How would you define an inertial frame of reference?

[Luffy]

I've researched about it and watched a few videos, but I can't seem to get my head around it. Would saying that "it's a marker that is fixed relative to your position, in which Newton's first law holds" be an accurate way to define it?

 

[HallsofIvy]

Not necessarily. If you are accelerating, then "fixed relative to your position" will not give an inertial frame of reference.

 

[tom.stoer]

How would you define an inertial frame of reference?

An inertial frame is a reference frame in which free particles (w/o any external force) move along geodesics (which is a generalization for arbitrary curved manifolds)

 

[vanhees71]

To make it very clear: What tom.stoer means are geodesics in fourdimensional general relativistic space-time.

This is, however, not the way an inertial frame is understood in general relativity. If there is a non-negligible gravitational field present, space-time manifold is curved and there is no global inertial frame for entire space time but according to the (weak) equivalence principle you can always find a local one, i.e., you restrict yourself to the neighborhood around your position in space-time which is small compared to the "curvature radii" of spac-time at this point. Then you can always find an equivalence class of reference frames where the local laws take the form of special-relativistic physics. These reference frames are given as the freely falling ones. In such a frame Newton's 1st law holds (locally): A body that is not subject to any forces is moving along a straight line with uniform velocity (or stays at rest).

 

[tom.stoer]

Yes, this is exactly what I mean: geodesic motion defines local inertial frames

 

[WannabeNewton]

Just for minor clarification, it's not enough to just have geodesic motion. That only constrains the frame to be non-accelerating. There is still residual freedom in how the spatial axes of the local Lorentz frame are oriented and more importantly how their orientation changes relative to local gyroscopes. One needs geodesic motion as well as Fermi-transported (which reduces to parallel transport for geodesics) spatial axes so as to keep the frame non-rotating and hence inertial.

 

[sathwik]

The co-ordinate system which you had considerd should have its acceleration i zero

 

[sathwik]

The co-ordinate system which you had considerd should have its acceleration i zero

EXAMPLE:Earth

 

[sathwik]

An inertial frame is a reference frame in which free particles (w/o any external force) move along geodesics (which is a generalization for arbitrary curved manifolds)

The co-ordinate system which you had considerd should have its acceleration zero

 

[D H]

The co-ordinate system which you had considerd should have its acceleration i zero

Acceleration with respect to what? And what exactly do you mean by "acceleration"?tome.stoer's answer, coupled with the modification suggested by WannabeNewton, suggest one way to define an inertial frame: Co-locate an accelerometer and rate gyro. From the perspective of general relativity, if the accelerometer reports zero acceleration and rate gyro reports zero rotation, a frame based on the accelerometer+gyro to within measurement error is a local inertial frame.

That's one of the nice things about GR: It provides an experimental method for defining an inertial frame, and the experiment is based solely on local measurements. One of the not so nice things: That inertial frame is local. The concept of an inertial frame has lost a lot of the meaning it had in Newtonian mechanics and special relativity.

Newtonian mechanics provides a means for determining whether a frame is inertial: Does any particle with no forces acting acting on it move along a straight line at constant speed? There's a slight downside here: Gravity acts on everything. There is no such thing as a particle with no forces acting on it.

 

[duordi134]

How about this.

How about this.

If Newtons laws of inertia hold within time length width and height limits within experimental accuracy then it is an inertial system.

As I remember it there is no true inertial frame of reference we can only approximate it.

Duordi

 

[D H]

There's a big difference between a Newtonian inertial frame and an inertial frame in general relativity. A frame based on a non-rotating object in free fall is a local inertial frame in general relativity but is not inertial in Newtonian mechanics.

 

[atyy]

I've researched about it and watched a few videos, but I can't seem to get my head around it. Would saying that "it's a marker that is fixed relative to your position, in which Newton's first law holds" be an accurate way to define it?

Newtonian mechanics provides a means for determining whether a frame is inertial: Does any particle with no forces acting acting on it move along a straight line at constant speed? There's a slight downside here: Gravity acts on everything. There is no such thing as a particle with no forces acting on it.

For particles acted on by forces, a more general definition of an inertial frame is one in which the laws of physics have their "standard form" (ie. no Christoffel symbols).

