How would you define an inertial frame of reference?

[pervect]

I'm having trouble parsing your first paragraph. The quoted part of the second paragraph seems clearly false to me. There is no observer who gets (t+ax,x) using clocks and rulers.

I believe that transform would correspond to an observer with a standard "grid" of clocks and rulers, but with a non-standard way of synchronizing the clocks.

 

[D H]

For instance the transformation from an inertial frame (t,x) to (t+ax,x).

So that's what you meant by a linear transformation that transforms away Newton's laws. You are treating time as if it were a coordinate. You can't do that in Newtonian mechanics. Time is not a coordinate in Newtonian mechanics. It is a parameter, distinct and independent from spatial coordinates. Time is *the* independent parameter in Newtonian mechanics.

 

[JustinLevy]

So that's what you meant by a linear transformation that transforms away Newton's laws. You are treating time as if it were a coordinate. You can't do that in Newtonian mechanics. Time is not a coordinate in Newtonian mechanics. It is a parameter, distinct and independent from spatial coordinates. Time is *the* independent parameter in Newtonian mechanics.

In that setup, distance is measured with a bunch of identical standard rulers and time with many identical standard clocks. This appeared to be your requirement for deciding what coordinate systems are eligible to be called a "reference frame".

In order to measure a velocity, one needs to be able to measure the time at a minimum of two separated events. Even in Newtonian mechanics, time is something that can be measured. So in addition to Newton's first laws, and in addition to all your requirements on coordinates systems to be a reference frame, you are now adding that clock synchronization must be done according to that implicit by Newton (ie. synchronize at a common location and then transport)?

You are shoving/hiding a lot of the definition into these extra requirements. Newton's first law is clearly not sufficient to define an inertial frame, otherwise you wouldn't need to add in all these extra conditions and requirements. This is the weirdest approach to defining an inertial frame I've seen yet.

Your requirement for a coordinate system to be considered a "frame" is still vague to me, but I think I get the gist of your views on the term inertial reference frame. Unless you think I'm still missing something, I guess there's nothing more to say. It was really interesting to hear all the varied takes on these terms. This ended up being a fun thread.

 

[Fredrik]

Time is not a coordinate in Newtonian mechanics. It is a parameter, distinct and independent from spatial coordinates. Time is *the* independent parameter in Newtonian mechanics.

Those last two sentences don't imply that it can't also be a coordinate. There's certainly nothing wrong with taking the spacetime of Newtonian mechanics to be $\mathbb R^4$.

 

[atyy]

There's nothing wrong in practice with that, it won't lead you to any troubles so certainly that is okay. It's really only in subtle cases that one must be careful in distinguishing Lorentz frames from coordinate systems. If it helps, one way to think of a frame is (as pervect noted) an ideal clock, a set of three mutually orthogonal meter sticks and a set of three mutually orthogonal gyroscopes carried by some observer. A coordinate system on the other hand would be a lattice of such rods, clocks (synchronized, at least locally, using some synchronization procedure), and gyroscopes laid out in a neighborhood of the observer.

A coordinate system talks about how a vector is represented. A reference frame talks about what a vector represents (and doesn't care so much about how the vector is represented). Whether you choose to represent a vector using cartesian coordinates, polar coordinates, or even some bizarre coordinate system with three non-coplanar basis vectors separated by 60 degrees and one unit vector representing one inch, another one meter, and the third one furlong, its still the same vector. A rose by any other name ...

Your statement seems to contradict WannabeNewton's that it is ok to think of a global inertial frame as a coordinate system in which the metric is diag(-1,1,1).

I would imagine that just as an unqualified "mass" refers to the invariant mass nowadays, an unqualified "inertial frame" refers to a global inertial frame, motivated by the "Principle of Relativity". (I think this is what you said in your post #3, since "inertial frames" in curved spacetime have to be qualified as "local inertial frames".)

 

[Fredrik]

My answer to the question in the thread title:

There's no theory-independent answer to questions like these. Physical terms are defined differently in different theories. The term "inertial frame of reference" is especially tricky because there are least two different definitions in GR (and I suppose, also in SR). It can refer to a coordinate system or a frame field. I will only be talking about coordinate systems.

The closest thing to a theory-independent answer that I can think of is this:

An inertial frame of reference is a coordinate system that the theory associates with a pair (p,C) where p is an event in spacetime, and C is a curve through p such that an accelerometer moving as described by C would show 0 acceleration at p.​

The exact details of how this association is made are however different in different theories. This answer works for pre-relativistic classical mechanics, SR and GR. The only caveat is that GR may associate many coordinate systems with a pair (p,C), not just one.

