Transformation Groups for a Schwarzschild-Type Geometry in f(R) Gravity

Emre Dil,Talha Zafer

doi:10.1155/2016/7636493

Abstract

We know that the Lorentz transformations are special relativistic coordinate transformations between inertial frames. What happens if we would like to find the coordinate transformations between noninertial reference frames? Noninertial frames are known to be accelerated frames with respect to an inertial frame. Therefore these should be considered in the framework of general relativity or its modified versions. We assume that the inertial frames are flat space-times and noninertial frames are curved space-times; then we investigate the deformation and coordinate transformation groups between a flat space-time and a curved space-time which is curved by a Schwarzschild-type black hole, in the framework of f(R) gravity. We firstly study the deformation transformation groups by relating the metrics of the flat and curved space-times in spherical coordinates; after the deformation transformations we concentrate on the coordinate transformations. Later on, we investigate the same deformation and coordinate transformations in Cartesian coordinates. Finally we obtain two different sets of transformation groups for the spherical and Cartesian coordinates.

Highlights

The coordinate transformations are of great importance in physics, such as nonrelativistic Galilean transformations between flat Euclidean spaces and special relativistic Lorentz transformations between flat Minkowski spacetimes
For the spherically symmetric Schwarzschild black hole solutions of general relativity (GR), we will refer to the spherically symmetric corresponding Schwarzschild-type black hole solutions of Capozziello et al [13]. In this curved and expanded space of Schwarzschild-type black hole, we will study the coordinate transformations from a noninertial frame to an inertial frame being the tangent vector space-time of the Schwarzschild-type space-time, and this flat space-time is represented by the local Lorentz coordinates (LLC)
The curvature of the space-time used here results from the Schwarzschild-type black hole whose solution for the metric tensor is obtained in the frameworks of modified f(R) gravity [7,8,9]

Summary

Introduction

The coordinate transformations are of great importance in physics, such as nonrelativistic Galilean transformations between flat Euclidean spaces and special relativistic Lorentz transformations between flat Minkowski spacetimes. For the spherically symmetric Schwarzschild black hole solutions of GR, we will refer to the spherically symmetric corresponding Schwarzschild-type black hole solutions of Capozziello et al [13] In this curved and expanded space of Schwarzschild-type black hole, we will study the coordinate transformations from a noninertial frame to an inertial frame being the tangent vector space-time of the Schwarzschild-type space-time, and this flat space-time is represented by the local Lorentz coordinates (LLC). From these transformations we will obtain the related effects, such as length contraction and time dilation, for the f(R) gravity of Schwarzschild-type black hole. We have the deformation transformation and the coordinate transformation groups in Cartesian coordinates for the same flat space-times and curved f(R) gravity Schwarzschild-type black hole space-time

Transformation Groups in Spherical Coordinates

Transformation Groups in Cartesian Coordinates

Conclusions