I Reference frame vs coordinate chart

[cianfa72]

TL;DR Summary

Reference frame vs coordinate system (or coordinate chart) in the context of SR and GR

Hello,

here on PF I've seen many threads about the concepts of 'reference frame' and 'coordinate system'.

In the context of SR my 'envision' about the concept of 'frame of reference' is basically the 'rods & clocks latticework' as introduced in the book Spacetime physics (Taylor, Wheeler). Clocks attached on rods run at a whatsoever rate and employing a whatsoever procedure we can assign three numbers to each rod in order to identify their spatial position in the 'grid'. The only requirement for this 'assignment procedure' is to be smooth.

As far as I can tell this way we have actually defined: (1) a frame of reference -- the latticework -- and (2) a coordinate system for it (aka a coordinate chart).

Next step is to take in account the conditions for which (1) + (2) actually define an 'inertial frame'.

Do you think it could be a reasonable way to introduce both the concepts of frame of reference and coordinate system (coordinate chart) for SR flat spacetime ? Thanks.

 

[PeterDonis]

In the context of SR my 'envision' about the concept of 'frame of reference' is basically the 'rods & clocks latticework' as introduced in the book Spacetime physics (Taylor, Wheeler).

That's a useful concept, yes, but note that it is not as "flexible" as you appear to think. See next comment.

Clocks attached on rods run at a whatsoever rate and employing a whatsoever procedure we can assign three numbers to each rod in order to identify their spatial position in the 'grid'. The only requirement for this 'assignment procedure' is to be smooth.

What you are now imagining is not a "reference frame" in the Taylor, Wheeler sense; it's a coordinate chart. The whole point of having the rods and clocks is that (a) the rods are assumed to be rigid--they always have the same proper length--and (b) the clocks are assumed to be synchronized using Einstein clock synchronization. Of course this means you don't just have a "reference frame", you have an inertial reference frame.

If you relax the conditions to allow your "reference frame" to be non-inertial, what you're really doing is throwing away the original justification for imagining the rods and clocks in the first place. Clocks running at "whatever" rate are not synchronized, so you no longer have any reason to assume that there is any meaning to two different clocks reading the same time. Allowing "whatever" procedure for assigning three numbers to a spatial position means you no longer have rigid rods whose proper lengths remain the same, so you no longer have any reason to assume that there is any meaning to the two ends of some object, measured "at the same time", having a particular difference in their sets of 3 spatial numbers. In other words, your assignments of numbers no longer have any physical meaning, and the term "reference frame" as you are using it is no longer a useful way to describe them; "coordinate chart" is better.

(Note, btw, that many sources use the term "reference frame" to mean the same thing as "coordinate chart", i.e., they don't assume that a "reference frame" has any anchoring in any kind of physical realization. You appear to favor a usage in which the term "reference frame" does imply some kind of physical anchoring, which is fine, but you should be aware that much of the literature does not use the term that way.)

 

[Dale]

There are (at least) three separate things that are often meant by the term “reference frame”

1) A physical system of actual clocks and rulers
2) A coordinate chart
3) A tetrad

I prefer 3) but in practice I use 2) and 3) interchangeably.

I think that using “reference frame” to refer to 1) is a bad idea because then “changing frames” requires physically getting different clocks and you cannot just analyze the situation in different reference frames. However, @vanhees71 is strongly of the opinion that 1) is the best (only?) meaning. So my preference is not universally shared.

I think that all you can do is to pick your preference and be clear about what meaning you personally are using.

 

[martinbn]

I think it would be better if the term reference frame was not used. If you mean coordinates, say that. If you mean a tetrad, say that. If you mean a timelike congruense, say that. If you mean...

 

[cianfa72]

That's a useful concept, yes, but note that it is not as "flexible" as you appear to think. See next comment.
What you are now imagining is not a "reference frame" in the Taylor, Wheeler sense; it's a coordinate chart. The whole point of having the rods and clocks is that (a) the rods are assumed to be rigid--they always have the same proper length--and (b) the clocks are assumed to be synchronized using Einstein clock synchronization. Of course this means you don't just have a "reference frame", you have an inertial reference frame.

Just to take it simple: consider a flat spacetime with one spatial dimension alone and a "discrete latticework" built using identical rods and clocks. As "spatial position assignment procedure" let's pick the number $x$ of rods counted starting from a fixed point in the latticework (the origin).

