously and isotropically, and time homogenously. In other words, the metric is (-1,1,1,1). This alone is sufficient to derive the form of the relation between inertial frames.

Some nonstandard clock synchronization procedures (external clock synchroniation, everyday clock synchroniozation make that in one of the involved frames the propagation of light is anisotropic. Does that mean that the reference frame is not inertial?
Do you consider that the definition: The reference frame in which for c going to infinity the relativistic laws become the classical ones (principle of correspondence) become the calssical ones.
Thanks to all who have posted on my thread.

DrGreg
Gold Member
Some nonstandard clock synchronization procedures (external clock synchroniation, everyday clock synchroniozation make that in one of the involved frames the propagation of light is anisotropic. Does that mean that the reference frame is not inertial?
I ask readers of this thread to consider: do you think the specification of a particular inertial frame includes the choice of spatial coordinates? Do you think it is compulsory to use Cartesian xyz coordinates, or would spherical polar $r\theta\phi$ coordinates be permissible too? Do you think the metric

$$ds^2 \, = c^2 \, dt^2 \, - \, dr^2 \, - \, r^2 (d\theta^2 \, + \, d\phi^2 \, sin^2 \theta)$$​

defines an inertial frame or not?

(I have just asked the same question in three different ways.)

If you insist that an inertial coordinate system must be cartesian, then by the same logic you ought to insist that Einstein's synchronisation convention must apply, and so the standard (1,-1,-1,-1) Minkowski metric form must apply.

If you allow non-cartesian coordinates in an inertial frame, then logically you ought to allow non-standard synchronisation and non-standard (but flat) metric forms. If you choose to go down this route then any statement of Newton's Laws will need to be appropriately worded to make sense (and be true) in your choice of coordinate system.

For an example of an abnormally synchronised system let txyz be Einstein-synchronised Minkowski coords in an inertial frame and define TXYZ by

T = t + vx/c2
X = x
Y = y
Z = z

In this coordinate system the coordinate speed ($\sqrt{(dX/dT)^2 + (dY/dT)^2 + (dZ/dT)^2}$) of light is anisotropic. Do you think TXYZ is an inertial coordinate system?

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Justin,

Regarding your comment, "Special relativity postulates the laws of physics have poincare symmetry."

I found a copy of Einstein's orginal paper and I don't see a refernce to poincare in the postulates. Do you have another reference?
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JustinLevy said:
The third law is not really true except at zero spatial separation (even if you want to consider v << c, it is still observable in electrodynamics, the forces between two spatially separated particles need not be equal and opposite).
I wouldn't say it this way. I would instead say that the third law is correct and all interactions are local. The third law is essentially the conservation of momentum, which always holds, so to me it seems confusing to phrase it the way you did: "not true except X" where X is a condition that always holds.
Yes, you could say it that way. But textbooks have worded it my way as well.

We will quickly devolve into semantics here if we try to continue this. Part of the problem is we know local realism to be false, and additionally we cannot, even in principle, measure if each vertex in a feynman diagram individually conserves momentum or if the process conserves momentum. I have seen annoying arguments over whether being "off shell" has the same mass but violates momentum conservation temporally, or whether "off shell" has a different mass and conserved momentum always... it can't be measured, it is just arguing math definitions.

So let me summarize with this: Yes you could say it that way. That is a consistent point of view. And while Griffith's EM textbook is saying the opposite to undergrads across the nation, your wording is probably the convention most physicists would take.

To clear up any confusion, if people wish to take that stance, it seems easier to just say: newton's third law is that momentum is conserved locally.

Justin,

Regarding your comment, "Special relativity postulates the laws of physics have poincare symmetry."

I found a copy of Einstein's orginal paper and I don't see a refernce to poincare in the postulates. Do you have another reference?
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I'm not quite sure what you are ultimately asking for here: a reference that explains poincare symmetry so that you can understand that comment, or if you are literally asking for another reference to Einstein's paper which has more in it or something.

