# Using _Maple_ and _GRTensorIII_ in relativistic spherical models

Medina V Departamento de Fisica - Facultad de Ingenieria - Universidad de
Carabobo. Valencia, Venezuela<EMAIL_ADDRESS>
###### Abstract

This article presents some aspects and experience in the use of algebraic
manipulation software applied to general relativity. Some years ago certain
results were reported using computer algebra platforms, but the growing
popularity of graphical platforms such as Maple allows us to approach the
problem of the simplifications of many expressions from another point of view.
Some simple algebraic programming procedures are presented (in Maple with the
GRTensorIII package) to obtain and study material distributions with spherical
symmetry and to search for exact solutions of the Einstein field equations.
The purpose is to show how useful a computer algebra system can be. All
calculations were performed using the GRTensorIII computer algebra package,
which runs on Maple 2017, along with several Maple routines that we have used
specifically for the simplification of many of the algebraic expressions that
are very common in this type of problem.

###### keywords:

Computer algebra, General relativity , Gravitation theory , Algorithms,
_Maple_ ,_GRTensorIII_

PACS: 95.30.Sf, 04.20.- q, 04.40.-b

##  1 Introduction

Computer algebra systems (CAS) have a wide variety of applications in fields
that require time-consuming, difficult-to-perform, and error-prone
calculations when done manually. They are used more and more frequently and
especially when it is necessary to complete a calculation in pages after pages
for many hours or perhaps several days. As early as the late 1950s and early
1960s, various programs appeared [1, 2, 3, 4] aimed at demonstrate that in the
scientific field it is possible to go beyond the purely numerical area and use
them to carry out a symbolic calculation. For reasons such as these, various
members of the computer algebra family of languages were created, and in fact
many were created to perform specific calculations of great complexity in
fields such as electronic optics [5], celestial mechanics [6], quantum
electrodynamics [7, 8, 9, 10, 11, 12] or general relativity [13, 14, 15, 16].
Computer algebra for general relativity (GR) has a long history, beginning
almost as early as computer algebra itself in the 1960s.

The first GR program was GEOM, written by J.G. Fletcher [17]. Its main ability
was to compute the Riemann tensor of a given metric. In 1969, R. A. d’Inverno
[18] developed ALAM (for Atlas Lisp Algebraic Manipulator) [19] and used it to
compute the Riemann and Ricci tensors of the Bondi metric. According to [20],
Bondi and his collaborators took 6 months to complete the original
calculations, while the ALAM calculation took 4 minutes and resulted in the
discovery of 6 errors in the original paper. Since then, numerous packages
have been developed and various investigations have been carried out using
manipulative algebraic software (CAS) and in many cases the calculations were
of such length that they would have been prohibitively expensive to complete
without the aid of a computer [21]. Its main advantage is the ability to
handle a large number of algebraic calculations and this particularity has
allowed advances in fields of theoretical physics such as GR or High Energy
Physics (HEP).

Many of these problems have used some free, open source, general purpose
software with an emphasis on tensor calculus for GR, such as Java-based
REDBERRY [22, 23] , SAGE [24] and SAGEMANIFOLDS written in Python, CADABRA[25,
26] developed in C++ and Python. Others, applied specifically for HEP, have
been designed based on special algorithms [27, 28, 12] and implemented in
programs like SCHOONSHIP [29, 30], designed by M. Veltman, ASHMEDAI [31, 32,
33] by M. Levine, REDUCE [7, 8, 9, 11] by A. Hearn, MACSYMA (to become MAXIMA
in 1998) [34, 35, 36] by J. Moses developed at MIT, or more recent FORM [37,
38, 39] by J. Vermaseren. Proprietary software such as MATHEMATICA [40, 41,
42, 43]by S. Wolfram or MAPLE [44, 45, 46, 47, 48, 49, 50, 51, 52] from B.
Char developed at the University of Waterloo. The CAS programs mentioned above
constitute only a very small part of the available applications, special
purpose and general systems that can be consulted in repositories and lists,
maintained and frequently updated 53, 54, 55, 56, 57, 58. Of all the CAS
applications mentioned above, we are going to refer in this work to MAPLE.

Maple is a general purpose CAS, initially developed at the University of
Waterloo as a result of discussions on the state of symbolic computing in the
1980s. At the time, large systems such as ALTRAN [59], CAMAL [13, 14, 15],
REDUCE [60] and MACSYMA [34], based on the computer technology of the 1960s,
they decided to design a new system from scratch, taking advantage of the
advances in software engineering available and the lessons of experience. Its
basic design features (for example, 61 elementary data structures,
input/output, number arithmetic, and elementary simplification) are encoded in
a low-level language for efficiency. An important property is that most of the
algebraic facilities of the system are implemented using the high-level user
language. The basic system, or kernel, is compact and efficient enough to be
practical to use for a shared environment or on personal computers with very
little main memory. Library functions are loaded into the system as needed,
adding features such as polynomial factorization, equation solving, indefinite
integration, and matrix manipulation to the system. The modularity of this
design allows users to demand available computing resources in proportion to
their actual use. It has specialized libraries for elementary and special
mathematical functions, it offers support for symbolic and numerical
computation with exact results, it can handle a wide set of equation systems,
including Diophantine equations, ordinary differential equations (ODE),
partial differential equations (PDE), Differential Algebraic Equations (DAE),
Delay Differential Algebraic Equations (DDE) and recurrence relations.
Initially the kernel of the system was written in macros that could be
translated by a locally developed macroprocessor (called Margay) into versions
of the kernel in the C programming language for various operating systems;
currently only C is used. The GUI was first released with version 5 (Maple V)
and continued to be numbered until Maple 18 and then changed to a yearly
label. Due to the low demand for main memory to run the kernel and the modular
design for many of the possible user applications in a high-level language, it
has become one of the main symbolic algebra systems used by many researchers
and engineering corporations. worldwide [62, 63]. These applications or
packages can be programmed by the user and many can be found on the web 64 as
_GRTensorII_ or its update _GRTensor III_ . (in short _GRTensor_).

