Are frames in physics necessary?

[pervect]

It's perfectly possible to do physics without using the words "frame" or "frame of reference" at all. The modern view of GR is based not on the idea of "frames", but rather on the idea of manifolds. Take your favorite modern text (Caroll and Wald come to mind), and count the number of times the various phrases "frames" and "manifolds" are used. Bettter yet , try to actually look at the significance of the terms to the presentation of the theory.

While arguing about the meaning of what constitutes a frame of reference may be fun for people who like to argue, it's not terribly productive in my opinion.

There is to my mind a dearth of definitive definitions in textbooks of precisely what a "frame of reference" is. As evidence of the lack of definition, I point ot the fact that we've had this argument before, and nobody has actually quoted an exact definition.

I believe this lack of definition is due precisely to the fact that the underlying concept is no longer central to the theory. Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.

 

[WannabeNewton]

Huh? Frames are still incredibly important in physics and in GR in particular. Since when are they not? Sorry but that statement is, if not phrased incorrectly, just misleading to people. Sure Wald barely mentions the word but that's a bad example because Wald is a book on mathematical GR, not the physics of GR. Basically every book that's about the latter will use the term frame or variants thereof profusely e.g. Hartle, MTW, Straumann, Schutz, Poisson, Thorne et al to name a few. At the end of the day most calculations that aren't in the realm of mathematical GR will involve frames in one way or another. Research in GR would be impossible without the utility of frames.

 

[Dale]

Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.

That sounds like a good insights article.

Do you by any chance know of a good test theory for SR which is expressed in terms of manifolds etc.?

 

[haushofer]

Maudlin's 'space and time' discusses Galilean and special relativity with the notion of inertial frame being derivative. Maybe that book gives some inspiration in a careful definition of relativity :)

 

[atyy]

It's perfectly possible to do physics without using the words "frame" or "frame of reference" at all. The modern view of GR is based not on the idea of "frames", but rather on the idea of manifolds. Take your favorite modern text (Caroll and Wald come to mind), and count the number of times the various phrases "frames" and "manifolds" are used. Bettter yet , try to actually look at the significance of the terms to the presentation of the theory.

While arguing about the meaning of what constitutes a frame of reference may be fun for people who like to argue, it's not terribly productive in my opinion.

There is to my mind a dearth of definitive definitions in textbooks of precisely what a "frame of reference" is. As evidence of the lack of definition, I point ot the fact that we've had this argument before, and nobody has actually quoted an exact definition.

I believe this lack of definition is due precisely to the fact that the underlying concept is no longer central to the theory. Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.

Huh? Frames are still incredibly important in physics and in GR in particular. Since when are they not? Sorry but that statement is, if not phrased incorrectly, just misleading to people. Sure Wald barely mentions the word but that's a bad example because Wald is a book on mathematical GR, not the physics of GR. Basically every book that's about the latter will use the term frame or variants thereof profusely e.g. Hartle, MTW, Straumann, Schutz, Poisson, Thorne et al to name a few. At the end of the day most calculations that aren't in the realm of mathematical GR will involve frames in one way or another. Research in GR would be impossible without the utility of frames.

Would the use of tetrads, especially for incorporating intrinsic spin, be an example supporting WannabeNewton's position?

 

[pervect]

I didn't mean to imply that frame-fields are "useless" by any means. But they're not as fundamental as the manifolds. I believe they have frame-fields have a relatively precise definition, too, though I haven't seen one in actual print. Frame fields can be regarded as the tangent space to a manifold, or perhaps sometimes they're the basis vectors of said tangent space.I beliee most of the discusson on "inertial frames" and "local inertial frames" and "global inertial frames" (which no longer exist in GR, so why worry about them?) are not talking about frame fields (tangent spaces to manifolds) at all. Which winds up in the usual muddle of an attempt on the discussion, whereby different people are interpreting the same words diffeently and basing their arguments about the difering defintions.

