How these notions relate to the usual SR approach?

[leo.]

In the context of General Relativity spacetime is a four-dimensional Lorentzian manifold $M$ with metric tensor $g$, its Levi-Civita connection $\nabla$ and a time orientation vector field $T \in \Gamma(TM)$.

In this context I've seem the following three definitions:

1. A coordinate system is a chart. Namely a pair $(U,\phi)$ where $\phi : M\to \mathbb{R}^4$ assigns coordinates to events. In that case $\phi$ gives rise to the so called coordinate functions $\phi^\mu$ and the coordinates of $x\in M$ are $\phi^\mu(x)$.
2. A reference frame is a set of four vector fields $e_\alpha \in \Gamma(TM)$ which form a basis at each tangent space. The reference is said orthonormal if the four vector fields are orthonormal with respect to the metric tensor $g$. In other words: $g(e_\alpha,e_\beta)=\eta_{\alpha \beta}$.
3. An observer is a pair $(\gamma,e_\mu)$ where $\gamma : \mathbb{R}\to M$ is a future-pointing timelike path and where $e_\mu : \mathbb{R}\to TM$ are four vector fields along $\gamma$, that is $e_\mu(\tau)\in T_{\gamma(\tau)}M$ such that (i) $\gamma'(\tau)=e_0(\tau)$ and (ii) $g_{\gamma(\tau)}(e_\alpha(\tau),e_\beta(\tau))=\eta_{\alpha\beta}$.

In that sense, in this approach coordinate systems, reference frames and observers are three different and completely independently defined concepts.

On the other hand, in the most basic approaches to Special Relativity, the distinction between these three concepts seems blurry. Textbooks seems to treat observers, coordinate systems and reference frames as all the same thing. It is quite common to see that "boost" example considering the relative motion of observers, thinking of axes being what is actually moving and end up relating coordinate systems - it is just all mixed up. To make this point clear, I quote Schutz:

It is important to realize that an 'observer' is in fact a huge information-gathering system, not simply one man with binoculars. In fact we shall remove the human element entirely from our definition, and say that an inertial observer is simply a coordinate system for spacetime, which makes an observation simply by recording the location $(x,y,z)$ and time $(t)$ of any event.

And it does make a difference. If we follow these three definitions from GR, we can use one observer to assign components to tensors located only at events on the observer's worldline. On the other hand in the traditional approach one observer may assign coordinates to events anywhere on spacetime. Furthermore, it is not even clear in this approach I presented how an observer would assign coordinates to anything by the way (he carries a basis of tangent space, not a chart).

What I want to know here is: first of all are these three definitions I've posted above standard among physicists in the context of General Relativity? If they are standard how do they relate to the standard Special Relativity of moving axes that allows one to register coordinates of events? I really can't bridge these two things, and the Special Relativity approach really should be a special case of the General Relativity approach.

I believe the main thing I'm missing is: in the SR approach, since an observer and a coordinate system are all the same, an observer can register coordinates of any events. In the GR approach I presented, an observer cannot give coordinates to events. Indeed an observer only knows of events on his wordline. This is quite different from the SR approach.

Anyway, how these approach actually relate? How can we bridge the gap between them so that the SR approach is really a special case?

 

[PeterDonis]

and a time orientation vector field $T \in \Gamma(TM)$.

Note that, strictly speaking, a spacetime in GR does not have to have such a vector field. But the spacetimes that are considered physically reasonable all do have one.

Also, what is $\Gamma(TM)$?

A coordinate system is a chart. Namely a pair $(U,\phi)$ where $\phi : M\to \mathbb{R}^4$ assigns coordinates to events

What is $U$ here?

A reference frame is a set of four vector fields$e_\alpha \in \Gamma(TM)$ which form a basis at each tangent space.

A basis of what?

In that sense, in this approach coordinate systems, reference frames and observers are three different and completely independently defined concepts.

Yes.

in the most basic approaches to Special Relativity, the distinction between these three concepts seems blurry. Textbooks seems to treat observers, coordinate systems and reference frames as all the same thing

Yes. That's a matter of definitions of words, not physics. All three concepts you describe above can be defined in SR, but since spacetime is flat, you can construct obvious isomorphisms between all three concepts, which makes it easy to conflate them.

If we follow these three definitions from GR, we can use one observer to assign components to tensors located only at events on the observer's worldline.

Yes, because of the way "observer" is defined in this approach.

