I had basically the same solution as Roberto for the last part,

except I put it in (what I feel) is a more explicit conservation-law form.

is a conservation law expressed in differential form, as applicable to differentiable tensor fields. (It also implies an integral form applicable to compact 4-volumes).

In the case of a set of point particles, however, we are dealing with a sum of discrete quantities, rather than fields, so there's no exactly analagous differential or integral conservation laws.

Instead, we can define a "time" parameter,

, by dividing spacetime into a continous stack of 3-space slices, parameterized by

. (Like slicing up a salami; it doesn't matter exectly how we do this - any stack of slices will do, so long as they are everywhere spacelike, and the stack covers all spacetime without any gaps).

Then we can phrase our conservation law as saying that the conserved quantity is "constant over time", i.e.

.

In particular, since particle trajectories are always timelike, they're never parallel to a slice, so

can always be used to parameterize the particle trajectories, i.e. the position and momentum of the

particle can be denoted by

and

(where the

is the particle index, and

is a tensor index).

Then the conservation law

(*) that we're after is:

Collision points are single spacetime points, so

is constant and since total (summed) 4-momentum is conserved, so is

for the particles involved in the collision at that point.

It remains to show that

is zero along the particle trajectories between collision points.

Since

parametrizes the particle trajectories, then along these trajectories:

for some positive scalar function

.

Then:

but

And we note that

along particle trajectories because they are geodesics (see p.304), and

because

is a Killing vector field, satisfying

.

So along particle trajectories the summed term in

(*) equates to zero as required, and we have thus demonstrated that

(*) holds everywhere (both at collision points and along the particle trajectories between collisions). This holds true for

any parameterization of time

and for

any spacetime that has a Killing vector field

on it, whether it's flat (i.e. Minkowski space) or not.