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A modification to the vis-viva equation that accounts for general relativistic effects is introduced to enhance the accuracy of predictions of orbital motion and precession. The updated equation reduces to the traditional vis-viva equation under Newtonian conditions and is a more accurate tool for astrodynamics than the traditional equation. Preliminary simulation results demonstrate the application potential of the modified vis-viva equation for more complex n-body systems. Spherical symmetry is assumed in this approach; however, this limitation could be removed in future research. This study is a pivotal step toward bridging classical and relativistic mechanics and thus makes an important contribution to the field of celestial dynamics.

Theories based on General Relativity or Quantum Mechanics have taken a leading position in macroscopic and microscopic Physics, but fail when used in the other extremity. Thus, we try to establish a new structure of united theory based on General Relativity by forming certain spacetime property and a new model of particle. This theory transforms the Riemann curvature tensor into spacetime density scalar so that gravitational field can be added to the Quantum Mechanics, and supposes the electromagnetic field in General Relativity to be a kind of spacetime fluid. Also the theory carries on Einstein-Cartan Theory about spacetime torsion to form the energy exchange in order that Hubble's law and dark mass can be simply explained. On the basis of the definition of signal, measurement and observation, the theory is able to construct a modified Quantum Equation in curved spacetime compatible with General Relativity, and hence unite the General Relativity and Quantum Mechanics in some range and puts forward a new idea of unification.

This year marks the hundredth anniversary of Einstein's 1915 landmark paper "Die Feldgleichungen der Gravitation" in which the field equations of general relativity were correctly formulated for the first time, thus rendering general relativity a complete theory. Over the subsequent hundred years physicists and astronomers have struggled with uncovering the consequences and applications of these equations. This contribution, which was written as an introduction to six chapters dealing with the connection between general relativity and cosmology that will appear in the two-volume book "One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity," endeavors to provide a historical overview of the connection between general relativity and cosmology, two areas whose development has been closely intertwined.

A generalized Robertson-Walker spacetime is the warped product with base an open interval of the real line endowed with the opposite of its metric and base any Riemannian manifold. The family of generalized Robertson-Walker spacetimes widely extends the one of classical Robertson-Walker spacetimes. In this article we prove a very simple characterization of generalized Robertson-Walker spacetimes; namely, a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field.

In this paper, by proposing a generalized $specific~volume$, we restudy the $P-V$ criticality of charged AdS black holes in the extended phase space. The results show that most of the previous conclusions can be generalized without change, but the ratio $\tildeρ_c$ should be $3 \tildeα/16$ in general case. Further research on the thermodynamical phase transition of black hole leads us to a natural interpretation of our assumption, and more black hole properties can be generalized. Finally, we study the number density for charged AdS black hole in higher dimensions, the results show the necessity of our assumption.

We consider dilaton--axion gravity interacting with $p\;\, U(1)$ vectors ($p=6$ corresponding to $N=4$ supergravity) in four--dimensional spacetime admitting a non--null Killing vector field. It is argued that this theory exibits features of a ``square'' of vacuum General Relativity. In the three--dimensional formulation it is equivalent to a gravity coupled $σ$--model with the $(4+2p)$--dimensional target space $SO(2,2+p)/(SO(2)\times SO(2+p))$. Kähler coordinates are introduced on the target manifold generalising Ernst potentials of General Relativity. The corresponding Kähler potential is found to be equal to the logarithm of the product of the four--dimensional metric component $g_{00}$ in the Einstein frame and the dilaton factor, independently on presence of vector fields. The Kähler potential is invariant under exchange of the Ernst potential and the complex axidilaton field, while it undergoes holomorphic/antiholomorphic transformations under general target space isometries. The ``square'' property is also manifest in the two--dimensional reduction of the theory as a matrix generalization of the Kramer--Neugebauer map.

The zeroth-order general Randić index $R^{0}_{a+1}$ of an $n$-vertices oriented graph $D$ is equal to the sum of $(d^{+}_{u_i})^{a}+(d^{-}_{u_j})^{a}$ over all arcs $u_iu_j$ of $D$, where we denote by $d^{+}_{u_i}$ the out-degree of the vertex $u_i$ and $d^{-}_{u_j}$ the in-degree of the vertex $u_j$, $a$ is an arbitrary real number. In the paper, we determine the orientations of cacti with the maximum value of the zeroth-order general Randić index for $a\geq 1$.

In this work we investigate the structure of white dwarfs using the Tolman-Oppenheimer-Volkoff equations and compare our results with those obtained from Newtonian equations of gravitation in order to put in evidence the importance of General Relativity (GR) for the structure of such stars. We consider in this work for the matter inside white dwarfs two equations of state, frequently found in the literature, namely, the Chandrasekhar and Salpeter equations of state. We find that using Newtonian equilibrium equations, the radii of massive white dwarfs ($M>1.3M_{\odot}$) are overestimated in comparison with GR outcomes. For a mass of $1.415M_{\odot}$ the white dwarf radius predicted by GR is about 33\% smaller than the Newtonian one. Hence, in this case, for the surface gravity the difference between the general relativistic and Newtonian outcomes is about 65\%. We depict the general relativistic mass-radius diagrams as $M/M_{\odot}=R/(a+bR+cR^2+dR^3+kR^4)$, where $a$, $b$, $c$ and $d$ are parameters obtained from a fitting procedure of the numerical results and $k=(2.08\times 10^{-6}R_{\odot})^{-1}$, being $R_{\odot}$ the radius of the Sun in km. Lastly, we point out that GR play

The quantum theory of General Relativity at low energy exists and is of the form called "effective field theory". In this talk I describe the ideas of effective field theory and its application to General Relativity.

