Non-gravitational Effects of the Metric Field over Complex Manifolds

Mapping Intimacies ◽

10.21203/rs.3.rs-163617/v1 ◽

2021 ◽

Author(s):

Swagatam Sen

Keyword(s):

General Relativity ◽

Complex Manifold ◽

Geodesic Equation ◽

Complex Manifolds ◽

Real Component ◽

Gravitational Effects ◽

Smooth Complex ◽

Straight Line ◽

Equations Of Motions

Abstract Objective of this work is to study whether some of the known non-gravitational phenomena can be explained as motion on a straight line as gravity is treated within General Relativity. To do that, we explore a metric field on a complexified manifold with holomorphic coordinates. Specifically we look into the behaviour of geodesics on such a smooth complex manifold and the path traced out by its real component. This yields a family of equations of motions in real coordinates which is shown to have deviations from usual geodesic equation and in that way expands the geodesic to capture contributions from additional fields and interactions beyond the mere gravitational ones as a function of the metric field.

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Real Projective Geodesics Embedded In Complex Manifolds}

10.31219/osf.io/g9jdc ◽

2020 ◽

Author(s):

Swagatam Sen

Keyword(s):

Complex Manifold ◽

Space Time ◽

Complex Manifolds ◽

The Real ◽

Christoffel Symbols ◽

Local Complex ◽

Mathematical Foundations ◽

Equations Of Motions

Focus of this study is to explore some aspects of mathematical foundations for using complex manifolds as a model for space-time. More specifically, certain equations of motions have been derived as a Projective geodesic on a real manifold embedded within a complex one. To that goal, first the geodesic on complex manifold has been computed using local complex and conjugate coordinates, and then its projection on the real sub-manifold has been studied. The projective geodesic, thus obtained, is shown to have additional terms beyond the usual Christoffel symbols, and hence expands the geodesic to capture effects beyond the mere gravitational ones.

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Complex Spacetimes and the Newman-Janis Trick

10.26686/wgtn.17013812 ◽

2021 ◽

Author(s):

◽

Del Rajan

Keyword(s):

General Relativity ◽

Black Hole ◽

Mathematical Theory ◽

Kerr Black Hole ◽

Complex Variables ◽

Complex Manifolds ◽

Schwarzschild Black Hole ◽

The Subject ◽

Short Method

<p>In this thesis, we explore the subject of complex spacetimes, in which the mathematical theory of complex manifolds gets modified for application to General Relativity. We will also explore the mysterious Newman-Janis trick, which is an elementary and quite short method to obtain the Kerr black hole from the Schwarzschild black hole through the use of complex variables. This exposition will cover variations of the Newman-Janis trick, partial explanations, as well as original contributions.</p>

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The precise calculations of the constant terms in the equations of motions of planets and photons of general relativity

Physics Essays ◽

10.4006/0836-1398-34.2.183 ◽

2021 ◽

Vol 34 (2) ◽

pp. 183-192

Author(s):

Mei Xiaochun

Keyword(s):

General Relativity ◽

Gravitational Field ◽

Deflection Angle ◽

Constant Term ◽

Equation Of Motion ◽

Time Dependent ◽

Newtonian Gravity ◽

Newtonian Theory ◽

The Deflection Angle ◽

Equations Of Motions

In general relativity, the values of constant terms in the equations of motions of planets and light have not been seriously discussed. Based on the Schwarzschild metric and the geodesic equations of the Riemann geometry, it is proved in this paper that the constant term in the time-dependent equation of motion of planet in general relativity must be equal to zero. Otherwise, when the correction term of general relativity is ignored, the resulting Newtonian gravity formula would change its basic form. Due to the absence of this constant term, the equation of motion cannot describe the elliptical and the hyperbolic orbital motions of celestial bodies in the solar gravitational field. It can only describe the parabolic orbital motion (with minor corrections). Therefore, it becomes meaningless to use general relativity calculating the precession of Mercury's perihelion. It is also proved that the time-dependent orbital equation of light in general relativity is contradictory to the time-independent equation of light. Using the time-independent orbital equation to do calculation, the deflection angle of light in the solar gravitational field is <mml:math display="inline"> <mml:mrow> <mml:mn>1.7</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> . But using the time-dependent equation to do calculation, the deflection angle of light is only a small correction of the prediction value <mml:math display="inline"> <mml:mrow> <mml:mn>0.87</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> of the Newtonian gravity theory with a magnitude order of <mml:math display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The reason causing this inconsistency was the Einstein's assumption that the motion of light satisfied the condition <mml:math display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in gravitational field. It leads to the absence of constant term in the time-independent equation of motion of light and destroys the uniqueness of geodesic in curved space-time. Meanwhile, light is subjected to repulsive forces in the gravitational field, rather than attractive forces. The direction of deflection of light is opposite, inconsistent with the predictions of present general relativity and the Newtonian theory of gravity. Observing on the earth surface, the wavelength of light emitted by the sun is violet shifted. This prediction is obviously not true. Practical observation is red shift. Finally, the practical significance of the calculation of the Mercury perihelion's precession and the existing problems of the light's deflection experiments of general relativity are briefly discussed. The conclusion of this paper is that general relativity cannot have consistence with the Newtonian theory of gravity for the descriptions of motions of planets and light in the solar system. The theory itself is not self-consistent too.

