# Particle Velocity parallel to p

Figure 2 Impact parameter b defined. A fast particle (mass m) approaches a black hole from a great distance with vector momentum p. Find a test particle with a parallel velocity that plunges radially—without deflection—into the black hole. The perpendicular distance b between their initially parallel paths (at this great distance) is called the impact parameter b this limit, motion of such a particle mimics that of the light flash. Now for the details.

A very fast particle—still of mass m—about to pass a black hole is initially at a great distance from it (Figure 2). Spacetime in the local neighborhood of the particle is effectively flat. The particle has a momentum vector p along its line of motion. By trial and error, find a second line of motion, initially parallel to the first, such that a test particle moving along this second line plunges radially into the black hole, deviating neither one way nor the other as it does so. Our original particle and the test particle start along parallel tracks from our remote location in flat spacetime. The perpendicular distance between these tracks at this remote location is called the impact parameter and given the symbol b (Figure 2).

The box on page 5-8 uses the impact parameter and the sneak-up-on-it strategy to derive the equations of motion for light.

4 Alternative Speeds of Light II

Bookkeeper: Different light speeds in different directions. Shell observer: Light speed always v = 1.

Equations  and  in the box on page 5-8 give the radial and tangential components of velocity as calculated by the remote bookkeeper, who reckons everything in reduced circumference r, far-away time t, and azi-muthal angle <|>. (For radial motion, b = 0, compare these results with equation .) Both radial and tangential velocity components get smaller and smaller as the light flash gets closer and closer to the event horizon at r = 2M. Now we can find the speed of light for any impact parameter. Square both sides of equations  and  in that box, add their respective sides, and take the square root of the result, giving equation :

Impact parameter b defined

Bookkeeper light speed is a function of impact parameter and r.

f light speed ^ reckoned by bookkeeper 