Are frames in physics necessary?

[WannabeNewton]

It's just terminology at the end of the day. Strictly speaking a reference frame does include a specification of an orthonormal basis (a tetrad) along with a transport law for the spatial triad. But in the literature this definition is much less common than the one Martin pointed to i.e. that of a reference as nothing more than a time-like congruence. The reason for this is in most calculations the latter definition is more than sufficient. In fact, in a project I'm working on now it is the definition of a reference frame as simply a time-like congruence that is being used.

 

[bcrowell]

One thing you would lose if you never said the naughty words "frame of reference" would be the ability to give the most common statement of the equivalence principle.

 

[vanhees71]

It's just terminology at the end of the day. Strictly speaking a reference frame does include a specification of an orthonormal basis (a tetrad) along with a transport law for the spatial triad. But in the literature this definition is much less common than the one Martin pointed to i.e. that of a reference as nothing more than a time-like congruence. The reason for this is in most calculations the latter definition is more than sufficient. In fact, in a project I'm working on now it is the definition of a reference frame as simply a time-like congruence that is being used.

Why must the basis be orthogonal or even orthonormal? These are convenient choices, but not necessary in principle. Of course, to properly define spinors, you need the tetrad formalism.

 

[bcrowell]

Why must the basis be orthogonal or even orthonormal? These are convenient choices, but not necessary in principle. Of course, to properly define spinors, you need the tetrad formalism.

To me the main reason that frames of reference are indispensable is that without them, you have no way of connecting the theory to measurements in the lab. Making the basis orthogonal correctly captures this. The orthogonality of t to the spatial axes captures the fact that any clocks in your lab are synchronized with each other in the lab's frame of reference. The orthogonality of the spatial axes to each other captures the fact that you're free to rotate your lab, but you can't distort its shape -- or you could say that it captures the fact that you can measure angles using a rigid protractor.

[EDIT: After further consideration, I want to recant this view. One can certainly, in principle, describe laboratory measurements without ever introducing the concept of a frame of reference.]

 

[pervect]

We've seen about three slightly different definitions of "frames of reference" in this thread - sprays, time-like congruences, and tetrads. I was finally able to track down the full text of one paper on the topic, which also mentioned three different approaches to defining "frames of reference". It had some discussion of the other literature (not that I had the time or the access to read any of the refrenced papers, alas).

Manoff, "Frames of reference in spaces with an affine connections and metrics"

There are at least (emphasis mine) three types of methods for defining a FR. They are based
on three different basic assumptions:
(a) Co-ordinate's methods. A frame of reference is identified with a local
(or global) chart (co-ordinates) in the differentiable manifold M

...

(b) Tetrad's methods. A frame of reference is identified with a set of basic contravariant vector fields [n linear independent vectors (called n-Beins,n-beams) f at every given point x of the manifold

...

(c) Monad's methods. A frame of reference is identified with a non-null (non-isotropic) (time-like) contravariant vector field interpreted as the velocity of an observer (material point).
...

I think we've seen pretty much the same three mentioned in this thread, though I'm not sure how many others besides myself have alluded to the first method, i.e. the notion that a frame of reference should "tell you where you are", a function that is arguably a property of coordinates.

The majority of these three definitions are phrased in terms of manifolds, with frequent mentions of the tangent spaces to said manifolds as well. So I continue to believe that a logical approach to teaching general relativity is to introduce manifolds and their tangent spaces. I also believe that this is the approach most textbooks take, though I'm open to counterexamples.

There is general agreement on the precise definition of just what a manifold is, which is a huge plus. It's not strictly necessary to actually give the precise definition of a manifold straight away, but it is necessary to have a common understanding, and being able to point to a precise definition is very helpful when quesitons arise. After introducing manifolds, then, at a later date, one can introduce one or more of the multiple existing concepts of "frames of reference" and optionally argue about "which one is the best".