A coordinate system is a function from a subset of spacetime into $\mathbb R^4$. If x is a coordinate system and p is an event in the domain of x, then x(p) is a 4-tuple of real numbers, called the coordinates of p in x.

I'm a big fan of the following approach: Suppose that we would like to find all theories of physics such that

• The theory's model of space and time is $\mathbb R^4$.
• The theory is consistent with the principle of relativity
• The theory is consistent with the principle of rotational invariance of space.
• Inertial coordinate systems are defined on all of spacetime.

Then we can make progress by trying to find all the functions of the form $x\circ y^{-1}$ where x and y are inertial coordinate systems. These are functions that change coordinates from one inertial coordinate system to another. (Note that we have $(x\circ y^{-1})(y(p))=x(p)$). To proceed, we must interpret the statements on the list as mathematical statements. Nothing could be more natural than to require that these functions are permutations of $\mathbb R^4$ that take straight lines to straight lines. This ensures that all inertial coordinate systems agree about which curves represent constant-velocity motion. We interpret the principle of relativity as saying that these functions should form a group, and we interpret the principle of rotational invariance as saying that this group should have the rotations of space as a subgroup.

One can then show (after some minor additional assumptions) that the group is either the group of Galilean transformations or the group of Poincaré transformations. Unfortunately the proof is very difficult. I think that this would have become a standard part of every introduction to SR if it had been easy.

So what does this have to do with this thread? Well, it suggests an answer to the question, at least for Newtonian mechanics and SR. Since there's exactly one inertial coordinate system for each element of the group, it makes perfect sense to think of the transformations themselves as inertial coordinate system. The identity map on $\mathbb R^4$ is my inertial coordinate system, and every other Galilean/Poincaré transformation is someone else's.

 

[Sugdub]

How would you define an inertial frame of reference?

I feel uncomfortable with defining a “frame of reference” as a “coordinate system”, merely because a physical concept cannot be reduced to a mathematical object.

I use to define an “inertial frame of reference” as a convention (set by the theoretician) whereby a peculiar collection of (presumably) non-accelerated physical objects are considered at rest across time”. With this definition the “inertial frame of reference” is a physical concept. Obviously the said convention may reveal inappropriate, so that it is part of the expertise of the theoretician to make a proper choice for those objects he/she considers being “non-accelerated”.

To this peculiar collection of objects and associated events in their life one can attach a space-time coordinate system via a function into R4, in such a way that their space coordinates will remain invariant across time. With this definition, the space-time “coordinate system” is a mathematical object.

One may decide to swap to a different space-time coordinate system, still attached to the same inertial frame of reference (i.e. related to the same convention). The transformation of the time coordinate (e.g. changing the origin of dates) shall ensure that the elapsed time between two events remains invariant (principle of homogeneity of the time flow), whereas the transformation of space coordinates (e.g. changing the origin and/or the orientation of axes) shall ensure that the relative position (distance, angles) between physical objects is unaffected (principle of homogeneity and isotropy of space). As a consequence, the relative speed between physical objects is unaffected by a change of the space-time coordinate system which targets the same inertial frame of reference.

Conversely, changing the frame of reference means adopting a different convention. By selecting another collection of physical objects which motion was uniform and identical in the first coordinate system, and further assigning those a conventional rest state, one defines a new “inertial frame of reference”.

The transformation of coordinates associated to a change of inertial frame of reference shall ensure that the class of non-accelerated physical objects is invariant: according to the principle of relativity of motion, the distinction between inertial and accelerated motion is objective, whereas the distinction between uniform motion and rest state is arbitrary. Assuming the space-time coordinate system has been defined in the same way on both sides, the requirement above translates into linearity constraints which determine two exclusive groups of mathematical transformations, depending on whether one imposes or not that the elapsed time between any pair of physical events should remain invariant. This is the way I read the difference between the Newtonian and SR physics theories.

Your comments are welcome.

 

[pervect]

How would you define an inertial frame of reference?

I feel uncomfortable with defining a “frame of reference” as a “coordinate system”, merely because a physical concept cannot be reduced to a mathematical object.

I think there are some subtle differences between a "frame of reference" and a coordinate system. But I don't really understand your point about "physical concepts" and "mathematical objects", perhaps you could explain what you think the distinction is?

I use to define an “inertial frame of reference” as a convention (set by the theoretician) whereby a peculiar collection of (presumably) non-accelerated physical objects are considered at rest across time”.