For the first, wikipedia gives the basics. You could start there. If you want more, maybe someone can recommend a book dealing more directly with this. Depending on your background, if you want to see how it is used in practice in modern physics, most QFT books approach relativity strictly from the symmetry it provides (often using the lie algebra of the group).

As for the second, no, einstein's paper does not refer explicitly to poincare symmetry. Science isn't a theology. Einstein's 1905 paper isn't the only word and last word on relativity, just as his 1916 paper isn't the only and last word on general relativity.

It is interesting to go back and read some original papers, but I would not recommend them for starting points of learning the material. Maxwell's original paper for electrodynamics is very difficult to read now, Einstein's GR is never taught now how it was presented there, the first paper on path integrals is also a more vague presentation than some of the more modern presentations, and even special relativity has been made more precise over the years (although I would say of the examples I listed here, it probably is the one that has held up best for 'accessibility').

I ask readers of this thread to consider: do you think the specification of a particular inertial frame includes the choice of spatial coordinates? Do you think it is compulsory to use Cartesian xyz coordinates, or would spherical polar $r\theta\phi$ coordinates be permissible too? Do you think the metric

$$ds^2 \, = c^2 \, dt^2 \, - \, dr^2 \, - \, r^2 (d\theta^2 \, + \, d\phi^2 \, sin^2 \theta)$$​

defines an inertial frame or not?

(I have just asked the same question in three different ways.)
Hmm... that's a neat point.
At this time I would say no, that is not an inertial coordinate system. But I have a feeling you may be changing my mind soon :)

The reason I would say no, is because the basis vectors themselves are now spatially dependent. This doesn't describe space in a homogenous way. The metric itself is even now described in a way that is spatially dependent. Of course it is still describing the same spacetime, and the spacetime is still flat, but that can be said of any coordinate system we choose.

Taking a step back though,
I would have to say that we have all jumped back and forth between coordinate representations for some problems, and at least personally, I didn't really think of it as going inertial <-> non-inertial when using non-cartesian coordinates. I guess it didn't really matter (since this is merely a semantic distinction), but I didn't think about it much. To include these, it seems like you would be allowing any coordinate system which preserves the same time slices as an "inertial" frame above, but rearranges the spatial coordinates.

If you allow non-cartesian coordinates in an inertial frame, then logically you ought to allow non-standard synchronisation and non-standard (but flat) metric forms. If you choose to go down this route then any statement of Newton's Laws will need to be appropriately worded to make sense (and be true) in your choice of coordinate system.

For an example of an abnormally synchronised system let txyz be Einstein-synchronised Minkowski coords in an inertial frame and define TXYZ by

T = t + vx/c2
X = x
Y = y
Z = z

In this coordinate system the coordinate speed ($\sqrt{(dX/dT)^2 + (dY/dT)^2 + (dZ/dT)^2}$) of light is anisotropic. Do you think TXYZ is an inertial coordinate system?
I agree, if you consider the spherical coordinate as 'inertial', then it would be difficult not to consider any coordinate system which uses planar time slices in spacetime as such as well. While you gave your coordinate system example above by changing the time coordinate, since it was a linear transformation, it still has a planar time slice, and can be written also as a coordinate transformation from an inertial coordinate system which only changes the spatial coordinates around.

Since this is just a semantics thing, does everyone agree they fall on one side or the other of DrGreg's delineation? If so, he's nailed the main distinction and there probably isn't anything more to discuss. But if someone actually is trying to pick and choose from each side ... I'd love to hear their "third choice".

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I ask readers of this thread to consider: do you think the specification of a particular inertial frame includes the choice of spatial coordinates? Do you think it is compulsory to use Cartesian xyz coordinates, or would spherical polar $r\theta\phi$ coordinates be permissible too? Do you think the metric

$$ds^2 \, = c^2 \, dt^2 \, - \, dr^2 \, - \, r^2 (d\theta^2 \, + \, d\phi^2 \, sin^2 \theta)$$​

defines an inertial frame or not?

(I have just asked the same question in three different ways.)

