How this definition of a reference frame is used?

[leo.]

In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field $Z$. In that sense a reference frame is a collection of observers, since its integral lines are all observers according to this defintiion. This definition is also highly employed by the brazilian physicist Waldyr Alves Rodrigues Jr in his publications.

I particularly like this definition from a mathematical standpoint, because it is extremely simple and can even be intuitive - usually we really consider intuitively a reference frame as a collection of observers at rest with respect to each other.

Following this definition one defines a naturaly adapted coordinate system $x^\mu$ to a reference frame $Z$ to be a chart on spacetime $M$ such that $\frac{\partial}{\partial x^0}$ is timelike, $\frac{\partial}{\partial x^i}$ is spacelike and the spacelike components of $Z$ with respect to this basis are zero.

In basic treatments of Special and General Relaitivity, one usually needs to resolve physical quantities relative to reference frames, and relate different reference frames, in order to convert measurements.

My question really becomes: how these definitions gets used in practice in order to (1) express physical quantities with respect to a reference frame and (2) convert results from different reference frames?

 

[PeterDonis]

how these definitions gets used in practice in order to (1) express physical quantities with respect to a reference frame and (2) convert results from different reference frames?

Does the book you are reading give any examples?

 

[leo.]

Does the book you are reading give any examples?

Not much examples really. After the definition it immediately gets to classify reference frames according to synchronizability. Actually in one exercise he defines a reference frame and asks to (i) show its a reference frame and (ii) show it is not synchronizable according to his definition. The said reference frame is defined in Minkowski spacetime and is: $$Q=\dfrac{1}{\sqrt{1-x^2}}\left(\dfrac{\partial}{\partial t} + x \dfrac{\partial}{\partial y}\right)$$
it is then quite easy to show that (i) $g(Q,Q)=1$ so that it is a timelike vector field and (ii) $g(Q,\partial_t)>0$ which shows it is future directed using the time direction given by the global vector field $\partial_t$. Synchronizability then is discussed computing the one-form $\alpha = g(Q,\cdot)$ and its derivative $d\alpha$.
I also computed the integral lines of this vector field imposing initial conditions $x\circ \gamma(0)=(t_0,x_0,y_0,z_0)$ and found out that the coordinate expression of the curves are $$x\circ\gamma(\tau)=\left(t_0+\dfrac{\tau}{\sqrt{1-x_0^2}}, x_0, y_0+\dfrac{x_0}{\sqrt{1-x_0^2}}\tau,z_0\right)$$

As for computations it is quite straightforward. However I still don't see how all of this gets used. For example (i) how does one define a reference frame in the first place, i.e., where such a definition for $Q$ comes from? (ii) it seems that to use a reference frame we need a naturaly adapted chart, is that true? And anyway how does one get such chart built out of a single vector field?

From a mathematical perspective the definitions are quite nice, I've seem no GR book define reference frame up to this one. But as for how this is used in practice, I'm quite unsure yet. Have you ever seem this approach before?

 

[PeterDonis]

where such a definition for QQ comes from?

From the idea that a reference frame is constituted by a family of observers. The simplest example is a global inertial frame in flat spacetime, which is constituted by a family of observers, all inertial and all at rest relative to each other.

it seems that to use a reference frame we need a naturaly adapted chart, is that true?

No. You have illustrated that by computing the worldlines of the reference frame in the exercise you describe, in a standard inertial chart on Minkowski spacetime. It should be obvious that this chart is not adapted to the reference frame in question.

how does one get such chart built out of a single vector field?

The timelike worldlines mark out the spatial coordinate positions in the chart (i.e., each worldline has a unique set of spatial coordinates $x_1, x_2, x_3$). The choice of how to mark out surfaces of constant time is not unique, but often there is a natural choice--for example, if all of the worldlines are orthogonal to a family of spacelike hypersurfaces, that family is a natural choice for the surfaces of constant time.

 

[pervect]

In physics textbooks, a unit future-directed timelike vector field is often called timelike congruences. Most of what I read in physics textbooks is specialized to the case of geodesic timelike congruences. Wiki has a treatment that covers non-geodesic timelike congruences, though. https://en.wikipedia.org/w/index.php?title=Congruence_(general_relativity)&oldid=737290097.

Properties of physical interest that one can calculate from a congruence are the expansion scalar, the shear tensor, and the vorticity tensor, which describes whether a small volume element grows in volume/srhinks, changes shape, or rotates. This is used in Raychaudhuri's equation, for instance, and is relevant to some focussing theorems IIRC.

There is a treatment of geodesic congruences in Poissons "A relativistis toolkit", and a very brief treatment of geodesic congruences in Wald's "General Relativity". (Neither of these treat the non-geodesic case as Wiki does).