 

[pervect]

I've got a bit more time, so I'll give my own take on what I think about the differences between Newtonian physics, special relativity, and general relativity are, without claiming that it's a particularly "novice friendly" way of thinking about things. It's rather off-the cuff, so it's a rough overview. Comments are welcome, I'm regarding this as an opportunity to clarify my own thoughts by putting them down in writing. I shall attempt to follow Bohr's dictum, and not write more precisely than I can think.

In Newtonian mechanics, positions are basically described by the mathematics of a vector space. Vector spaces have a definition of dimensionality, and the sort of vector space that describes positions in Newtonian mechanics is a 3d vector space. Time is regarded as being totally separate from space. You can think of time as being a vector space too , albeit a one-dimensional one - you can represent time with a time-line.

My view is that one can regard the vector space describing position in Newtonian mechanics as a "frame of reference". The spatial geometry of this vector space is described by the concept of "distance", which is mathematically the dot product of vectors in the vector space, and is independent of the observer as a geometrical concept should be.

When one move up to SR, one keeps the same overall vector space structure, but instead of having one 3d vector space for space and another totally separate 1d vector space for time, one has a single 4 dimensional vector space for space-time, where space and time are regarded as a unified concept. The Newtonian concept of distance and the Newtonian concept of simultaneity becomes observer dependent as part of unification process. As a result, we describe the geometry of space-time in observer-independent terms by the observer-independent Lorentz interval , rather than in terms of "distance" and "time". Note, however, that one can regard a space-like Lorentz interval as being a "proper distance", and a time-like Lorentz interval as being "proper time". Also note that the reasons for the unification boil down, in the end, to compatibility with experimental results. I am not attempting to motivate the "why" of SR at this point, just trying to give a very brief high-level abstract description of the end result.

When one goes from SR to GR, fundamentally one no longer have a vector space structure for space-time. Instead one replaces it with a manifold structure. The tangent space to the manifold, however, has essentially the very same vector space structure as the vector space in special relativity.

The basic issue as I see it is that the "frame of reference" idea is applicable to a vector space. One can regard the basis vectors of the vector space constitute the "frame" part of the "frame of reference". And the "frame of reference" regarded in this manner does not directly apply to the underlying manifold structure of space-time, but rather to the tangent space of the space-time manifold.

 

[PAllen]

I find myself generally disagreeing with parts of Pervect's approach. The part I agree with is that for doing GR 'in the large' (gobally, or over a large region), frames are useless. You need to use manifold geometry either coordinate free for general results, or with coordinates (whatever is easiest for a particular problem) for concrete computation.

However, the utility of local frames that are not just the tetrad at a point or the tangent space at a point in GR comes from asking about the physics a small region. You don't have to introduce any notion of frame for this, but it is very useful (IMO) for understanding. Already in SR, if you ask about physics in a rocket or spinning space station, it becomes convenient to introduce a frame in the sense of local coordinate system built from local measurements. This passes directly to GR. In fact, if you compute observables for such a situation in general global (or large scale) coordinates, you find yourself implicitly building the local frame, if you go beyond a point to a small region (e.g. a space station around a spinning neutron star). By building the local frame (approximately; you typically can't do it exactly, but you don't need to) you do one substantive calculation from which many observables can be efficiently derived rather than starting from scratch for each observable. So far as I know, it is well accepted that the local coordinates that directly describe small region measurements are Fermi-Normal coordinates, optionally generalized for rotation.

 

[haushofer]

If you want to view GR as a gauge theory of the Poincaré algebra, then one of your gauge fields are tetrads. Dito for Newton-Cartan theory view as a gauge theory of the Bargmann algebra.

 

[vanhees71]

I didn't mean to imply that frame-fields are "useless" by any means. But they're not as fundamental as the manifolds. I believe they have frame-fields have a relatively precise definition, too, though I haven't seen one in actual print. Frame fields can be regarded as the tangent space to a manifold, or perhaps sometimes they're the basis vectors of said tangent space.I beliee most of the discusson on "inertial frames" and "local inertial frames" and "global inertial frames" (which no longer exist in GR, so why worry about them?) are not talking about frame fields (tangent spaces to manifolds) at all. Which winds up in the usual muddle of an attempt on the discussion, whereby different people are interpreting the same words diffeently and basing their arguments about the difering defintions.

