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The term observer has a number of non-equivalent uses in science.
The term observer in special relativity refers most commonly to an inertial reference frame. In such cases an inertial reference frame may be called an "inertial observer" to avoid ambiguity. Note that these uses differ significantly from the ordinary English meaning of "observer". Reference frames are inherently nonlocal constructs, covering all of space and time or a nontrivial part of it; thus it does not make sense to speak of an observer (in the special relativistic sense) having a location, except for denoting the origin of his reference frame. Also, an inertial observer cannot accelerate at a later time, nor can an accelerating observer stop accelerating.
In general relativity the term "observer" refers more commonly to a person (or a machine) making passive local measurements, a usage much closer to the ordinary English meaning of the word. Commonly used are FIDO (fiducial observer, stationary), FFO (free falling observer), ZAMO (zero angular momentum observer, corotating).
Thermodynamics and information theory
See for example Maxwell's demon.