AAWSAP DIRD, Antigravity for Aerospace Applications, March 2010

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(15)
whereby this term now behaves like the stress-energy tensor of the vacuum, Tv~; ,
which acts as a gravitational source:
T r,v -
Ac4
r,v
(16)
vac - 87tG g
One should note that the absence of a preferred frame in special relativity means that
r;.; must be the same (that is, isotropic or invariant) for all observers. There is only
one isotropic tensor of rank 2 that meets this requirement: riµv (the Minkowski flat
spacetime metric tensor in locally inertial frames). So in order for r;.; to remain
invariant under Lorentz transformations, the only requirement is that it must be
proportional to ri µv . But this generalizes in a straightforward way from inertial
coordinates to arbitrary coordinates by replacing ri µv with gµv, thus justifying the curved
spacetime metric tensor in Equation (16). By comparing Equation (16) with the perfect­
fluid stress-energy tensor in Equation (12), one finds that the vacuum looks like a
perfect fluid with an isotropic pressure P vac opposite in sign to the energy density P v•c •
Therefore, the vacuum must possess a negative-pressure equation of state (according
to the first law of thermodynamics):
P vac = -Pvac
(17)
The vacuum energy density should be constant throughout spacetime, since a gradient
would not be Lorentz invariant. So by substituting Equation ( 17) into PE + 3p, the
following is produced
P vac + 3 P vac = P vac + 3 ( - P vac )
(18)
=-2Pvac
< 0.
The vacuum equation of state is therefore manifestly negative. Last, when incorporating
pvac into the Einstein field equation as a gravitational source term, and comparing its
corresponding (Lorentz invariant) stress-energy tensor P vacg•" ' with Equation (16), then
the usual identification (or definition) is made that:
Ac4
(19)
P vac = S1tG
Thus the terms "cosmological constant" and "vacuum energy" are essentially
interchangeable in this perspective and mean the same thing (whereupon P vac = p.,..),
which is seen in the present-day cosmological literature.
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14