New Gravity Field Equation is derived by Einstein Field Equation in General Relativity Theory

We found the 4-order curvature term satisfied the co-variant derivative. Einstein gravity fieldequation is consist of 2-order curvature terms. Hence, the 4-order curvature term and 2-order curvature termsmake new gravity field equation. In this point, Einstein’s gravity field equation can be modified by new 4-order curvature term because gravity field equation’s term doesn’t have to be 2-order term. Indeed, Einsteinhimself was like that, 0-order term, the cosmological term. Therefore, our theory is based on legitimate facts.

Radial dynamics of an encapsulated microbubble with interface energy

In this work a mathematical model based on interface energy is proposed within the framework of surface continuum mechanics to study the dynamics of encapsulated bubbles. The interface model naturally induces a residual stress field into the bulk of the bubble, with possible expansion or shrinkage from a stress-free configuration to a natural equilibrium configuration. The significant influence of interface area strain and the coupled effect of stretch and curvature is observed in the numerical simulations based on constrained optimization. Due to the bending rigidity related to additional terms, the dynamic interface tension can become negative, but not due to the interface area strain. The coupled effect of interface strain and curvature term observed is new and plays a dominant role in the dominant compression behaviour of encapsulated bubbles observed in the experiments. The present model is validated by fitting the experimental data of $1.7\,\mathrm {\mu }$ m, $1.4\,\mathrm {\mu }$ m and $1\,\mathrm {\mu }$ m radii bubbles by calculating the optimized parameters. This work also highlights the role of interface parameters and natural configuration gas pressure in estimating the size-independent viscoelastic material properties of encapsulated bubbles with interesting future developments.

Universe as Klein–Gordon eigenstates

We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.

The Kerr-Schild double copy in Lifshitz spacetime

The Kerr-Schild double copy is a map between exact solutions of general relativity and Maxwell’s theory, where the nonlinear nature of general relativity is circumvented by considering solutions in the Kerr-Schild form. In this paper, we give a general formulation, where no simplifying assumption about the background metric is made, and show that the gauge theory source is affected by a curvature term that characterizes the deviation of the background spacetime from a constant curvature spacetime. We demonstrate this effect explicitly by studying gravitational solutions with non-zero cosmological constant. We show that, when the background is flat, the constant charge density filling all space in the gauge theory that has been observed in previous works is a consequence of this curvature term. As an example of a solution with a curved background, we study the Lifshitz black hole with two different matter couplings. The curvature of the background, i.e., the Lifshitz spacetime, again yields a constant charge density; however, unlike the previous examples, it is canceled by the contribution from the matter fields. For one of the matter couplings, there remains no additional non-localized source term, providing an example for a non-vacuum gravity solution corresponding to a vacuum gauge theory solution in arbitrary dimensions.

The Casimir effect in the presence of infrared transparency

We revisit the Casimir effect perceived by two surfaces in the presence of infrared (IR) transparency. To address this problem, we study a model, where such a phenomenon naturally arises: the DGP model with two parallel 3-branes, each endowed with a localized curvature term. In that model, the ultraviolet modes of the 5-dimensional graviton are suppressed on the branes, while the IR modes can penetrate them freely. First, we find that the DGP branes act as “effective” (momentum-dependent) boundary conditions for the gravitational field, so that the (gravitational) Casimir force between them emerges. Second, we discover that the presence of an IR transparency region for the discrete modes modifies the standard Casimir force — as derived for ideal Dirichlet boundary conditions — in two competing ways: i) The exclusion of soft modes from the discrete spectrum leads to an increase of the Casimir force. ii) The non-ideal nature of the boundary conditions gives rise to a “leakage” of hard modes. As a result of i) and ii), the Casimir force becomes weaker. Since the derivation of this result involves only the localized kinetic terms of a quantum field on parallel surfaces (with codimension one), the derived Casimir force is expected to be present in a variety of setups in arbitrary dimensions.

Extended General Relativity for a Curved Universe

The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background, which endorses the early universe positive curvature with a confidence level exceeding 99%. Although general relativity performs accurately in the present universe where spacetime is almost flat, the necessity of dark matter/energy and the lost boundary term might be signs of its incompleteness. Utilising Einstein–Hilbert action, I present extended field equations considering the pre-existing universal curvatures. The new extended field equations are inclusive of Einstein field equations in addition to the boundary term and the conformal curvature term contributions.