Now because our (toy) two-dimensional spacetime is flat we know we'll be able to Einstein synchronize clocks bolted on the latticework to bring the metric into:

$$ s^2 = \alpha(x)x^2 - t^2$$ or better $$ ds^2 = \alpha(x)dx^2 - dt^2$$

That actually should mean rods proper lengths remain the same and the difference between $x$ values of the two ends of some object measured "at the same time" --as defined using Einstein clock synchronization procedure-- have the meaning of proper length provided it is "adjusted/scaled" by $\alpha(x)$ factor

 

[cianfa72]

There are (at least) three separate things that are often meant by the term “reference frame”

1) A physical system of actual clocks and rulers
2) A coordinate chart
3) A tetrad

Could you be more specific about 3) ? What basically is a 'tetrad' in this context ? Thanks

 

[Dale]

Could you be more specific about 3) ? What basically is a 'tetrad' in this context ? Thanks

A tetrad is a set of four orthonormal vector fields on some region of spacetime, one timelike and three spacelike. They are interpreted as the axes carried by a family of observers.

Tetrads do not establish a synchronization convention, so they can be used to describe a congruence of rotating observers without getting into the well known synchronization problems.

 

[martinbn]

A tetrad is a set of four orthonormal vector fields on some region of spacetime, one timelike and three spacelike. They are interpreted as the axes carried by a family of observers.

Sometimes by a tetrad people mean any four vector fields indendent at each point. For example the Newman Penrose null tetrad has four null vector fields. I am saying this just to point to the OP that the context is important.

 

[cianfa72]

A tetrad is a set of four orthonormal vector fields on some region of spacetime, one timelike and three spacelike. They are interpreted as the axes carried by a family of observers.

Tetrads do not establish a synchronization convention, so they can be used to describe a congruence of rotating observers without getting into the well known synchronization problems.

Thus orbits of the timelike vector field are basically the paths the observers themselves are following through spacetime while the three spacelike vector fields represent physically the three orthonormal axis carried by them.

 

[vanhees71]

There are (at least) three separate things that are often meant by the term “reference frame”

1) A physical system of actual clocks and rulers
2) A coordinate chart
3) A tetrad

I prefer 3) but in practice I use 2) and 3) interchangeably.

I think that using “reference frame” to refer to 1) is a bad idea because then “changing frames” requires physically getting different clocks and you cannot just analyze the situation in different reference frames. However, @vanhees71 is strongly of the opinion that 1) is the best (only?) meaning. So my preference is not universally shared.

I think that all you can do is to pick your preference and be clear about what meaning you personally are using.

I'm of the opinion that 1) is the physical definition and has to come first, because it defines physical space-time observables to begin with and 2) and 3) are mathematical descriptions thereof within a space-time model, but it's of course interrelated, because to define clocks and rulers you already need a spacetime model to define quantitatively the measures of space and time.

I thought finally we came, after a long and heated debate, to the conclusion that you need all 1)-3) to have a complete description of what a reference frame is as a physical object.

For sure 1) is not a bad idea, because without it you have an empty mathematical scheme (called a differentiable manifold for 2) or a Lorentzian manifold for 3)) but no connection to the physical world. It's also clear that "changing frames" in this physical sense indeed implies to set up the clocks and rulers in a different way (e.g., setting up new clocks and rulers that move wrt. the previous ones).

 

[Dale]

For sure 1) is not a bad idea, because without it you have an empty mathematical scheme

This is the usual false characterization of my position that got me so upset. Please cease further repetition of this false point. I do not need to use the word "reference frame" to refer to a physical collection of clocks and rulers in order to do experimental science, and using different words to refer to physical objects and mathematical constructs in no way implies that what I am doing is "an empty mathematical scheme." You should know this by now, so please stop misrepresenting my position.

 

[vanhees71]

It's my opinion, and it's just an obvious statement implicit in all physics texts, but I don't want to get in this fruitless discussion again, and there is no need to get upset only because you are of a different opinion than I.

 

[Dale]

I definitely agree with you...we can talk about spacetime from a mathematical point of view (its properties, structure and so on), however to do physics we actually need a 'connection' with the physical world.

Obviously. But that connection does not need to be called by the name "reference frame".