_GRTensor_ is a computer algebra package for performing calculations in the
general area of differential geometry 65. Its main objective is the
computation of tensor components in curved spacetimes specified in terms of a
metric or set of basis vectors. The library relies on a series of special
commands starting with $\mathit{"gr"}$ (for example,
$\mathit{grcalc,grdisplay,gralter,grdefine}$, etc.) to deal with a series of
(pre) geometric objects. defined as the metric tensor, Ricci tensor and
scalar, Einstein tensor, Chrisstoffell symbols, etc. This library of objects
can be extended to define new tensors, or use the Newman-Penrose formalism.
Although originally designed for use in the field of general relativity,
GRTensor is useful in many other fields 66 There is a version for MATHEMATICA
called _GRTensorM_. The _GRTensorII_ package was originally developed for
_Maple V_ and can be run with versions from _Maple V Release 3_ to _Maple 13_.
The _GRTensor III_ version runs as of _Maple 15_. All documentation and
software are distributed free of charge to help both for research and
teaching.

The final objective of this article is to present the calculations, especially
in Sec. 2 that have been carried out with an emphasis on the methods, packages
and techniques of computational algebra that we use in a spreadsheet developed
for the _GRTensorIII_ package running on the _Maple 17_ platform. Initially,
it was written in _GRTensorII_ running on top of _Maple 13_ and when upgrading
_GRTensor_ to version III, it became necessary to upgrade the _Maple_ version
as well. The spreadsheet follows the algorithm described in 67, 68 starting
from a spherically symmetric perfect fluid distribution of density
$\hat{\rho}$, radial pressure $\hat{P}$, tangential pressure $\hat{P}_{t}$, a
flux of unpolarized radiation moving in the radial direction with density
$\hat{\varepsilon}$ and $3\hat{\mu}$ the isotropic radiation of the energy
density, in a local Minkowskian system. By performing the corresponding
coordinate transformations, to a radiative (Bondi) coordinate system, we can
construct an Impulse Energy Tensor that must satisfy both the Einstein field
equations $\left(G_{ab}=\kappa T_{ab}\right)$ as the conservation equations
$\left(T_{\>;\,b}^{ab}=0\right)$. When calculating these equations, it is
necessary to simplify them and for this, some 69, procedures and functions
have been used, as in 70. In Sec. 3 some comments on the calculations made are
presented and in the last section the conclusions and the possibility of
extending this procedure to other astrophysical scenarios are detailed.

##  2 Algorithm structure

As indicated in the previous section, the created spreadsheet basically
follows the procedure used in [71] and for the charged case in [72] with the
modifications made in 67, 68. In the original sheet, the commands are grouped
by sections and each of the sections has a name that suggests the calculation
to be carried out. In this build for the article, commands (in red) and output
(in blue) from the Maple interpreter will be reproduced. All the commands to
perform the calculations of the sheet are present in this work, however, not
all the outputs (in blue) were placed and the most extensive ones were edited.

###  2.1 Login

To start the Maple session we loaded the _GRTensorIII_ libraries, the
expression simplification routines were taken from Davies [70] and the
directory from where the metrics will be loaded. We start with a reset
statement to ensure that you have a new session of the sheet,

  * >restart;

>grtw();

$\begin{array}[b]{c}"\\\ \mathit{GRTensorIII\>v2.2\>Oct\,1,\>2018}"\\\
"\mathit{Copyright\>2018,\>Peter\>Musgrave,\>Denis\>Pollney,\>Kayll\>Lake}"\\\
\mathit{"Latest\>version\>is\>at\>http://github.com/grtensor/grtensor"}\\\
\mathit{"For\>help\>?grtensor"}\\\
<EMAIL_ADDRESS>\mathit{\boldsymbol{module()\>...end\>module()}}\end{array}$ (1)

  * >with(grtensor): groptions():

The expression simplification routines were taken from Davies [70]

  * >read ‘/home/victor/relatividad/maple/loadmpl/pertutils.mpl‘;

###  2.2 Minkowskian local flat space

It is assumed that the region of the space to be considered is composed of
energy density anisotropic material $\hat{\rho}$, radial pressure$\hat{P}$,
tangential pressure $\hat{P}_{t}$, radiation isotropic energy density
$3\hat{\mu}$ and non-polarized radiation of energy density $\hat{\xi}$
propagating in a radial direction. We are going to establish the metric of
this region of space.

  * >qload(minkowski);

Calculated ds for minkowski (0.001000 sec.)

$\begin{array}[b]{c}\mathit{Default\,spacetime=minkowski}\\\
\mathit{For\,the\,minkowski\,spacetime:}\\\ \mathit{Coordinates}\\\
\mathit{x(up)}\\\ \mathit{x^{\alpha}=\left[t\;x\;y\;z\right]}\\\ \mathit{Line\
element}\\\ \mathit{ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2}}\\\ \mathit{The\
Minkowski\>metric-plana-}\end{array}$ (2)

The covariant components of the metric tensor of the local Minkowski system
are calculated and displayed. You can use two _GRTensorIII_ instructions:
$\mathit{grcalc()}$ to calculate and $\mathit{grdisplay()}$ to show the
result. However, $\mathit{grcalcd()}$, allows you to do the calculation and
shows the result, combining the two previous instructions.

  * >grcalcd(g(dn,dn));

Let’s set the values for the contravariant unit vectors in the Minkowski
reference system. We will use them to calculate the components of the tensor
in the local system

  * >grdef(‘u0{^a }:=kdelta{$t ^a} ‘): grdef(‘u1{^a }:=kdelta{$x ^a} ‘):

grdef(‘u2{^a }:=kdelta{$y ^a} ‘): grdef(‘u3{^a }:=kdelta{$z ^a} ‘):

And to establish the Stress-energy tensor of this region, let’s start with:
Energy density anisotropic material $\rho$, Radial pressure $P$ and Tangential
pressure $P_{t}$

  * >grdef(‘TM{^a b}:= (rho_M + P_t)*u0{^a }*u0{b }- P_t *g{^a b} + (P_M - P_t)*u1{^a}*u1{b} ‘):

>grcalcd(TM(dn, dn));

Isotropic radiation of energy density $3\mu$

  * >grdef(‘Tp{^a b}:= 3*u0{^a }*u0{ b} +u1{^a }*u1{ b}+u2{^a }*u2{ b}+u3{^a }*u3{ b}‘);

The component corresponding to Polarized radiation

  * >grcalcd(Tp(dn, dn));

Unpolarized radiation of energy density $\xi$ propagating in the radial
direction

  * > grdef(‘ v{a }:=[1, -1, 0, 0 ]‘);

> grdef(‘Tnp{a b}:= xi *v{a }*v{b } ‘);

> grcalcd(Tnp(dn, dn));

Bringing together the three parts of the Stress-Energy tensor, we get the
stress-energy tensor Matter + radiation:

  * > grdef(‘T0{a b}:=TM{a b}+mu*Tp{a b}+Tnp{a b}‘);

> grcalcd(T0(dn, dn));

Calculated T0(dn,dn) for minkowski (0.003000 sec.)

