The Erez–Rosen Solution Versus the Hartle–Thorne Solution

In this work, we investigate the correspondence between the Erez−Rosen and Hartle−Thorne solutions. We explicitly show how to establish the relationship and find the coordinate transformations between the two metrics. For this purpose the two metrics must have the same approximat...

Full description

Bibliographic Details
Main Authors:	Kuantay Boshkayev, Hernando Quevedo, Gulmira Nurbakyt, Algis Malybayev, Ainur Urazalina
Format:	Article
Language:	English
Published:	MDPI AG 2019-10-01
Series:	Symmetry
Subjects:	vacuum solutions quadrupole moment coordinate transformations
Online Access:	https://www.mdpi.com/2073-8994/11/10/1324

id	doaj-d7972a6db8c746f8a07235d62419494d
record_format	Article
spelling	doaj-d7972a6db8c746f8a07235d62419494d2020-11-25T01:56:45ZengMDPI AGSymmetry2073-89942019-10-011110132410.3390/sym11101324sym11101324The Erez–Rosen Solution Versus the Hartle–Thorne SolutionKuantay Boshkayev0Hernando Quevedo1Gulmira Nurbakyt2Algis Malybayev3Ainur Urazalina4National Nanotechnology Laboratory of Open Type, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanNational Nanotechnology Laboratory of Open Type, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanNational Nanotechnology Laboratory of Open Type, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanNational Nanotechnology Laboratory of Open Type, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanNational Nanotechnology Laboratory of Open Type, Department of Theoretical and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040, KazakhstanIn this work, we investigate the correspondence between the Erez−Rosen and Hartle−Thorne solutions. We explicitly show how to establish the relationship and find the coordinate transformations between the two metrics. For this purpose the two metrics must have the same approximation and describe the gravitational field of static objects. Since both the Erez−Rosen and the Hartle−Thorne solutions are particular solutions of a more general solution, the Zipoy−Voorhees transformation is applied to the exact Erez−Rosen metric in order to obtain a generalized solution in terms of the Zipoy−Voorhees parameter <inline-formula> <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mi>q</mi> </mrow> </semantics> </math> </inline-formula>. The Geroch−Hansen multipole moments of the generalized Erez−Rosen metric are calculated to find the definition of the total mass and quadrupole moment in terms of the mass <i>m</i>, quadrupole <i>q</i> and Zipoy−Voorhees <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> parameters. The coordinate transformations between the metrics are found in the approximation of ∼q. It is shown that the Zipoy−Voorhees parameter is equal to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mrow> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. This result is in agreement with previous results in the literature.https://www.mdpi.com/2073-8994/11/10/1324vacuum solutionsquadrupole momentcoordinate transformations
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Kuantay Boshkayev Hernando Quevedo Gulmira Nurbakyt Algis Malybayev Ainur Urazalina
spellingShingle	Kuantay Boshkayev Hernando Quevedo Gulmira Nurbakyt Algis Malybayev Ainur Urazalina The Erez–Rosen Solution Versus the Hartle–Thorne Solution Symmetry vacuum solutions quadrupole moment coordinate transformations
author_facet	Kuantay Boshkayev Hernando Quevedo Gulmira Nurbakyt Algis Malybayev Ainur Urazalina
author_sort	Kuantay Boshkayev
title	The Erez–Rosen Solution Versus the Hartle–Thorne Solution
title_short	The Erez–Rosen Solution Versus the Hartle–Thorne Solution
title_full	The Erez–Rosen Solution Versus the Hartle–Thorne Solution
title_fullStr	The Erez–Rosen Solution Versus the Hartle–Thorne Solution
title_full_unstemmed	The Erez–Rosen Solution Versus the Hartle–Thorne Solution
title_sort	erez–rosen solution versus the hartle–thorne solution
publisher	MDPI AG
series	Symmetry
issn	2073-8994
publishDate	2019-10-01
description	In this work, we investigate the correspondence between the Erez−Rosen and Hartle−Thorne solutions. We explicitly show how to establish the relationship and find the coordinate transformations between the two metrics. For this purpose the two metrics must have the same approximation and describe the gravitational field of static objects. Since both the Erez−Rosen and the Hartle−Thorne solutions are particular solutions of a more general solution, the Zipoy−Voorhees transformation is applied to the exact Erez−Rosen metric in order to obtain a generalized solution in terms of the Zipoy−Voorhees parameter <inline-formula> <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mi>q</mi> </mrow> </semantics> </math> </inline-formula>. The Geroch−Hansen multipole moments of the generalized Erez−Rosen metric are calculated to find the definition of the total mass and quadrupole moment in terms of the mass <i>m</i>, quadrupole <i>q</i> and Zipoy−Voorhees <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> parameters. The coordinate transformations between the metrics are found in the approximation of ∼q. It is shown that the Zipoy−Voorhees parameter is equal to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mrow> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. This result is in agreement with previous results in the literature.
topic	vacuum solutions quadrupole moment coordinate transformations
url	https://www.mdpi.com/2073-8994/11/10/1324
work_keys_str_mv	AT kuantayboshkayev theerezrosensolutionversusthehartlethornesolution AT hernandoquevedo theerezrosensolutionversusthehartlethornesolution AT gulmiranurbakyt theerezrosensolutionversusthehartlethornesolution AT algismalybayev theerezrosensolutionversusthehartlethornesolution AT ainururazalina theerezrosensolutionversusthehartlethornesolution AT kuantayboshkayev erezrosensolutionversusthehartlethornesolution AT hernandoquevedo erezrosensolutionversusthehartlethornesolution AT gulmiranurbakyt erezrosensolutionversusthehartlethornesolution AT algismalybayev erezrosensolutionversusthehartlethornesolution AT ainururazalina erezrosensolutionversusthehartlethornesolution
_version_	1724978091712839680

The Erez–Rosen Solution Versus the Hartle–Thorne Solution

Similar Items