I think that it is OK to use the same word to refer to a physical object and a mathematical object only when there is a unique unambiguous mapping between the two. In the case of sets of physical clocks and rulers and coordinate systems or tetrads there is no unique mapping between them, so I think that separating the terminology is best. To use the same term for both risks "confusing the map with the territory".

 

[Dale]

It's my opinion, and it's just an obvious statement implicit in all physics texts, but I don't want to get in this fruitless discussion again, and there is no need to get upset only because you are of a different opinion than I.

I get upset because you repeatedly misrepresent my opinion, not because our opinions differ. It is ludicrous for you to continue to distort my actual position on the matter, and now you are spreading that falsehood to other people.

 

[vanhees71]

For me it's ludicrous that you deny to use the usual definition of a "reference frame" as a physical setup to quantify space-time positions. I cannot remember all the time that you don't want to use "reference frame" in this sense and insist to use this notion only for mathematical abstract idealizations thereof.

 

[cianfa72]

I think that it is OK to use the same word to refer to a physical object and a mathematical object only when there is a unique unambiguous mapping between the two. In the case of sets of physical clocks and rulers and coordinate systems or tetrads there is no unique mapping between them, so I think that separating the terminology is best. To use the same term for both risks "confusing the map with the territory".

So using your analogy the "map" basically is the "reference frame" whereas the "territory" is just the set of physical objects the latticework is built of.

 

[Dale]

So using your analogy the "map" basically is the "reference frame" whereas the "territory" is just the set of physical objects the latticework is built of.

Yes, exactly.

The thing is that you can analyze a set of clocks and rulers using any coordinate system. So there is not a unique link between the two. Particularly in a system like GPS where none of the physical clocks used are at rest in the coordinate system it implements. So for me, it is best to refer to the coordinate system and the collection of satellite clocks by different words. It is confusing (IMO) to call the physical GPS satellites themselves a reference frame when they are not even at rest with respect to each other.

 

[Dale]

For me it's ludicrous that you deny to use the usual definition of a "reference frame" as a physical setup

Your preferred definition is not THE usual definition. It is merely one of many usual definitions.

 

[vanhees71]

Yes, exactly.

The thing is that you can analyze a set of clocks and rulers using any coordinate system. So there is not a unique link between the two. Particularly in a system like GPS where none of the physical clocks used are at rest in the coordinate system it implements. So for me, it is best to refer to the coordinate system and the collection of satellite clocks by different words. It is confusing (IMO) to call the physical GPS satellites themselves a reference frame when they are not even at rest with respect to each other.

Well, the physical GPS satellites do not define a reference frame, and I never claimed that they do. Since for the GPS we need to use GR we have to use some locally defined reference frame (which the satellites are obviously not). It's in fact pretty complicated (which is not surprising given the necessary accuracy of the definition of the reference frame needed to make the GPS work):

http://www.nbmg.unr.edu/staff/pdfs/Blewitt_Encyclopedia_of_Geodesy.html

 

[PeterDonis]

Just to take it simple: consider a flat spacetime with one spatial dimension alone and a "discrete latticework" built using identical rods and clocks.

Yes, this forms an inertial reference frame, which can only be built in flat spacetime. In a curved spacetime it is impossible to build such a construction, even in principle.

 

[PeterDonis]

3) A tetrad

Strictly speaking, this should be called a tetrad field; a tetrad is an element of a tetrad field (i.e., a set of one timelike and three spacelike orthonormal vectors) at one particular point in the region of spacetime that the field covers.

 

[cianfa72]

Yes, this forms an inertial reference frame, which can only be built in flat spacetime. In a curved spacetime it is impossible to build such a construction, even in principle.

Thus it actually forms an inertial reference frame even though in the coordinate system (aka coordinate chart) chosen (see my post #5) the metric is not in the standard Minkowski form

$$ds^2 = dx^2 - dt^2$$

 

[PeterDonis]

the metric is not in the standard Minkowski form

First, rescaling the $x$ coordinate is just a way of confusing yourself; the metric you wrote down can be trivially transformed back to Minkowski form by undoing the rescaling.

Second, you are the one who wanted to draw a distinction between "reference frame" and "coordinate chart". So of course you should not be surprised that you can have an inertial reference frame without having to use a standard inertial coordinate chart. Which of course you can; there are plenty of other coordinate choices you can make on flat spacetime that do not trivially transform back to standard Minkowski coordinates, and you can use any of them to describe your inertial reference frame; the description won't be as simple, but it will be perfectly valid.

