For an affinely parameterised geodesic we can form the Lagrangian: $$ \mathcal L = g_{ab}\dot x^a\dot x^b = \text{constant} $$ The Lagrangian is constant by the fact that the geodesic parallel transforms it's tangent vector, $\dot x^a$, thereby preserving it's length. My question in then this: If...

It can be proven that if one considers a separable Hilbert space to define the theory on, then the orthonormal basis for this space is countable. If one considers a discrete set of harmonic oscillators with a Hamiltonian: $$ H=\sum_{i=1}^{n}\hbar w\left(a^\dagger_ia_i+\frac{1}{2}\right). $$ Then ...

Well, when you study special relativity in a "covariant way" (using the formalism of Lorentzian geometry and so on...) you realize that the structure of the spacetime isn't a mere vector space, but rather, that of affine space $[1]$. But, a small conceptual gap is annoying me. General Relativity ...