I think, this is just a "clash of civilizations". For the mathematical foundation, the manifold is the fundamental notion. But for physics it's an empty game with a mathematical axiom system. You can just do the math, defining differentiable manifolds, add some further structures (an affine connection, a (pseudo-)metric, constrain it to a (pseudo-)Riemannian manifold with or without torsion etc. etc.). It's all nice, but it's no physics. To make these mathematical structures useful for theoretical physics, i.e., the description of the observable world, there's no way out: You must introduce reference frames, which are realized by material objects like rulers to measure distances, clocks to measure time intervals. It's a trap to overemphasize the math, we theoreticians easily walk into, but finally as a theorist you have to make predictions with all your fancy math that can be tested by experiments, otherwise we do (usually not too rigorous) math but no physics. That's why I made this comment already earlier in this thread, and it seem to be important to emphasize although Dalespam and I agreed to take it as a somewhat "semantic issue".

 

[PWiz]

I agree with vanhees71 over here. One can do all of topology and differential geometry to describe spacetime, but to actually do some physics, frames are essential.

Riemannian manifold with or without torsion

A small point here though - what exactly do you mean by this? I've only heard of the term 'torsion' being applied to connections on a tangent bundle.

 

[atyy]

Some more thoughts on pervect's comments. I brought up tetrads, and I think it is true that those are only part of why "local inertial frames" are important. I think the other assumption that is important is the equivalence principle, which is heuristically stated as local physics is special relativity. One way to implement the EP in a theory of gravity a spacetime metric is to use minimal coupling. So I guess it is that we have manifold (yes, that's basic in all forms), then the spacetime metric which is equivalent to tetrads or "frames", then we have the equivalence principle or minimal coupling which is what makes "inertial" frames.

In a way, pervect's view is that the geodesic equation is not fundamental. That is indeed true in one sense, and in another sense it is not true - this goes back to the history of the EP as being heuristic as well as formalizable.

 

[bcrowell]

There is to my mind a dearth of definitive definitions in textbooks of precisely what a "frame of reference" is. As evidence of the lack of definition, I point ot the fact that we've had this argument before, and nobody has actually quoted an exact definition.

I don't see what the problem is with using a frame field as a definition: https://en.wikipedia.org/wiki/Frame_fields_in_general_relativity . Probably the reason that not many texts introduce them in full generality is that such a definition isn't often very useful. In most cases of interest, we only care about a frame field defined along one observer's world-line, or just at one point in spacetime. If the rotational degrees of freedom aren't of interest (e.g., if you can analyze your problem in 1+1 dimensions), then a frame field at one point is fully determined by an observer's velocity vector.

So the idea can be presented in simpler or more complicated ways, but you can't conclude from that that textbook authors are somehow confused about the topic, or that the topic is shrouded in mystery or logically ill-founded.

I believe this lack of definition is due precisely to the fact that the underlying concept is no longer central to the theory. Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.

When you're learning SR, the concept of a manifold is irrelevant; it's a generalization that isn't needed. It's not realistic to imagine that every freshman biology major in college is going to learn about manifolds when they get the one week's worth of exposure to SR that is all they'll ever see in their lifetimes.

The notion of a frame of reference is a natural and indispensable one, because it encapsulates what measurement means. If you omit the description of how your theory connects to observables, then it isn't a scientific theory. [EDIT: On second thought, I don't really believe this.]

From my experience in teaching SR at a variety of levels, and GR at the physics-for-poets level, the main problem that students have with the concept of a frame of reference in SR is that because a frame of reference can be made into a global thing in SR, they lose track of what a sophisticated object a global frame of reference is. For example, they slip into thinking that statements made in a particular frame are what you *see*. This kind of thinking also leads to issues like attempts to analyze the twin paradox in which one tries to assign the time difference to the brief period in which the traveling twin accelerates.