Attempting to build on top of an existing concept of a "frame of reference" doesn't seem to me to be advisable considering the lack of agreement over just what a frame of reference is in the literature. Furthermore, I suspect the lack of agreement among people who haven't studied the literature on frames of references is even broader than the amount of disagreement in the literature.

As far as eliminating frames of reference goes , I feel I should mention one of the precedents for this in the literature.

Misner, "Precis of General Relativity"

A method for making sure that the relativity effects are specified correctly
(according to Einstein’s General Relativity) can be described rather briefly.
It agrees with Ashby’s approach but omits all discussion of how, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles.

One first banishes the idea of an “observer”. This idea aided Einsteinin building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape into two categories.

One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments and the data tables they provide. For GPS the measuring instruments can be taken to be either ideal SI atomic clocks in trajectories determined by known forces, or else electromagnetic signals describing the state of the clock that radiate [from the clocks].

...

The conceptual model for a relativistic system is a spacetime map or diagram plus some rules for its interpretation.

My interpretation of Misner's "conceptual model" is that one has a set of coordinate labels that assign labels to space-time events, plus a mathematical structure that allows one to calculate the invariant lorentz interval between (nearby) events, though the associated text is a bit too long to quote in full.

My interpretation of Misner's motivation for writing the paper was the amount of argument generated by Neil Ashby's paper that talked about "the need to use the ECI frame of reference" when talking about GPS, which caused some debate and confusion.

I also feel that frames of reference are still very useful, in any of the various forms we've seen in this thread and/or in the literature. I think that the logical place to introduce one (or more, but I can't see burdening a new student with the details of more than one) is after one has done as much as one can without them - which as Misner points out is really quite a bit.

 

[bcrowell]

We've seen about three slightly different definitions of "frames of reference" in this thread - sprays, time-like congruences, and tetrads.

You seem to have started the discussion with two assumptions: (1) that the existence of multiple definitions was a problem, and (2) that such a problem would indicate that there was a problem with the whole notion of a frame of reference. The thread has gotten long, so maybe I've missed something, but I don't see where you've ever justified either of these assumptions. We have many concepts in physics that have multiple definitions, and often this is a good thing. It gives us the flexibility to use the correct tool for the job. For example, we usually use the term "mass" without any modifiers, because the context makes it clear which we need, but when precision is required we can specify a particular definition, such as gravitational mass. And of course one can have multiple definitions that turn out to be equivalent, or one definition that's a special case of another definition, as I pointed out in #13.

I also feel that frames of reference are still very useful, in any of the various forms we've seen in this thread and/or in the literature. I think that the logical place to introduce one (or more, but I can't see burdening a new student with the details of more than one) is after one has done as much as one can without them - which as Misner points out is really quite a bit.

Again, I may have missed something in this lengthy thread, but you never seem to have stated what audience or what level of presentation you have in mind. Are you advocating that high school kids learning Newtonian mechanics for the first time should never be exposed to the notion of a frame of reference? Are you stating your preferred pedagogy for a graduate course in GR, so that students would already know about frames of reference, but frames would be deemphasized and delayed in the presentation? FactChecker also asked about this in #20, but you haven't addressed the issue.

 

[atyy]

In line with bcrowell's point about the equivalence principle, the notion of "locality" in that principle ranges from ill-defined to quite precise. Initially, it was ill-defined as a heuristic before we had GR. After we have GR, we can define "local" within GR.

 

[pervect]

You seem to have started the discussion with two assumptions: (1) that the existence of multiple definitions was a problem, and (2) that such a problem would indicate that there was a problem with the whole notion of a frame of reference. The thread has gotten long, so maybe I've missed something, but I don't see where you've ever justified either of these assumptions. We have many concepts in physics that have multiple definitions, and often this is a good thing. It gives us the flexibility to use the correct tool for the job. For example, we usually use the term "mass" without any modifiers, because the context makes it clear which we need, but when precision is required we can specify a particular definition, such as gravitational mass. And of course one can have multiple definitions that turn out to be equivalent.