On the other hand in the traditional approach one observer may assign coordinates to events anywhere on spacetime.

Yes, because of the different way "observer" is defined in this approach.

it is not even clear in this approach I presented how an observer would assign coordinates to anything by the way (he carries a basis of tangent space, not a chart)

But if the observer (in the GR sense--a single worldline) is inertial, which is usually assumed in this context, there is an obvious coordinate chart in which he is at rest and his set of four vectors at every event are the coordinate basis vectors at that event. Furthermore, there will be an infinite family of similar observers covering the entire spacetime for which the same is true. Taking all of the observers together, we therefore have an obvious global reference frame and coordinate chart, and an obvious isomorphism between both of these and the family of observers. So, as I said above, it is easy to conflate the three concepts in this specific case.

are these three definitions I've posted above standard among physicists in the context of General Relativity?

Fairly standard, yes, although not all treatments will be careful to distinguish all three concepts.

how do they relate to the standard Special Relativity of moving axes that allows one to register coordinates of events?

I'm not sure what you mean by "moving axes". But for how to relate the two approaches, see above.

 

[leo.]

@PeterDonis concerning the initial questions: in my writing $\Gamma(TM)$ is the set of smooth sections of the tangent bundle, that is, the set of vector fields. As for the chart I really wasn't careful when writing. In what I wrote a chart is a pair $(U,\phi)$ with $U\subset M$ open and $\phi : U\to \mathbb{R}^4$ a homeomorphism which assigns coordinates to events.

As for the basis, I mean a basis of tangent space. I mean in this definition a reference frame on $U\subset M$ are four vector fields $e_\alpha \in \Gamma(TM)$ such that if $x\in M$ then $e_\alpha(x)$ form a basis of the tangent space $T_x M$.

But if the observer is inertial, which is usually assumed in this context, there is an obvious coordinate chart in which he is at rest and his set of four vectors at every event are the coordinate basis vectors at that event.

Are you talking about the Fermi-Walker coordinates built with the exponential map? I've heard about them, although not much yet. In that case this would be they key construction to build the gap between the two approaches?

Also how would we define inertial observer in this context? Would it be just an observer whose worldline $\gamma$ is a geodesic?

I'm not sure what you mean by "moving axes". But for how to relate the two approaches, see above.

I mean that traditional example where we have one observer represented as a set of axes and another observer represented by another set of axes moving with respect to the first along some axes. When one considers this picture one usually relates "coordinates" measures by each of them, which doesn't seem possible in this more general context, since observers can only deal with events on their worldlines. But if I understood your point, what actually is being done is just the description of the motion of one observer in the Fermi-Walker coordinates of the other. Is this the right way to bridge the gap?

 

[PeterDonis]

Are you talking about the Fermi-Walker coordinates built with the exponential map?

No, I'm talking about the standard inertial coordinates $t, x, y, z$ in flat spacetime, chosen such that the inertial observer in question is at constant $x, y, z$ for all $t$.

how would we define inertial observer in this context? Would it be just an observer whose worldline $\gamma$ is a geodesic?

That's the standard definition of an inertial observer, yes.

I mean that traditional example where we have one observer represented as a set of axes and another observer represented by another set of axes moving with respect to the first along some axes.

Ah, ok. That's not what I was trying to describe; see below.

When one considers this picture one usually relates "coordinates" measures by each of them, which doesn't seem possible in this more general context, since observers can only deal with events on their worldlines.

Not if "observer" is defined the way SR texts usually define it. A better term using the terminology from GR that you are describing might be "family of observers". See below.

what actually is being done is just the description of the motion of one observer in the Fermi-Walker coordinates of the other.

No. I was describing an infinite family of observers, all of whom are inertial, and all of whom are at rest relative to each other. In the global inertial coordinate chart I described above, all observers would be at constant $x, y, z$ for all $t$, and each unique triple $x, y, z$ would correspond to a different observer. So we have a family of observers (in the GR sense) that fills the entire spacetime, and each observer's set of four vectors at each event is the same as the basis vectors of the global inertial coordinate chart at that event. And it should be obvious that, taken all together, the basis vectors of all the observers form a reference frame (in the GR sense) that covers the entire spacetime. That is how you bridge the gap.

 

[Orodruin]

I am not sure I would call the concepts completely independent. Given a coordinate system you will automatically be associated with a reference frame (albeit a non-orthonormal one) on the corresponding chart. Given a reference frame at a single event, you can construct local coordinates by the exponentiation map based and the parallel transported frame along these exponential maps will be the corresponding frame in the entire chart.

