Recent content by cianfa72

Quoting Wikipedia - kinematical decomposition In the notation used there, ##X^a_{;b}## should be the (1,1) tensor ##\nabla_b X^a## written in abstract index notation (note the Latin indices). Sometimes I've seen ##\nabla_b X^a := (\nabla X)^{a}{}_{b}## rather than ##\nabla_b X^a := (\nabla...

Even in the context of relativity, relative vector velocities entering the definition of separation rate as vector sum, are not 4-velocities, right?

You said ##h_{ab}## is a spatial 3-metric tensor. Your definition of it is ##h_{ab} = g_{ab} + {u_a}{u_b}##. However the indices ##a,b## still run from 0 to 3. Yes, since zero vorticity implies (iff) hypersurface orthogonality. Ok.

Ah ok, basically your point is that if we look at 1d lines, then for any timelike congruence there always exists such an orthogonal (spacelike) line (this boils down to the fact that when looking at 1d submanifolds, the Frobenius integrability condition is always met - basically what is required...

But, what if the timelike congruence representing fixed points on the object (say a ruler) hasn't zero vorticity ? In that case there isn't a spacelike hypersurface orthogonal to the congruence's worldlines. So there isn't any such spacelike path you were talking about.

I'd ask for clarification on the definition of Born rigidity, see for instance Born Rigidity. In the context of SR (flat spacetime) consider a ruler moving through spacetime. Its points define a timelike congruence in the region of flat spacetime it occupies. The general definition of Born...

Just to provide a recipe to check their effective clock rates actually match: Consider a GPS satellite clock sending an electromagnetic signal that encodes the time shown by itself at the departure (say ##t_1##). It then sends an encoded signal later when it reads ##t_2##. The clock at rest on...

Let me try to recap my understanding about this. In the context of category theory, given two objects ##X_1## and ##X_2## of a given category (say topological spaces), one can define/construct different products. A product consists of two pieces of information: an object ##X## (of the same...

As far as I can understand, your point is that a trivial bundle includes two pieces of information: a trivializable bundle plus a specific trivialization of it. On the other hand, a trivializable bundle has some (hence infinite) trivialization of it, however no specific trivialization is...

You mean, take a topological space ##B## that is homeomorphic to the Cartesian product ##A \times F##. The choice of the homeomorphism ##\varphi## basically defines a product decomposition of ##B##. Now combining ##\varphi## with the projection on the first factor one defines a trivial fiber...

Yes, but for the case of product bundle the projection mapping is canonically/naturally given by the structure of the product (it is one of the two projections on factors). Note that, following JM. Lee - ISM, a product bundle is not the same as a trivial fiber bundle (a product bundle is a...

Ok, you are assuming the inertial rest frame of the table. By Newton 3rd law, the book acts with a force on the table, so there must be a force from the floor (friction) acting to the table to get zero net force on it. The table stays continuously at rest in that inertial frame, so no mechanical...

You mean the table doesn't move w.r.t. the rest frame of "book + table" system's Center-of-Mass.

Yes, for instance in the rest frame of the "Earth + object" system's Center-of-Mass, the kinetic energy is basically associated to the rising object, i.e. we can neglect the kinetic energy term associated to the Earth.

I'd like to point out that the increase in mechanical energy associated with lifting an object up, goes into the potential energy of the "Earth + object" system. In other words potential energy is a property of a system, not of an object (book) alone.