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MCRF -- Momentarily Comoving Reference Frame -- An inertial frame of reference which happens to be moving in the same direction, at the same speed, as an object or an accelerated frame which we're examining. Many questions in relativity can only be addressed in an inertial frame. In many cases, when the frame under consideration isn't inertial, such questions can still be addressed by considering the MCRF of the accelerated frame.

- http://www.physicsinsights.org/glossary.html

I'm often unsure whether people are using "frame" to mean a chart, or a basis field of tangent vectors, or a basis for an individual tangent space. Have I got this (more or less) right?

(1) A MCRF of a pointlike object is a function that assigns to each event

*p*in the object's worldline an orthonormal basis for the corresponding tangent space,

*T*, such that the timelike basis tangent vector

_{p}M*E*is equal to the proper velocity of the object at that event. At each event, the choice of spacelike tangent vectors is arbitrary, so there's no unique choice of MCRF for a given worldline.

_{0}|_{p}(2) A MCRF of a fluid is defined similarly but over a 4-dimensional region, with

*E*equal to the limit as V --> 0 of any function which outputs the average proper velocities of fluid particles in a suitably well-defined (to allow the necessary integration) set of events containing

_{0}|_{p}*p*and whose input is some measure V of the volume of that set.

(3) A MCRF of a chart x is a function which assigns an orthonormal basis to the tangent space at each event

*p*in the chart's domain such that

[tex]E_0 |_p = \frac{\partial_0}{\sqrt{|g(\partial_0,\partial_0)|}},[/tex]

supposing the chart induces one and only one timelike coordinate basis tangent vector at each point; otherwise I suppose there is no such family of MCRFs associated with the chart.