Now, for an inertial observer in SR, all of these things will blur together because the reference frame defined by Minkowski coordinates is orthonormal and you get exactly Minkowski coordinates if you exponentiate an orthonormal frame at a given event. Of course, you can do SR in curvilinear coordinates but this is usually not the case in most introductory SR textbooks.

 

[vanhees71]

I would say the Minkowski space is an affine pseudoeuclidean space, and as such a (pretty) special case of a pseudo-Riemannian manifold.

 

[PeterDonis]

Given a coordinate system you will automatically be associated with a reference frame (albeit a non-orthonormal one) on the corresponding chart

Only if your "reference frame" does not have to have one timelike and three spacelike vectors at every event, since the basis vectors of a coordinate chart do not have to meet that requirement. The usual definition, at least as I understand it, requires that.

Given a reference frame at a single event, you can construct local coordinates by the exponentiation map based and the parallel transported frame along these exponential maps will be the corresponding frame in the entire chart

This won't work in the fully general case, since the coordinate curves constructed by this method, in a general curved spacetime, might cross, so they won't be a valid chart for the entire spacetime. Of course it will work in a sufficiently small neighborhood of the chosen event.

 

[Orodruin]

Only if your "reference frame" does not have to have one timelike and three spacelike vectors at every event, since the basis vectors of a coordinate chart do not have to meet that requirement. The usual definition, at least as I understand it, requires that.

I was going off the definition in the OP:

A reference frame is a set of four vector fields eα∈Γ(TM)eα∈Γ(TM)e_\alpha \in \Gamma(TM) which form a basis at each tangent space. The reference is said orthonormal if the four vector fields are orthonormal with respect to the metric tensor ggg. In other words: g(eα,eβ)=ηαβg(eα,eβ)=ηαβg(e_\alpha,e_\beta)=\eta_{\alpha \beta}.

This does not explicitly prohibit reference frames with a number of timelike directions different from 1, just that it is said to be orthonormal if it is orthonormal everywhere. Of course I agree that if you add the requirement of having one timelike direction, it excludes some coordinate bases.

This won't work in the fully general case, since the coordinate curves constructed by this method, in a general curved spacetime, might cross, so they won't be a valid chart for the entire spacetime.

Hence "local coordinates" instead of "global coordinates" in #5.

 

[Dale]

n that sense, in this approach coordinate systems, reference frames and observers are three different and completely independently defined concepts

Yes, this does seem to be a surprise to most people, including myself. As @PeterDonis mentioned already, it arises just because in SR there is a "natural" or at least a "well established conventional" way to map between the three.

In principle it would be nice to introduce the concepts separately from the beginning, but there is a considerable amount of pedagogical inertia involved. Particularly with the common "thought experiment" approach of presenting SR.

 

[PeterDonis]

I was going off the definition in the OP

Yes, but I'm not sure that definition matches how "reference frame" is actually used in the GR literature. (I'm not even sure that specific term is used very much; the term I'm more used to seeing is "frame field", which does, as I have seen it defined, explicitly require orthonormal vectors.)

 

[leo.]

No, I'm talking about the standard inertial coordinates $t,x,y,z$ in flat spacetime, chosen such that the inertial observer in question is at constant $x,y,z$ for all $t$.

But how does one actually build this coordinate system? I mean, given spacetime $M$ and a timelike future-pointing worldline $\gamma : \mathbb{R}\to M$ which carries a reference frame along it, it is not at all obvious to me how does one construct a chart $(U,\phi)$ such that $\phi(\gamma(\tau))=(\tau, x_0,y_0,z_0)$. I mean, obviously $\gamma$ has to be a coordinate line of $t$ constant, and also it is a geodesic, but that's all we know.

Actually thinking about this physically, the answer should not be unique. There could be lots of coordinate systems adapted to the observer. The only question I pose is: how do we actually construct these coordinate systems? After all in practice we would need to do so. Or I'm not really getting the point here about this coordinate system?

 

[PeterDonis]

how does one actually build this coordinate system?

Pick a geodesic worldline $\gamma$. Proper time $\tau$ along that worldline defines the $t$ coordinate on the worldline. Label all points on the worldline with $x = y = z = 0$. In other words, this worldline is the $t$ axis.