 

[WannabeNewton]

But they're not as fundamental as the manifolds.

Neither is the metric so I don't agree with this premise. The manifold in and of itself is not what GR is concerned with, rather it is the metric. And the metric is just the "square" of a tetrad. The local physics, which is directly tied to tetrads and the metric, is what contains the interesting conceptual aspects. Obviously the global structure of space-time is important but it does so through the various quantities derived from and defined through the metric and tetrads (e.g. gravitational charges).

With the exception of mathematical physicists, I would be surprised if practicing relativists ever cared about the manifold itself outside of niche topics like censorship theorems. It's just not conceptually important at all as far as GR is concerned. The manifold is only important insofar as a person learning GR should know what a manifold is at a basic level.

 

[Mentz114]

It's perfectly possible to do physics without using the words "frame" or "frame of reference" at all. The modern view of GR is based not on the idea of "frames", but rather on the idea of manifolds. Take your favorite modern text (Caroll and Wald come to mind), and count the number of times the various phrases "frames" and "manifolds" are used. Bettter yet , try to actually look at the significance of the terms to the presentation of the theory.

While arguing about the meaning of what constitutes a frame of reference may be fun for people who like to argue, it's not terribly productive in my opinion.

There is to my mind a dearth of definitive definitions in textbooks of precisely what a "frame of reference" is. As evidence of the lack of definition, I point ot the fact that we've had this argument before, and nobody has actually quoted an exact definition.

I believe this lack of definition is due precisely to the fact that the underlying concept is no longer central to the theory. Rather than get stuck in rehashing "frames", it's better to realize they aren't terribly important anymore, and proceed onwards and learn something about manifolds.

I hope you don't include tetrad fields in the concept of 'frame field' or 'frame of reference' you are deprecating. Tetrads are an essential tool and are a rigorously defined expression of the equivalence principle.

The problem is that people who try to understand SR without understanding coordinate systems first, think a frame of reference is a state of motion.

 

[vanhees71]

I think, I still haven't made my point clear. With "frame", I don't mean simply a mathematical structure defining space time and then introducing the one or other kind of coordinates (or more generally abstract concepts like a "time slicing" in special or general relativity) but a concrete realization of a reference frame. It's hard to define, but it's underlying all of physics. To measure distances you need a concrete realization of a "ruler". This is at the very foundations of the system of units we define based on the best of our knowledge of the natural laws, and for these you need a (mathematical) model, which introduces the problem that it's not possible to separate the theoretical considerations from experimental realization of tests of these theoretical concepts.

In the SI of units the "rulers", are based on a relativistic space-time concept. It's not clear to me in how far one considers general relativitivity, but the definition is formulated pretty much using SRT concepts, and thus from the point of view of GR, define local distances only. For practical purposes of metrology one starts with a definition of time ("duration") measurements based on the observation that atomic systems deliver very accurate "clocks" in terms of their transitions. At the moment, again mainly for practical metrological reasons as far as I know, one uses the frequenzy of a hyperfine-transition in Cesium:

"The second is the duration of 9 192 631 770 periods of the radiation
corresponding to the transition between the two hyperfine levels of the
ground state of the cesium 133 atom."

(from http://physics.nist.gov/Pubs/SP330/sp330.pdf)

This units have to be realized with concrete apparati and must be used to calibrate concrete real-world measurement devices, and these themselves define a "frame". In theory we always talk about "observers" defining a frame, but what's meant by this are such very concrete realizations of a measurement of the quantities of interest (here time and space intervals).