 

[bcrowell]

The Newtonian and relativistic answers to this question are not the same. This was asked in the relativity forum, but the OP's question actually sounds like it's being asked in a Newtonian context. In the Newtonian context, an inertial frame is one in which Newton's first law holds. For anyone who wants to understand this concept better in the Newtonian context, I'd suggest watching the classic PSSC Frames of Reference films:

For the relativistic case, I have a discussion in ch. 5 of my SR book: http://www.lightandmatter.com/sr/ .

An inertial frame is a reference frame in which free particles (w/o any external force) move along geodesics (which is a generalization for arbitrary curved manifolds)

This is both incorrect and at the wrong level for the OP, who just seems confused about the Newtonian notion of an inertial frame. It's incorrect because neither the definition of a geodesic nor the question of whether a test particle moves along a geodesic is dependent on the coordinates used.

 

[tom.stoer]

Sorry for the confusion I started.

 

[JustinLevy]

In the Newtonian context, an inertial frame is one in which Newton's first law holds.

No. That is a necessary, but not sufficient condition. Otherwise any linear transformation from an inertial coordinate system would give another inertial coordinate system.

---

It is incredibly hard to define an inertial frame precisely. Everyone 'knows' what we mean, so most books just describe some properties of it (and sometimes the symmetries relating different inertial frames) and then move on. Heck, even Einstein in his 1905 paper defined inertial frames as those in which the laws of mechanics hold, then went on to show that the laws of mechanics in inertial frames needed to be changed slightly. There was even a debate for awhile if parity transformations should be considered as relating inertial frames. Because there was no precise definition, this was more a philosophical debate ... until it turned out the weak force violated parity symmetry. So our concept of inertial frames changed slightly.

I think the Landau and Lifshitz definition wikipedia uses is fairly decent (given the alternatives). That definition is basically saying an inertial frame of reference is a choice of coordinates such that our description of space and time is maximally uniform. Check out the length between two coordinate points (x,y,z,t) and (x+1,y,z,t) ... it is the same regardless of x (or of y, z, or t as well). Similarly if we had looked at points that differed in y, or z. Same with t, but now measuring time. Do some experiment which fires off a bullet with some consistent energy, and it will have the same velocity regardless of the direction we point it (with the uniformity in spatial coordinates already given, this shows we have the 'most uniform' clock synchronization), alternatively you could check clock synchronization by slow transport of clocks and see you get the same result in all directions. In inertial frames our description of space and time are maximally uniform.

This holds fine for GR as well, but an inertial frame can now only be defined locally and becomes more approximate the larger the extent of the "local" frame is.

 

[ghwellsjr]

How about: an inertial frame of reference is what the Lorentz Transformation operates on?

 

[sathwik]

Here is the another defination for inertial frame : an imaginary system which is either rest or in uniform motion and where Newton's law are valid

 

[sathwik]

Hey can anyone help me how to post the question

 

[ghwellsjr]

Hey can anyone help me how to post the question

Near the top of this page where it says:

Physics Forums > Physics > Special & General Relativity > How would you define an inertial frame of reference?

click on Special & General Relativity

Then click on the button that says New Thread

 

[Andrew Mason]

I've researched about it and watched a few videos, but I can't seem to get my head around it. Would saying that "it's a marker that is fixed relative to your position, in which Newton's first law holds" be an accurate way to define it?

Welcome to PF Luffy! I bet you thought you had asked a simple question...

It is difficult to define an inertial reference frame in Newtonian physics without reference to the concept of a force. And it is difficult to define force without the concept of an inertial reference frame. So definitions tend to be somewhat circular.

First of all, one has to define: "reference frame". A reference frame is the coordinate system that one would use to measure positions in space and time. In Newtonian mechanics, time is the same for all observers regardless of how they are moving. So a Newtonian reference frame is really the coordinate system that an observer would use to measure positions in space i.e. relative to the observer.

An inertial reference frame would be the coordinate system that an inertial observer would use to measure positions in space relative to that inertial observer.

So the real issue is how to define an inertial observer without using the concept of force. One way to do that might be to define the inertial observer as one that is not interacting, directly or indirectly, with any other body (matter).

AM

 

[bcrowell]

No. That is a necessary, but not sufficient condition. Otherwise any linear transformation from an inertial coordinate system would give another inertial coordinate system.

My definition didn't say anything about transformations. It just defined what was an inertial frame. That's actually the hard part of the whole process, the "getting started" part, of finding at least one frame that's inertial. It's an operational definition; it requires using clocks and rulers to measure the motion of test particles, and verify or falsify Newton's first law.