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8. Gravitational waves

Waves: A Very Short Introduction ◽

10.1093/actrade/9780198803782.003.0008 ◽

2018 ◽

pp. 117-126

Author(s):

Mike Goldsmith

Keyword(s):

General Relativity ◽

Gravitational Waves ◽

Time And Space ◽

Gravitational Effects ◽

New Ideas ◽

The Universe

In 1916, Einstein published his theory of general relativity, which incorporated fundamentally new ideas about the nature of gravity, including that gravitational effects take time to travel. He also showed that under some circumstances, objects lose energy by emitting ‘ripples’ in time and space: gravitational waves. ‘Gravitational waves’ explains how these waves are very weak and only the most powerful events in the Universe generate strong enough versions to be detected. Gravitational waves differ from other kinds of waves as their only effect is to cause objects to move together and then apart again. They provide a unique new window on the Universe, allowing us to look deeper and further than ever before.

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OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

Forum of Mathematics Sigma ◽

10.1017/fms.2018.16 ◽

2019 ◽

Vol 7 ◽

Author(s):

A. ASOK ◽

J. FASEL ◽

M. J. HOPKINS

Keyword(s):

Vector Bundle ◽

Complex Manifold ◽

Algebraic Variety ◽

Vector Bundles ◽

Necessary Condition ◽

Chern Classes ◽

Smooth Complex ◽

Steenrod Operations ◽

Complex Algebraic Variety ◽

Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

Georgian Mathematical Journal ◽

10.1515/gmj.2006.7 ◽

2006 ◽

Vol 13 (1) ◽

pp. 7-10

Author(s):

Edoardo Ballico

Keyword(s):

Vector Bundle ◽

Complex Manifold ◽

Vector Bundles ◽

Open Neighborhood ◽

Holomorphic Vector Bundle ◽

Stein Manifold ◽

Complex Manifolds ◽

Holomorphic Vector Bundles

Abstract Let 𝑋 be a holomorphically convex complex manifold and Exc(𝑋) ⊆ 𝑋 the union of all positive dimensional compact analytic subsets of 𝑋. We assume that Exc(𝑋) ≠ 𝑋 and 𝑋 is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle 𝐸 on 𝑋 such that is not holomorphically trivial for every open neighborhood 𝑈 of Exc(𝑋) and every integer 𝑚 ≥ 0. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood 𝑈, which are not extendable across a 2-concave point of ∂(𝑈).

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The Double Complex of a Blow-up

International Mathematics Research Notices ◽

10.1093/imrn/rnz139 ◽

2019 ◽

Cited By ~ 2

Author(s):

Jonas Stelzig

Keyword(s):

Blow Up ◽

Differential Forms ◽

Complex Manifolds ◽

Double Complex ◽

Smooth Complex ◽

Projective Bundles ◽

De Rham ◽

Suitable Notion ◽

Complex Valued ◽

Compact Complex Manifolds

AbstractWe compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for “all” cohomologies naturally associated with this complex (in particular, de Rham, Dolbeault, Bott–Chern, and Aeppli).

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Does general relativity explain gravitational effects?

Theoretical and Mathematical Physics ◽

10.1007/bf01017615 ◽

1986 ◽

Vol 69 (3) ◽

pp. 1179-1187 ◽

Cited By ~ 5

Author(s):

A. A. Logunov ◽

Yu. M. Loskutov ◽

Yu. V. Chugreev

Keyword(s):

General Relativity ◽

Gravitational Effects

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Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

Scientific Reports ◽

10.1038/s41598-020-75211-5 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Xiaobo Zhai ◽

Changyu Huang ◽

Gang Ren

Keyword(s):

General Relativity ◽

Quantum Physics ◽

Harmonic Maps ◽

Classical Mechanics ◽

Harmonic Mapping ◽

Geodesic Equation ◽

Lagrange Equations ◽

Least Action ◽

Principle Of Least Action ◽

Fundamental Equations

Abstract One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

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