$\begin{array}[t]{c}\mathit{\mathit{\mathit{`CPUTime`=0.005}}}\\\
\mathit{For\,the\,minkowski\,spacetime:}\\\ \mathit{T0(dn,dn)}\\\
\mathit{T0(dn,dn)}\end{array}$
$T0_{ab}=\left[\begin{array}[]{cccc}3\mu+\mathit{rho_{M}}+\xi&-\xi&0&0\\\
-\xi&P_{M}+\mu+\xi&0&0\\\ 0&0&P_{t}+\mu&0\\\
0&0&0&P_{t}+\mu\end{array}\right]$ (3)

Redefining the radiation and matter variables of the stress-energy tensor:

  * > grmap(T0(dn, dn), subs, rho_M = rho[0]-3*mu, ‘x‘);

> grmap(T0(dn, dn), subs, P_M = P[0]-mu, ‘x‘);

> grmap(T0(dn, dn), subs, P_t = P[t]-mu, ‘x‘);

> grdisplay(T0(dn,dn));

$\begin{array}[t]{c}\mathit{For\,the\,minkowski\,spacetime:}\\\
\mathit{T0(dn,dn)}\\\ \mathit{T0(dn,dn)}\end{array}$
$T0_{a\,b}=\left[\begin{array}[]{cccc}\rho_{0}+\xi&-\xi&0&0\\\
-\xi&P_{0}+\xi&0&0\\\ 0&0&P_{t}&0\\\ 0&0&0&P_{t}\end{array}\right]$ (4)

###  2.3 Lorentz transformation

Let’s suppose that you have an observer moving in relation to the local system
Minkowskian, with a radial velocity $\omega$. The components of the Stress-
Energy tensor in this new system of Lorentz, will be given by the
relationship:

$\bar{T}_{\mu\nu}=\Lambda_{\>\mu}^{\alpha}\Lambda_{\>\nu}^{\beta}\hat{T}_{\alpha\beta}$

where $\Lambda_{\>\mu}^{\alpha}$ is the transformation matrix

  * > grdef(‘umsqrto := 1/sqrt(1-omega^2) ‘);

grdef(‘Lambda{^a b}:= umsqrto *kdelta{^a $t}*kdelta{$t b}

-omega* umsqrto *kdelta{^a $x}* kdelta{$t b} -omega* umsqrto *kdelta{^a $t}* kdelta{$x b}

\+ umsqrto *kdelta{^a $x}* kdelta{$x b}+ kdelta{^a $y}* kdelta{$y b}+

kdelta{^a $z}* kdelta{$z b}‘);

> grcalcd(Lambda(up, dn));

$\begin{array}[t]{c}\mathit{CPU\>Time=0.011}\\\
\mathit{\mathit{For\,the\,minkowski\,spacetime:}}\\\ \mathit{Lambda(up,dn)}\\\
\mathit{\Lambda(up,dn)}\end{array}$
$\Lambda_{\>\mu}^{\alpha}={\color[rgb]{0,0,1}}\left[\begin{array}[]{cccc}\frac{1}{\sqrt{1-\omega^{2}}}&-\frac{\omega}{\sqrt{1-\omega^{2}}}&0&0\\\
-\frac{\omega}{\sqrt{1-\omega^{2}}}&\frac{1}{\sqrt{1-\omega^{2}}}&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right],$ (5)

Let’s carry out the transformation operation now. To do it, we will establish
this transformation as:

  * > grdef(‘T1{a b}:=Lambda{^c a}*Lambda{^d b}*T0{c d}‘);

> grcalcd(T1(dn, dn));

$\begin{array}[t]{c}\mathit{CPU\>Time=0.012}\\\
\mathit{For\,the\,minkowski\,spacetime:}\\\ \mathit{T1\left(dn,dn\right)}\\\
\mathit{\Lambda(up,dn)}\end{array}$
$T1_{a\,b}=\left[\begin{array}[]{cccc}-\frac{\rho_{0}+2\omega\xi+\omega^{2}P_{0}+\omega^{2}\xi}{-1+\omega^{2}}&\>\frac{\omega_{0}\rho_{0}+2\omega\xi+\xi+\omega
P_{0}}{-1+\omega^{2}}&0&0\\\ \frac{\omega_{0}\rho_{0}+2\omega\xi+\xi+\omega
P_{0}}{-1+\omega^{2}}&-\frac{\omega^{2}\rho_{0}+\omega^{2}\xi+2\omega\xi+P_{0}+\xi}{-1+\omega^{2}}&0&0\\\
0&0&P_{t}&0\\\ 0&0&0&P_{t}\end{array}\right]$ (6)

We obtain the expression of the Stress-energy tensor in the local system, with
radial velocity $\omega$.

###  2.4 Bondi Radiative Coordinate System

As the study we are doing, is related to radiation, then it is logical to
assume that we must use a coordinate system according to the theme. Therefore,
we are going to use Bondi’s radiative coordinate system as in [73].

  * >qload(bondi);

Calculated ds for bondi (0.001000 sec.)

$\begin{array}[b]{c}\mathit{Default\,space\,time=bondi}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{Coordinates}\\\
\mathit{x\left(up\right)}\\\
\mathit{x^{\alpha}=\bar{\left[u,r,\theta,\phi\right]}}\\\
\mathit{Line\>elemen}t\\\ \mathit{ds^{2}=\frac{V\left(u,r\right)\cdot
e^{2\beta\left(u,r\right)}}{r}du^{2}+2e^{2\beta\left(u,r\right)}dr\cdot
du-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d\phi^{2}}\\\
\mathit{The\,Bondi\,metric\,(Proc.\,Roy.\,Soc.\,A\,269\;21)}\end{array}$ (7)

Changing the expression of the Stress-energy tensor of the local Minkowskiana
metric to its structure in the Bondi radiation coordinate system.