The definition I gave does make it possible to have, e.g., two inertial frames that differ by a parity transformation. One might or might not prefer to further restrict the definition to avoid this.

The classic, careful treatment of this sort of thing is The Science of Mechanics, 1902 English Edition, tr McCormack, http://www.archive.org/details/sciencemechanic00machgoog . The nontrivial issues aren't issues like parity, they're issues revolving around the fact that if we want to test Newton's first law, we need to be able to determine the forces acting on a test particle *independently* of our measurement of the motion of the particle. The Newtonian answer is that we have an omniscient observer who knows where all the other particles are in the universe at any given instant, and can use force laws to predict the total force.

 

[JustinLevy]

My definition didn't say anything about transformations. It just defined what was an inertial frame.

I apologize if I am being dense here, but I honestly can't tell if you missed my point.

The point was, Newton's first law does not define an inertial frame.

It is a necessary property, but not a sufficient property. I brought up that any linear transformations preserves Newton's first law to help people understand that. There are many coordinate systems in which Newton's first law holds, but which are not an inertial frames.

 

[D H]

JustinLevy, it appears you are confusing the concept of a reference frame and a coordinate system.

bcrowell had it exactly right. The modern interpretation of Newton's first law is that it defines the concept of a Newtonian inertial frame of reference.

 

[JustinLevy]

JustinLevy, it appears you are confusing the concept of a reference frame and a coordinate system.

I am interested to hear what distinction you feel there is.

From wikipedia, on "frame of reference":
"In physics, a frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at different moments of time."

I do not wish to "argue by wikipedia" (because it of course has errors in it), but hopefully make it clear that my use of the phrases is at least not uncommon. From my physics upbringing a "reference frame" was usually shorthand for coordinate system. I have also heard reference frame used to mean a class of coordinate systems (for example they would consider there to be a unique inertial rest frame of some object, leaving room for choosing the axes, etc.). This rarely caused problems, as the meaning was usually clear from context and could of course easily be clarified.

It is unclear to me what distinction you want me to make between a "reference frame" and a "coordinate system". For you don't want me to use a coordinate system, but you still wish to refer to velocities. But velocities are a coordinate system dependent quantity. If you are assigning positions vs time to points on an object's path, is that not referring to a coordinate system?

So I'd like to hear more about the distinction you make.
When you use the phrase "reference frame" and "coordinate system", what is the relation of those terms to you?

And to help find common ground, would you at least agree that there are coordinate systems in which Newton's first law holds, but that are not inertial coordinate systems?

 

[pervect]

Using slightly more formal language, I'd say that a frame of reference consists of an observer, and a set of "basis vectors". It's common to say that the observer "carries" the basis vectors.

The "basis vectors" are envisioned in the PSSC film mentioned by bcrowell as a set of three rods welded together at right angles, but not given the name "basis vectors" in that film, at least not that I noticed.

Then, using this definition, the requirements for a frame of reference to be inertial are that the observer not be accelerating, and that the basis vectors carried by the observer don't rotate.

I'd say that in flat space-time it's straightforwards to go from the reference frame (which is defined by the basis vectors) to coordinates, as you can describe the position of an object by adding together multiples of the basis vectors (as described by the PSSC film).

In curved space-time, it turns out to be important to be more precise, the basis vectors when multiplied and added naturally span a flat tangent space rather than the curved space-time. This starts to get into territory not relevant to the original question, questions (if any) about this point probably belong in another thread.

[change] A non-flat space-time that otherwise meets the criterion of an inertial frame is usually called a locally inertial frame, or a locally Lorentz frame, and not just an "inertial frame". This is a bit more relevant to the current thread than the remarks above. I'm not sure it would be downright wrong to omit the "locally" qualifier when talking about a "local inertial frame", but at minimum it would be confusing.

 

[WannabeNewton]

I am interested to hear what distinction you feel there is.

That many people see them as the same leads to a handful of the common conceptual confusions in GR regarding the unphysical coordinate quantities and measurable physical observables tied to tetrads/vierbeins.

A Lorentz frame is an orthonormal basis for the tangent space at a given event on an observer's world-line such that the time-like basis is the observer's 4-velocity, such that the frame is then transported along the world-line through some transport law (parallel transport, Fermi-transport, Lie transport etc.). A congruence of observers filling all of space-time then defines a frame field. A special type of coordinate system is the exponential map of a Lorentz frame onto a normal neighborhood of each event on the observer's world-line. When applied to a frame field we get a collection of coordinate systems that together cover all of space-time. However an arbitrary coordinate system on some open subset of space-time need not come from a frame at all. In fact the vast majority of coordinate systems do not stem from frames. The two are truly distinct concepts.

