#004: See how much of relativity is already there in the particle equation

Category: Classical mechanics - Tags: Differential geometry, Riemannian geometry, Symplectic geometry, General relativity

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Following the conventions set up here, apply the Hamiltonian constraint on the symplectic form to see whether there is a dependence between the EM field and the metric tensor.

Mathematical problem. In the linked note, we find the expression of the symplectic form in terms of position and velocity. The expression lives on the eight dimensional phase space (i.e. \(\{t, q^i, E, p_i\}\)), which is redundant given the Hamiltonian constraint \(\mathcal{H} = \frac{1}{2} m u^\alpha g_{\alpha\beta} u^\beta\). The idea is to use the constraint to lower the dimensionality of the form to a seven dimensional non-symplectic form, and to understand whether, on that submanifold, there is a relationship between the \(A_\alpha\) and \(g_{\alpha\beta}\) fields.

Physical significance. The Hamiltonian constraint plays a double role: it is the generator of the affine parameter (i.e. “time” evolution) and is a conserved quantity during evolution (specifically, the rest energy of the particle). What we want to understand is whether this double role puts a constraint on how the \(A_\alpha\) and \(g_{\alpha\beta}\) fields can vary across phase-space.

If there is indeed a connection, it would mean that there are some hints of general relativity in plain particle dynamics. If there isn’t, then the link between curvature and energy is a separate assumption from those required to derive particle dynamics under scalar and potential forces.