In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field [itex]Z[/itex]. In that sense a reference frame is a collection of observers, since its integral lines are all observers according to this defintiion. This definition is also highly employed by the brazilian physicist Waldyr Alves Rodrigues Jr in his publications.(adsbygoogle = window.adsbygoogle || []).push({});

I particularly like this definition from a mathematical standpoint, because it is extremely simple and can even be intuitive - usually we really consider intuitively a reference frame as a collection of observers at rest with respect to each other.

Following this definition one defines a naturaly adapted coordinate system [itex]x^\mu[/itex] to a reference frame [itex]Z[/itex] to be a chart on spacetime [itex]M[/itex] such that [itex]\frac{\partial}{\partial x^0}[/itex] is timelike, [itex]\frac{\partial}{\partial x^i}[/itex] is spacelike and the spacelike components of [itex]Z[/itex] with respect to this basis are zero.

In basic treatments of Special and General Relaitivity, one usually needs to resolve physical quantities relative to reference frames, and relate different reference frames, in order to convert measurements.

My question really becomes: how these definitions gets used in practice in order to (1) express physical quantities with respect to a reference frame and (2) convert results from different reference frames?

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# I How this definition of a reference frame is used?

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