Again, I may have missed something in this lengthy thread, but you never seem to have stated what audience or what level of presentation you have in mind. Are you advocating that high school kids learning Newtonian mechanics for the first time should never be exposed to the notion of a frame of reference? Are you stating your preferred pedagogy for a graduate course in GR, so that students would already know about frames of reference, but frames would be deemphasized and delayed in the presentation?

I've probably missed more than one thing in this long thread. I do regard the ambiguity in the frame of reference as a problem, if nothing else a problem for communication.

My focus and primary concern is with students and PF posters making the transition from Newtonian physics to General Relativity which I would characterize as occurring at the undergraduate level. However, I'll have to admit that I don't always think and/or phrase my arguments at that level, even if that's where I want to wind up.

In Newtonian physics, we have the concept of displacement vectors, nicely formalized by the notion of an "affine space" as vanheer71 pointed out. So it makes perfect sense to define a set of basis vectors, and use the concept of displacement vectors to determine the location of a point, so the notions of a frame of reference as something that tells you "where you are" and the notion of a frame of reference as a tetrad are basically the same. The tetrad notion does tell you "were you are", directly.

I view this as the basic undergraduate approach to understanding of the concept of a frame of reference, as evidenced by various undergraduate level teaching aids such as the PSSC films, though I'm not in this post trying to use undergraduate level language to describe the underlying concepts.

Now, when we make the transition to General relativity, we notice that we no longer have an affine space, because the concept of a displacement vectors would require that displacement operators to commutes. And in curved space-time, they don't. To make a less abstract and more specific example, we can ask "What is a "frame of reference" on a curved manifold, like the surface of a sphere?" We have numerous examples of a lattice structure constituting a frame of reference in Newtonian physics (note that lattice motif in the PSSC film for one example, or the rigid famework of clocks and rods that is also often used). But it's not clear how to generalize such a lattice structure that would fit on the surface of the sphere.

So basically I regarded the Newtonian notion of a frame of reference of being a conflation of two different concepts, the tetrad concept (that specifies a set of basis vectors) and the coordinate concept (that tells you where you are). The issue is that these two concepts are not quite the same. I'm open to argumements that the two concepts are in some abstract manner "the same concept", but at the moment I don't see it this way, in spite of the fact that the standard terminology of physics has chosen to give these IMO different, but related, concepts the same name, which I regard as a bit unfortunate and confusing.

The closest I can come is to unifying the concepts is to say that in a small area around any point in the manifold, we can use the tetrad concepts to specify a location, and vica versa. The process involves the usual Newtonian techniques for finding a point in the tangent space given a set of basis vectors and their coordinates , then using the exponential map which creates a 1:1 map from a subset of the tangent space to a subset of the manifold- but only (as I understand it) in the local convex region of the manifold.

[add]Additionally, consider the state of mind of the student, who has come to regard "frames of reference" as absolutely and fundamental to doing physics, because of their training, when one confronts them with the idea that "frames of reference" are limited in the extent of space-time that they cover. I don't have to imagine it so much because I've seen the reaction of various and sundry PF posters who've experienced it. Most of them, I'm pretty sure, never do believe that it makes sense to talk about a "frame of reference" that only covers a limited extent of space-time.

 

[vanhees71]

Again, in my opinion, one cannot stress the physical, rather than mathematical, definition of reference frames enough. It's true that to define a physical frame of reference (in the most simple way a set of three "rigid rods" and a "clock" used by a "pointlike observer") you need a geometric model of space-time, which is in the case of GR a pseudo-Riemannian manifold with a fundamental form of signature (1,3) or (3,1) depending on your preference of the west- or east-coast convention. Then it's clear, how with help of the three rods and the clock the observer can make measurements of space-time intervals in his neighborhood. Defining space-time distances for larger scales is very difficult and if I understood this right (I'm not an expert in GR) it's possible only for pretty symmetric space-time manifolds, particularly for static ones. Already for cosmology (FLRW metric) there are various distance measures, which must be clearly defined by operational physical measurement procedures (like, e.g., the luminosity distance of supernovae etc.). Such physical definitions then can, of course, formalized in the theory as various abstract structures, but after all we are doing physics and not pure math. So we have to define measurable quantities in a clear way and map them (usually in idealized form) to the formalism used in theory to make a comparison between theory and experiment possible.