  * >grdef(‘Vsr:=sqrt(V(u,r)/r)‘);

>grdef(‘Mu{^a b}:= exp(beta(u,r))*Vsr*kdelta{^a $u}*kdelta{$u b}+

exp(beta(u,r))/Vsr*kdelta{^a $u}*kdelta{$r b}+ exp(beta(u,r))/

Vsr*kdelta{^a $r}*kdelta{$r b}+r*kdelta{^a $theta}*kdelta{$theta b}+

r*sin(theta)*kdelta{^a $phi}*kdelta{$phi b}‘);

Defining and showing the matrix of the Minkowski local system transformation
to the Lorentz system

  * >grcalcd(Mu(up, dn));

Calculated Vsr for bondi (0.002000 sec.)

Calculated grtensor:-kdelta(dn,dn) for bondi (0.002000 sec.)

Calculated grtensor:-kdelta(up,dn) for bondi (0.002000 sec.)

Calculated Mu(up,dn) for bondi (0.011000 sec.)

$\begin{array}[t]{c}\mathit{CPU\,Time=0.021}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{Mu\left(up,dn\right)}\\\
\mathit{M\left(up,dn\right)}\end{array}$

$M_{\hskip
7.00002ptb}^{a}=\left[\begin{array}[]{cccc}e^{2\beta\left(u,r\right)}\sqrt{\frac{V\left(u,r\right)}{r}}&\frac{e^{2\beta\left(u,r\right)}}{\sqrt{\frac{V\left(u,r\right)}{r}}}&0&0\\\
0&\frac{e^{2\beta\left(u,r\right)}}{\sqrt{\frac{V\left(u,r\right)}{r}}}&0&0\\\
0&0&r&0\\\ 0&0&0&r\sin\theta\end{array}\right]$ (8)

This matrix of transformation, allows us to obtain the expression of the
stress-energy tensor of the local system of Minkowski to the system of
radiative coordinates with

$\bar{T}_{ab}=M_{\hskip 7.00002pta}^{\alpha}M_{\hskip
7.00002ptb}^{\beta}\hat{T}_{\>\alpha\beta}$ (9)

where $M_{\ b}^{a}$ is the transformation matrix. As we are performing an
operation between two expressions with different metrics, we must specify the
space - or metric - corresponding to each term of the multiplication

  * > grdef(‘TB<2>{a b}:=Mu{^c a} *Mu{^d b}*T1<1>{c d}‘);

by defining what is the scope of the definition of each term

  * > grcalcd(1 = minkowski, 2 = bondi, TB(dn, dn));

$\begin{array}[t]{c}\mathit{CPU\>Time=0.202}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{TB\left(dn,dn\right)}\\\
\mathit{TB\left(dn,dn\right)}\end{array}$

$TB_{ab}=\left[\begin{array}[]{cccc}-\frac{e^{2\beta\left(u,r)\right)}V\left(u,r\right)\left(\rho_{0}+2\omega\xi+\omega^{2}P_{0}+\omega^{2}\xi\right)}{r\left(-1+\omega^{2}\right)}&-\frac{\left(\omega
P_{0}-\rho_{0}\right)}{\omega+1}e^{2\beta\left(u,r\right)}&0&0\\\
-\frac{\left(\omega
P_{0}-\rho_{0}\right)}{\omega+1}e^{2\beta\left(u,r\right)}&-\frac{\left(\omega\rho_{0}+\omega
P_{0}-\rho_{0}-P_{0}\right)r}{V\left(u,r\right)\left(\omega+1\right)}&0&0\\\
0&0&r^{2}P_{t}&0\\\ 0&0&0&r^{2}\sin^{2}\theta\,P_{t}\end{array}\right]$ (10)

###  2.5 Effective Variables

We are going to define the effective variables in a similar way to the case of
[71, 67]. However, here we will denote it as $\rho{}_{1}$ and $P_{1}$, without
a bar at the top:

  * > grmap(TB(dn, dn), subs, rho[0] = (rho[1]+omega*P[1])/(1-omega), ‘x‘);

> grmap(TB(dn, dn), subs, P[0] = (P[1]+omega*rho[1])/(1-omega), ‘x‘);

Changing the term a bit for radiation

  * > grmap(TB(dn, dn), subs, xi = (1-omega)*epsilon/(1+omega), ‘x‘);

> gralter(TB(dn, dn), expand, factor);

> grdisplay(TB(dn, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{TB\left(dn,dn\right)}\\\ \mathit{TB\left(dn,dn\right)}\end{array}$

$TB_{ab}=\left[\begin{array}[]{cccc}\frac{e^{2\beta\left(u,r)\right)}V\left(u,r\right)\left(\omega^{2}\varepsilon+\omega^{2}\rho_{1}+\omega
P_{1}-2\varepsilon\omega-\omega\rho_{1}+\varepsilon+\rho_{1}\right)}{r\left(-1+\omega\right)^{2}}&\rho_{1}\,e^{2\beta\left(u,r\right)}&0&0\\\
\rho_{1}\,e^{2\beta\left(u,r\right)}&\frac{re^{2\beta\left(u,r)\right)}\left(\rho_{1}+P_{1}\right)}{V\left(u,r\right)}&0&0\\\
0&0&r^{2}P_{t}&0\\\ 0&0&0&r^{2}\sin^{2}\theta\,P_{t}\end{array}\right]$ (11)

Using the routines defined in [70] we can obtain an alternate expression for
$TB_{uu}$:

  * > TBdndnuu := kfactor(hcollect(grcomponent(TB(dn, dn), [u, u]), {P[1], rho[1], V(u, r)},

{r, omega, beta(u, r)}), V(u, r)*exp(2*beta(u, r))/r);

$\displaystyle\mathit{TBdndnuu}$ $\displaystyle\mathit{=}$
$\displaystyle\mathit{\frac{V\left(u,r\right)}{r}e^{2\beta\left(u,r\right)}\left(\frac{\omega
P_{1}}{\left(-1+\omega\right)^{2}}+\frac{\left(-\omega+\omega^{2}+1\right)\rho_{1}}{\left(-1+\omega\right)^{2}}+\varepsilon\right)}$
(12)

or equivalently

$\mathit{TBdndnuu=\frac{V\left(u,r\right)}{r}e^{2\beta\left(u,r\right)}\left[\frac{\omega\left(\rho_{1}+P_{1}\right)}{\left(1-\omega\right)^{2}}+\rho_{1}+\varepsilon\right]}$

