Quantum Foam, Spacetime and Non-Linear Multigravity Theory

Credit-address for the header photo. In this post, I will discuss and use non-linear multigravity theory to model quantum foam and probe solutions to the cosmological constant cosmic/Planck-scales ‘discrepancy paradox’, related to the hierarchy problem: namely, the 10-47GeV 4/EZP E ≈ 1071GeV cut-off one. Note first that spacetime/quantum randomness-foamy-chaos can be interpreted as a large N composition of Schwarzschild wormholes with a scalar curvature R in n-dimensions being

 

eq1

 

with \Sigma the Regge-Wheeler hypersurface and let me begin with the action involving N massless gravitons without matter fields

 

eq2.2

 

with {\Lambda _i} and {G_i} being the cosmological constant and the Newton constant corresponding to the i-th universe, respectively, and the total action takes the following form

    \[{S_{tot}} = \sum\limits_{i = 1}^N {S\left[ {{g_i}} \right]} + \lambda {S_{{\mathop{\rm int}} }}\left( {{g_1},...,{g_N}} \right)\]

In this way, the action {S_0} describes a Bose-Einstein condensates of gravitons. Start with the N = 1 Einstein field equations

 

eq3

 

with {G_{\mu \nu }} the Einstein tensor and injecting a time-like unit vector {u^\mu } such that u \cdot u = - 1 yields

    \[{G_{\mu \nu }}{u^\mu }{u^\mu } = {\Lambda _c}\]

which is the Hamiltonian constraint expressed in terms of the equation of motion. So far, we are at the classical ‘level’. The discrepancy between the observed classically-regimed cosmological constant and the quantum-numerical result is in its quantum version: that is, one can derive the expectation value

    \[\left\langle {{\Lambda _c}} \right\rangle \]

and given that

eq4.1

 

dimensional analysis for 3-D gives us

 

eq4.2

with the r.h.s. equal to

 

eq4.3

A

 

and we integrated over the Regge-Wheeler hypersurface and divided by its volume, and it can be derived starting with the Wheeler-De Witt equation which represents invariance under time reparametrization: that is the Sturm-Liouville cosmological constant problem

The boundary conditions are given by the choice of the quantum fluctuational Gaussian wavefunctionals. Extracting the TT tensor, second order in perturbation, contribution from A, of the spatial part of the metric into a background term, {\bar g_{ij}} and a perturbation, {h_{ij}} , yields

 

eq4.4

B

 

with {G^{ijk}} the inverse DeWitt metric and with the following definition

    \[{K^ \bot }{\left( {\vec x,\vec y} \right)_{iakl}} = \sum\limits_\tau {\frac{{h_{ia}^{(\tau ) \bot }(\vec x)h_{kl}^{(\tau ) \bot }(\vec y)}}{{2\lambda (\tau )}}} \]

for the propagator

    \[{K^ \bot }{\left( {x,x} \right)_{iakl}}\]

with h_{ia}^{\left( \tau \right) \bot }\left( {\vec x} \right) the eigenfunctions of {\Delta _2}.

Now, the expectation value of \hat \Lambda \begin{array}{*{20}{c}} \bot \\\Sigma \end{array} is obtained by inserting the form of the propagator into A and minimizing with respect to the variational function \lambda \left( \tau \right). So, the total one loop energy density for TT tensors is

    \[\frac{\Lambda }{{8\pi G}} = - \frac{1}{{4V}}\sum\limits_\tau {\left[ {\sqrt {\omega _1^2\left( \tau \right)} + \sqrt {\omega _2^3\left( \tau \right)} } \right]} \]

and its contribution to the spin-two operator for the Schwarzschild metric is

 

eq6

 

and {\Delta _S} is the scalar curved Laplacian, given by

 

eq7

with

    \[R_i^a = \left\{ { - \frac{{2MG}}{{{\tau ^3}}},\frac{{MG}}{{{\tau ^3}}},\frac{{MG}}{{{\tau ^3}}}} \right\}\]

the mixed Ricci tensor

hence, the scalar curvature is traceless

Thus, we must analyse the eigenvalue equation

    \[\left( {{\Delta _2}{h^{TT}}} \right)_i^j = {\omega ^2}h_j^i\]

