On the Splitting of the Einstein Field Equations with respect to General (1+3) Threading of Spacetime by Aurel Bejancu, Hani Reda Farran

“…Based on general (1+3) threading of the spacetime (M,g), we obtain a new and simple splitting of both the Einstein field equations (EFE) and the conservation laws in (M,g). …”
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Analysis of the vacuum solution of the five-dimensional Einstein field equations with negative cosmological constant via variational symmetries by Fatemeh Ahangari

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Schwarzschild-Levi-Civita black hole by S. Habib Mazharimousavi

“…Inspired by the geometry of the Ernst black hole which interpolates the static spherically symmetric Schwarzschild black hole and the static axially symmetric magnetic Melvin spacetime, in this paper, we aim to present a solution to Einstein's field equation in vacuum to interpolate the Schwarzschild black hole and the Levi-Civita spacetime. …”
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A Relativistic Algorithm with Isotropic Coordinates by S. A. Ngubelanga, S. D. Maharaj

“…We generate new exact solutions to the Einstein field equations which also contain isotropic pressures. …”
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Spacetime Junctions and the Collapse to Black Holes in Higher Dimensions by Filipe C. Mena

“…The models are constructed through the junction of two exact solutions of the Einstein field equations: an interior collapsing fluid solution and a vacuum exterior solution. …”
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A study of classical and quantum cosmology in modified teleparallel gravity and the role of Noether symmetry by Dipankar Laya, Shriton Hembrom, Sourav Dutta, Subenoy Chakraborty

“…In the background of homogeneous and isotropic flat Friedmann–Lema $$\hat{\text{ i }}$$ i ^ tre–Robertson–Walker (FLRW) model, the Noether symmetry vector with the associated conserved charge identifies a cyclic variable so that the modified Einstein field equations get simplified and becomes solvable in the present model. …”
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Axially Symmetric-dS Solution in Teleparallel f(T) Gravity Theories by Gamal G. L. Nashed

“…We apply a tetrad field with six unknown functions to Einstein field equations. Exact vacuum solution, which represents axially symmetric-dS spacetime, is derived. …”
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Axially Symmetric Null Dust Space-Time, Naked Singularity, and Cosmic Time Machine by Faizuddin Ahmed

“…We present a gravitational collapse null dust solution of the Einstein field equations. The space-time is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. …”
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Generalized Phenomenological Models of Dark Energy by Prasenjit Paul, Rikpratik Sengupta

“…The more conventional one is to modify the matter part of the Einstein field equations, and the second one is to modify the geometry part. …”
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Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry by Ahmad T. Ali

“…For this, Lie group analysis method is used to identify the generators that leave the given system of nonlinear partial differential equations (NLPDEs) (Einstein field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions provide nontrivial solutions of the original system. …”
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Cosmologies with Scalar Fields from Higher Dimensions Applied to Bianchi Type VIh=-1 Model: Classical and Quantum Solutions by J. Socorro, L. Toledo Sesma, Luis O. Pimentel

“…The classical Einstein field equations give us a hidden symmetry, corresponding to the equality between two radii B=C, which permits us to solve exactly the equations of motion. …”
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Regular black holes from Kiselev anisotropic fluid by L. C. N. Santos

“…We extend this model by allowing w in the expression $$p_t(r)/\rho (r) = (3w + 1)/2$$ p t ( r ) / ρ ( r ) = ( 3 w + 1 ) / 2 to vary as a function of the radial coordinate, and derive new solutions to the Einstein field equations for this configuration. In particular, we study solutions that describe regular black holes. …”
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Metric-affine cosmological models and the inverse problem of the calculus of variations. Part II: Variational bootstrapping of the $$\Lambda $$ Λ CDM model by Ludovic Ducobu, Nicoleta Voicu

“…Starting from an “educated guess” that formally resembles the Einstein field equations with a cosmological “constant” (actually, a scalar function built from the metric and the connection) and a dark matter term, the method then allows to find “corrected” metric equations and to “bootstrap” the connection field equations. …”
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Constructing stellar solutions with spherical symmetry through quadratic anisotropy in f(Q) gravity by R. Kumar, S. K. Maurya, A. Errehymy, A. Jaiswal, K. Myrzakulov, S. Sharma

“…A physically meaningful metric potential $$g_{rr}$$ g rr is considered, and a quadratic form of anisotropy is utilized to solve the Einstein field equations in closed form. This class of solutions is then applied to characterize observed pulsars from various perspectives. …”
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