###  2.6 Electromagnetic Component

If we are interested in the possibility that the material subject to the study
presents an electric charge, it is necessary to include it in the tensor and
therefore establish the expression for the Faraday tensor

$T_{\alpha\beta}=\frac{1}{4\pi}\left[F_{\alpha\mu}F_{\negmedspace\beta}^{\mu}+\frac{1}{4}g_{\alpha\beta}\left(F_{\mu\nu}F^{\mu\nu}\right)\right]$
(13)

where $F^{ab}$satisfies the equations

$\left(\sqrt{g}F^{\alpha\beta}\right)_{,\beta}=4\pi\sqrt{g}\,J^{\alpha}\qquad
F_{\left[\alpha\beta;\gamma\right]}=0$

Assuming spherical symmetry, the only non-zero component is $F$ and we have
$F^{01}$

$\displaystyle\left(e^{2\beta}r^{2}F^{01}\right)_{,1}$ $\displaystyle=$
$\displaystyle 4\pi e^{2\beta}r^{2}J^{0}$
$\displaystyle\left(e^{2\beta}r^{2}F^{10}\right)_{,0}$ $\displaystyle=$
$\displaystyle 4\pi e^{2\beta}r^{2}J^{1}$

and when integrating the first:

$\left(e^{2\beta}r^{2}F^{01}\right)=\int_{0}^{r}\left(4\pi
e^{2\beta}r^{2}J^{0}\right)dr\equiv Q\left(u,r\right)$

Because of this, we can write

$F^{01}=\frac{e^{-2\beta}}{r^{2}}\cdot Q\left(u,r\right)$

where $Q\left(u,r\right)$, plays the role of electric charge and the
electromagnetic component of the stress tensor is

  * > grdef(‘F{^a ^b}:= F01*(kdelta{$u ^a}*kdelta{$r ^b} -kdelta{$r ^a}*kdelta{$u ^b})‘);

> grdef(‘Faraday2:=F{^a ^b}*F{a b} ‘);

> grdef(‘Tem{a b}:= (F{a c}* F{^c b}+(1/4)*g{a b}*F{c d}*F{^c ^d })/(4*Pi) ‘);

> grcalcd(Faraday2);

> grcalcd(F(up, up));

$\begin{array}[t]{c}\mathit{CPU\,Time=0}.\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{F(up,up)}\\\
\mathit{F(up,up)}\end{array}$ $F^{ab}=\left[\begin{array}[]{cccc}0&F01&0&0\\\
-F01&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right],$ (14)

In terms of $F^{01}$, the electromagnetic component of the stress tensor is

  * > grcalcd(Tem(dn, dn));

Calculated g(up,up) for bondi (0.004000 sec.)

Calculated F(up,dn) for bondi (0.004000 sec.)

Calculated Tem(dn,dn) for bondi (0.000000 sec.)

$\begin{array}[t]{c}\mathit{CPU\,Time=0.011}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{Tem(dn,dn)}\\\
\mathit{Tem(dn,dn)}\end{array}$
$Tem_{ab}=\left[\begin{array}[]{cccc}\frac{1}{8}\frac{V\left(u,r\right)\left(e^{2\beta\left(u,r\right)}\right)^{3}F01^{2}}{\pi
r}&\frac{1}{8}\frac{\left(e^{2\beta\left(u,r\right)}\right)^{3}F01^{2}}{\pi}&0&0\\\
\frac{1}{8}\frac{\left(e^{2\beta\left(u,r\right)}\right)^{3}F01^{2}}{\pi}&0&0&0\\\
0{\color[rgb]{0,0,1}}&0&\frac{1}{8}\frac{r^{2}\left(e^{2\beta\left(u,r\right)}\right)^{3}F01^{2}}{\pi}&0\\\
0&0&0&\frac{1}{8}\frac{r^{2}\sin\left(\theta\right)^{2}\left(e^{2\beta\left(u,r\right)}\right)^{3}F01^{2}}{\pi}\end{array}\right],$
(15)

The electromagnetic component of the stress tensor is expressed in terms of
the electric charge as:

  * > grmap(Tem(dn, dn), subs, F01 = exp(-2*beta(u, r))*Q(u, r)/r^2, ‘x‘);

> grmap(Faraday2, subs, F01 = exp(-2*beta(u, r))*Q(u, r)/r^2, ‘x‘);

> gralter(Tem(dn, dn), expand, factor);

Component simplification of a GRTensorIII object:

Applying routine expand to object Tem(dn,dn)

Applying routine factor to object Tem(dn,dn)

$\mathit{CPUTime=0.007}$

  * > grdisplay(Tem(dn, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{Tem(dn,dn)}\\\ \mathit{Tem(dn,dn)}\end{array}$
$\mathit{Tem_{ab}}=\left[\begin{array}[]{cccc}\frac{1}{8}\frac{V\left(u,r\right)\left(e^{2\beta\left(u,r\right)}\right)^{2}Q^{2}\left(u,r\right)}{\pi
r^{5}}&\frac{1}{8}\frac{\left(e^{2\beta\left(u,r\right)}\right)^{2}Q^{2}\left(u,r\right)}{\pi
r^{4}}&0&0\\\
\frac{1}{8}\frac{\left(e^{2\beta\left(u,r\right)}\right)^{2}Q^{2}\left(u,r\right)}{\pi
r^{4}}&0&0&0\\\ 0&0&\frac{1}{8}\frac{Q^{2}\left(u,r\right)}{\pi r^{2}}&0\\\
0&0&0&\frac{1}{8}\frac{\sin\left(\theta\right)^{2}Q^{2}\left(u,r\right)}{\pi
r^{2}}\end{array}\right].$ (16)

###  2.7 Stress tensor

The stress tensor total that includes matter + radiation + electric charge is:

  * > grdef(‘T{a b} := TB{a b}+Tem{a b}‘);

If we want to study neutral or uncharged cases, we can turn OFF the
contribution of the electric charge

  * > #grmap(T(dn, dn), subs, Q(u, r) = 0, ‘x‘)

Calculating the covariant components

  * > grcalc(T(dn, dn));

> gralter(T(dn, dn), expand, factor);

