Formal definition of an observer? What is the formal definition of an observer in special relativity? I have seen a few:


*

*The actual coordinate system.

*The collection of synchronised clocks that cover the coordinate system.

*A well reasoned person in the system.
But what is the actual definition? and can you give your source, thanks
 A: 
What is the formal definition of an
  observer
  in special relativity? I have seen a few:
  
  
*
  
*The actual coordinate system.
  
*The collection of synchronised clocks that cover the coordinate system.
  
*A well reasoned person in the system.
But what is the actual definition?

In #3, what is missing is that the observer's state of motion, i.e., normalized velocity four-vector, needs to be defined, and what shouldn't be included is the stuff about it being an actual person. An "observer" in the SR sense can be a totally impersonal mathematical description, not even necessarily a physical object.
If the observer is inertial, then there is a one-to-one correspondence between normalized velocity vectors and Minkowski coordinate systems. Therefore it doesn't matter in this case whether you do a definition in the style of 1 or 3. However, SR is compatible with an arbitrary coordinate system, which can be defined by any change of coordinates that is well behaved (a diffeomorphism). (GR isn't required for this.) For a generic choice of coordinates, there is no way to define a corresponding observer. Therefore if you're allowing non-inertial observers, approach #3 is the only one that works. In that situation, the observer could be both rotating and accelerating, so you end up with the definition given in FenderLesPaul's answer.
Number 2 doesn't work, because it doesn't define anything about the observer's spatial coordinate system, e.g., it doesn't tell us how the observer is oriented.
A: An observer is a timelike worldline with 4-velocity $u^{\mu}$ and an orthonormal basis $e_{\hat{\alpha}}$ with $e_{\hat{0}} = u$ such that $e_{\hat{\alpha}}$ is transported along the worldline under some transport law e.g. Lie transport, Fermi transport, or parallel transport. Physically the Lorentz frame represents a local set of three orthogonal meter sticks or gyroscopes and an ideal clock. 
An observer can use $e_{\hat{\alpha}}$ to define a comoving local coordinate system (e.g. a Fermi-normal coordinate system) with clocks that are e.g. Einstein synchronized but the coordinate system isn't necessary to define the observer. 
C.f. chapter 6 of MTW, section 13.6 of MTW, section 2.1 of Sachs and Wu "General Relativity for Mathematicians", and most importantly chapter 3 of Eric Gourgoulhon "Special Relativity in General Frames". 
A: I guess there is no single definition, since it varies according to author. Personally, I think the term should just be abandoned. It's too ambiguous. It mixes up two ideas that need to be cleanly separated, but often aren't: the frame of reference as a grid of sensors, and a single sensor that receives incoming information, like a camera.
Mixing up these two ideas is a recipe for misunderstanding. 
A: An "observer" in the theory of relativity means any individual who is capable, or, in the context of thought experiments, any material point or principal identifiable point to which the capability is attributed, 


*

*to collect perceptions,

*to distinguish the perceptions collected, and

*to judge the coincidence (or otherwise: the sequence) of distinct collected perceptions.


Reference:
A. Einstein, "Relativity. The Special and General Theory", chap. 8: 

 If the observer perceives the two flashes of lightning at the same time, then [...] 

Like all of Physics and Geometry, the "observer" notion of RT as well as the derivation of geometric relations between observers (to characterize sets of distinct observers as a "system" or "reference frame") stand of course without any assignment or even particular choice of coordinates.
