Schwarzschild–Droste solution

2021 ◽  
pp. 229-248
Author(s):  
Andrew M. Steane
Keyword(s):  
Field Equation ◽  
Vacuum Solution ◽  
Einstein Field Equation ◽  
Accretion Disc ◽  
Circular Motion ◽  
Spherically Symmetric ◽  
De Sitter ◽  
Christoffel Symbols ◽  
Symmetric Vacuum ◽  
Physical Phenomena

The spherically symmetric vacuum solution to the Einstein field equation (Schwarzschild-Droste solution) is derived and associated physical phenomena derived and explained. It is shown how to obtain the Christoffel symbols by the Euler-Lagrange method, and hence the metric for the general spherically symmetric vacuum. Equations for general orbits are presented, and their solution for radial motion and for circular motion. Geodetic (de Sitter) precession is calculated exactly for circular orbits. The null geodesics (photon worldlines) are obtained, and the gravitational redshift. Emission from an accretion disc is calculated.

SPHERICALLY SYMMETRIC VACUUM SOLUTION IN ASHTEKAR'S FORMULATION OF GRAVITY

1991 ◽  
Vol 06 (33) ◽  
pp. 3047-3053 ◽  
Author(s):  
KIYOSHI KAMIMURA ◽  
SHINOBU MAKITA ◽  
TAKESHI FUKUYAMA
Keyword(s):  
Einstein Equation ◽  
Vacuum Solution ◽  
De Sitter Space ◽  
North Pole ◽  
Spherically Symmetric ◽  
South Pole ◽  
De Sitter ◽  
Similar Form ◽  
The North ◽  
Symmetric Vacuum

The Schwarzschild–de Sitter solution of Einstein equation is discussed in the Ashtekar formalism. The gauge connections have a similar form to those non-Abelian monopole solutions. In the de Sitter space we find a monopole at the North pole and an anti-monopole at the South pole of S3.

Further spherically symmetric solutions

2021 ◽  
pp. 249-259
Author(s):  
Andrew M. Steane
Keyword(s):  
Black Hole ◽  
Field Equation ◽  
Cosmological Constant ◽  
Electromagnetic Fields ◽  
Stellar Structure ◽  
Spherically Symmetric ◽  
De Sitter ◽  
Schwarzschild Solution ◽  
Spherically Symmetric Solutions ◽  
Structure Equations

We obtain the interior Schwarzschild solution; the stellar structure equations (Tolman-Oppenheimer-Volkoff); the Reissner-Nordstrom metric (charged black hole) and the de Sitter-Schwarzschild metric. These both illustrate how the field equation is tackled in non-vacuum cases, and bring out some of the physics of stars, electromagnetic fields and the cosmological constant.

scholarly journals A Brief Observational History of the Black-Hole Spacetimes

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Wolfgang Kundt
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Vacuum Solution ◽  
Albert Einstein ◽  
Spherically Symmetric ◽  
100Th Birthday ◽  
Black Hole Spacetimes ◽  
History Of ◽  
Symmetric Vacuum

In this year (2015), black holes (BHs) celebrate their 100th birthday, if their birth is taken to be triggered by a handwritten letter from Martin Schwarzschild to Albert Einstein, in connection with his newly found spherically symmetric vacuum solution.

scholarly journals UNIQUENESS OF STATIC SPHERICALLY SYMMETRIC VACUUM SOLUTIONS IN THE IR LIMIT OF HOŘAVA–LIFSHITZ GRAVITY

2011 ◽  
Vol 20 (01) ◽  
pp. 111-118 ◽  
Author(s):  
TOMOHIRO HARADA ◽  
UMPEI MIYAMOTO ◽  
NAOKI TSUKAMOTO
Keyword(s):  
Quantum Gravity ◽  
Cosmological Constant ◽  
Hamiltonian Constraint ◽  
Spherically Symmetric ◽  
De Sitter ◽  
Nonrelativistic Quantum ◽  
Schwarzschild Solution ◽  
Flat Rotation Curves ◽  
Symmetric Vacuum ◽  
Vacuum Solutions

We investigate static spherically symmetric vacuum solutions in the IR limit of projectable nonrelativistic quantum gravity, including the renormalizable quantum gravity recently proposed by Hořava. It is found that the projectability condition plays an important role. Without the cosmological constant, the spacetime is uniquely given by the Schwarzschild solution. With the cosmological constant, the spacetime is uniquely given by the Kottler (Schwarzschild–(anti) de Sitter) solution for the entirely vacuum spacetime. However, in addition to the Kottler solution, the static spherical and hyperbolic universes are uniquely admissible for the locally empty region, for positive and negative cosmological constants, respectively, if its nonvanishing contribution to the global Hamiltonian constraint can be compensated by the nonempty or nonstatic region. This implies that static spherically symmetric entirely vacuum solutions would not admit the freedom to reproduce the observed flat rotation curves of galaxies. On the other hand, the result for locally empty regions implies that the IR limit of nonrelativistic quantum gravity theories do not simply recover general relativity but include it.

ON A PERFECT FLUID K¨ AHLER SPACETIME

2016 ◽  
Vol 12 (3) ◽  
pp. 4350-4355
Author(s):  
VIBHA SRIVASTAVA ◽  
P. N. PANDEY
Keyword(s):  
Field Equation ◽  
Perfect Fluid ◽  
Sectional Curvature ◽  
Einstein Field Equation ◽  
Cosmological Term ◽  
Conformal Killing ◽  
Killing Vectors ◽  
Projectively Flat ◽  
Conformal Killing Vectors

The object of the present paper is to study a perfect fluid K¨ahlerspacetime. A perfect fluid K¨ahler spacetime satisfying the Einstein field equation with a cosmological term has been studied and the existence of killingand conformal killing vectors have been discussed. Certain results related to sectional curvature for pseudo projectively flat perfect fluid K¨ahler spacetime have been obtained. Dust model for perfect fluid K¨ahler spacetime has also been studied.

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