> grdisplay(T(dn, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{T(dn,dn)}\\\ \mathit{T(dn,dn)}\end{array}$
$\begin{array}[b]{c}T_{uu}=\frac{1}{8}\frac{1}{(-1+\omega)^{2}r^{5}\pi}\left(V(u,r)\cdot\left(e^{\beta\left(u,r\right)}\right)^{2}\left(8r^{4}\pi\varepsilon\omega^{2}+8r^{4}\pi\omega^{2}\rho_{1}+8r^{4}\pi\omega
P_{1}-16r^{4}\pi\varepsilon\omega\right.\right.\\\
\left.\left.-8r^{4}\pi\omega\rho_{1}+8r^{4}\pi\varepsilon+8r^{4}\pi\rho_{1}+\left(Q\left(u,r\right)\right)^{2}-2\left(Q\left(u,r\right)\right)^{2}\omega+\omega^{2}\left(Q\left(u,r\right)\right)^{2}\right)\right)\\\
T_{ru}=\frac{1}{8}\frac{\left(e^{\beta\left(u,r\right)}\right)^{2}\left(8r^{4}\pi\rho_{1}+Q^{2}\left(u,r\right)\right)}{r^{4}\pi}\\\
T_{ur}=\frac{1}{8}\frac{\left(e^{\beta\left(u,r\right)}\right)^{2}\left(8r^{4}\pi\rho_{1}+Q^{2}\left(u,r\right)\right)}{r^{4}\pi}\\\
T_{rr}=\frac{r\left(e^{\beta\left(u,r\right)}\right)^{2}(P_{1}+\rho_{1})}{V(u,r)}\\\
T_{\theta\theta}=\frac{1}{8}\frac{8r^{4}P_{t}\pi+Q\left(u,r\right)^{2}}{\pi
r^{2}}\\\
T_{\phi\phi}=\frac{1}{8}\frac{\sin\theta^{2}\left(8r^{4}P_{t}\pi+Q\left(u,r\right)^{2}\right)}{\pi
r^{2}}\end{array}$ (17)

As an example of the use of simplification routines, we can factor the time
covariant component

  * >tempTdndnuu := kfactor(hcollect(grcomponent(T(dn, dn), [u, u]), {P[1], rho[1], V(u, r)},

{r, omega, beta(u, r)}), V(u, r)*exp(2*beta(u, r))/r);

$\mathit{tempTdndnuu:=\frac{V\left(u,r\right)e^{2\beta\left(u,r\right)}\left(\frac{\omega
P_{1}}{\left(-1+\omega\right)^{2}}+\frac{\left(-\omega+\omega^{2}+1\right)\rho_{1}}{\left(-1+\omega\right)^{2}}+\frac{1}{8}\left(\frac{8r^{4}\pi\varepsilon+Q\left(u,r\right)^{2}}{r^{4}\pi}\right)\right)}{r}}$
(18)

The contravariant radial component can be simplified with the help of the
previously loaded factoring functions

  * > tempTupuprr := kfactor(hcollect(hcollect(grcomponent(T(up, up), [r, r]), {epsilon,

Q(u, r), V(u, r), exp(2*beta(u, r))}, {epsilon, P[1], rho[1], Q(u, r)}),
{epsilon,

P[1], rho[1], Q(u, r)}, {V(u, r), beta(u, r)}), exp(-2*beta(u, r))*V(u, r/r)

$\mathit{tempTupuprr=\frac{V\left(u,r\right)e^{-2\beta\left(u,r\right)}\left(\varepsilon+\frac{\left(-\omega+\omega^{2}+1\right)P_{1}}{\left(-1+\omega\right)^{2}}-\frac{1}{8}\frac{Q\left(u,r\right)^{2}}{r^{4}\pi}+\frac{\omega\rho_{1}}{\left(-1+\omega\right)^{2}}\right)}{r}}$
(19)

that we can simplify as

$\mathit{tempTupuprr=\frac{V\left(u,r\right)e^{-2\beta\left(u,r\right)}}{r}\left[\frac{\omega\left(\rho_{1}+P_{1}\right)}{\left(1-\omega\right)^{2}}+\varepsilon+\left(P_{1}-\frac{Q\left(u,r\right)^{2}}{8\pi
r^{4}}\right)\right]}$

###  2.8 Temporary and radial dependency of the Stress Tensor

To calculate the conservation equations, it is necessary to establish that the
density $\rho=\rho\left(u,r\right)$, pressure $P=P\left(u,r\right)$, radiation
$\varepsilon=\varepsilon\left(u,r\right)$ and tangential pressure
$P_{t}=P_{t}\left(u,r\right)$ depend on that of the temporal and radial
coordinates.

  * > grmap(T(up, dn), subs, rho[1] = rho(u, r),‘x‘);

> grmap(T(up, dn), subs, P[1] = P(u, r), ‘x‘);

> grmap(T(up, dn), subs, P[t] = P[t](u, r), ‘x‘);

> grmap(T(up, dn), subs, epsilon = epsilon(u, r), ‘x‘);

> grmap(T(up, dn), subs, omega = omega(u, r), ‘x‘);

> gralter(T(up, dn), simplify);

> grdisplay(T(up, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{T(up,dn)}\\\ \mathit{T(up,dn)}\end{array}$ $\displaystyle
T_{\>b}^{a}=$
$\displaystyle\left[\>\begin{bmatrix}\frac{1}{8}\left(\frac{8r^{4}\pi\rho(u,r)+Q(u,r)^{2}}{r^{4}\pi}\right),&r\left(\frac{P(u,r)+\rho(u,r)}{V\left(u,r\right)}\right),&0,&0\end{bmatrix},\right.$
$\displaystyle\begin{bmatrix}\frac{V\left(u,r\right)}{r}\frac{\left(\varepsilon\left(u,r\right)\omega\left(u,r\right)^{2}+\omega\left(u,r\right)P\left(u,r\right)-2\varepsilon\left(u,r\right)\omega\left(u,r\right)+\omega\left(u,r\right)\rho\left(u,r\right)+\varepsilon\left(u,r\right)\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}},&\frac{-Q(u,r)^{2}+8r^{4}\pi
P(u,r)}{r^{4}\pi}&0,&0\end{bmatrix},$
$\displaystyle\begin{bmatrix}0,&0,&-\frac{1}{8}\left(\frac{8r^{4}\pi
P_{t}(u,r)+Q(u,r)^{2}}{r^{4}\pi}\right),&0\end{bmatrix},$
$\displaystyle\left.\begin{bmatrix}0,&0,&0,&-\frac{1}{8}\left(\frac{8r^{4}\pi
P_{t}(u,r)+Q(u,r)^{2}}{r^{4}\pi}\right)\end{bmatrix}\,\right]$ (20)

###  2.9 Conservation equations (TOV)

These equations are important since the Einstein field equations are composed
of two structures: One is the geometry of the $G_{ab}$ system and the other is
basically the energy $T_{ab}$ . From the expressions that we have taken the
geometry is linear in the second derivative and non-linear in the first.
However, the impulse energy tensor that we have used is linear. Presumably,
this non-linearity, of the geometrical part must have a non-linear version in
some expression of the impulse energy tensor. It is therefore necessary to
calculate the components of the conservation equation since it is possible
that they contain the _non-linear_ expression necessary to correct our choice
of the structure of the Stress Tensor.