 

[cianfa72]

Second, you are the one who wanted to draw a distinction between "reference frame" and "coordinate chart". So of course you should not be surprised that you can have an inertial reference frame without having to use a standard inertial coordinate chart. Which of course you can; there are plenty of other coordinate choices you can make on flat spacetime that do not trivially transform back to standard Minkowski coordinates, and you can use any of them to describe your inertial reference frame; the description won't be as simple, but it will be perfectly valid.

ok, got it.

Yes, this forms an inertial reference frame, which can only be built in flat spacetime. In a curved spacetime it is impossible to build such a construction, even in principle.

I take it that if our (toy) two-dimensional spacetime was curved than we could still build the latticework of rods & clocks however this time bolted clocks at first Einstein synchronized would not keep in synch and at the end of the day it wouldn't be possibile to bring the metric in the standard Minkowski form.

 

[PeterDonis]

I take it that if our (toy) two-dimensional spacetime was curved than we could still build the latticework of rods & clocks

Not with rigid rods--they would stretch or squeeze--and not with clocks that all run at the same rate and stay synchronized.

at the end of the day it wouldn't be possibile to bring the metric in the standard Minkowski form.

Obviously this is impossible for any curved spacetime.

 

[cianfa72]

Not with rigid rods--they would stretch or squeeze--and not with clocks that all run at the same rate and stay synchronized.

I've some "problem/misunderstanding" with this kind of statements. What do you mean with rigid rods ? Respect to what they would stretch or squeeze ?

 

[PeterDonis]

What do you mean with rigid rods ?

Rods whose proper length does not change.

Respect to what they would stretch or squeeze ?

Their proper length would change. You could test this by putting a mirror on one end of the rod and a light source with a clock on the other, and measuring the round-trip light travel time. If the round-trip travel time changes, the rod is being stretched or squeezed.

 

[cianfa72]

Their proper length would change. You could test this by putting a mirror on one end of the rod and a light source with a clock on the other, and measuring the round-trip light travel time. If the round-trip travel time changes, the rod is being stretched or squeezed.

If we restrict to a static (or stationary ?) two dimensional curved spacetime that does mean that at a given place the round-trip travel time will stay unchanged even though its value would change from place to place along the unidimensional latticework.

 

[PeterDonis]

If we restrict to a static (or stationary ?) two dimensional curved spacetime that does mean that at a given place the round-trip travel time will stay unchanged even though its value would change from place to place along the unidimensional latticework.

With an appropriate definition of "place", yes, this will be the case for a stationary spacetime.

 

[cianfa72]

With an appropriate definition of "place", yes, this will be the case for a stationary spacetime.

"place" as simple as the position of the rod having the fixed spatial coordinate value assigned to it (see post #5)

 

[PeterDonis]

"place" as simple as the position of the rod having the fixed spatial coordinate value assigned to it (see post #5)

With an appropriate choice of coordinates, yes.

 

[cianfa72]

With an appropriate choice of coordinates, yes.

Maybe I'm missing your point...

 

[PeterDonis]

Maybe I'm missing your point...

How so?

 

[Dale]

Maybe I'm missing your point...

The appropriate choice of coordinates is one where one coordinate basis vector matches the timelike killing vector.

 

[cianfa72]

With an appropriate choice of coordinates, yes.

Based on what @Dale said in post #34 my understanding is as follow:
take our two-dimensional curved stationary spacetime and a one-dimensional latticework built of rods & clocks. Since there exist a timelike killing vector field then we'll be able to assign spatial coordinates to the rods and "adjust" clocks rates in order to put the metric in the form -- note that it does not depend on coordinate time $t$
$$ ds^2 = \alpha(x)dx^2 - \beta(x)dt^2$$
Basically what we have done is "select/define" a "special" global coordinate chart for our toy spacetime. However since it (the toy spacetime) is not flat it will be impossible to globally put the metric in the standard Minkowski form of SR.

 

[DrGreg]

$$ ds^2 = \alpha(x)dx^2 - \beta(x)dt^2$$

This is correct for a static spacetime, but for a stationary spacetime it will be
$$ ds^2 = \alpha(x) \, dx^2 - \beta(x) \, dt^2 +\delta(x) \, dx \, dt $$
which is a non-orthogonal coordinate system. This is related to the issue Dale mentioned earlier:

Tetrads do not establish a synchronization convention, so they can be used to describe a congruence of rotating observers without getting into the well known synchronization problems.

