In General Relativity, there exist two objects of limiting compactness, one with a null boundary defining the horizon of a black hole and the other with a timelike boundary defining a Buchdahl star. The two are characterized by gravitational energy equal to or half the mass. Since non-gravitational mass-energy is the source of gravitational energy, these configurations suggest an underlying energetic balance. In this letter, we demonstrate, in a simple and general manner, that the corresponding equilibrium state always represents a minima of an appropriate quasi-local energy functional, thereby establishing energetic (variational) stability within the quasi-local configuration space.

India

Soumya Chakrabarti

Chiranjeeb Singha

South Africa

Naresh Dadhich

On the stability of the objects of limiting compactness: Black hole and Buchdahl star

We explore the connection of a general relativistic matter-energy momentum tensor with the polynomial degeneracies of curvature invariants defined in Riemannian geometry. The degeneracies enforce additional constraints on the energy-momentum tensor components. Due to these constraints the formation of a curvature singularity, for instance during a gravitational collapse can no longer be treated as inevitable. We find that there can be a formation of singularity iff the interior fluid evolves into ( <italic>i</italic>) a pressure-less dust, <math altimg='si1.svg'><mo stretchy='false'>(</mo><mi>i</mi><mi>i</mi><mo stretchy='false'>)</mo></math> an isotropic sphere or <math altimg='si2.svg'><mo stretchy='false'>(</mo><mi>i</mi><mi>i</mi><mi>i</mi><mo stretchy='false'>)</mo></math> a distribution with negative pressure.

On the polynomial degeneracy of Ricci invariants and spacetime singularity

We discuss the notion of generating a cosmic inflation without any big bang singularity. It has recently been proved by Good and Linder ( [gr-qc]) that such an expansion of the universe can be driven by quantum fluctuations embedded in vacuum. The rate of expansion is guided by a cosmological sum rule defined through the Schwarzian derivative. We explore the thermodynamic roots of Schwarzian and connect it with the surface gravity associated with an apparent horizon. In General Relativity the cosmological sum rule can be enforced only if the early universe is a Milne vacuum. We show that this restriction can be removed by considering an entropic source term in the Einstein–Hilbert action.

Acceleration from a phase of entropic balance

We present a simple toy model of cosmic acceleration driven purely by a self-interacting scalar field embedded in theory of grand unification. The scalar self-interaction is Higgs-like and provokes a spontaneous symmetry breaking. The coefficient of the quadratic term in the self-interaction potential has an evolution and it leads to a cosmic variation of proton-to-electron mass ratio, $$\mu $$ . We perform a cosmological reconstruction from the kinematic parameter jerk and discuss a few cosmological consequences of the theory. We also compare the theoretically calculated $$\mu $$ variation with the observations of molecular absorption spectra from Cesium Atomic Clock data.

V. Anagha

Selva Ganesh

Vivek Menon

A cosmological reconstruction of the Higgs vacuum expectation value

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

Self-similar collapse in Painlevé–Gullstrand coordinates

We show that the transformation of a time-evolving spherically symmetric metric tensor into a Painlevé–Gullstrand–Lemaître form brings forth a few curious consequences. The time evolution describes a non-singular gravitational collapse, leading to a bounce and dispersal of all the clustered matter, or a wormhole geometry for certain initial conditions. The null convergence condition is violated only at the onset of bounce or the wormhole formation. As an example, the requirements to develop a Simpson–Visser wormhole/regular black-hole geometry is discussed. The solution can be regarded as a new time-evolving twin of sonic dumb holes found in analog gravity.

The river model of gravitational collapse

We show that it is possible to steer clear of a spacetime singularity during gravitational collapse by considering time-variation of a fundamental coupling, in this case, the fine structure constant  $$\alpha $$  <math> <mi>α</mi> </math>  . We study a spherical distribution of cold dark matter coexisting with other fluid elements, collapsing under its own gravity. The dark matter is written as a scalar field interacting with electrically charged matter. This leads to a time variation of  $$\alpha $$  <math> <mi>α</mi> </math>   and as a consequence, a breakdown of local charge conservation within the sphere. The exterior has no such field and therefore, Einstein’s GR and standard equivalence principles remain valid. We derive the lowest possible bound on the collapse of this sphere beyond which there is a bounce and dispersal of most of the accumulated matter. We discuss the critical behavior of the system around this point and show that the bound is connected to a length scale of the order of Planck, introduced in the theory for dimensional requirements.

Can a variation of fine structure constant influence the fate of gravitational collapse?

The manuscript deals with an interacting scalar field that mimics the evolution of the so-called axion scalar dark matter or axion like particles with ultra-light masses. It is discussed that such a scalar along with an ordinary fluid description can collapse under strong gravity. The end state of the collapse depends on how the axion interacts with geometry and ordinary matter. For a self-interacting axion and an axion interacting with geometry the collapse may lead to a zero proper volume singularity or a bounce and total dispersal of the axion. However, for an axion interacting with the ordinary fluid description, there is no formation of singularity and the axion field exhibits periodic behavior before radiating away to zero value. Usually this collapse and dispersal is accompanied by a violation of the null energy condition for the ordinary fluid description.

Collapse of an axion scalar field

The role of the Raychaudhuri equation in studying gravitational collapse is discussed. A self-similar distribution of a scalar field along with an imperfect fluid in a conformally flat spacetime is considered for the purpose. The general focusing condition is found out and verified against the available exact solutions. The connection between the Raychaudhuri equation and the critical phenomena is also explored.

Shibendu Choudhury

Ananda Dasgupta

Narayan Banerjee

Self similar collapse and the Raychaudhuri equation

Exact solutions are studied in the context of modified Brans–Dicke theory. The non-linearity of the modified Brans–Dicke field equations is treated with the Euler–Duarte–Moreira method of integrability of anharmonic oscillator equation. While some solutions show a forever accelerating nature, in some cases there is a signature flip in the evolution of deceleration parameter in recent past. Importance of these latter models are studied in the context of late time acceleration of the universe. Constraints on the model parameters are obtained from Markov Chain Monte Carlo (MCMC) analysis using the Supernova distance modulus data, observational measurements of Hubble parameter, Baryon acoustic oscillation data and the CMB Shift parameter data.