#003: Characterize the expression \( G_{\alpha\beta\gamma} = \partial_\alpha g_{\beta \gamma} - \partial_\beta g_{\alpha \gamma} \)

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

Characterize the geometry and the physics captured by the expression \(G_{\alpha\beta\gamma} = \partial_\alpha g_{\beta \gamma} - \partial_\beta g_{\alpha \gamma}\). This expression appears when expressing the symplectic form in terms of position and velocity (see here for more details).

Mathematical problem. The expression appears with the EM tensor in the spatial-temporal part of the symplectic form when it is expressed in terms of position and velocity. It also appears as a “correction” for the raised anti-symmetrized covariant derivative. This means the expression should have both geometrical and physical significance, which merit further study.

As an example of the type of things we are looking for:

  • Is the expression a tensor? It would seem so as the spatial-temporal part of the symplectic form should transform like a tensor.
  • How is it related to curvature? Is it always zero on flat spaces? Can it be zero on curved spaces?
  • Can it be interpreted as a gravitational force?
  • From calculation, the tensor can be understood as the difference between taking an antisymmetrized covariant derivative and an antisymmetrized partial derivative: what does that say geometrically?

Physical significance. Note that the setting is not really that of space-time (i.e. Riemannian manifold) but that of phase-space (i.e. symplectic manifold). The general idea is to reunderstand the metric tensor not as a geometrical property of space-time but as a geometrical property of phase-space.