Aside: In my mind I associate "stationary but not static" with spatial rotation. But you can't have spatial rotation in a two-dimensional spacetime, which begs the question, is it possible for a two-dimensional spacetime to be stationary but not static?

 

[martinbn]

This is correct for a static spacetime, but for a stationary spacetime it will be
$$ ds^2 = \alpha(x) \, dx^2 - \beta(x) \, dt^2 +\delta(x) \, dx \, dt $$
which is a non-orthogonal coordinate system. This is related to the issue Dale mentioned earlier:Aside: In my mind I associate "stationary but not static" with spatial rotation. But you can't have spatial rotation in a two-dimensional spacetime, which begs the question, is it possible for a two-dimensional spacetime to be stationary but not static?

For the given metric you can make the change of variables, where $x$ is the same and $t=T+f(x)$ with the choice for $f(x)$ so that there are no mixed terms. If I did it right, $f'(x)=\frac{\delta}{2\beta}$ works.

Or you can think of it this way. If $\omega$ is the one form that gives the orthogonal distribution to the Killing field. Frobenious says that the Killinf field is hypersurface orthogonal if $\omega\wedge d\omega=0$, but a three form on a two dimensional manifold is always zero.

 

[cianfa72]

For the given metric you can make the change of variables, where $x$ is the same and $t=T+f(x)$ with the choice for $f(x)$ so that there are no mixed terms. If I did it right, $f'(x)=\frac{\delta}{2\beta}$ works.

For our (toy) two-dimensional stationary curved spacetime that basically means we should be able to assign spatial coordinate $x$ to latticework rods and "adjust" the rate of clocks bolted on it to put the metric in the form

$$ ds^2 = \alpha(x) \, dx^2 - \beta(x) \, dt^2$$

the proper length of the rods each at spatial position $x$ will stay unchanged in coordinate time $t$.

 

[cianfa72]

Reading the Landau book "The classic theory of fields" at the end of section 82 we find his definition of 'system of reference' in GR:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

What do you think about ? It could be a reasonable definition of what we meant with 'system of reference' ?

 

[martinbn]

Reading the Landau book "The classic theory of fields" at the end of section 82 we find his definition of 'system of reference' in GR:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

What do you think about ? It could be a reasonable definition of what we meant with 'system of reference' ?

Yes, it is a reasonable definition.

 

[pervect]

A reference from an old thread.

From Manoff, "Frames of reference in spaces with an affine connections and metrics" , https://arxiv.org/abs/gr-qc/9908061

There are at least three types of methods for defining a FR [frame of reference]. 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).

Definition c above falls in the category of the time-like congruence that I've mentioned - oh, a few times now.

Manoff's paper is a pretty hard read, but I found the summary of different types of things "a frame of reference" could mean very abstract, but useful.

Examples of frames of reference on the surface of the earth, considered as a 3d manifold, 2 spatial and one time.

1) Lattitude, longitude, and a coordinate time such as TAI. These is the coordinate chart defintion. The coordinates tell you where you are on the Earth.

2) Defining a set of unit basis vectors (north, east, and time) at every point on the Earth's surface. This is an example of the second definition. Unlike the first definition, it doesn't directly tell you where you are located, but it allows you to specify vectors at any point by the components of the frame-field. The method is still very useful for doing physics, it's commonly called "frame fields".

3) A set of "material points" at rest on the Earth's surface. This is definition c. The wordlines of the material points define a time-like congruence of worldlines.

I

 

[cianfa72]

Or you can think of it this way. If $\omega$ is the one form that gives the orthogonal distribution to the Killing field. Frobenious says that the Killing field is hypersurface orthogonal if $\omega\wedge d\omega=0$, but a three form on a two dimensional manifold is always zero.

Sorry to resume this topic, just to check my understanding: $\omega$ is one-form hence $d\omega$ is a 2-form then the wedge product $\omega \wedge d\omega$ is a 3-form. Then a whatever 3-form on a two dimensional manifold is always zero.

Right ? Thanks all.

