A gravity-anti-gravity (GaG) odd linear dilaton action offers an eternal inflation evolution governed by the unified (cosmological constant plus radiation) equation of state $rho-3P=4Lambda$. At the mini superspace level, a two-particle variant of th
e no-boundary proposal, notably one-particle energy dependent, is encountered. While a GaG-odd wave function can only host a weak Big Bang boundary condition, albeit for any $k$, a strong Big Bang boundary condition requires a GaG-even entangled wave function, and singles out $k=0$ flat space. The locally most probable values for the cosmological scale factor and the dilaton field form a grid ${a^2,aphi}simsqrt{4n_1+1}pmsqrt{4n_2+1}$.
A quantum Schwarzschild black hole is described, at the mini super spacetime level, by a non-singular wave packet composed of plane wave eigenstates of the momentum Dirac-conjugate to the mass operator. The entropy of the mass spectrum acquires then
independent contributions from the average mass and the width. Hence, Bekensteins area entropy is formulated using the $langle text{mass}^2 rangle$ average, leaving the $langle text{mass} rangle$ average to set the Hawking temperature. The width function peaks at the Planck scale for an elementary (zero entropy, zero free energy) micro black hole of finite rms size, and decreases Doppler-like towards the classical limit.
We discuss the cosmological constant problem, at the minisuperspace level, within the framework of the so-called normalized general relativity (NGR). We prove that the Universe cannot be closed, and reassure that the accompanying cosmological constan
t $Lambda$ generically vanishes, at least classically. The theory does allow, however, for a special class of $Lambda ot=0$ solutions which are associated with static closed Einstein universe and with Eddington-Lema^{i}tre universe.
We prove the existence of general relativistic perfect fluid black hole solutions, and demonstrate the phenomenon for the $P=wrho$ class of equations of state. While admitting a local time-like Killing vector on the event horizon itself, the various
black hole configurations are necessarily time dependent (thereby avoiding a well known no-go theorem) away from the horizon. Consistently, Hawkings imaginary time periodicity is globally manifest on the entire spacetime manifold.
If general relativity is spontaneously induced, the black hole limit is governed by a phase transition which occurs precisely at the would have been horizon. The exterior Schwarzschild solution then connects with a novel core of vanishing spatial vol
ume. The Kruskal structure, admitting the exact Hawking imaginary time periodicity, is recovered, with the conic defect defused at the origin, rather than at the horizon. The entropy stored inside textbf{any} interior sphere is universal, equal to a quarter of its surface area, thus locally saturating the t Hooft-Susskind holographic bound. The associated Komar mass and material energy functions are non-singular.
Hawking-Bekenstein entropy formula seems to tell us that no quantum degrees of freedom can reside in the interior of a black hole. We suggest that this is a consequence of the fact that the volume of any interior sphere of finite surface area simply
vanishes. Obviously, this is not the case in general relativity. However, we show that such a phenomenon does occur in various gravitational theories which admit a spontaneously induced general relativity. In such theories, due to a phase transition (one parameter family degenerates) which takes place precisely at the would have been horizon, the recovered exterior Schwarzschild solution connects, by means of a self-similar transition profile, with a novel hollow interior exhibiting a vanishing spatial volume and a locally varying Newton constant. This constitutes the so-called hollowgraphy driven holography.
A gravitational extension of Diracs Extensible model of the electron is presented. The Dirac bubble, treated as a 3-dim electrically charged brane, is dynamically embedded within a 4-dim $Z_{2}$-symmetric Reissner-Nordstrom bulk. Crucial to our analy
sis is the gravitational extension of Diracs brane variation prescription; its major effect is to induce a novel geometrically originated contribution to the energy-momentum tensor on the brane. In turn, the effective potential which governs the evolution of the bubble exhibits a global minimum, such that the size of the bubble stays finite (Planck scale) even at the limit where the mass approaches zero. This way, without fine-tuning, one avoids the problem so-called classical radius of the electron.
