First Integrals of Shear-Free Fluids and Complexity

A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics&g...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Sfundo C. Gumede, Keshlan S. Govinder, Sunil D. Maharaj
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	shear-free fluids Einstein field equations first integrals Science Q Astrophysics QB460-466 Physics QC1-999
Acceso en línea:	https://doaj.org/article/05483e2da13c419c8c1751dda5819836
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:05483e2da13c419c8c1751dda5819836
record_format	dspace
spelling	oai:doaj.org-article:05483e2da13c419c8c1751dda58198362021-11-25T17:30:46ZFirst Integrals of Shear-Free Fluids and Complexity10.3390/e231115391099-4300https://doaj.org/article/05483e2da13c419c8c1751dda58198362021-11-01T00:00:00Zhttps://www.mdpi.com/1099-4300/23/11/1539https://doaj.org/toc/1099-4300A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>y</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∼</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>5</mn></msup></mfrac><msup><mfenced separators="" open="(" close=")"><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mrow><mo>−</mo><mn>15</mn><mo>/</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.Sfundo C. GumedeKeshlan S. GovinderSunil D. MaharajMDPI AGarticleshear-free fluidsEinstein field equationsfirst integralsScienceQAstrophysicsQB460-466PhysicsQC1-999ENEntropy, Vol 23, Iss 1539, p 1539 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	shear-free fluids Einstein field equations first integrals Science Q Astrophysics QB460-466 Physics QC1-999
spellingShingle	shear-free fluids Einstein field equations first integrals Science Q Astrophysics QB460-466 Physics QC1-999 Sfundo C. Gumede Keshlan S. Govinder Sunil D. Maharaj First Integrals of Shear-Free Fluids and Complexity
description	A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>y</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∼</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>5</mn></msup></mfrac><msup><mfenced separators="" open="(" close=")"><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mrow><mo>−</mo><mn>15</mn><mo>/</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.
format	article
author	Sfundo C. Gumede Keshlan S. Govinder Sunil D. Maharaj
author_facet	Sfundo C. Gumede Keshlan S. Govinder Sunil D. Maharaj
author_sort	Sfundo C. Gumede
title	First Integrals of Shear-Free Fluids and Complexity
title_short	First Integrals of Shear-Free Fluids and Complexity
title_full	First Integrals of Shear-Free Fluids and Complexity
title_fullStr	First Integrals of Shear-Free Fluids and Complexity
title_full_unstemmed	First Integrals of Shear-Free Fluids and Complexity
title_sort	first integrals of shear-free fluids and complexity
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/05483e2da13c419c8c1751dda5819836
work_keys_str_mv	AT sfundocgumede firstintegralsofshearfreefluidsandcomplexity AT keshlansgovinder firstintegralsofshearfreefluidsandcomplexity AT sunildmaharaj firstintegralsofshearfreefluidsandcomplexity
_version_	1718412266008739840

First Integrals of Shear-Free Fluids and Complexity

Ejemplares similares