Rigorous definition of a family of observers in relativity

[WannabeNewton]

Does anyone have a textbook reference for the rigorous definition of a family of observers? Wald uses the term a lot but never actually defines it i.e. is it a congruence of unit time - like world lines defined on a proper open subset of the space - time or is it a congruence defined over the entire space - time or does it even have to be a congruence at all (i.e. can two worldlines in the family intersect)? To this entire family we give an orthonormal frame field that is supposed to characterize how the measuring apparatuses of each observer is oriented but what is the rigorous link between the mathematical and physical definition here (i.e. what constitutes the measuring apparatuses - are they just a physical way of characterizing the orientation of the orthonormal frame field?). I believe MTW actually gives proper definitions of these things but the book is huge I got lost trying to find anything pertinent so I'm hoping someone else possibly knows the relevant sections or has other texts to look at. Thanks!

 

[robphy]

Wald gives a definition on p. 69.
A family of observers is represented by a unit timelike vector field v^a.

Alternate resources
www.google.com/search?tbm=bks&hl=en&q="family+of+observers"
check out the search results in de Felice.

 

[WannabeNewton]

Ooooh I must have missed it; thanks a ton robphy! Also can you explain the part about measuring apparatuses to me? Is the orientation of the measuring apparatus carried by an observer in the family at a point in his world line just a physical way of characterizing the orientation of the orthonormal triad making up the spatial part of the basis for the tangent space at that point in the world line of said observer? Thanks a lot for the de Felice link that one looks like it will be quite helpful!

 

[Ben Niehoff]

A measuring apparatus is an orthonormal frame: a set of 3 measuring rods 1 meter long, oriented at right angles to each other, combined with a clock that ticks once every (1 m/c) seconds.

 

[WannabeNewton]

A measuring apparatus is an orthonormal frame: a set of 3 measuring rods 1 meter long, oriented at right angles to each other, combined with a clock that ticks once every (1 m/c) seconds.

Ok that makes sense but is there a reason why Wald only refers to the 3 meter sticks when talking about the orientation of the apparatus? He attributes to every point on the worldline of the observer an orthonormal basis $(e_{\alpha })^{a}$ and says $(e_{0 })^{a}$ is the unit tangent vector to the world line which corresponds to your statement about the ticking clock and then says the $(e_{i })^{a}$'s characterize the orientation of the apparatus held by the observer, which it would seem corresponds to the orientation of the 3 meter sticks. Is he just excluding the time - like basis vector from the statement about orientation of the apparatus because the orientation of this is already fixed along the world line of the observer (future - pointing)? Thanks ben!

EDIT: The relevant page I got this from in the book was 342 if you are interested. Cheers!