 

[martinbn]

Sorry to resume this topic, just to check my understanding: $\omega$ is one-form hence $d\omega$ is a 2-form then the wedge product $\omega \wedge d\omega$ is a 3-form. Then a whatever 3-form on a two dimensional manifold is always zero.

Right ? Thanks all.

Right.

 

[pervect]

Could you be more specific about 3) ? What basically is a 'tetrad' in this context ? Thanks

The following is more or less my interpretation. I can't point to a specific reference that says exactly what I am about to say, but I think it may be helpful.

I think of a tetrad as living in (being an element of) some particular tangent space on a manifold. Every point (event) on the manifold has its own, distinct tangent space.

Near some particular point (event) of interest, the tangent space and the manifold itself can be related to each other through the exponential map, but they are not the same concept.

The basic issue is that vectors, by definition, must add commutatively. This is true in the tangent space, and it's why vectors "live" in the tangent space rather than in the manifold.

Physical clocks "live" (are elements of) physical space, which is represented mathematically by the manifold. Philosophically, physical space and the manifold are two different concepts as well, as the map is not the territory. However, I often conflate the two in my thinking, it usually doesn't cause any problems for me.

When we consider tetrads as a reference frame, they are slightly different than considering physical clocks and rods, because the tetrads exist in the tangent space rather than the physical space.

Many aspects of my thinking are strongly influenced by Misner's "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043

A couple of relevant quotes.

Instead one divides the theoretical landscape into two categories.
One category is the mathematical/conceptual model of whatever is happen-
ing that merits our attention. The other category is measuring instruments
and the data tables they provide.

What is the conceptual model? It is built from Einstein’s General Rel-
ativity which asserts that spacetime 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?] But one can draw
a spacetime map and give unambiguous interpretations. [On a Mercator
projection of the Earth, one minute of latitude is one nautical mile every-
where, but the distance between minute tics varies over the map and must
be taken into account when reading off both NS and EW distances.] There
is no single best way to draw the spacetime map, but unambiguous choices
can be made and communicated, as with the Mercator choice for describing
the Earth.

 

[PeterDonis]

I think of a tetrad as living in (being an element of) some particular tangent space on a manifold.

More precisely, it's a set of four such elements, since the elements of the tangent space are tangent vectors and a tetrad is a set of four tangent vectors, one timelike and three spacelike, that are all mutually orthogonal.

 

[vanhees71]

Yes, and observables are local observables and defined via the local inertial frames at the location of the observation.

 

[martinbn]

More precisely, it's a set of four such elements, since the elements of the tangent space are tangent vectors and a tetrad is a set of four tangent vectors, one timelike and three spacelike, that are all mutually orthogonal.

Sometimes a tetrad can be something more general without the restriction of one timelike and three spacelike all orthogonal. For example a null tetrad.

 

[Dale]

Sometimes a tetrad can be something more general without the restriction of one timelike and three spacelike all orthogonal. For example a null tetrad.

I have always seen the restricted definition of one timelike, three spacelike and orthonormal.

 

[martinbn]

I have always seen the restricted definition of one timelike, three spacelike and orthonormal.

In the Newman-Penrose formalism for example null tetrads are used.

https://en.wikipedia.org/wiki/Newman–Penrose_formalism

 

[cianfa72]

I'm aware of the discussion about the possible interpretations for the term 'reference frame' - see for instance What is an inertial frame? A conflict of two definitions

As pointed out by @Dale a reference frame is actually the mathematical object 'map' that assign coordinates to the physical world (to summarize in one sentence: reference frame like a map is not the territory).

Said that, we can define such a map (aka reference frame) starting from a set of physical bodies -- note that there is no one-to-one correspondence between the set of physical bodies chosen and the maps (reference frames).

Thus the reference frame chosen could be one in which the set of given physical objects are not actually at rest (i.e. their worldlines have non constant spatial coordinates in that reference frame).

Now I was thinking how to build such frame as follows:

1. assign fixed spatial coordinates to each of the given set of physical bodies plus a coordinate time $t$ to label events along their worldlines; starting from this reference frame (in which by construction those bodies are at rest w.r.t it) make a mathematical transformation mixing spatial and time coordinates to get a new reference frame in which those bodies are not at rest anymore
2. for each body in a given set of bodies assign non constant values of spatial coordinates plus a coordinate time $t$ to label events along their worldlines

What do you think, is it a reasonable way to build a such frame ?