 

[atyy]

That many people see them as the same leads to a handful of the common conceptual confusions in GR regarding the unphysical coordinate quantities and measurable physical observables tied to tetrads/vierbeins.

A Lorentz frame is an orthonormal basis for the tangent space at a given event on an observer's world-line such that the time-like basis is the observer's 4-velocity, such that the frame is then transported along the world-line through some transport law (parallel transport, Fermi-transport, Lie transport etc.). A congruence of observers filling all of space-time then defines a frame field. A special type of coordinate system is the exponential map of a Lorentz frame onto a normal neighborhood of each event on the observer's world-line. When applied to a frame field we get a collection of coordinate systems that together cover all of space-time. However an arbitrary coordinate system on some open subset of space-time need not come from a frame at all. In fact the vast majority of coordinate systems do not stem from frames. The two are truly distinct concepts.

But an inertial frame in special relativity is said to be global. In that case, couldn't an inertial frame be said to be a coordinate system? Eg. I've heard definitions that a global inertial frame in special relativity is one in which the metric is diag(-1,1,1,1). Would that be ok?

 

[JustinLevy]

The two are truly distinct concepts.

Let me try to summarize to see if I understand your use of the terms.

To you a frame is nothing more than the basis vectors for the tangent space at some point. Where-as coordinates are just the labels used for the points. It is possible, but not necessary, to construct a coordinate system from a collection of frames across spacetime (frame field).

All that seems reasonable. We can have bizarre coordinate systems, or non-coordinate basis.
I'm unclear if you are then saying "frames of reference" should then be a "frame field". Of if in your terminology a "frame of reference" is just a single 'frame' at a point. The former seems more appropriate to me, so I'm guessing that is what you mean?

Thus, whether a "frame of reference" is inertial or not then comes down to whether on all geodesic paths through the frame field, the frame is parallel transported (or some other 'constant' transport law)? Is that really only possible in flat space-time?

 

[WannabeNewton]

The former seems more appropriate to me, so I'm guessing that is what you mean?

When I think of a single reference frame or Lorentz frame, I think of a set of orthonormal basis vectors transported (through some transport law) along a single world-line with the time-like one equal to the 4-velocity of the world0line. A frame field would be a set of vector field of such basis vectors, one set along each world-line in a congruence of world-lines. The following excellent page may be of use: http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

Thus, whether a "frame of reference" is inertial or not then comes down to whether on all geodesic paths through the frame field, the frame is parallel transported (or some other 'constant' transport law)? Is that really only possible in flat space-time?

So I think this ties back to the above. A single inertial frame is as defined above with the additional constraints that it must be non-accelerating and non-rotating which amounts to saying that all the basis vectors of the frame are parallel transported along the world-line. We can also have an entire set of vector fields of inertial frames which would indeed correspond to an entire congruence of time-like geodesics with the frame attached to each geodesic being parallel transported along it. An example of such a congruence in curved space-time is that of the observers in Schwarzschild space-time freely falling radially from rest at infinity i.e. the Painleve observers. One can easily write down a non-rotating frame field for this congruence.

But an inertial frame in special relativity is said to be global. In that case, couldn't an inertial frame be said to be a coordinate system? Eg. I've heard definitions that a global inertial frame in special relativity is one in which the metric is diag(-1,1,1,1). Would that be ok?

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.

 

[D H]

I am interested to hear what distinction you feel there is.

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 ...

And to help find common ground, would you at least agree that there are coordinate systems in which Newton's first law holds, but that are not inertial coordinate systems?

No, I wouldn't. Please explain why you think this is the case.

You have to define a metric and a time derivative. This is easiest in a cartesian coordinate system in which the spatial bases are orthogonal, isotropic, and homogeneous and time is "uniform" (Newtonian mechanics isn't quite specific on this, but then again, neither is GR. Time is what ideal clocks measure. And what do they measure? Passage of time.) The metric and time derivative for some bizarre coordinate system must agree with this because that bizarre system is just a rose by some other (bizarre) name.

 

[JustinLevy]

I feel WannabeNewton explained his stance well.

