The following is a simplified description of the analysis underlying the quantum vacuum inertia hypothesis. This is an XHTML document. The equations should appear properly if you are using Internet Explorer 5 or Netscape 6.

BASIC CONCEPT: In an accelerating reference frame, the random electromagnetic zero-point fluctuations of the quantum vacuum acquire a Rindler-horizon asymmetry resulting in a non-zero energy and momentum flux in a direction opposite to the acceleration, call this the Rindler energy-momentum flux (see Rueda and Haisch 1998a, 1998b.). This is related to, but not identical with, the well-known Unruh-Davies effect -- investigated in the context of Hawking radiation from black holes -- in which an accelerating detector experiences a pseudo-thermal radiation spectrum whose "temperature" is proportional to acceleration. An interaction of the Rindler energy-momentum flux with the quarks and electrons comprising an accelerating object would generate a reaction force which is found to be proportional to and opposing the acceleration. This suggests a quantum vacuum basis for, or at least contribution to, the inertia of matter.

BASIC ANALYSIS: The electromagnetic field is quantized by equating each E and B mode with a harmonic oscillator. The Heisenberg uncertainty relation tells us that the minimum (zero-point) energy of a harmonic oscillator is hν/2=hω/4π (unfortunately there is no h-bar symbol in XHTML). The electromagnetic field therefore has a minimum quantum energy state consisting of zero-point fluctuations having an average energy per mode of hω/4π. In the semi-classical approach known as stochastic electrodynamics (SED) the quantum fluctuations of the electric and magnetic fields are treated as random plane waves summed over all possible modes with each mode having this hω/4π energy. The electric and magnetic zero-point field fluctuations in the SED approximation are thus (where the sum is over the two polarization states, ε is a unit electric field vector, and θ is a completely random phase term).

Ezp(r,t) = λ=1,2 d3k (hω/4π3)1/2 ε (k,λ) cos[k·r - ωt - θ (k,λ)]

Bzp(r,t) = λ=1,2 d3k (hω/4π3)1/2 [k × ε (k,λ)]/k cos[k·r - ωt - θ (k,λ)]

Since the Ezp and Bzp field fluctuations are entirely random, there is no net energy flux across any surface, or in other words the value of the time-averaged Poynting vector must be zero:

Nzp= (c /4π) Ezp × Bzp = 0

It is straightforward to transform the Ezp and Bzp field fluctuations from a stationary frame to one undergoing constant acceleration, customarily called a Rindler frame. Such a frame will have an asymmetric event horizon leading to a non-zero electromagnetic energy and momentum flux. The velocity, β(=v/c) and the Lorentz factor for such a frame are (where τ is the proper time):

βτ= tanh (aτ/c)

γτ= cosh (aτ/c)

The Lorentz transformation of electromagnetic fields is (cf. eqn. 11.149 in "Classical Electrodynamics" by Jackson, 1999)

E′= γ(E + β × B) - (γ2/γ+1) β (β · E)

B′= γ(B - β × E) - (γ2/γ+1) β (β · B)

Transforming Ezp and Bzp to the Rindler frame we find (where x, y, z and k all designate unit vectors since XHTML has no unit vector "hats"):

Ezp(0,τ) = λ=1,2 d3k (hω/4π3)1/2 { x εx + ycosh(aτ/c) [ εy - tanh(aτ/c) (k × ε)z ] + zcosh(aτ/c) [ εz + tanh(aτ/c) (k × ε)y ] } cos [ kx(c2/a) cosh(aτ/c) - (ωc /a) sinh(aτ/c) - θ (k,λ)]

Bzp(0,τ) = λ=1,2 d3k (hω/4π3)1/2 { x (k × ε)x + ycosh(aτ/c) [ (k × ε)y + tanh(aτ/c) εz ] + z cosh(aτ/c) [ (k × ε)z - tanh(aτ/c) εy ] } cos [ kx(c2/a) cosh(aτ/c) - (ωc /a) sinh(aτ/c) - θ (k,λ)]

The Poynting vector is no longer zero in this accelerating frame. This has to do with the fact that there is now an asymmetry in the Rindler horizon. For uniform acceleration in the x-direction, the Poynting vector is

Nzp =-x(2c /3) sinh(2aτ/c) (hω3/4π3c3) dω =-x(2c /3) sinh(2aτ/c) ρ(ω) dω

where ρ(ω) is the well-known spectral energy density of the zero-point fluctuations. An accelerating observer will see a non-zero energy and momentum flux arising out of the electromagnetic zero-point fluctuations. If some fraction of the Rindler energy-momentum flux interacts (via scattering, for example) with the quarks and electrons in an accelerating object, a reaction force will be generated. We parametrize that interaction via a dimensionless parameter, η(ω), which can be interpreted as the fraction of the electromagnetic zero-point momentum flux transiting an object which interacts with the particles constituting the object and the strength of that interaction. The time rate of change of the transiting momentum flux can be calculated from the Poynting vector, and this results in a force which turns out to be proportional to acceleration:

fzp=-[V0/c2 η(ω) ρ(ω) dω] a

(A fully covariant analysis eliminates a factor of 4/3 and also yields a proper relativistic four-vector force expression; see Rueda and Haisch 1998a, 1998b.) In order to maintain the acceleration of such an object, a motive force must continuously be applied to balance the electromagnetic reaction force

f = -fzp= [V0/c2 η(ω) ρ(ω) dω] a

Which is strongly suggestive that inertial mass is

mi=V0/c2 η(ω) ρ(ω) dω

One possible interpretation of this equation is that the inertial mass of an object of volume V0 can be traced back to the energy density of electromagnetic zero-point flux instantaneously transiting through and interacting with that object. Inertial mass of this sort would in reality be a reaction force that arises upon acceleration as a result of the Rindler energy-momentum flux. We suspect that η(ω) involves some kind of resonance at the Compton frequency (ωC=2πmc2/h), since this suggests a close connection between the origin of mass and the de Broglie wavelength, both stemming from interactions of matter with the quantum vacuum (see Haisch and Rueda 2000.)