Energy of a Particle in Free Fall

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In this page an expression is found for the energy of a stream of photons originating at infinity as measured by an observer in a spaceship in free-fall. The observer falls from rest at infinity. The observer is assumed to be falling radially inward toward a massive spherical body. The Schwarzschild Metric describes the space-time geometry near a such a body and is given by

Since the Schwarzschild metric is not an explicit function of time the energy of the falling observer (i.e. the spaceship), E_obs, is conserved. Let P_obs º m₀U_obs equal the four-momentum of the spaceship having a four-velocity and proper mass which are, respectively, U_obs and m₀.

where m = gm₀. Since the motion is radial

The 4-momentum of the free-fall observer is therefore

The energy of the observer is defined as E_obs = (P_obs)₀ where

Since E_obs is constant and has the value m₀c² at infinity and at rest, it follows that E_obs = m₀c². To find g we use the Schwarzschild metric and set df = dq = 0 and since the geodesic will be timelike we set ds² = c²dt² in Eq. (1) to give

where s º 1 - GM/c²r. Divide Eq. (5) through by c²dt² to give

where br º dr/d(ct) and

Solving for b_r gives

Let the four-momentum of the photon be P º (mc, pr, 0, 0). Then the energy of the photon as measured by the observer in free-fall, E_p, is given by

The quantity P₀ is the energy of the photon. Since this is a constant it has the same value as it has at infinity and therefore

The quantity P1 has the value given by

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