Similarly, in Newtonian gravitation, the acceleration of a timelike curve must always be spacelike, and so the total force on a particle at a point must be spacelike as well. A vector ξa at a point in a classical spacetime is timelike if ξata ≠ 0; otherwise it is spacelike. The required result thus follows by observing that given a curve with unit tangent vector ξa, ta(ξn∇nξa) = ξn∇n(ξata) = 0, again because ξa has constant (temporal) length along the curve. Note that one cannot say simply “orthogonal” (as in the relativistic case) because in general, the classical metrics do not provide an unambiguous inner product between timelike and spacelike vectors.
A “force field,” meanwhile, is a field on spacetime that may give rise to forces on par- ticles/bodies at a given point, where the force produced by a given force field may depend on factors such as the charge or velocity of a body. We understand force fields to generate forces on bodies, and so there can be a force associated with a given force field at a point just in case the force field is non-vanishing at that point. (The converse need not hold: a force field may be non-vanishing at a point and yet give rise to forces for only some particles at that point.) A canonical example of a force field is the electromagnetic field in relativity theory. Fix a relativistic spacetime (M, gab). Then the electromagnetic field is represented by the Faraday tensor, which is an anti-symmetric rank 2 tensor field Fab on M. Given a particle of charge q, the force experienced by the particle at a point p of its worldline is given by qFabξb, where ξa is the unit tangent vector to the particle’s worldline at p. Note that since Fab is anti-symmetric, this force is always orthogonal to the worldline of the particle, because Fabξaξb = 0.