Reference frames in General Relativity seem simple to understand and hard to come out with one agreed definition. In the intuition it represents a certain "point of view" of several observers. In practice, the best approach seemed to me at first to consider reference frame to be a frame field: that is, four orthonormal vector fields which form a basis at each tangent space.

Recently in some papers by the brazilian physicist Waldyr Alves Rodrigues Jr., I've found another definition of reference frame that is also presented on the book "General Relativity for Mathematicians" by Sachs and Wu, which can actually be concisely stated:

Definition 1:

  1. One observer in spacetime $M$ is a smooth timelike future-pointing curve $\gamma : I\subset \mathbb{R}\to M$.
  2. A reference frame in $U\subset M$ is a vector field $Q : U\to TU$ such that each of its integral lines is one observer. In other words, $Q$ is timelike and future-pointing. One also requires $Q$ to be normalized.

The first point establishes what is one observer. The second point establishes that a reference frame is a collection of observers.

On the other hand, watching the International Winter School on Gravity and Light General Relativity course, the following definition was made:

Definition 2:

One observer in spacetime $M$ is a pair $(\gamma,e_\mu)$ being $\gamma :I\subset \mathbb{R}\to M$ a smooth timelike future-pointing curve and $e_\mu : I\subset \mathbb{R}\to M$ four vector fields over $\gamma$, that is, $e_\mu(\lambda)\in T_{\gamma(\lambda)}M$ such that

  1. $e_0(\lambda) = \gamma'(\lambda)$
  2. $g_{\gamma(\lambda)}(e_{\mu}(\lambda),e_{\nu}(\lambda))=\eta_{\mu\nu}$, that is, the four vector fields are orthonormal.

In these lectures, reference frame was left undefined.

Now, there is some overlap between these two definitions. In both of them observers are associated to worldlines of massive particles.

On the other hand the major difference is: the definition 1 considers that an observer carries just a timelike vector that a reference frame is comprised by specifying just a timelike direction. The definition 2 requires that an observer carries a whole moving frame with it.

Intuitively thinking, it does make more sense to carry an entire basis rather than just a timelike vector, since in order to express physical quantities and write down equations "from the point of view of an observer" one requires the spatial vectors too.

Indeed it also seems natural that the rigorous definition of reference frames requires a whole frame field, and I already questioned this here.

So my question is: how should we actually understand this definition 1? Why would one define it like that, requiring both for observers and reference frames just the timelike vector? Is definition 1 somewhat standard, or the standard definition of reference frame is to define it as a frame field?

I'm intrigued by that definition because it seems to lack information. I mean, if we define as a frame field, the reference frame also include the spatial vectors. If we just give one timelike field, we could have an infinite number of basis since we could pick any three linearly independent spacelike vectors. I don't know, I'm confused with this.

  • $\begingroup$ user1620696: "[...] requiring both for observers and reference frames just the timelike vector? Is definition 1 somewhat standard [...]?" -- This restriction seems to recall the property of Lorentzian distance (cmp. google.com/#q=Beem+Ehrlich+Easley+distance&*) $$\ell : \mathcal S \times \mathcal S \rightarrow [0 ... \infty]$$ which provides quantitative distinction of event pairs which are timelike related to each other, while assigning the same value "$\ell = 0$" to all pairs of events which are spacelike or lightlike related to each other. $\endgroup$ – user12262 Mar 21 '17 at 22:39

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