---
source_pdf: Supplemental_Material__Density_Field_Dynamics_Letter.pdf
title: "Supplemental Material"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
format_note: "Markdown extracted from the source PDF for clean AI ingestion. No editorial changes; mathematical typography in the PDF is authoritative."
---

Supplemental Material

I.

Relative to flat space, the one-way excess is

GRAVITATIONAL REDSHIFT
2

With n = eψ and Φ = − c2 ψ, the frequency of a cavity
mode scales as
∆ν
∆Φ
= −∆ψ = − 2 .
ν
c

(1)

∆t =

ds
.
r

(7)



2GM
r1 + r2 + L
∆t =
ln
,
c3
r1 + r2 − L

(8)

GRAVITATIONAL LIGHT DEFLECTION
2

In the weak field, ψ(r) = 2GM/(c r) and n(r) ≃
1+ψ(r). For a light ray with impact
√ parameter b and coordinate z along the path, r = b2 + z 2 . The deflection
angle α follows from Fermat’s principle:
Z ∞
α≃
∂b n(r) dz
(2)
−∞

=

2GM
c2

Z ∞
−∞


∂b √

1
b2 + z 2


dz

 Z ∞

2GM
dz
−b
=
2
2 3/2
c2
−∞ (b + z )
=

4GM
.
c2 b

(4)
(5)

SHAPIRO TIME DELAY (RADAR ECHO
DELAY)

Photon travel time is

Z
Z 
1
1
2GM
T =
n(r) ds ≃
1+ 2
ds.
c
c
c r

p
r12 + r22 − 2r1 r2 cos θ. The two-way radar
with L =
delay doubles this to the GR value 4GM/c3 .
IV.

PERIHELION PRECESSION

In isotropic gauge the effective metric to O(Φ/c2 ) is




2Φ
2γΦ
ds2 = − 1 + 2 c2 dt2 + 1 − 2
dx2 .
c
c

(9)

(3)

This reproduces the full Einstein value, including the factor of two that Einstein’s 1911 calculation missed. (Historical note: Einstein’s original 1911 prediction gave only
half this value.)

III.

Z

For endpoints at r1 , r2 with impact parameter b, this
gives

This reproduces the standard gravitational redshift relation of GR.

II.

2GM
c3

Here γ is the standard PPN parameter that quantifies
spatial curvature per unit Newtonian potential. Its value
is fixed by the light-deflection result (5), hence γ = 1.
Minimal coupling of matter to this metric yields the
Newtonian limit with 1PN corrections corresponding to
β = 1. The perihelion shift per orbit in the PPN formalism is
∆ϖ =

(10)

which reduces to the GR value ∆ϖ = 6πGM/[c2 a(1−e2 )]
when β = γ = 1.

V.

(6)

6πGM
2 − β + 2γ
·
,
c2 a(1 − e2 )
3

COMPARISON TABLE

2
TABLE I. Weak-field predictions. All classical tests coincide with GR in the weak-field limit; the cavity–atom ratio provides
the decisive discriminator.
Observable
Gravitational redshift
Light deflection
Shapiro delay (two-way)
Perihelion precession
Cavity–atom slope

GR
−∆Φ/c2
4GM/(c2 b)
4GM/c3
6πGM/[c2 a(1 − e2 )]
0

DFD
−∆Φ/c2
4GM/(c2 b)
4GM/c3
6πGM/[c2 a(1 − e2 )]
2∆Φ/c2