Recent content by cianfa72

Ok, indeed. My point was to stress the notion of parallel and FW transport: assign a curve C, a point P on it and a vector V at P (i.e. an element of the tangent space at P). Then the parallel or FW transport of V along C from P is well-defined. Lie dragging is a different matter since it...

Yes, of course. I'm not asking about specific properties of frame field. I'm asking about the simple case of FW transport of a set of vectors (one timelike and three spacelike mutually orthogonal) from point P along a known curve C where the timelike vector is the tangent vector of the curve C...

No. Given any curve C, a point P on it and a vector V at P one can solve the first-order differential equation for the parallel transport of V along C from P.

Ah ok, maybe I wasn't very clear in post #22. In the general case (not geodesic curve) for both parallel and FW transport we need to know the curve and the vectors involved in the transport only at the point P from which the transport is carried out.

I think the same is true for parallel transport since it is defined for any curve including non-geodesic ones. However I never said that for FW transport the curve C isn't needed. What I said was that to perform the FW transport of a tetrad along the curve C from P, we need to know only the...

Ok yes. Yes, however the curve C is known (it is the observer's worldline) hence the tangent vector ##\mathbf u## along the curve (4-velocity) is also known and proper acceleration is ##\nabla_{\mathbf u} \mathbf u##. Therefore, to perform the FW transport of a tetrad (not a tetrad/frame field)...

I'm not sure whether the following argument applies to FW transport as well. Take the parallel transport equation along a curve C from point P and fix a chart. It becomes a first-order differential equation, hence the solution depends only on the components of the vector being parallel...

the curve C is the timelike worldline of an observer carrying three mutually orthogonal rulers.

Very well, next step: starting from a tetrad at point P (i.e. from just the set of four vectors given by evaluating the frame field at point P) one defines its Fermi-Walker (FW) transport along the curve C from P. All the additional information needed are the connection coefficients in a...

Are you referring to the condition that the local 3d spacelike hyperplane picked is orthogonal to the observer's 4-velocity ?

I meant the way/implication curve -> directional derivative not the other way around (i.e. a curve defines a unique directional derivative at point P). Consider a ruler with zero tickness. Every point on it describes a timelike worldline in spacetime. The set of ruler worldtube's worldlines is...

Yes, exactly (directional derivative aka derivation at P). Yes, indeed. My point was on the other way around: any curve (even a non-geodesic one) through P defines there an unique directional derivative/derivation (i.e. an element of the tangent space at P). My point was related how the ruler...

Sorry to insist. An element (vector) of the tangent space at point P is a function derivative operator (derivation). Take the set of smooth curves C passing though P and consider the composite map for any smooth function ##f## and curve C $$f(x(t))$$ where ##x(t)## is the representative of C in...

Ok, that's my point though: a ruler is a physical thing described by its worldtube in the spacetime model. My difficulty in understanding this clearly is: how can a physical thing define/locate/identify an element in the tangent space at event A (the spacelike direction the ruler points to)? My...

Ok. Let me use a more heuristic/intuitive description. Let's fix an event A along the observer's (timelike) worldline. Consider the worldtube's worldlines of one of the observer's carried rulers in a "small" neighborhood of A. Take the (unique) local spacelike hyperplane orthogonal to the...