# Why so many definitions of an inertial frame?

• peter46464

#### peter46464

A Newtonian inertial frame is one where objects obey Newton's first law.

Schutz (A first course in general relativity) says an inertial frame cannot be constructed in a gravitational field because it's then impossible to synchronize the frame's clocks? For the same reason an inertial frame cannot be accelerating.

Wikepedia defines an inertial frame as one "that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner."

I'm getting very confused about all these, undoubtedly correct, descriptions of inertial frames. Which are the chickens and which are the eggs?

Can anyone explain, simply, (1) what's the difference and (2) why there needs to be a difference between a Newtonian inertial frame and an inertial frame in special relativity.

Thank you

I don't think there needs to be a difference definition of inertial frame between special relativity and Galilean relativity. There is a difference in how to correlate observations between such frames, but no disagreement about what frames are inertial.

For general relativity, there is a big difference. I would interpret what Schutz is saying is that there are no global inertial frames in general relativity. There are local inertial frames, and they are free fall frames.

As for the wikipedia definition, I assume they define additional context. It is certainly not true that an inertial frame can detect no anisotropy - at a given location, all inertial frames except one can detect anisotropy in the cosmic background radiation.

Inertial frames in Newtonian mechanics and in special relativity are one and the same thing. Einstein used Newtonian inertial frames in his formulation of special relativity. Note that special relativity is specialized to situations where gravitation is not present (or where its presence can be ignored). How gravity factors in with inertial frames is not an issue in Newtonian mechanics vs. special relativity.

How gravity factors in is a huge issue in general relativity. The equivalence principle indicates that something is wrong with the Newtonian concept of an inertial frame. A person in an enclosed room cannot conduct any local experiment to distinguish between sitting still on a non-rotating planet versus being on an accelerating rocket, nor between orbiting a planet versus being in deep space. Newtonian mechanics says sitting still on a non-rotating planet is inertial which orbiting the planet is not. General relativity says the reverse.

A person in an enclosed room cannot conduct any local experiment to distinguish between sitting still on a non-rotating planet versus being on an accelerating rocket
Why only a non-rotating planet?

Why only a non-rotating planet?

focault pendulum?

Or a ring laser gyro. I was thinking of a spacecraft that is undergoing translational acceleration only.

Inertial reference frame is a base everything else is build on. Defining inertial reference frames is a bit like pulling ourselves up from the bootstraps.

In mathematics we often define things by some core property they fix by their symmetries.

A reference frame can be defined by the velocity it calls 0 at each point. An inertial reference frame is one where there is no acceleration if we parallel transport in the spatial directions.

Clearly if we change our treatment of velocity our concept of a reference frame will be altered. Galilean transforms are not equivalent to Lorentz transforms. This is the root of why inertial frames in SR differ from classical reference frames.

If our reference frame is not Euclidean it may be possible to return to a point through bulk parallel transports with a different orientation than we started with. It may even be possible for a change in magnitude to occur, making a 0 velocity poorly defined globally. Since GR is not Euclidean, this is why inertial frames in GR are problematic.

Inertial frames in Newtonian mechanics and in special relativity are one and the same thing.
I wouldn't say that. To me that would imply that the Galilei group is equal to the Poincaré group. I suppose you could say that they're used for the same things.

A person in an enclosed room cannot conduct any local experiment to distinguish between sitting still on a non-rotating planet versus being on an accelerating rocket...

Doesn't the equivalence principle assume locality approaches zero?

Wouldn't the planet gravity have a gradient, and have radial normal directions from apparent point source so that a vertically oriented ring of particles beginning as circular would decrease radius laterally as it falls, and become egg shaped with the pointy end toward the gravitational source because of the gradient?

Within the rocket, wouldn't the circle of particles remain in a circle ?

Doesn't the equivalence principle assume locality approaches zero?

Wouldn't the planet gravity have a gradient, and have radial normal directions from apparent point source so that a vertically oriented ring of particles beginning as circular would decrease radius laterally as it falls, and become egg shaped with the pointy end toward the gravitational source because of the gradient?

Within the rocket, wouldn't the circle of particles remain in a circle ?

DH said 'local'. What do you think that means? It means, precisely, that curvature is below measurement precision over the region of interest.

I'm thinking there may be three "locals":

The "casual local" in the context of "A person in an enclosed room cannot conduct any local experiment to distinguish..." meaning to me enough space to conduct an experiment, even if a small space.

The "null local" of your suggestion, which I had not thought of, that the assumption of locality is met when insufficient measurement precision fails to measure a difference.

The "absolute local" of EP locality of d/dx

Casual local defines with experimental space size and assumes some level of insufficient measurement precision. Null local more strictly defines on insufficient measurement precision first and infers the corresponding space size. Absolute local defines on the math, which implies both space size d/dx and absolute measurement precision.

In the case in question, the assumption of null local implies casual local, so I think I see your point.

OK, now I've defined multiple definitions of local in a thread asking why so many definitions of IRF...

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A Newtonian inertial frame is one where objects obey Newton's first law.

Schutz (A first course in general relativity) says an inertial frame cannot be constructed in a gravitational field because it's then impossible to synchronize the frame's clocks? For the same reason an inertial frame cannot be accelerating.

Wikepedia defines an inertial frame as one "that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner."
I wouldn't consider these statements "definitions". They are just attempts to explain the concept without defining it. If they are actual attempts to define it, they are terrible.

By far the simplest way to define inertial coordinate systems in SR and pre-relativistic classical mechanics is to identify them with members of the restricted Poincaré group and the restricted Galilei group respectively. It would take more time than I have right now to explain why that's a good way to do it, but it's definitely what I would do. In GR, I would say that any "normal" coordinate system (e.g. Riemannian normal coordinates, Fermi normal coordinates, etc.) would qualify as a "local inertial coordinate system".

There are no theory-independent definitions of "inertial frame" or "inertial coordinate system". They have to be defined within the framework of each theory.

(I'm choosing to talk about coordinate systems instead of frames, because "frame" is a fairly sophisticated concept).

Thanks everyone - that is clearer now.

Just a couple of points that are still puzzling me.

1. The problem with a Newtonian inertial frame in GR is that (because of the equivalence principle) you don't know whether it's in a gravitational field or if the frame is accelerating). That's why the definition of an inertial frame in GTR is one that is freely falling. Is that more or less correct?

2. Is it possible to have useful approximations of an inertial frame in STR with a weak gravitational field? I haven't phrased that very well. I think I mean is STR still useful on Earth (for example) even though there is a gravitational field on Earth?

Thank you

1. The problem with a Newtonian inertial frame in GR is that (because of the equivalence principle) you don't know whether it's in a gravitational field or if the frame is accelerating). That's why the definition of an inertial frame in GTR is one that is freely falling. Is that more or less correct?

Yes. It is inertial to first order in spacetime derivatives on the free-falling worldline, so local means, in part, at a point. Even at this point, it is not inertial at second order. The technical implementation of this is called "Fermi normal coordinates". The restriction to first order at a point is the technical implementation of "local" in the statement of the equivalence principle.

2. Is it possible to have useful approximations of an inertial frame in STR with a weak gravitational field? I haven't phrased that very well. I think I mean is STR still useful on Earth (for example) even though there is a gravitational field on Earth?

Yes. For points near a free-falling wordline, Fermi normal coordinates shows in an order by order expansion how they deviate from inertiality. For points near the free-falling worldline, the deviations may be so small as to be negligible. In practice, "near the worldline" is large enough to include entire particle accelerators, which only take SR into account, not GR (unless they use GPS, like OPERA ).

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