Momentarily Comoving Reference Frame

[Rasalhague]

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_pM, such that the timelike basis tangent vector E₀|_p 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.

(2) A MCRF of a fluid is defined similarly but over a 4-dimensional region, with E₀|_p 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 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

$$E_0 |_p = \frac{\partial_0}{\sqrt{|g(\partial_0,\partial_0)|}},$$

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.

 

[Dale]

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.

My understanding is that technically a frame is a field of orthonormal vectors at each point in the manifold:
http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

So the correct usage is essentially your second one. However, (and I am particularly guilty of this), it is pretty common to mis-use "frame" as a synonym for "coordinate system", particularly in SR where the distinction is less important. So your first usage is incorrect but very common mis-usage. I have never seen someone use frame as a basis for a tangent space at a single point in the manifold.

 

[Rasalhague]

Thank you, DaleSpam. I've been parsing "frame field" like "vector field"; that's what led me to imagine people sometimes meant a basis for the tangent space at a point. And, of course, the word "vector" is often used as short-hand for "vector field". But reading the Wikipedia article now, it seems like they really are using "frame" and "frame field" as synonyms.

 

[Rasalhague]

Also relevant: http://en.wikipedia.org/wiki/Proper_frame

In Semi-Riemannian Geometry, O'Neill defines a "frame field" as an orthonormal basis field of tangent vectors (he also sometimes calls it simply a "frame"). He defines a Lorentz (=inertial) coordinate system as a time preserving isometry from Minkowsi space (conceived as an affine space) to the corresponding vector space R⁴₁. A frame (field) determines a family of such charts by the exponential map; they're the normal charts each specified by the orthonormal basis at a point. I wonder if these are the "frames of reference" which Wikipedia: Proper frame mentions, as it explicitly defines "frame of reference" as a coordinate system, or perhaps that article is just conflating the concepts of chart and frame.

 

[Rasalhague]

Ah, and here's more from O'Neill. It seems "field of reference" can also be used for a certain kind of vector field, rather than a chart:

"An observer field on an arbitrary spacetime M is a timelike, future-pointing, unit vector field U. Each integral curve of U is indeed an observer paramatrized by proper time. Thus U describes a family of U observers filling M. [...] (Observer fields are called reference frames in [SW], [...]"

([SW] = Sachs & Wu: General Relativity for Mathematicians.)