p2

where ρ is the energy density of the modified vacua under consideration such that ρ → ρ E ~ E 2 for the
electric field vacuum, ρ → ρ B ~ B 2 for the magnetic field vacuum, and ρ → ρ T ~ π 2 T 4 for the thermal
vacuum. If the vacuum is a FRW gravitational vacuum, then one has to substitute one factor of α in
(2.20) by −m e 2 G and ρ → ρ r . Equation (2.13) for the Casimir Effect vacuum studied earlier is recovered
when ρ → ρ Casimir = −(π 2 /240)a −4 .
Let us recast (2.20) into a more useful form. We subtract one from both sides of (2.20), do some
algebra, and thus define the ratio of the change in the speed of light ∆c in a modified vacuum to the speed
of light in free space c 0 :
c ∗
0
1 c ∗
− c ∆
− = ≡
c
c0 c0 c0
∆ c 44 2 ρ
=− α (
4 = c0 = ε
0 = µ
0 = 1) (2.21).
c 135 me
0
Equations (2.20) and (2.21) are in quantum field theory natural units, which is completely undesirable for
estimating physically measurable values of ∆c/c 0 . We thus transform or “unwrap” (2.20) and (2.21) back
into MKS or CGS units by making the following substitutions (Puthoff, 2003)
ρ
ρ (natural units) → (MKS or CGS units)
c
mc
e
m (natural units) → (MKS or CGS units) ,
e
and after some algebra and rearranging we arrive at the final result:
c
∗ 44 2 ρ
1 α
⎛
= −
2 ⎜ ⎞
⎟
c0 135 mec0 ⎝mec0
⎠
3
(2.22)
and
3
∆ c 44 2 ρ ⎛ ⎞
=− α
2 ⎜ ⎟
c0 m c0 m c0
135
e ⎝ e
⎠
(2.23),
where all quantities are now in MKS or CGS units. We chose the former units so that c 0 = 3×10 8 m/s, ħ =
1.055×10 −34 J-s, m e = 9.11×10 −31 kg, and α = 1/137. Note that the ratio of the modified vacuum energy
density to the electron rest-mass energy has the dimension of (volume) −1 while the quantity in the bracket
is the cubed Compton wavelength of the electron having the dimension of (volume), and the product of
these is dimensionless.
An excellent example for estimating the magnitude of the change in the speed of light (in a modified
vacuum) is the Casimir Effect vacuum, since Casimir Effect experiments are common and widespread
such that this would be ideal to experimentally test (2.23). We substitute the Casimir vacuum energy
density ρ Casimir = −(π 2 ħc 0 /240)a −4 (in MKS units) into (2.23), do the algebra, insert the MKS values for the
physical constants, and make further simplifications to get:
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