 

[bcrowell]

My focus and primary concern is with students and PF posters making the transition from Newtonian physics to General Relativity which I would characterize as occurring at the undergraduate level.

I see. That makes it much more clear what you're advocating. My knowledge of curricula may be out of date, but my impression is that most physics majors never receive any formal instruction in GR as undergrads, and that even among people who get a PhD in physics, quite a few never take a GR course. On the other hand, we do have less mathematical treatments of GR in books like Hartle (for upper-division physics majors) and Hewitt (gen ed students).

In any case, I think you may be telescoping things a little by describing a transition from Newtonian physics to GR. For most people SR is an intermediate step. Would you advocate not discussing frames in SR? By the way, have you seen this book?

Bertel Laurent, Introduction to spacetime: a first course on relativity , https://www.amazon.com/dp/9810219296/?tag=pfamazon01-20

I don't 100% love the book (you can see my reasons in my amazon review), but it does have a very nice coordinate-free approach to SR, which might resemble what you have in mind.

As a teacher, one thing that I find happens when I teach SR is that students really need to cycle back to the Newtonian notion of frames of reference, which they often haven't fully absorbed the first time around. IIRC this is a point that Mermin makes in a preface to his gen ed book on SR (It's About Time: Understanding Einstein's Relativity, https://www.amazon.com/dp/0691141274/?tag=pfamazon01-20 ). They need practice, for example, in giving a verbal description of a process in one frame and then converting into the description in another frame. So with respect to real-world students, I would say that a frame of reference is already a hard enough concept. Describing things in terms of manifolds and tangent spaces is IMO inherently more difficult and abstract, so I'm not convinced it would be wise to skip ahead to such a description.

So basically I regarded the Newtonian notion of a frame of reference of being a conflation of two different concepts, the tetrad concept (that specifies a set of basis vectors) and the coordinate concept (that tells you where you are). The issue is that these two concepts are not quite the same. I'm open to argumements that the two concepts are in some abstract manner "the same concept", but at the moment I don't see it this way, in spite of the fact that the standard terminology of physics has chosen to give these IMO different, but related, concepts the same name, which I regard as a bit unfortunate and confusing.

I agree with you that they are different concepts, but they're clearly very closely related. Once you have a set of coordinates $x^\mu$, you immediately have a natural choice of basis vectors, which is $\partial_\mu$ if you use Cartan's notational trick.

 

[stevendaryl]

What seems to be meant by "frame of reference" when it is used in discussing SR or the equivalence principle in GR is something like: A decomposition of spacetime into space + time, with the constraint that the metric tensor be time-independent. Tetrads don't really seem to line up with the way "frame of reference" in discussions of SR or GR thought experiments. For one thing, it seems to me that "frame of reference" in discussions of SR does not imply a choice of basis vectors. It's a standard for simultaneity and rest. Perhaps a frame reference is an equivalence class of tetrad fields?

 

[Mentz114]

What seems to be meant by "frame of reference" when it is used in discussing SR or the equivalence principle in GR is something like: A decomposition of spacetime into space + time, with the constraint that the metric tensor be time-independent. Tetrads don't really seem to line up with the way "frame of reference" in discussions of SR or GR thought experiments. For one thing, it seems to me that "frame of reference" in discussions of SR does not imply a choice of basis vectors. It's a standard for simultaneity and rest. Perhaps a frame reference is an equivalence class of tetrad fields?