If you insist that an inertial coordinate system must be cartesian, then by the same logic you ought to insist that Einstein's synchronisation convention must apply, and so the standard (1,-1,-1,-1) Minkowski metric form must apply.

If you allow non-cartesian coordinates in an inertial frame, then logically you ought to allow non-standard synchronisation and non-standard (but flat) metric forms. If you choose to go down this route then any statement of Newton's Laws will need to be appropriately worded to make sense (and be true) in your choice of coordinate system.

For an example of an abnormally synchronised system let txyz be Einstein-synchronised Minkowski coords in an inertial frame and define TXYZ by

T = t + vx/c2
X = x
Y = y
Z = z

In this coordinate system the coordinate speed ($\sqrt{(dX/dT)^2 + (dY/dT)^2 + (dZ/dT)^2}$) of light is anisotropic. Do you think TXYZ is an inertial coordinate system?
Thanks for having restated my three questions in such an elegant way. The "abnormally synchronized" system is the starting point in
Hans C. Ohanian, “The role of dynamics in the synchronization problem,” Am.J.Phys. 72, 141-148 (2004)

Since there is no boost between the xyzt and XYZT systems they
are the same inertial frame with an arbitrary, spatially-dependent,
definition of time for XYZT. Such a definition is obviously
mathematically possible but has no physical significance

That is an answer received from a friend I appreciate very much.

Whether a reference frame is inertial is completely determined by whether Newton's laws are applicable for particles moving at low (that is, nonrelativistic) speeds in that reference frame.

Of course not...that wording includes acceleration....

Dale
Mentor
2020 Award
There is nothing wrong with an object accelerating wrt an inertial reference frame.

cinci: Justin, Regarding your comment, "Special relativity postulates the laws of physics have poincare symmetry."

I found a copy of Einstein's orginal paper and I don't see a reference to Poincare in the postulates. Do you have another reference?

Justin: I'm not quite sure what you are ultimately asking for here: a reference that explains poincare symmetry so that you can understand that comment, or if you are literally asking for another reference to Einstein's paper which has more in it or something.

cinci: No a big deal, your comment is that "relativity postulates Poincare symmetry". It seems to me it would be fair to say that relativity is consistent with Poincare symmetry but Einstein didn't postulate symmetries as far as I can tell. My question was whether you had a reference where Einstin did refer to Poincare symmetry or perhaps redevelped the theory based on it.
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Fredrik
Staff Emeritus
Gold Member
My question was whether you had a reference where Einstin did refer to Poincare symmetry or perhaps redevelped the theory based on it.
Are you asking because you're interested in the history of relativity, or do you think that this has a significance that goes beyond history? (I'm not very interested in the history myself, so I can't help you find a reference).

cinci: No a big deal, your comment is that "relativity postulates Poincare symmetry". It seems to me it would be fair to say that relativity is consistent with Poincare symmetry but Einstein didn't postulate symmetries as far as I can tell. My question was whether you had a reference where Einstin did refer to Poincare symmetry or perhaps redevelped the theory based on it.
Are you saying that if I can't find a paper or textbook where Einstein specifically referred to poincare symmetry that somehow that means relativity doesn't posulate poincare symmetry in the fundemental laws of nature?

That is not how science works. Einstein doesn't have the final say on relativity. In fact many students who have learned GR now arguably have a better understanding of relativity than Einstein did.

Einstein's original paper was a wonderful paper, yet it does have many flaws (for one, the amazing lack of references to previous work and two, the modern understanding of relativity is more precise). Science is not an aristocracy or dogmatic. Science is continually learning and evolving, and becoming more precise.

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Science history is interesting, for the ways people solved problems and motivated approaches can be useful guides to solving new problems. But I agree with Fredrik here, if you want to learn more about relativity it would be better if you asked questions about the science which we can try to answer directly for you.

Do you understand how stating relativity postulates poincare symmetry maintains the same essence as the physics in Einstein's original paper, and also understand how it is more precise and why such precision is necessary?