"At rest across time" sounds very vague, and not particularly standard. It also doesn't seem to me to capture the idea of a frame of referene, either.

It seems rather difficult to find any really definitive definition for "inertial frame of reference", but I'm sure we can do better then "at rest across time".

 

[Fredrik]

I feel uncomfortable with defining a “frame of reference” as a “coordinate system”, merely because a physical concept cannot be reduced to a mathematical object.

And yet all physical concepts (work, momentum, temperature,...) have mathematical definitions. The same term often has different definitions in different theories.

The "cannot be reduced to" part of your statement is true in the sense that no theory of physics is perfectly accurate. People often say that theories are approximate descriptions of reality, but I don't like that view, as it suggests that the differences are numerical rather than conceptual. I prefer to think of a theory as an exact description of a fictional universe that resembles our own. The mathematical definition of a term should be viewed as a perfect description of something in that fictional universe, rather than a flawed description of something in the real world.

That definitions refer to mathematical things is not a bad thing. It's the right way to handle these things, since that fictional universe is the only thing we can really describe anyway.

 

[vanhees71]

Why not just sticking to Newton's definition? A reference frame is inertial if a body stays at rest or uniform motion with constant velocity if no force acts on it. It's an axiom of the mathematical description of the physical world that such a frame of reference always exists as long as you stick to either the Galilei-Newton or the Einstein-Minkowski description of spacetime.

Nowadays we know that this is always an approximation, because the so far most comprehensive mathematical description of spacetime is that of General Relativity, according to which the idea of an inertial reference frame can only defined as a local concept but not globally to describe the universe as a whole.

I also think, it is quite important to distinguish between the mathematical models (or physics theories) from the real world. The chalenge of physics is to describes as comprehensively and precisely as we can in terms of a consistent mathematical model. The connection between the mathematical picture and the phenomena in the real world is what distinguishes a construct of pure thought, as which any system of axioms in pure mathematics can be formulated, from theoretical/mathematical physics.

The history of physics and all natural science shows that constructs of pure thought have never been successful in finding good physical theories. The great fundamental findings of the 19th and 20th century, electrodynamics and the theory of relativity (both special and general), quantum theory, and statistical physics, have had all their solid foundation in the empirical findings and accurate experiments (electrodynamics: Faraday's comprehensive empirical basis was mandatory for Maxwell to find his famous equations; later the model had to be developed further, last but not least driven by technological challenges like telegraphy and undersea cables; quantum theory: atomic spectra (known but ununderstood since the mid 18ths, black-body radiation (Rubens, Kurlbaum et al at the Physikalisch Technische Reichsanstalt trying to fix an accurate and objective standard for the radiation strength for light) etc.). Also general relativity after all is based on the observation of the equivalence principle, i.e., the strict and universal proportionality of what was known as inertial and gravitational mass. Of course, with the development of general relativity by Einstein (and Hilbert one should say to be just although the fundamental ideas go back to Einstein and the final equations were found independently by Einstein and Hilbert in 1915) the idea of gravitational mass had to be refined, and was substituted by the universal coupling of gravity to the energy-momentum tensor of matter, which after all is also a gauge-theoretical concept, describing the equivalence principle.

There are examples for great predictions based on models of what is empirically known like the prediction of anti-particles (Dirac) or the prediction of the neutral electroweak currents and the W- aund Z- bosons (Glashow, Salam, Weinberg) etc. But all these predictions were based on a solid empirical basis (in this case about high-energy particle physics) and a good paradigm for model building (renormalizable local relativistic quantum field theory).

 

[DrGreg]

Why not just sticking to Newton's definition? A reference frame is inertial if a body stays at rest or uniform motion with constant velocity if no force acts on it. It's an axiom of the mathematical description of the physical world that such a frame of reference always exists as long as you stick to either the Galilei-Newton or the Einstein-Minkowski description of spacetime.

In terms of relativity, it depends on whether you consider Einstein's 2nd postulate to be part of the definition of "inertial frame" or not. If you do, then there is this objection...

If so, note that there are still references frames (by this definition) in which Newton's first law holds, but are not inertial frames. For instance the transformation from an inertial frame (t,x) to (t+ax,x). Since it is linear transformation, Newton's first law will still hold, but when you start to throw interactions / dynamics in there you will see that the other laws of mechanics will not always hold now. It will appear as if there is a fictitious force. Thus, even with all those extra restrictions, one still cannot define an inertial frame by Newton's first law.

Newton's first law is a necessary, but not sufficient, condition for a coordinate system to be an inertial coordinate system.

