Reference frame vs coordinate chart

[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.

 

[cianfa72]

Yes, it is a reasonable definition.

My personal impression is that this definition actually relies on physical "grounds" since physical objects -- namely a continuum of physical bodies and clocks (better wristwatches !) -- are involved in the definition itself.

 

[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 ?

 

[vanhees71]

I hope this will not end again in a battle about words.

From a formal point of view, I'd say a family of local reference frames in GR is defined by a coordinate chart together with a tetrad field. Local inertial frames across this chart are then further defined by a tetrad field parallel transported from an arbitrary point within the chart along arbitrary time-like geodesics (a time-like congruence) covering (at least a part of) the chart.

A nice book elucidating this from both the theoretical and observational point of view is

https://www.amazon.de/dp/B08BWQGXH6/

 

[cianfa72]

Any other point of view about the claims in post #51 ? Thanks all.

 

[ergospherical]

don’t poke this sleeping bear again babes

 

[Dale]

The bear is too busy this week.

 

[cianfa72]

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.

Sorry to go back to this topic: suppose you have 'labeled' each clock in the set with different spatial coordinate values and employed as the coordinate time the proper time measured by each of them (it is in fact a timelike coordinate). So far so good.

Then, as you said, suppose we want to analyze the same set of physical clocks using another coordinate chart. As an example of it we could re-label the spatial coordinate values assigned to each of them and attach to each of them a wristwatch with an arbitrary time rate.

This way we have defined a new coordinate chart in which physical clocks are at rest as well.

Does it make sense ? Thank you.

 

[Dale]

Sorry to go back to this topic: suppose you have 'labeled' each clock in the set with different spatial coordinate values and employed as the coordinate time the proper time measured by each of them (it is in fact a timelike coordinate). So far so good.

Then, as you said, suppose we want to analyze the same set of physical clocks using another coordinate chart. As an example of it we could re-label the spatial coordinates assigned to each of them and attach to each of them a wristwatch with an arbitrary time rate.

This way we have defined a new coordinate chart in which physical cloks are at rest as well.

Does it make sense ? Thank you.

Yes, that makes sense to me.

 

[cianfa72]

So the claim "take the rest frame of such and such body" is actually not uniquely defined. Namely there exist infinite rest reference frames defined up to re-labeling of their spatial coordinates and re-definition of timelike coordinate.

 

[Dale]

So the claim "take the rest frame of such and such body" is actually not uniquely defined. Namely there exist infinite rest reference frames defined up to re-labeling of their spatial coordinates and re-definition of timelike coordinate.

That is true. However, if such and such body is inertial then there is a standard convention that is implied and understood.

 

[cianfa72]

That is true. However, if such and such body is inertial then there is a standard convention that is implied and understood.

Which is that implied standard convention ?

 

[Dale]

Which is that implied standard convention ?

The standard inertial frame with Einstein synchronization convention where such and such body is at rest.

 

[cianfa72]

The standard inertial frame with Einstein synchronization convention where such and such body is at rest.

So that does mean: take a family of free bodies (zero proper acceleration) at rest w.r.t. the given (free/inertial) such and such body in a region surrounding it (just to fix ideas we can imagine a wristwatch attached to each of them).

Note that 'at rest' does actually mean that the round-trip time of 2-way light signals exchanged between those bodies does not change. Then, as pointed out in a recent PF thread, we can consistenly apply the Einstein synchronization convention to synchronize such wristwatches (the resulting one-way speed of light in the frame being defined is the universal constant value c).

Label every wristwatch (or body) with fixed different spatial coordinate values and take the proper time of each of them as the coordinate time of the frame being defined.

The map defined as above is the standard inertial frame you were talking about. Btw we're aware of we can do that just in a limited spacetime region in the context of GR. In flat spacetime instead (SR) there is no limit in principle to extend such standard inertial frame to the entire spacetime.

Make sense ? Thank you.

 

[Dale]

Make sense ? Thank you

Yes, that makes sense.