Then the unit of lengths/distances are defined by defining the speed of light as an exact value:

"The meter is the length of the path traveled by light in vacuum during a
time interval of 1/299 792 458 of a second."
(from http://physics.nist.gov/Pubs/SP330/sp330.pdf)

 

[pervect]

Basically, what I see happening here is a lot of disagreement on what constitutes a "frame of reference" and what the term means. What I'm reading from the comments is that people do not, generally, accept my idea that a frame of reference is a vector space. Fair enough - that's not something that's spelled out in textbooks, though so far I've yet to run into a situation that I'd regard as a counterexample to the proposed definition.

So the first point is to see if someone else can point to a precise mathematical definition of the necessary properties of a "frame of reference", or whether it is (as it appears to me) a rather fuzzy term, whose exact properties are a bit unclear. Does anyone have a precise textbook definition of what a frame of reference is (one that is as precise as the definition of a manifold)?

The second point I wish to repeat, which may have gotten lost as a result of other issues, is a rather specific notion about an important difference between SR and GR, due to the effects of curvature in GR.

In GR, it has been said (http://arxiv.org/abs/gr-qc/9508043)

... Einstein’s General Relativity which asserts that space-time is curved. This means that there is no precise intuitive significance for time and position. [Think of a Caesarian general hoping to locate an outpost. Would he understand that 600 miles North of Rome and 600 miles West could be a different spot depending on whether one measured North before West or visa versa?

Well, it seems to me that the notion of latitude and longitude is both precise and reasonably intuitive, so the remark is a bit a bit vague. The example given, though, clarifies the underlying issue. Finite changes in position in General relativity (or on non-flat manifolds, such as positions on the surface of a sphere) do not add commutatively. It matters whether you go west then north, or north than west. (There are a couple of ways to interpret the idea of what it means to "go west" or "go north", I would assume that the notion of geodesic motion is preferred in this example over the notion of coordinate motion, the difference between the notions would mean that going west is not along a geodesic in the coordinate case, but along a line of contant lattitude. Regardless of the interpretation, howver, the observation that one winds up in a different spot remains true).

This consequence of this observation is that finite changes in position on a manifold do not form a vector space. Representing changes of position as a vector is something we routinely do in Newtonian physics, but it won't work in GR.

A meta-point is that if manifolds have a relatively precise definition that one can look up and quote, and frames of reference do not, then one can avoid a lot of semantic arguments by talking about manifolds rather than "frames of references". If someone can point to a precise and authoritative definition of a "frame of reference" I would have to reconsider this. I don't expect this to happen, though.

 

[pervect]

Another comment on "frames of reference". It seems to me that one of the required functions, I'd go so far as to say the main function, of a frame of reference is to tell you where you are, what your position is.

The other common aspect of frames of reference is that they involve triad (or tetrad, in the case of SR) of vectors. For instance, in the PSSC film "Frames of Reference"

at 1 minute 57 seconds into the video the narrator holds up a framework of 3 wooden rods and says "This represents a frame of reference".

Now, a somewhat leading question. Given a triad of vectors (which I assume is what the framework the narrator is holding up represents), how does one go about using this concept to determine one's position in space?

 

[Mentz114]

Another comment on "frames of reference". It seems to me that one of the required functions, I'd go so far as to say the main function, of a frame of reference is to tell you where you are, what your position is.

I think its function is to provide clocks and rulers for that are valid at every point on an integral curve of a timelike 4-velocity. Without these any attempts to establish a position would be futile.

See section II Coordinate systems ( especially Fermi Normal) of

The motion of point particles in curved spacetime
Eric Poisson, Adam Pound, and Ian Vega
arXiv:1102.0529v2 [gr-qc] 8 Feb 2011

 

[FactChecker]

Certainly from a mathematical point of view, you can jump right to manifolds and skip reference frames. But from a teaching, pedagogy, point of view that may be a mistake. There is always a conflict between teaching / learning the abstract approach that applies in the general case versus the more concrete example that applies in a particular special case. I think that many people will be happy with SR in terms of reference frames. There is much of value to learn and understand without going further. Trying to do SR with manifolds is unnecessarily abstract. It would add nothing and would confuse a lot of people -- not just because manifolds are more abstract, but also because a student would always be looking for a reason that the manifold description is necessary, when it is not necessary at all. It should be easy enough to introduce manifolds when they are needed in GR.