I don't agree with any of that. Choosing a frame is exactly choosing a set of basis vectors. In SR this is well hidden, but in curved spacetime the 'coordinate frame' is 'all Salavdor Dali' ( Andrew Hamilton) so it is necessary to use the tangent space of a timelike curve as a frame of reference.

Nearly all the publications I read use a frame field. Physics is independent of the choice and a suitable choice can make the computations clearer and more meaningful.

 

[stevendaryl]

I don't agree with any of that. Choosing a frame is exactly choosing a set of basis vectors. In SR this is well hidden, but in curved spacetime the 'coordinate frame' is 'all Salavdor Dali' ( Andrew Hamilton) so it is necessary to use the tangent space of a timelike curve as a frame of reference.

I'm not disputing that other notions of "frame of reference" are useful in GR, or even that they are more useful than what I described. I'm just saying that they are not what is meant when people talk about "frame of reference" in SR or the equivalence principle.

 

[PAllen]

To me, the limitation of the tangent space approach is that a spinning rocket (e.g. to counter weightlessness) flying near Jupiter has a certain internal experience over time - not just at one moment and not just at one point. It does seem there isn't clear consensus on this, but MTW introduce the notion of a proper reference frame for such a case, built geometrically, and I have seen plenty of papers adopting this approach. It does boil down to Fermi-Normal coordinates extended to allow rotation of the tetrad along the origin world line relative to Fermi-Walker transport of a starting tetrad. You get a description within a world tube where it is believed that coordinate quantities will correspond to measurements made in the spinning rocket using reasonable measuring devices for time, distance, and position. To me, this is the most physical notion of frame in GR - local, not global, but substantially extended in time, less so in distance from the 'origin'.

 

[Mentz114]

I'm not disputing that other notions of "frame of reference" are useful in GR, or even that they are more useful than what I described. I'm just saying that they are not what is meant when people talk about "frame of reference" in SR or the equivalence principle.

Of course, I didn't mean to sound so robust but it seemed a good place to push my POV on this.

Calculations in SR rely on assigning coordinates to events. Coordinates means vectors and vector spaces and basis vectors. A Lorentz transformation is a change of basis.
One does not have to know this to do the calculations but that is what is is happening.

To me, the limitation of the tangent space approach is that a spinning rocket (e.g. to counter weightlessness) flying near Jupiter has a certain internal experience over time - not just at one moment and not just at one point. It does seem there isn't clear consensus on this, but MTW introduce the notion of a proper reference frame for such a case, built geometrically, and I have seen plenty of papers adopting this approach. It does boil down to Fermi-Normal coordinates extended to allow rotation of the tetrad along the origin world line relative to Fermi-Walker transport of a starting tetrad. You get a description within a world tube where it is believed that coordinate quantities will correspond to measurements made in the spinning rocket using reasonable measuring devices for time, distance, and position. To me, this is the most physical notion of frame in GR - local, not global, but substantially extended in time, less so in distance from the 'origin'.

That is a good example of finding a frame field (basis vectors) adapted to a particular situation. Unfortunately I don't have MTW but I've seen frame fields adapted and used in spin-2 boson theories.

 

[pervect]

I pretty much think of the tetrad approach as being called "frame fields", the coordinate approach as being called "local lorentz frames", usually with the constraint on the metric when this term is used being that the metric be Minkowskii at a point. With this structure, local Lorentz frames don't rotate, making them a bit stricter than the suggestion that we allow "rotating frame of reference", and also stricter than the suggestion that the metric coefficients be time-independent. It's possible to constrain the metric even further, i.e. Fermi-normal coordinates are always a local Lorentz frame near the reference worldline used to construct them, but the reverse isn't true, MTW points out that there's an interesting relationship between the Riemann tensor and the metric nearby the reference worldline in the case of Fermi-normal coordinates that doesn't hold with more general coordinate choices. I don't really think of timelike congruences as being called "frames", but apparently there's some precedent.

