When is the Spacetime Temporally Orientable?

space-time

In both general relativity and Newtonian gravitation, forces are represented by vectors at a point. We assume that the total force acting on a particle at a point (computed by taking the vector sum of all of the individual forces acting at that point) must be proportional to the acceleration of the particle at that point, as in F = ma, which holds in both theories. We understand forces to give rise to acceleration, and so we expect the total force at a point to vanish just in case the acceleration vanishes. Since the acceleration of a curve at a point, as determined relative to some derivative operator, must satisfy certain properties, it follows that the vector representing total force must also satisfy certain properties. In particular, in relativity theory, the acceleration of a curve at a point is always orthogonal to the tangent vector of the curve at that point, and thus the total force on a particle at a point must always be orthogonal to the tangent vector of the particle’s worldline at that point.

More precisely, we take a model of relativity theory to be a relativistic spacetime, which is an ordered pair (M, gab), where M is a smooth, connected, paracompact, Hausdorff 4-manifold and gab is a smooth Lorentzian metric. A model of Newtonian gravitation, meanwhile, is a classical spacetime, which is an ordered quadruple (M, tab, hab, ∇), where M is again a smooth, connected, paracompact, Hausdorff 4-manifold, tab and hab are smooth fields with signatures (1, 0, 0, 0) and (0, 1, 1, 1), respectively, which together satisfy tabhbc = 0, and ∇ is a smooth derivative operator satisfying the compatibility conditions ∇atbc = 0 and ∇ahab = 0. The fields tab and hab may be interpreted as a (degenerate) “temporal metric” and a (degenerate) “spatial metric”, respectively. Note that the signature of tab guarantees that locally, we can always find a field ta such that tab = tatb. In the special case where this field can be smoothly extended to a global field with the stated property, we call the spacetime temporally orientable.

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