D.H., you also are distinguishing reference frame and coordinate system, but I'm still having trouble understanding your stance. Maybe we can cut this short -- Are you trying to say the same thing as WannabeNewton?

...
The following only applies if you are using the terminology different than WN (it appears to me that you are).

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 ...

Given some vector, we could discuss it as a mathematical object, independent of our choice of coordinate system. But, since you want to define an inertial frame with Newton's first law, how do you intend to determine that velocity vector without referring to a coordinate system? The velocity vector is clearly coordinate system dependent in Newtonian mechanics. That's why it can be zero in some coordinate systems and non-zero in others.

Since you are bringing metrics into this, it sounds like you are trying to redefine velocity for use in Newton's first law. So working in Galilean spacetime, it sounds like you would like velocity to be something like dS / dT where S is the spatial path length (calculated from the metric), and T is the time path length (calculated from the metric). I am not very comfortable working with the degenerate metrics of Galilean spacetime, so that may not be what you are suggesting. But what is clear is you are trying to get at something that removes our dependence on how we describe it with a coordinate system.

But then your definition becomes meaningless. Even a rotating coordinate system is now an inertial coordinate system. Even an accelerating coordinate system is now an inertial coordinate system. ANY transformation from an inertial coordinate system gives you another inertial coordinate system. That definition is meaningless.

Hopefully I've misunderstood what you are trying to say with the metrics. If so, please do explain further.

And to help find common ground, would you at least agree that there are coordinate systems in which Newton's first law holds, but that are not inertial coordinate systems?

No, I wouldn't. Please explain why you think this is the case.

I have explained.

Said a slightly different way:
Newton's first law has the symmetry of arbitrary linear transformations. Inertial coordinate systems however are related only by the more restrictive relations of the Galilean group in Newtonian mechanics. There is a mismatch there.

Maybe at this point I should turn the question around.
What would you consider a non-inertial coordinate system?
How bizarre can the transformation from an inertial coordinate system be before you decide, yep Newton's Laws don't hold in that coordinate system? Or do you really feel it doesn't matter as it is still an inertial coordinate system (rose by another name).

 

[bcrowell]

Maybe at this point I should turn the question around.
What would you consider a non-inertial coordinate system?
How bizarre can the transformation from an inertial coordinate system be before you decide, yep Newton's Laws don't hold in that coordinate system? Or do you really feel it doesn't matter as it is still an inertial coordinate system (rose by another name).

Suppose that A is an inertial observer, i.e., one for whom clock-and-ruler measurements verify Newton's first law. A can use clocks and rulers to define coordinates (t,x).

Any transformation of the form (t,x)->(t+c,x+vt+d) gives another frame of an inertial observer. These are called Galilean transformations.

You can augment this set of transformations is several ways. You can allow a change of units, which is valid but not very exciting. You can allow discrete transformations such as parity and time-reversal, which are not things you can physically *do* to an observer in the sense that you can change an observer's position or state of motion. However, it would not violate the laws of physics to have a parity-reversed observer, or even a time-reversed one. You can have transformations like (t,x)->(x,t); Newton's laws do not really allow such transformations, since the laws don't make sense, for example, if a particle's position isn't a single-valued function of time.

Another kind of linear transformation you can imagine is something like a Lorentz transformation, in which simultaneity isn't preserved. If A is an inertial observer, then according to Newton, the result of such a transformation is not a frame of reference at all, because it doesn't correspond to the clock-and-ruler measurements of any possible observer B. The question of whether it's inertial doesn't even arise, since it's not even a frame of reference.

Nonlinear transformations will take the inertial frame A to something that either isn't a frame of reference or that is a frame of reference but is noninertial.

 

[JustinLevy]

The question of whether it's inertial doesn't even arise, since it's not even a frame of reference.

Wow, and yet a different distinction someone is making between a coordinate system and a frame of reference. It seems the answers are almost as numerous as the people. Fascinating.

Thank you for sharing your take on it.

If I have a 'grid' of rulers and clocks at the intersections to mark off events with labels / coordinates corresponding directly with the ruler markings and clock time by the event, does that mean this coordinate system is special enough to be a "reference frame" by your use of the word?

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.

 

[bcrowell]

If I have a 'grid' of rulers and clocks at the intersections to mark off events with labels / coordinates corresponding directly with the ruler markings and clock time by the event, does that mean this coordinate system is special enough to be a "reference frame" by your use of the word?

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).

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.

 

[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.