Recent content by cianfa72

As @Ibix showed, the orthogonal grey grid lines in the diagram actually correspond/represent straight lines orthogonal in spacetime (note that the notion of straight line in spacetime is a geometric notion). The isotropy of coordinate one way speed of light corresponds to pick orthogonal grid...

I'd add the qualifier coordinate speed in both SR and GR is invariant in inertial frames since the 4-velocity of light, like any 4-vector, is a geometric quantity/object therefore invariant by its very definition (like the light cones in any spacetime).

In the above, I believe the only sensibile way to check/ensure what I highlighted in bold, is to use light beams sent forth and back from O to O' and the other way around measuring that the round-trip travel time stays constant (as measured by the same clock).

In Anderson coordinates the expression of the flat spacetime metric ##ds^2## includes mixed term in ##\kappa dtdx##. For ##\kappa=0## (i.e. standard inertial coordinates) they vanish, hence a distinctive property/feature of the standard inertial coordinates is that the timelike coordinate is...

Ah ok, I think what I put in bold is the key point. What is actually invariant are the light cones and therefore the light's velocity as tangent vector to light cones' boundary in spacetime. Yes, as special case Anderson coordinates (underlying flat spacetime) can be classified as inertial...

I think you actually mean the one-way speed of light (OWSOL) is only invariant in the standard inertial coordinates (assuming of course flat spacetime).

Of course the relevant gluing map glues "corresponding points" on disks ' boundary (e.g. point ##(1,0)## of ##D_{+}## is glued with point ##(6,5)## on ##D_{-}##). By the same logic, one could take as definition of topological 2-sphere, any manifold homeomorphic to a specific pair of closed...

Very good, congratulations !

Yes of course. From the definition of ##f## given in #2 it takes the same value (##f=0##) on the pair of points in ##D_+## and ##D_-## identified by the quotient building (i.e. gluing pair of "corresponding" points on the boundaries of the disks). Yes, definitely. P.s. pls note that in #2 the...

Ok, coming back to the OP, just to be clear, what is a bijective continuos map is the function ##\tilde f## resulting by passing ##f## to the quotient.

I was thinking as follows: call ##D_{+}## the disk of radius 1 centered at the origin of ##\mathbb R^2## and ##D_{-}## the disk of radius 1 centered at ##(5,5)##. Define ##f## as: $$f(x,y) = \begin{cases} (x,y,+\sqrt{1 - x^2 - y^2}) , & x,y \in D_{+} \\ (x,y,-\sqrt{1 - (x-5)^2 - (y-5)^2}), &...

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of...

Eventually I found a counterexample to my question in the OP. Take the following qubit state given in the abstract ##\mathcal H_2## standard basis ##\{ \ket{\uparrow}, \ket{\downarrow} \}## as $$\begin{bmatrix} \frac {1} {\sqrt{2}} \\ \frac {1} {\sqrt{2}} \end{bmatrix}$$ Now applying to it the...

Here my argument: a point on the Bloch sphere $$\ket{\psi} = \cos {(\theta /2)}\ket{\uparrow} + e^{j\phi}\sin {(\theta /2)} \ket{\downarrow}$$ represents a full ray in ##\mathbb C^2##, i.e. all vectors ##\lambda \ket{\psi}, \lambda \neq 0## are represented by the same point on it. Restricting to...

Ah yes, you are right :rolleyes: . So, which is the difference in applying an element of SU(2) vs an element of U(2) to the state $$\ket{\psi} = \cos {(\theta /2)}\ket{\uparrow} + e^{j\phi}\sin {(\theta /2)} \ket{\downarrow}$$ Both are unitary operators hence the norm 1 of the transformed...