There is NO ZERO POINT ENERGY in ordinary non-exotic vacuum. It is all absorbed into the Higgs Ocean that is then completely calm without any surfy foam of breaking waves.
Exotic vacuum means foamy breaking waves in the Higgs Ocean for UFOs with warp drive to SURF on! These foamy waves are not to be confused with quantum gravity foam, which does not exist in this theory!
The exotic vacuum effective gravity potential energy per unit test particle mass induced by its zero point energy is in the spherically symmetric S wave case of the static Green's function multipole expansion of the Poisson equation weak field slow speed limit of Einstein's field equation is
Vzpf ~ c^2/\zpfr^2
Example 1: Pioneer Anomaly with local SINGLE-VALUED Higgs Ocean Order Parameter in S^2 space;
In the case of the Pioneer anomaly, the Higgs Ocean has an exotic vacuum hedgehog point topological defect at the center of the Sun.
/\zpf = 0 from r = 0 at center of the Sun out to r ~ 20AU at the orbit of Uranus.
From that point on out to some outer limit r* yet to be measured by future spinning space-probes
/\zpf = H(t)/cr
H(t) = R(t)^-1dR(t)/dt
R(t) is the DIMENSIONLESS FRW scale factor for the currently accelerating expansion of 3D space.
Therefore, in this zero point energy exotic vacuum hedgehog region between the inner and outer spherical surfaces centered at the Sun
V(zpf) = cHr = (speed of light)(Hubble recession speed)
The observed Pioneer anomalous acceleration is
a_P = cH ~ 10^-7 cm/sec^2
This is known to a precision of ~ 10^-8 cm/sec^2 therefore all claims of a minimal acceleration of c^2/(Hubble scale) from the quantum gravity foam of geometry are falsified.
Example 2: Quantum Gravity Foam does not exist! - prediction
There is no quantum gravity foam in the Higgs Ocean! Therefore, anomalous dispersion of cosmic gamma rays from
E^2 = [(pc)^2 + (mc^2)^2]/[1 + (pc)^2/Ep^2]
With E --> Ep as pc/mc^2 >> 1
should NOT be seen. So far they haven't been seen.
Example 3: Dark Matter Galactic Halo of "Dark Matter" surrounding the central large black hole keeping the outer stars in the galaxy
There may also be a hole in the zero point energy exotic vacuum halo there as well, but it is not a hedgehog, the flat circulating tangential velocity v* rotation curves imply
/\zpf ~ 1/r^2
V(zpf) independent of r
We can also write
V(zpf) = cv*
where v* is the constant circulating tangential speed of the stars in the dark matter Galactic Halo
Reply to Zielinski:
We are spinning our wheels here. You ask questions that make no sense to me. You seem to think that P_k being "frame dependent" is a problem. Are you saying P_kP^-k is not frame invariant?
Second of all WHAT set of "frames" do you mean? Define them! When you do you will see they must be global frames in the asymptotically flat region outside the finite region of curvature.
Diff(4) only has LOCAL FRAMES. Therefore the frame of the integrated quantity P_k is NOT part of Diff(4) but is for the asymptotically flat approximate Poincare group! Yet you say Pauli does not mention that. That is a contradiction. Pauli probably assumed that was so obvious that he was not explicit about it.
Let's put this on hold because I would have to do a careful reread of Pauli, MTW etc. on this that will take a lot of time.
The physical reason for the use of the pseudo-tensor for stress-energy density currents is the splitting of the dynamical degrees of freedom of the full metric field into off-mass-shell nonpropagating near field background + propagating on-mass-shell far field gravity waves. Also you need to put in the EM field explicitly as part of the detectors - a point made by Rovelli BTW.
Einstein's GR Diff(4) field equations TRIVIALLY conserve stress-energy density currents LOCALLY. GR is only a local field theory. The P_k are NOT local Diff(4) objects. They only have meaning in special space-times that are asympotically flat with a DIFFERENT emergent approximate asymptotic GLOBAL Poincare group that has no Diff(4) anymore. The "global frames" for P_k are NOT Diff(4) local frames! If you confuse apples with oranges you get in trouble.
Doing global integrals in curved space-time is tricky business and involves topology - global constraints beyond the local field equations. So what you have is a new set of theories i.e.
Local GR (1915) + Global Topological Constraints
The latter introduced "nonlocality" - not same as "quantum"
Einstein's local field equations are simply
Sum of all relevant stress-energy density current tensors = 0
In the 1915 theory
tuv(geometry) + Tuv(matter) = 0
tuv(geometry) = (c^4/8piG)Guv
Guv = Ruv - (1/2)Rguv
Now in 1915 theory the Bianchi identities give the vanishing covariant divergence
tuv(geometry)^;v = 0
Therefore, the "wood" matter currents are locally CONSERVED separately without any direct exchange with geometry.
The "marble geometry" is in a separate compartment from the "wood matter".
Tuv(matter)^;v = 0
ALL BY ITSELF!
Metric engineering of warp, wormhole and weapon is IMPOSSIBLE in this limit!
Indeed, metric engineering requires the violation of the 1915 Bianchi identities.
Furthermore in this limit in VACUUM defined by
Tuv(matter) = 0
tuv(geometry) = 0
Therefore the total local stress-energy current density Diff(4) tensor is WELL DEFINED and is EXACTLY ZERO!
P_u is a horse of a different color - an integral of a far field pseudo-tensor piece of tuv(geometry) = 0
Now in my theory I DERIVE Einstein's GR theory of gravity as a limiting case of the variations in the Goldstone Phase Field of the post-inflationary macro-quantum coherent calm partially random highly-ordered massive Higgs Ocean that emerges from the spontaneous symmetry-breaking instability in the micro-quantum incoherent stormy completely random massless clothed Dirac Sea including w = -1 zero point virtual bosons of positive energy density and equal and opposite negative pressure as well as w = -1 zero point virtual fermion-antifermion pairs of negative energy density and equal and opposite positive pressure.
Also I throw away Tuv(matter) completely in first approximation as a small perturbation because in the FRW limit of cosmology
Omega(matter) ~ 0.04 with w = 0 and w = 1/3 components
w = pressure/(energy density)
Omega(exotic vacuum) ~ 0.96 with w = -1
Dark matter detectors will not click with The Right Stuff, i.e. dark matter is virtual exotic vacuum not real exotic particle on mass shell.
Exotic vacuum can either anomalously gravitate or ANTI-GRAVITATE depending on TWO FACTORS
1. Sign of the virtual energy density
2. Space distribution of the energy density.
The exotic vacuum field equation is
Guv + /\zpfguv = 0
Guv^;v =/= 0
This is necessary for metric engineering.