 

[Fredrik]

Why not just sticking to Newton's definition? A reference frame is inertial if a body stays at rest or uniform motion with constant velocity if no force acts on it.

This sounds very ambiguous to me, and possibly circular. To say that the velocity is constant, we must first use a coordinate system, and which one should we use if not the inertial one that we're trying to define? And how is "force" defined here?

I think that if we are careful to avoid ambiguity and circularity, and are clear about what's defined mathematically and what's defined operationally, we end up with something like what I said here:

The closest thing to a theory-independent answer that I can think of is this:

An inertial frame of reference is a coordinate system that the theory associates with a pair (p,C) where p is an event in spacetime, and C is a curve through p such that an accelerometer moving as described by C would show 0 acceleration at p.​

 

[Dale]

I feel uncomfortable with defining a “frame of reference” as a “coordinate system”, merely because a physical concept cannot be reduced to a mathematical object.

You really shouldn't feel uncomfortable with that. All physical theories include a mapping between physical concepts and mathematical objects.

 

[Sugdub]

You really shouldn't feel uncomfortable with that. All physical theories include a mapping between physical concepts and mathematical objects.

I see no problem with the mapping itself, but with the identification or the definition of a physical concept as a mathematical object.


Although I concur to the need to avoid circularity as pointed out in #47 by Fredrik, I think his suggestion does not make it. The mere reference to an “accelerometer” which measures “acceleration” is introducing circularity in his definition, in the same way as a reference to “forces” does. One may of course rename the “accelerometer” using a new word like “ABCD-meter”, still providing a connection to an operational protocol, but the statement he proposed will no longer hold as a definition for the concept of an “inertial frame of reference”. It will only provide an operational protocol for assessing whether the coordinate system attached to the measurement device itself can be considered “inertial” according to a degree of accuracy which will need to be evaluated through a different protocol (a calibration process), and this brings the perspective of an open-ended regression.


I also concur to the statement by Vanhees71 in #45 whereby “The connection between the mathematical picture and the phenomena in the real world is what distinguishes a construct of pure thought,...” . In this respect, I think that any attempt to propose a formal (mathematical) definition for the word “inertial” will induce either circularity or an open-ended regression as shown above. Otherwise physics could be a "construct of pure thought". The only way is to connect the word "inertial" to everyone's intuitive sense of “acceleration”. The absence of this sensation in our body is what best defines the meaning of “inertial”. Stating that an object is "non-accelerated" or in "inertial" motion means that we would not sense an "acceleration" in our body, should we remain collocated with it.


Finally I've given examples (change of origin for dates or space, change of orientation of space axes) which show that a continuous family of mathematical frameworks (space-time coordinate systems) can be used to account for the same physical convention (one inertial frame of reference, I mean one convention whereby a given collection of non-accelerated physical objects are considered being “at rest”). I think it shows that there is a one-to-many mapping between the choice of an inertial frame of reference and a continuous family of coordinate systems which share the following: the class of non-accelerated objects is the same, the relative position of objects between themselves is the same, the space distance and the time gap between any pair of events is the same.


Overall, changing the space-time coordinate system does not necessarily imply a change of the frame of reference (i.e. changing the set of objects which are considered “at rest”). For me these are two different concepts.

 

[Dale]

I see no problem with the mapping itself, but with the identification or the definition of a physical concept as a mathematical object.

I have never seen anyone here do that. The fact that it is a map is well understood, even if it is not expressly stated in every post which uses math.

 

[D H]

This sounds very ambiguous to me, and possibly circular. To say that the velocity is constant, we must first use a coordinate system, and which one should we use if not the inertial one that we're trying to define?

You don't need a coordinate system. All you need is a nice combination of very ancient and rather modern measuring devices. A protractor, a laser, and a good clock do the trick. (Adding something to measure Doppler shift is a nice to have, but is not essential.) Upon taking a number of measurements of some object with those devices over time, it's easy to deduce whether those measurements are statistically consistent with the hypothesis that the object has been moving at a constant velocity.

And how is "force" defined here?

Newton's definition of force was perhaps circular. Or even better, an undefined term, which is how I like to look at it. Then again, my table might just have one too many beer mugs placed on it.

 

[Sugdub]

I have never seen anyone here do that. The fact that it is a map is well understood, even if it is not expressly stated in every post which uses math.

Well, I'm not going to fight on that.

More important for me is whether physicists agree or not with my last statement whereby changing the coordinate system does not necessarily capture the essence of a change of the frame of reference, which consists in changing the assignment of which objects are considered at rest.