 

[strangerep]

It seems to me that one of the required functions, I'd go so far as to say the main function, of a frame of reference is to tell you where you are, what your position is.

Er,... who is the "you"? But perhaps you meant something like "...establish relative correspondence(s) between observed event(s) and oneself" ?

As for how to express "frame of reference" more precisely, have you considered Sprays ?

Informally, a spray on a manifold $M$ (being the configuration space of a dynamical system) is (associated with) a family of compatible systems of 2nd order ordinary differential equations, expressed in local coordinates as
$$
\frac{d^2 c^\mu}{ds^2} ~+~ 2 G^\mu(\dot c) ~=~ 0 ~,~~~~~~~~~
\mbox{where}~~~ \dot c ~:=~ \frac{dc}{ds} ~~\in~ TM~.
$$Here, $c^\mu(s)$ denotes the coordinates of a curve $c(s)$ with arbitrary parameter $s$. (The $G^\mu$ must be 2-homogeneous so that unphysical rescalings of $s$ have no effect on physical results.)

More rigorously, a spray on a manifold $M$ is a smooth vector field $S$ on the tangent bundle $TM$, expressed in a local coordinate system $(x^\mu, u^\mu)$ on $TM$ as follows:
$$
S(x,u) ~:=~ u^\mu \left. \frac{\partial}{\partial x^\mu} \right|_{(x,u)}
-~ 2 G^\mu(x,u) \left. \frac{\partial}{\partial u^\mu} \right|_{(x,u)} ~.
$$The $G^\mu(x,u)$ are known as the spray coefficients, and induce a (possibly nonlinear) connection on $TM$.

Although I've jumped to a coordinate representation in the above, it's possible to express the concept more abstractly -- see the Wiki link above. But the basic idea is that a frame of reference is the set of integral curves of the spray -- one can (metaphorically) "move" along them in a sensible fashion, using the nonlinear connection induced by the $G^\mu$. This establishes the desired "relative correspondence" mentioned at the start.

In this sense, a frame of reference is not a vector space in general, but rather a particular type of vector field.

 

[1977ub]

If one has no need of stating that two remote events are simultaneous, is there some other need for inertial frames?

 

[atyy]

To make these mathematical structures useful for theoretical physics, i.e., the description of the observable world, there's no way out: You must introduce reference frames, which are realized by material objects like rulers to measure distances, clocks to measure time intervals.

But what is a ruler and what is a clock? They are not fundamental things in the theory, are they (this is classical physics, not quantum mechanics)? If rulers and clocks are not fundamental, but only the fields are, then the rulers and the clocks should be considered specific field configurations.

Actually, the issue is tricky. We have frame fields = metric as fundamental mathematical objects. However, in Riemannian geometry, one does indeed give the metric physical meaning because we have "external" rigid bodies and rulers that do not bend the metric being measured. However, since there is no "external" rigid body in GR, maybe the metric does not deserve the name "metric" as a physical concept, only as a mathematical concept. Maybe this is what pervect is saying.

A very good argument that the metric of GR is not a metric is (IIRC PAllen said this is the same Anderson mentioned in MTW):

http://arxiv.org/abs/gr-qc/9912051
Does General Relativity Require a Metric
James L. Anderson

 

[vanhees71]

Sure, that's analogous to the notion of test charges in classical electrodynamics, used to define the electromagnetic field. It's only valid in a limiting sense, letting the test charge go to 0 (arbitrarily small). The same holds for the clocks and rulers in GR. You neglect the electromagnetic field of the test charges and the back reaction of the charges creating the field to be measured or the gravitational field of the clocks and rulers and its back reaction to the energy-momentum-stress distribution creating the gravitational field to be measured.

