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result(s) for
"Einstein equations"
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General relativistic self-similar waves that induce an anomalous acceleration into the standard model of cosmology
We prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form
a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical (
eBook
On gravity : a brief tour of a weighty subject
\"In On Gravity, physicist A. Zee combines profound depth with incisive accessibility to take us on an original and compelling tour of Einstein's general theory of relativity. Inspired by Einstein's audacious suggestion that spacetime could ripple, Zee begins with the stunning discovery of gravity waves. He goes on to explain how gravity can be understood in comparison to other classical field theories, presents the idea of curved spacetime and the action principle, and explores cutting-edge topics, including black holes and Hawking radiation. Zee travels as far as the theory reaches, leaving us with tantalizing hints of the utterly unknown, from the intransigence of quantum gravity to the mysteries of dark matter and energy. Concise and precise, and infused with Zee's signature warmth and freshness of style, On Gravity opens a unique pathway to comprehending relativity and gaining deep insight into gravity, spacetime, and the workings of the universe\"--Publisher's website.
Book
General Relativity and the Einstein Equations
General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. This book overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.
eBook
Why does e=mc2 : (and why should we care?)
Looks at Einstein's famous equation; explaining and simplifying the notions of energy, mass, and light and exploring commonly held misconceptions.
Book
Gravidynamics, spinodynamics and electrodynamics within the framework of gravitational quantum field theory
By noticing the fact that the charged leptons and quarks in the standard model are chirality-based Dirac spinors since their weak interaction violates maximally parity symmetry though they behave as Dirac fermions in electromagnetic interaction, we show that such a chirality-based Dirac spinor possesses not only electric charge gauge symmetry U(1) but also inhomogeneous spin gauge symmetry WS(1,3)=SP(1,3)⋊W
1,3
, which reveals the nature of gravity and spacetime. The gravitational force and spin gauge force are governed by the gauge symmetries W
1,3
and SP(1,3), respectively, and a biframe spacetime with globally flat Minkowski spacetime as base spacetime and locally flat gravigauge spacetime as a fiber is described by the gravigauge field through emergent non-commutative geometry. The gauge-geometry duality and renormalizability in gravitational quantum field theory (GQFT) are carefully discussed. A detailed analysis and systematic investigation on gravidynamics and spinodynamics as well as electrodynamics are carried out within the framework of GQFT. A full discussion on the generalized Dirac equation and Maxwell equation as well as Einstein equation and spin gauge equation is made in biframe spacetime. New effects of gravidynamics as extension of general relativity are particularly analyzed. All dynamic equations of basic fields are demonstrated to preserve the spin gauge covariance and general coordinate covariance due to the spin gauge symmetry and emergent general linear group symmetry GL(1,3,R), so they hold naturally in any spinning reference frame and motional reference frame.
Journal Article
One-Dimensional Subspaces of the SL(n,R) Chiral Equations
In this work we find solutions of the (
n
+
2
)-dimensional Einstein Field Equations (EFE) with
n
commuting Killing vectors in vacuum. In the presence of
n
Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation
(
α
g
,
z
¯
g
-
1
)
,
z
+
(
α
g
,
z
g
-
1
)
,
z
¯
=
0
with
g
∈
S
L
(
n
,
R
)
. The other block reduces to the differential equation
(
ln
f
α
1
-
1
/
n
)
,
z
=
1
/
2
α
tr
(
g
,
z
g
-
1
)
2
and its complex conjugate. We use the ansatz
g
=
g
(
ξ
)
, where
ξ
satisfies a generalized Laplace equation, so the chiral equation reduces to a matrix equation that can be solved using algebraic methods, turning the problem of obtaining exact solutions for these complicated differential equations into an algebraic problem. The different EFE solutions can be chosen with desired physical properties in a simple way.
Journal Article
Signatures of quantum geometry from exponential corrections to the black hole entropy
It has been recently shown in Chatterjee and Ghosh (Phys Rev Lett 125:041302, 2020, https://doi.org/10.1103/PhysRevLett.125.041302) that microstate counting carried out for quantum states residing on the horizon of a black hole leads to a correction of the form
exp
(
-
A
/
4
l
p
2
)
in the Bekenstein-Hawking form of the black hole entropy. In this paper, we develop a novel approach to obtain the possible form of the spacetime geometry from the entropy of the black hole for a given horizon radius. The uniqueness of this solution for a given energy-momentum tensor has also been discussed. Remarkably, the black hole geometry reconstructed has striking similarities to that of noncommutative-inspired Schwarzschild black holes (Nicolini et al. in Phys Lett B 632:547, 2006). We also obtain the matter density functions using Einstein field equations for the geometries we reconstruct from the thermodynamics of black holes. These also have similarities to that of the matter density function of a noncommutative-inspired Schwarzschild black hole. The conformal structure of the metric is briefly discussed and the Penrose–Carter diagram is drawn. We then compute the Komar energy and the Smarr formula for the effective black hole geometry and compare it with that of the noncommutative-inspired Schwarzschild black hole. We also discuss some astrophysical implications of the solutions. Finally, we propose a set of quantum Einstein vacuum field equations, as a solution of which we obtain one of the spacetime solutions obtained in this work. We then show a direct connection between the quantum Einstein vacuum field equations and the first law of black hole thermodynamics.
Journal Article
Mathematical Theory of the Expanding Universe Based on the Principle of Least Action
In classical works, equations for gravitation and electromagnetic fields are proposed without deriving their right-hand sides. In this paper, we derive the right-hand sides and analyze the energy–momentum tensor in the framework of the Vlasov–Maxwell–Einstein equations. Additionally, cosmological models of the Milne–McCrea and Friedmann type are considered.
Journal Article