 

[Dale]

More important for me is whether physicists agree or not with my last statement whereby changing the coordinate system does not necessarily capture the essence of a change of the frame of reference, which consists in changing the assignment of which objects are considered at rest.

I have mixed feelings about that statement:

Overall, changing the space-time coordinate system does not necessarily imply a change of the frame of reference (i.e. changing the set of objects which are considered “at rest”). For me these are two different concepts.

I agree that frame of reference and coordinate system are two different concepts, but not for the reason that you give. There are a few main differences between frame of reference and coordinate system. While understanding that there is a mapping between the physical and mathematical aspects of a theory, the differences between a frame and coordinate systems are easiest to see on the mathematical side.

The mathematical object used to represent a frame of reference is called a tetrad or vierbein. It is a set of four orthonormal vector fields that are used to represent local rods and clocks at each event.

A coordinate system (or coordinate chart) is only a mathematical object and has no physical counterpart. It is simply a mapping (smooth and invertible) between events in spacetime and points in R4.

A tetrad must be orthonormal, whereas there is no such restriction on a coordinate chart. Also, a coordinate chart (usually) includes a notion of simultaneity, whereas a tetrad does not. A coordinate chart must be defined on an open subset of the spacetime, whereas a tetrad can be defined on any subset.

 

[Fredrik]

Although I concur to the need to avoid circularity as pointed out in #47 by Fredrik, I think his suggestion does not make it. The mere reference to an “accelerometer” which measures “acceleration” is introducing circularity in his definition, in the same way as a reference to “forces” does.

There's no circularity there. The term "accelerometer" in that sentence isn't supposed to be defined a "any device that measures acceleration". It's supposed to be defined by instructions on how to build such a device.

I don't consider what I wrote there a "definition". It's just a statement about what inertial coordinate systems in different theories have in common. A definition would specify what coordinate systems are considered "inertial" in one theory.

 

[pervect]

The mathematical object used to represent a frame of reference is called a tetrad or vierbein. It is a set of four orthonormal vector fields that are used to represent local rods and clocks at each event.

That's close to my understanding of the 4d relativistic case, though I thought of the objects as vectors, not vector fields. As vectors, the needed mathematical properties are to be able to be multiplied by scalars and added together. The space spanned by the vectors is in general the tangent space - this is equal to the manifold itself only when the manifold is flat.

I don't quite see how to re-interpret my understanding in terms of vector fields, I was wondering if we really had different views or whether it's just a semantic and word-choice issue. I'm hoping that are views are basically the same.

On another issue:

As near as I can tell, the Newtonian case has 3 basis vectors, time isn't handled explicitly (it is assumed to be separate from space, and handled as what is usually called absolute time, which is the usual Newtonian way of handling time).

Another important difference is the treatment of gravity, as the PSSC film quoted by bcrowell mentions, (I think bcrowell's book does too).

in the Newtonian case a frame fixed to a non-rotating Earth would be considered to be inertial, while in the GR case the frame anchored to the non-rotating Earth would be considered non-inertial, and inertial frame would be free-falling.

 

[Dale]

That's close to my understanding of the 4d relativistic case, though I thought of the objects as vectors, not vector fields. As vectors, the needed mathematical properties are to be able to be multiplied by scalars and added together. The space spanned by the vectors is in general the tangent space - this is equal to the manifold itself only when the manifold is flat.

I don't quite see how to re-interpret my understanding in terms of vector fields, I was wondering if we really had different views or whether it's just a semantic and word-choice issue. I'm hoping that are views are basically the same.

A vector field is simply a set of vectors which are located somewhere in the manifold. Since the rods and clocks represented by a tetrad are somewhere in the universe, they form a vector field.

Note, the vector field doesn't need to be defined everywhere on the manifold, just somewhere. Even along just a single worldline or at a single event (although that would be hard to arrange physically). I suspect that is the thing that was bothering you.

As near as I can tell, the Newtonian case has 3 basis vectors, time isn't handled explicitly (it is assumed to be separate from space, and handled as what is usually called absolute time, which is the usual Newtonian way of handling time).

Another important difference is the treatment of gravity, as the PSSC film quoted by bcrowell mentions, (I think bcrowell's book does too).

in the Newtonian case a frame fixed to a non-rotating Earth would be considered to be inertial, while in the GR case the frame anchored to the non-rotating Earth would be considered non-inertial, and inertial frame would be free-falling.

Yes, all of that is correct according to my understanding also.