I read the first part of the introduction to the cited paper, and I'm already stuck, because it seems to be based on a contradictory argument. On the one hand the author claims, one can derive the Einstein field equations without the concept of a (pseudo-)Riemannian space and without the interpretation of the gravitational-field potentials $g_{\mu \nu}$ as a (pseudo-)metric, but on the other hand he invokes the famous uniqueness theorem (which probably is due to Hilbert, but I have to check this) that then the action has to be given by the Riemann scalar modulo a cosmological constant, but this very idea needs the assumption of a pseudo-Riemannian manifold in a strict sense (particularly the assumption to be torsion-free). Giving up this very constraining assumptions, you can have more general space-time descriptions then the original GR one by Einstein (and Hilbert).

 

[vanhees71]

Basically, what I see happening here is a lot of disagreement on what constitutes a "frame of reference" and what the term means. What I'm reading from the comments is that people do not, generally, accept my idea that a frame of reference is a vector space. Fair enough - that's not something that's spelled out in textbooks, though so far I've yet to run into a situation that I'd regard as a counterexample to the proposed definition.

If you say "affine space" instead of vector space, I fully agree (the point being that for SR the full symmetry group is the proper orthochronous Poincare group and not only the proper orthochronous Lorenz group). In GR it's the tangent-space at each space-time point, where the basic notions of (local!) space-time measurements are defined.

But that's incomplete! It's the mathematical structure used to describe the space-time manifold (and it begins with the assumption that nature is described well by a space-time manifold at all, but this seems to be justified by experience to a hight degree of accuracy). It's indeed the foundation for such space-time measurements, but these must be realized in the lab with real-world clocks and rulers (or rods). In a way you define local maps in the sense of differentiable manifolds to local space-time coordinates but, more importantly, a physically realizable frame of reference.

This goes together with a local time-slicing of space time, and you define the local measurement of time intervals and distances via light signals. A very good discussion of this, using a minimum of math, you can find in volume 2 of Landau+Lifshitz' Course of Theoretical Physics, which in my opinion is among the best books on classical field theory (electromagnetism and GR) ever written.

 

[atyy]

Sure, that's analogous to the notion of test charges in classical electrodynamics, used to define the electromagnetic field. It's only valid in a limiting sense, letting the test charge go to 0 (arbitrarily small). The same holds for the clocks and rulers in GR. You neglect the electromagnetic field of the test charges and the back reaction of the charges creating the field to be measured or the gravitational field of the clocks and rulers and its back reaction to the energy-momentum-stress distribution creating the gravitational field to be measured.

I read the first part of the introduction to the cited paper, and I'm already stuck, because it seems to be based on a contradictory argument. On the one hand the author claims, one can derive the Einstein field equations without the concept of a (pseudo-)Riemannian space and without the interpretation of the gravitational-field potentials $g_{\mu \nu}$ as a (pseudo-)metric, but on the other hand he invokes the famous uniqueness theorem (which probably is due to Hilbert, but I have to check this) that then the action has to be given by the Riemann scalar modulo a cosmological constant, but this very idea needs the assumption of a pseudo-Riemannian manifold in a strict sense (particularly the assumption to be torsion-free). Giving up this very constraining assumptions, you can have more general space-time descriptions then the original GR one by Einstein (and Hilbert).

Yes, it's basically the test mass issue. If you read the Anderson paper, that's the second argument he gives, so even if one doesn't agree with his first argument, the paper is bringing up the test mass issue which you and I and Anderson agree about. So again the difference between the two extremes (yours and pervect's - I would guess WannabeNewton's is intermediate?) has the history in the various statements of the equivalence principle and how one defines the notion that the equivalence principle is only "locally" valid.

 

[martinbn]

Does anyone have a precise textbook definition of what a frame of reference is (one that is as precise as the definition of a manifold)?

I don't know how universally accepted this is, but Sachs and Wu have a precise definition of a reference frame: a vector field whose integral curves are normalized future pointing timelike curves. In other words a family of observers that fill space-time.

