dorsal/arxiv
View SchemaForm Invariance of Differential Equations in General Relativity
| Authors | Luis P. Chimento |
|---|---|
| Categories | |
| ArXiv ID | physics/9702029 |
| URL | https://arxiv.org/abs/physics/9702029 |
| DOI | 10.1063/1.531996 |
| Journal | J.Math.Phys. 38 (1997) 2565-2576 |
Abstract
Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second order nonlinear ordinary differential equation $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int{f(y) dy}+\gamma f(y)=0$. Also, it appears in the generalized statistical mechanics for the most interesting value q=-1. The invariant form of this equation is imposed and the corresponding nonlocal transformation is obtained. The linearization of that equation for any $\alpha, \beta$ and $\gamma$ is presented and for the important case $f=by^n+k$ with $\beta=\alpha ^2 (n+1)/((n+2)^2)$ its explicit general solution is found. Moreover, the form invariance is applied to yield exact solutions of same other differential equations.
{
"annotation_id": "cb8d0dec-4abf-40fd-8690-0be2391bafc4",
"date_created": "2026-03-02T18:01:18.193000Z",
"date_modified": "2026-03-02T18:01:18.193000Z",
"file_hash": "11550f5aec8eb4e578721dde868cab9c43d08bd400baf9983ffd3d7ba8d5aa9a",
"private": false,
"record": {
"abstract": "Einstein equations for several matter sources in Robertson-Walker and Bianchi\nI type metrics, are shown to reduce to a kind of second order nonlinear\nordinary differential equation $\\ddot{y}+\\alpha f(y)\\dot{y}+\\beta f(y)\\int{f(y)\ndy}+\\gamma f(y)=0$. Also, it appears in the generalized statistical mechanics\nfor the most interesting value q=-1. The invariant form of this equation is\nimposed and the corresponding nonlocal transformation is obtained. The\nlinearization of that equation for any $\\alpha, \\beta$ and $\\gamma$ is\npresented and for the important case $f=by^n+k$ with $\\beta=\\alpha ^2\n(n+1)/((n+2)^2)$ its explicit general solution is found. Moreover, the form\ninvariance is applied to yield exact solutions of same other differential\nequations.",
"arxiv_id": "physics/9702029",
"authors": [
"Luis P. Chimento"
],
"categories": [
"math-ph",
"math.MP"
],
"doi": "10.1063/1.531996",
"journal_ref": "J.Math.Phys. 38 (1997) 2565-2576",
"title": "Form Invariance of Differential Equations in General Relativity",
"url": "https://arxiv.org/abs/physics/9702029"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "d7a658ae-6a3a-470f-ba18-5f9d9647727f",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}