 

[PAllen]

I pretty much think of the tetrad approach as being called "frame fields", the coordinate approach as being called "local lorentz frames", usually with the constraint on the metric when this term is used being that the metric be Minkowskii at a point. With this structure, local Lorentz frames don't rotate, making them a bit stricter than the suggestion that we allow "rotating frame of reference", and also stricter than the suggestion that the metric coefficients be time-independent. It's possible to constrain the metric even further, i.e. Fermi-normal coordinates are always a local Lorentz frame near the reference worldline used to construct them, but the reverse isn't true, MTW points out that there's an interesting relationship between the Riemann tensor and the metric nearby the reference worldline in the case of Fermi-normal coordinates that doesn't hold with more general coordinate choices. I don't really think of timelike congruences as being called "frames", but apparently there's some precedent.

A few clarifications. IMO, local lorentz coordinates in GR correspond to Riemann Normal coordinates at a point, which not only require that the metric is Minkowski at the origin, but that the connection vanishes at the origin. That is, they are local free fall coordinates at a point (event). Fermi-Normal coordinates are also Riemann-Normal at each point of the origin world line if and only if the the Fermi-Normal origin world line is a inertial. Otherwise, they have the feature that while the metric is Minkowski along the origin, the connection's first order terms at the origin describe proper acceleration in the same manner as the first order terms in the connection for a local frame with the same proper acceleration in flat spacetime (e.g. the translation of the origin of Rindler coordinates to a particular 'position' world line). MTW extends this to also allow rotation of the tetrad in relation to Fermi-Walker transport of the tetrad (which formalizes the meaning of non-rotation). In this case, the rotation contributions to the connection at the origin first order match the connection of a rotating frame in flat spacetime.

 

[bcrowell]

Some recent discussions on PF have given me a strong philosophical kick toward agreeing with pervect on this. I've seen some cases where people really tied themselves up in knots by insisting that frames of reference were fundamental. As part of my religious conversion, I want to recant the view expressed in #13 and 34 to the effect that we can't discuss measurements without discussing frames of reference. In principle, all measurements in relativity can be reduced to incidence relations. You can have, e.g., clocks without having a frame of reference.

 

[strangerep]

[...] In principle, all measurements in relativity can be reduced to incidence relations. [...]

I find it both puzzling and fascinating how a set of incidence relations gets extrapolated into a manifold, by "filling in the blanks", so to speak. Puzzling, because we somehow lose intrinsic causality by doing so, and must put it back in by hand later.

 

[vanhees71]

Some recent discussions on PF have given me a strong philosophical kick toward agreeing with pervect on this. I've seen some cases where people really tied themselves up in knots by insisting that frames of reference were fundamental. As part of my religious conversion, I want to recant the view expressed in #13 and 34 to the effect that we can't discuss measurements without discussing frames of reference. In principle, all measurements in relativity can be reduced to incidence relations. You can have, e.g., clocks without having a frame of reference.

I agree with you that there's no way to measure anything without the careful reference to a reference frame. Only the last sentence contradicts this obvious fact again. It must be "You can NOT have, e.g., clocks without having a frame of reference." Otherwise, please describe a clock which does not define a reference frame. I cannot even imagine something like this. Note that I talk about a real thing not a coordinate in the mathematical world!

 

[bcrowell]

I agree with you that there's no way to measure anything without the careful reference to a reference frame. Only the last sentence contradicts this obvious fact again. It must be "You can NOT have, e.g., clocks without having a frame of reference." Otherwise, please describe a clock which does not define a reference frame. I cannot even imagine something like this. Note that I talk about a real thing not a coordinate in the mathematical world!

You've discussed this extensively in various threads on the site, and I disagree. I don't think there is much to add to that discussion.

 

[vanhees71]

No, this is utmost important. I'd like to understand how you define a clock without a frame!