  * > grdef(‘cero{a }:=[0, 0, 0, 0 ]‘);

> grdef(‘TOV{b} := T{^a b;a}= cero{b}‘);

Calculating the covariant components of the conservation equation:

  * > grcalc(TOV(dn));

> gralter(TOV(dn), simplify);

> grdisplay(TOV(dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{TOV(dn)}\\\ \mathit{TOV(dn)}\end{array}$ $\displaystyle TOV_{u}$
$\displaystyle=\left(\frac{1}{4}\frac{1}{r^{4}V\left(u,r\right)\left(-1+\omega\left(u,r\right)\right)^{3}}\left(8\left(\omega\left(u,r\right)^{2}\varepsilon\left(u,r\right)+\left(P\left(u,r\right)+\rho\left(u,r\right)-2\cdot\varepsilon\left(u,r\right)\right)\omega\left(u,r\right)+\varepsilon\left(u,r\right)\right)\cdot\right.\right.$
$\displaystyle
V\left(u,r\right)^{2}r^{3}\pi\left(-1+\omega\left(u,r\right)\right)+\cdots=0$
(21) $\displaystyle TOV_{r}$
$\displaystyle=\left(\frac{1}{4}\frac{1}{r^{4}V\left(u,r\right)^{2}\pi}\left(4r^{5}\pi
V\left(u,r\right)\left(\frac{\partial}{\partial
u}\rho\left(u,r\right)\right)+4r^{5}\pi
V\left(u,r\right)\left(\frac{\partial}{\partial
u}\rho\left(u,r\right)\right)-\cdots\right.\cdots\right.$
$\displaystyle-4r^{5}\pi
P\left(u,r\right)r^{4}\pi\left(\frac{\partial}{\partial
u}V\left(u,r\right)\right)-4r^{5}\pi\rho\left(u,r\right)+\cdots=0$ (22)

We can use the simplification routines loaded at the beginning of the sheet to
compare the terms observed in the conservation equations. The following
simplifications allow us to establish that some of the identities are not
really independent:

  * >TOVn[0] := hcollect(simplify(exp(2*beta(u, r))*lhs(grcomponent(TOV(up), [u]))),

{P(u, r), rho(u, r), diff(P(u, r), r), diff(Q(u, r), r), P[t](u, r)},

{diff(V(u, r), r), diff(V(u, r), u), diff(beta(u, r), r), diff(beta(u, r),
u)});

$\displaystyle TOVn_{0}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}-\frac{r\left(\frac{\partial}{\partial
u}V\left(u,r\right)\right)}{V\left(u,r\right)^{2}}-\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)+\frac{2r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}-\frac{3}{2r}\right)P\left(u,r\right)+$
(23) $\displaystyle-\frac{\partial}{\partial
r}P\left(u,r\right)+\frac{1}{4}\frac{Q\left(u,r\right)\frac{\partial}{\partial
r}Q\left(u,r\right)}{\pi
r^{4}}+\left(-\frac{1}{2}\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}-\frac{r\left(\frac{\partial}{\partial
u}V\left(u,r\right)\right)}{V\left(u,r\right)^{2}}+\right.$
$\displaystyle\left.-\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)+\frac{2r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}+\frac{1}{2r}\right)\rho\left(u,r\right)+\frac{2P_{t}\left(u,r\right)}{r}+\frac{r\left(\frac{\partial}{\partial
u}P\left(u,r\right)+\frac{\partial}{\partial
u}\rho\left(u,r\right)\right)}{V\left(u,r\right)}$

  * >TOVn[1] := hcollect(hcollect(simplify(r*exp(2*beta(u, r))*lhs(grcomponent(TOV(up),

[r]))/V(u, r)), {V(u, r), beta(u, r), diff(V(u, r), r), diff(V(u, r), u),
diff(

beta(u, r), r), diff(beta(u, r), u)},{diff(epsilon(u, r), r), diff(epsilon(u,
r),

u), diff(omega(u, r), r), diff(omega(u, r), u), epsilon(u, r), omega(u, r)}),

{P(u, r), Q(u, r), rho(u, r), diff(P(u, r), r), diff(Q(u, r), r),

diff(Q(u, r), u), diff(epsilon(u, r), r), epsilon(u, r), P[t](u, r)}, {V(u,
r),

beta(u, r), diff(V(u, r), r), diff(V(u, r), u), diff(beta(u, r), r), diff(beta

(u, r), u)});

$\displaystyle TOVn_{1}$ $\displaystyle:=$
$\displaystyle\left(\frac{\left(\left(\omega\left(u,r\right)\right)^{2}+1\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\right.$
(24) $\displaystyle+\frac{1}{2}\frac{-2\,r\frac{\partial}{\partial
r}\omega\left(u,r\right)+3\,\left(\omega\left(u,r\right)\right)^{3}-3+7\,\omega\left(u,r\right)-7\,\left(\omega\left(u,r\right)\right)^{2}-\cdots}{r\left(-1+\omega\left(u,r\right)\right)^{3}}$
$\displaystyle-\frac{1}{2}\frac{-\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\omega\left(u,r\right)\right)^{2}+2\,r\left(\frac{\partial}{\partial
u}\beta\left(u,r\right)\right)\left(\omega\left(u,r\right)\right)^{2}-\cdots}{V\left(u,r\right)\left(-1+\omega\left(u,r\right)\right)^{2}}+$
$\displaystyle\left.\frac{-\frac{\partial}{\partial
r}V\left(u,r\right)+2\,r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{1}{2}\,\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}\right)P\left(u,r\right)+$
$\displaystyle+\left(-\frac{1}{4}\,\frac{\frac{\partial}{\partial
r}Q\left(u,r\right)}{\pi\,r^{4}}+\frac{1}{4}\,\frac{\frac{\partial}{\partial
u}Q\left(u,r\right)}{V\left(u,r\right)\pi\,r^{3}}\right)Q\left(u,r\right)\cdots$