At the event on the worldline $\gamma$ at which $\tau = 0$ (which will also have the coordinate $t = 0$), use the three spacelike frame vectors (assumed to be orthonormal) to pick out three spacelike geodesics (a vector at a given event uniquely determines a geodesic through that event). Label these geodesics as the $x$, $y$, and $z$ axes, and use distance along them to label all points on the axes with appropriate $x$ or $y$ or $z$ coordinates. All points on the axes have $t = 0$.

Given the axes, it should be straightforward to see how to extend the coordinates to every event in the spacetime. But there are two key observations that you should also be able to convince yourself of:

(1) In the hyperplane $t = 0$, at each point $x, y, z$, there will be a timelike geodesic that passes through that point and is parallel to the $t$ axis $\gamma$. This is the worldline of an inertial observer at rest relative to our original observer.

(2) At every event on any such observer's worldline, his reference frame vectors are also the coordinate basis vectors at that event, in the coordinates we have constructed.

 

[Dale]

But how does one actually build this coordinate system? I mean, given spacetime

In addition to what @PeterDonis mentioned, since the spacetime is flat you can also parallel transport the observer's local frame to any other event to get a local frame at that event, and the result is path-independent. You can use that to decompose any such displacement into a path composed of a series of displacements only along the frame vectors.

 

[Orodruin]

In addition to what @PeterDonis mentioned, since the spacetime is flat you can also parallel transport the observer's local frame to any other event to get a local frame at that event, and the result is path-independent. You can use that to decompose any such displacement into a path composed of a series of displacements only along the frame vectors.

Note that with the definition of "observer" in the OP, this might be dependent on which frame on the observer worldline you chose to parallel transport as nothing in that definition forbids rotating frames. (In other words, the frame itself might not be parallel transported along the worldline - of course, you could add that as a requirement.)

 

[vanhees71]

This leads to the idea of Fermi-Walker transport and, among other things, an elegant derivation of the Thomas precession.

 

[Dale]

this might be dependent on which frame on the observer worldline you chose to parallel transport as nothing in that definition forbids rotating frames

Yes, good point. I should have specified "local frame at any given event" and then we would construct the momentarily comoving inertial frame.

 

[leo.]

@PeterDonis, I believe this construction is actually what I saw other people calling "Fermi-Walker" coordinates. One first defines the exponential map at the point $p\in M$ to be $\exp_p : T_p M \to M$ by setting $\exp_p(v) = \gamma_p(1)$ where $\gamma_p$ is the geodesic with $\gamma(0)=p$ and $\gamma'(0)=v$. In that setting, given the observer's frame $e_\mu$, one defines $\phi : \mathbb{R}^4\to M$ by setting

$$\phi(x^0,x^1,x^2,x^3)= \exp_{\gamma(x^0)}\left( x^i e_i \right)$$

So given the coordinates $x^0,x^1,x^2,x^3$ to find the matching event, we pick the event $\gamma(x^0)$, build the "spatial vector" $x^i e_i$ and follow the geodesic in its direction. If I'm not wrong, this is exactly the construction you outlined right?

Now, all of this gives a quite nice and well defined notions of observers and coordinate systems and their relations: observers are worldlines carrying orthonormal basis, coordinate systems are charts. We can use charts to describe the motion of observers and for each inertial observer there are charts on which they are at rest. Of course, this also allows for a precise description of inertial observers as those whose worldlines are geodesics.

On the other hand we still have frames and there's something I didn't undesrtand yet.

Note that with the definition of "observer" in the OP, this might be dependent on which frame on the observer worldline you chose to parallel transport as nothing in that definition forbids rotating frames. (In other words, the frame itself might not be parallel transported along the worldline - of course, you could add that as a requirement.)

Yes, good point. I should have specified "local frame at any given event" and then we would construct the momentarily comoving inertial frame.

I believe by these comments, that you are saying that a reference frame in general relativity is really local in the sense I described: it exists only along the worldline of an observer. Is that right?

This still makes me confused with the idea of reference frame from traditional Special Relativity. I mean how can one use a reference frame if it is only defined along the worldline of an observer? Sometimes people talk about "analyse something in a particular frame", but how could one do this if the frame is defined just on a worldline? For example: if we consider a particle moving and we want to talk about its four momentum, if the particle's worldline doesn't coincide with the observer's worldline the frame can't be used at all.

On the other hand the "frame fields" defined on spacetime, seems to be disconnected from the notion of observers and coordinate systems, since coordinate basis need not be orthonormal.