{\omega ^2} the eigenvalue of the corresponding equation. Following Regge-Wheeler, the 3-D gravitational perturbation is represented by its even-parity form

    \[\left( {{h^{even}}} \right)_j^i\left( {\tau ,\vartheta ,\phi } \right) = {\rm{diag}}\left[ {\not H(\tau ),K(\tau ),L(\tau )} \right]{\Upsilon _{lm}}\left( {\vartheta ,\phi } \right)\]

Hence, the system

    \[\left( {{\Delta _2}{h^{TT}}} \right)_i^j = {\omega ^2}h_j^i\]

from the throat of the bridge, becomes

 

eq8

 

Hence, we have, for \tau \equiv \tau \left( x \right) and

    \[\left\{ {\begin{array}{*{20}{c}}{m_1^2\left( \tau \right) = {U_1}\left( \tau \right) = m_1^2\left( {\tau ,M} \right) - m_2^2\left( {\tau ,M} \right)}\\{m_2^2\left( \tau \right) = {U_2}\left( \tau \right) = m_1^2\left( {\tau ,M} \right) + m_2^2\left( {\tau ,M} \right)}\end{array}} \right.\]

and

 

eq9

 

So, one can write

    \[\left\{ {\begin{array}{*{20}{c}}{m_1^2\left( \tau \right) \simeq - \,m_2^2\left( {{\tau _0},M} \right)}\\{m_2^2\left( \tau \right) \simeq + \,m_2^2\left( {{\tau _0},M} \right)}\end{array}} \right.\]

where we have

    \[\left\{ {\begin{array}{*{20}{c}}{{\tau _0} > 2MG}\\{m_0^2\left( {{\tau _0},M} \right) = 3MG/\tau _0^3}\end{array}} \right.\]

We can now explicitly evaluate:

    \[\frac{\Lambda }{{8\pi G}} = - \frac{1}{{4V}}\sum\limits_\tau {\left[ {\sqrt {\omega _1^2\left( \tau \right)} + \sqrt {\omega _2^3\left( \tau \right)} } \right]} \]

in terms of the effective mass. Via the ‘t Hooft BWM method, one can derive

    \[\begin{array}{c}{\rho _i}\left( \varepsilon \right) = \frac{{m_i^4\left( \tau \right)}}{{256{\pi ^2}}}\left[ {\frac{1}{\varepsilon } + {\rm{In}}\left( {\frac{{{\mu ^2}}}{{m_i^2\left( \tau \right)}}} \right) + 2\,{\rm{In}}\,2 - \frac{1}{2}} \right]\\i = 1,2\end{array}\]

by using the zeta function regularization method to numerically analyse the energy densities {\rho _i} and by introducing the mass parameter \mu so as to restore the correct dimension for the regularized quantities. So the energy density is renormalized due to the absorption divergences, yielding the classical constant

eq10

 

removing the dependence on the mass scale \mu, it is appropriate to use the renormalization group equation, which means imposing:

 

eq11

 

After solving, one realizes that the renormalized constant {\Lambda _0} should be treated as a running one

eq12

and the cosmological constant takes the form

 

eq13

 

with a minimum

    \[\frac{{m_0^2\left( {{\tau _0},M} \right)\sqrt e }}{{4\mu _0^2}} = \frac{1}{{\sqrt e }}\]

and the condition

    \[\frac{{{\Lambda _0}\left( {{\mu _0},\tau } \right)}}{{8\pi G}} = - \frac{{\mu _0^4}}{{16{e^2}{\pi ^2}}}\]

We are …