 

[PeterDonis]

a vector field whose integral curves are normalized future pointing timelike curves. In other words a family of observers that fill space-time.

Is this sufficient by itself? This defines the worldlines of the observers, but not their references for spatial measurements. In other words, for a complete "reference" frame, you would need a frame field--an assignment of four orthonormal basis vectors, one timelike and three spacelike, to each point of spacetime, such that the timelike basis vectors form the vector field you describe (i.e., its integral curves are the worldlines of a family of observers that fills spacetime). For usefulness, you would probably also want to impose some kind of continuity requirement on the spatial basis vectors.

 

[vanhees71]

But isn't this the standard definition? Following Landau-Lifshitz vol. 2 we have the following physically motivated definitions of local measurements of "infinitesimal" space-time intervals. A definition of finite spatial distances is pretty tricky and can only be defined unambigously for the case of static pseudometrics.

Having some coordinates $q^{\mu}$ and a pseudometric
$$\mathrm{d}s^2=g_{\mu \nu} \mathrm{d} q^{\mu} \mathrm{d} q^{\nu},$$
This defines a set of "observers" at any space-time point (I'm assuming the local coordinates are realizable by some real-world "rulers" and "clocks"). An observer at a fixed point in his space, i.e., for $\mathrm{d}q^j=0$ with $j \in \{1,2,3\}$ measures time in terms of the proper time
$$\mathrm{d} \tau^2=g_{\mu \nu} \mathrm{d} q^{\mu} \mathrm{d} q^{\nu} = g_{00} (\mathrm{d} q^0)^2.$$
Further this observer measures (by definition!) the distance to an infinitesimally close point in space by sending a light signal to it and getting it reflected. The distance is then given as half the travel time of his light signal.

This physical definition can now be used to define a local spatial metric for this observer in the following way: The ligh$t signals move on null geodesics in space-time, i.e., they fulfill
$$\mathrm{d} s=g_{\mu \nu} \mathrm{d} q^{\mu} \mathrm{d} q^{\nu} = g_00 (\mathrm{d} q^0)^2+2 g_{0 j} \mathrm{d} q^0 \mathrm{d} q^j + g_{jk} \mathrm{d} q^{j} \mathrm{d} q^{k},$$
with the greek indices running from 0 to 3 and the latin ones from 1 to 3.

The time difference is given by solving this equation first for $\mathrm{d} q^0$, giving two solutions, where one is for the forward-moving and the other for the back-ward moving ligth signal:
$$\mathrm{d} q^0 = \frac{1}{g_{00}} \left [-g_{0j} \mathrm{d} q^j \pm \sqrt{(g_{0j} g_{0k}-g_{jk} g_{00}) \mathrm{d} q^{j} \mathrm{d} q^k} \right].$$
The proper time is given by half the difference of these $\mathrm{d} q^{0}$ coordinates times $\sqrt{g_{00}}$. Finally we get for the proper distance
$$\mathrm{d} L^2 = \gamma_{jk} \mathrm{d} q^j \mathrm{d} q^k=\left (\frac{g_{0j} g_{0k}}{g_{00}}-g_{jk} \right ).$$

 

[PeterDonis]

his observer measures (by definition!) the distance to an infinitesimally close point in space by sending a light signal to it and getting it reflected. The distance is then given as half the travel time of his light signal.

By "infinitesimally close point in space", I assume you mean "infinitesimally close worldline in the same family"? If so, yes, this will work as a definition of "distance". But distance alone isn't enough to define a reference frame; see below.

This physical definition can now be used to define a local spatial metric for this observer

More precisely, it picks out surfaces of "constant time" using the radar definition of simultaneity and defines a spatial metric in those surfaces (which might be different for different surfaces). It still doesn't pick out three mutually orthogonal spatial basis vectors, though; you appear to be using the spacetime metric to define the "spatial directions", but there is no guarantee that the spacetime metric is orthonormal.

 

[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 defintion 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.