In this note we give a general definition of the gravitational tension in a given asymptotically translationally-invariant spatial direction of a space-time. The tension is defined via the extrinsic curvature in analogy with the Hawking-Horowitz definition of energy. We show the consistency with the ADM tension formulas for asymptotically-flat space-times, in particular for Kaluza-Klein black hole solutions. Moreover, we apply the general tension formula to near-extremal branes, constituting a check for non-asymptotically flat space-times.

In the first part of this thesis, Kerr-Schild metrics and extended Kerr-Schild metrics are analyzed in the context of higher dimensional general relativity. Employing the higher dimensional generalizations of the Newman-Penrose formalism and the algebraic classification of spacetimes based on the existence and multiplicity of Weyl aligned null directions, we establish various geometrical properties of the Kerr-Schild congruences, determine compatible Weyl types and in the expanding case discuss the presence of curvature singularities. We also present known exact solutions admitting these Kerr-Schild forms and construct some new ones using the Brinkmann warp product. In the second part, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these quadratic gravities are studied in arbitrary dimension. We investigate classes of Einstein spacetimes and spacetimes with a null radiation term in the Ricci tensor satisfying the vacuum field equations of quadratic gravity and provide examples of these metrics.

An energy-momentum tensor for general relativistic spinning fluids compatible with Tulczyjew-type supplementary condition is derived from the variation of a general Lagrangian with unspecified explicit form. This tensor is the sum of a term containing the Belinfante-Rosenfeld tensor and a modified perfect-fluid energy-momentum tensor in which the four-velocity is replaced by a unit four-vector in the direction of fluid momentum. The equations of motion are obtained and it is shown that they admit a Friedmann-Robertson-Walker space-time as a solution.

In the Palatini action of general relativity the connection and the metric are treated as independent dynamical variables. Instead of assuming a relation between these quantities, the desired relation between them is derived through the Euler-Lagrange equations of the Palatini action. In this manuscript we construct an extended Palatini action, where we do not assume any a priori relationship between the connection, the covariant metric tensor, and the contravariant metric tensor. Instead we treat these three quantities as independent dynamical variables. We show that this action reproduces the standard Einstein field equations depending on a single metric tensor. We further show that in this formulation the cosmological constant has an additional theoretical significance. Normally the cosmological constant is added to the Einstein field equations for the purpose of having general relativity be consistent with cosmological observations. In the formulation presented here, the nonvanishing cosmological constant also ensures the self-consistency of the theory.

We explore whether generalised Brans -- Dicke theories, which have a scalar field $Φ$ and a function $ω(Φ)$, can be the effective actions leading to the effective equations of motion of the LQC and the LQC--inspired models, which have a massless scalar field $σ$ and a function $f(m) \;$. We find that this is possible for isotropic cosmology. We relate the pairs $(σ, f)$ and $(Φ, ω)$ and, using examples, illustrate these relations. We find that near the bounce of the LQC evolutions for which $f(m) = sin \; m$, the corresponding field $Φ\to 0$ and the function $ω(Φ) \propto Φ^2 \;$. We also find that the class of generalised Brans -- Dicke theories, which we had found earlier to lead to non singular isotropic evolutions, may be written as an LQC--inspired model. The relations found here in the isotropic cases do not apply to the anisotropic cases, which perhaps require more general effective actions.

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le σ\le 1$, where $σ$ is related to the mass per unit length of the shell.

In a class of generalized gravity theories with general couplings between the scalar field and the scalar curvature in the Lagrangian, we describe the quantum generation and the classical evolution processes of both the scalar and tensor structures in a simple and unified manner.

Considering the existence of nonconformal stochastic fluctuations in the metric tensor a generalized uncertainty principle and a deformed dispersion relation (associated to the propagation of photons) are deduced. Matching our model with the so called quantum kappa--Poincare group will allow us to deduce that the fluctuation--dissipation theorem could be fulfilled without needing a restoring mechanism associated with the intrinsic fluctuations of spacetime. In other words, the loss of quantum information is related to the fact that the spacetime symmetries are described by the quantum kappa--Poincare group, and not by the usual Poincare symmetries. An upper bound for the free parameters of this model will also be obtained.

We implement a spatially fixed mesh refinement under spherical symmetry for the characteristic formulation of General Relativity. The Courant-Friedrich-Levy (CFL) condition lets us deploy an adaptive resolution in (retarded-like) time, even for the nonlinear regime. As test cases, we replicate the main features of the gravitational critical behavior and the spacetime structure at null infinity using the Bondi mass and the News function. Additionally, we obtain the global energy conservation for an extreme situation, i.e. in the threshold of the black hole formation. In principle, the calibrated code can be used in conjunction with an ADM 3+1 code to confirm the critical behavior recently reported in the gravitational collapse of a massless scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios studied, the fixed mesh refinement offers improved runtime and results comparable to code without mesh refinement.