- #1

leo.

- 96

- 5

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

- A
*coordinate system*is a chart. Namely a pair [itex](U,\phi)[/itex] where [itex]\phi : M\to \mathbb{R}^4[/itex] assigns coordinates to events. In that case [itex]\phi[/itex] gives rise to the so called coordinate functions [itex] \phi^\mu[/itex] and the coordinates of [itex]x\in M[/itex] are [itex]\phi^\mu(x)[/itex]. - A reference frame is a set of four vector fields [itex]e_\alpha \in \Gamma(TM)[/itex] 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 [itex]g[/itex]. In other words: [itex]g(e_\alpha,e_\beta)=\eta_{\alpha \beta}[/itex].
- An observer is a pair [itex](\gamma,e_\mu)[/itex] where [itex]\gamma : \mathbb{R}\to M[/itex] is a future-pointing timelike path and where [itex]e_\mu : \mathbb{R}\to TM[/itex] are four vector fields along [itex]\gamma[/itex], that is [itex]e_\mu(\tau)\in T_{\gamma(\tau)}M[/itex] such that (i) [itex]\gamma'(\tau)=e_0(\tau)[/itex] and (ii) [itex]g_{\gamma(\tau)}(e_\alpha(\tau),e_\beta(\tau))=\eta_{\alpha\beta}[/itex].

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 [itex](x,y,z)[/itex] and time [itex](t)[/itex] 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?