Energy conservatuion in TEP paper due to increased gratiatioanl pull of light pencil
I've been going through the 1931 paper by Tolman, Ehrenfest and Podolsky (Physical Review 37, 602) on the gravitational field produced by light and I've hit something I can't resolve. Posting here to see if I'm making an obvious mistake or if this is genuinely subtle.
Background
The paper proves using linearized GR that a stationary test particle placed midway alongside a finite pencil of light — at position (l/2, y) where y is perpendicular distance from the beam — experiences gravitational acceleration toward the beam given by their Equation 22:
d²y/dt² = 2Gρlm / c² × 1/√[(l/2)² + y²]
where ρ is linear energy density and l is beam length. This is exactly twice the Newtonian value you get by treating radiation mass as E/c². I'm not disputing this — I've checked the derivation and it follows cleanly from the geodesic equation.
My setup
I'm using the exact geometry of the paper. Source at x = 0 containing stored energy E = ρl as rest mass M = ρl/c². Absorber at x = l. Test particle of mass m starting at (l/2, 0) — on the beam axis, midway between source and absorber.
Before the beam turns on I move the test particle from (l/2, 0) to (l/2, y). The energy sits entirely as rest mass at x = 0. The work done against the source gravity is:
W = G(ρl/c²)m × [ 2/l − 1/√((l/2)² + y²) ]
Particle is now held at (l/2, y) by a rigid support.
Beam turns on. Source converts its stored energy into a steady pencil of light to the absorber. Total energy E = ρl is conserved. Nothing enters or leaves the system.
Particle is released. It falls from (l/2, y) back toward (l/2, 0) keeping x = l/2 throughout — always at the beam midpoint so TEP Equation 22 applies at every point along the path. The force at height y' is:
F(y') = 2G(ρl/c²)m / √[(l/2)² + y'²]
Integrating this from y down to 0 gives the kinetic energy gained on arrival:
U = (2Gρlm/c²) × ln[ (y + √((l/2)²+y²)) / (l/2) ]
The problem
W and U have completely different functional forms. W goes as 1/r. U goes as ln(r). Plugging in numbers with l = 2m and G(ρl/c²)m = 1:
y = 0.25m → W = 0.039, U = 0.490, surplus = 0.451 (U is 12× larger) y = 0.50m → W = 0.106, U = 0.962, surplus = 0.856 (U is 9× larger) y = 1.00m → W = 0.293, U = 1.763, surplus = 1.470 (U is 6× larger) y = 2.00m → W = 0.634, U = 2.877, surplus = 2.243 (U is 4× larger)
The surplus is not a small correction — it's massive across all values of y. The total energy of the system hasn't changed. The particle never moved during the conversion. I just switched the source energy from rest mass to light and the potential energy available to the test particle jumped enormously.
What I think the mistake might be but can't pin down
Either the path integral is wrong — maybe TEP Equation 22 can't be integrated along the fall path the way I've done it, but I can't see why since the particle stays at x = l/2 the whole time where the formula applies.
Or converting rest mass to radiation changes the gravitational potential at (l/2, y) in a way I'm not accounting for — the field configuration is completely different before and after conversion and maybe that difference swallows the surplus. But then what is the correct potential energy before and after, explicitly?
Or there's energy stored in the gravitational field configuration itself during conversion that I'm missing from both W and U entirely.
I've searched and can't find this specific cycle discussed anywhere. Has anyone seen this before or can point to where the accounting closes?
Reference
Tolman R.C., Ehrenfest P., Podolsky B. (1931). On the gravitational field produced by light. Physical Review, 37, 602. https://doi.org/10.1103/PhysRev.37.602
I feel the contradiction energy must somwhwo assocaited with spacetuime disturbance vreated by phton pencuil