Finally today I've read something about another definition that basically says that a reference frame is just a timelike future-pointing vector field not needing the other three.

What is actually the standard definition of a reference frame? And if a frame is tied to the worldline of an observer how is it used, since obviously one needs to expand tensors outside the observer's worldline? How these "local frames" from GR relate to the traditional idea of "frame of an observer" from SR which are considered the axes extending over the whole spacetime?

 

[PeterDonis]

I believe this construction is actually what I saw other people calling "Fermi-Walker" coordinates.

The term I've seen is "Fermi Normal Coordinates" (or sometimes just "Fermi Coordinates"). There is also a term "Fermi-Walker transport", which is a generalization of parallel transport that applies to non-geodesic worldlines, which we aren't considering here. However, Fermi Normal Coordinates as a concept are also much more general than we need to consider here, because they can be used to construct a chart in a neighborhood of any worldline (geodesic or not) in any spacetime (curved as well as flat). We are talking here about the very special case of flat spacetime and a geodesic worldline, in which the construction is much simpler. Also, only in this very special case will we have the global threefold correspondence between a coordinate chart, a reference frame, and a family of observers.

I'm also not sure I understand the notation you are using. Here is how I would describe the construction, using somewhat more mathematical notation than I did before. Remember we are working in flat Minkowski spacetime.

(1) Pick an event. Label this event with coordinates $(t, x, y, z) = (0, 0, 0, 0)$.

(2) Pick a timelike geodesic passing through this event. Parameterize the geodesic by proper time $\tau$ such that $\tau = 0$ at the event we chose in step 1. Label all events on the geodesic with coordinates $(t, x, y, z) = (\tau, 0, 0, 0)$.

(3) Pick a set of three orthonormal spacelike vectors at the event we chose in step 1. These will determine three mutually orthogonal spacelike geodesics. Parameterize the three geodesics by proper distance, labeling the three parameters as $x$, $y$, and $z$. Label all events on these geodesics, respectively, with coordinates $(0, x, 0, 0)$, $(0, 0, y, 0)$, and $(0, 0, 0, z)$.

(4) We now have a labeling of all events in the $t = 0$ hyperplane. For each event in that hyperplane, which we have labeled with coordinates $(0, x, y, z)$, find the unique timelike geodesic through that event that is parallel to the geodesic we picked in step 2. Parameterize each such geodesic by proper time as we did for the geodesic in step 2. Then label all events on each geodesic with coordinates $(t, x, y, z)$ with $t$ being given by the proper time $\tau$ and $x, y, z$ being given by their values at the event in the $t = 0$ hyperplane.

This completes the construction of the coordinate chart for the entire spacetime. Now we must show that we have also constructed (a) a reference frame, and (b) a family of observers. Here are the steps:

(1a) At the event we picked in step 1 above, the unit tangent vector to the timelike geodesic we picked in step 2 above (i.e., its 4-velocity) is the timelike vector of the reference frame at that event. The three spacelike vectors we picked in step 3 above are the spacelike vectors of the frame at that event. It should be obvious by inspection that these four vectors are the coordinate basis vectors at this event, in the chart we constructed above.

(2a) From the coordinate chart construction above, we have a piecewise geodesic path from the event we picked in step 1 to every event in the spacetime (just go, in sequence, along the $t$, $x$, $y$, and $z$ axes a length equal to the coordinates of the event). By parallel transporting the vectors from step 1a above, we construct a set of 4 orthonormal vectors at every event; this forms a reference frame covering the entire spacetime. And it should again be obvious by inspection that, at every event, the 4 vectors so constructed are in fact the coordinate basis vectors at that event, in the chart we constructed above.

(1b) The timelike geodesics we picked in steps 2 and 4 above form a family of timelike geodesics which are non-intersecting and which fill the entire spacetime (i.e., every event in the spacetime lies on one and only one such geodesic). And the reference frame vectors at each event on each geodesic will satisfy all the properties required for the geodesic and those vectors to define an observer (as you originally defined it).

This completes the entire construction.

 

[PeterDonis]

What is actually the standard definition of a reference frame?

There isn't one; the term is used in many different ways. For this discussion, I have been using the definition you gave in the OP (and similarly for "coordinate chart" and "observer"--the latter term is also used in many different ways and doesn't have a standard definition). More precisely, I've been using your OP definition plus the requirement for the vectors to be orthonormal.