  * >TOVn[2] := hcollect(simplify(r*lhs(grcomponent(TOV(dn), [u]))/V(u, r)), {P(u, r), rho(u, r),

diff(P(u, r), r), diff(Q(u, r), u), diff(V(u, r), r), diff(V(u, r), u),
diff(rho(u, r), r),

diff(rho(u, r), u), diff(beta(u, r), r), diff(beta(u, r), u), diff(epsilon(u,
r), r),

diff(omega(u, r), r), epsilon(u, r), omega(u, r)}, {P(u, r), Q(u, r),
epsilon(u, r),

omega(u, r)});

$\displaystyle\mathit{TOVn_{2}}$ $\displaystyle=$
$\displaystyle\left(\frac{\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\omega\left(u,r\right)}{V\left(u,r\right)\left(-1+\omega\left(u,r\right)\right)^{2}}-\frac{1}{2}\,\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}+2\,\frac{\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)\omega\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}\right.$ (25)
$\displaystyle\left.-\frac{\left(1+\omega\left(u,r\right)\right)\frac{\partial}{\partial
r}\omega\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{3}}+\frac{\omega\left(u,r\right)}{r\left(-1+\omega\left(u,r\right)\right)^{2}}\right)P\left(u,r\right)+\left(\frac{\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\omega\left(u,r\right)}{V\left(u,r\right)\left(-1+\omega\left(u,r\right)\right)^{2}}-\frac{1}{2}\,\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}\right.$
$\displaystyle\left.+2\,\frac{\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)\omega\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}-\frac{\left(1+\omega\left(u,r\right)\right)\frac{\partial}{\partial
r}\omega\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{3}}+\frac{\omega\left(u,r\right)}{r\left(-1+\omega\left(u,r\right)\right)^{2}}\right)\rho\left(u,r\right)+$
$\displaystyle+\frac{\left(\frac{\partial}{\partial
r}P\left(u,r\right)\right)\omega\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{1}{2}\,\frac{Q\left(u,r\right)\frac{\partial}{\partial
u}Q\left(u,r\right)}{r^{3}\pi\,V\left(u,r\right)}+\frac{\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\epsilon\left(u,r\right)}{V\left(u,r\right)}+\frac{\omega\left(u,r\right)\frac{\partial}{\partial
r}\rho\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{r\frac{\partial}{\partial
u}\rho\left(u,r\right)}{V\left(u,r\right)}$
$\displaystyle+2\,\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right){\color[rgb]{0,0,1}+}\epsilon\left(u,r\right)+\frac{\partial}{\partial
r}\epsilon\left(u,r\right)$

  * >TOVn[3] := hcollect(simplify(lhs(grcomponent(TOV(dn), [r]))), {P(u, r), rho(u, r),

diff(P(u, r), r), diff(Q(u, r), r), P[t](u, r)}, {diff(V(u, r), r), diff(V(u,
r), u),

diff(beta(u, r), r), diff(beta(u, r), u)});

$\displaystyle\mathit{TOVn_{3}}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}\,\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}-\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}-\frac{\partial}{\partial
r}\beta\left(u,r\right)+2\,\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}-\,\frac{3}{2r}\right)P\left(u,r\right)+\left(-\frac{1}{2}\,\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}\right.$ (26)
$\displaystyle\left.-\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}-\frac{\partial}{\partial
r}\beta\left(u,r\right)+2\,\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}+\,\frac{1}{2r}\right)\rho\left(u,r\right)-\frac{\partial}{\partial
r}P\left(u,r\right)+\frac{1}{4}\,\frac{Q\left(u,r\right)\frac{\partial}{\partial
r}Q\left(u,r\right)}{\pi\,r^{4}}$
$\displaystyle+\frac{2P_{t}\left(u,r\right)}{r}+\frac{r\left(\frac{\partial}{\partial
u}P\left(u,r\right)+\frac{\partial}{\partial
u}\rho\left(u,r\right)\right)}{V\left(u,r\right)}$

By comparing term by term, a certain regularity can be observed, and we can
verify that the two equations are the same:

  * > temp03 := collect(simplify(TOVn[0]-TOVn[3]), r);

$\mathit{temp03:=0}$ (27)

With the previous expression we can conclude that

$e^{2\beta}T_{\;;a}^{a0}=T_{\;1;a}^{a}=0$

and the same conservation equation is obtained. We have in this way 3
independent conservation equations.

  * > temp21 := hcollect(hcollect(hcollect(simplify(TOVn[2]-TOVn[1]), {V(u, r), beta(u, r)},

{diff(P(u, r), u), diff(V(u, r), r)}), {P(u, r), rho(u, r), diff(P(u, r), r),
diff(Q(u, r), u)},

{diff(rho(u, r), u), diff(beta(u, r), r), diff(beta(u, r), u)}), {P(u, r),
rho(u, r),

P[t](u, r)}, {Q(u, r), diff(Q(u, r), r)});

$\displaystyle temp21$ $\displaystyle:=$
$\displaystyle\left(-\frac{1}{2}\,\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}-\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}-\frac{\partial}{\partial
r}\beta\left(u,r\right)+2\,\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}-\,\frac{3}{2r}\right)P\left(u,r\right)+\left(-\frac{1}{2}\,\frac{\frac{\partial}{\partial
r}V\left(u,r\right)}{V\left(u,r\right)}\right.$ (28)
$\displaystyle\left.-\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(V\left(u,r\right)\right)^{2}}-\frac{\partial}{\partial
r}\beta\left(u,r\right)+2{\color[rgb]{0,0,1}}\,\frac{r\frac{\partial}{\partial
u}\beta\left(u,r\right)}{V\left(u,r\right)}+\,\frac{1}{2r}\right)\rho\left(u,r\right)-\frac{\partial}{\partial
r}P\left(u,r\right)+\frac{1}{4}\,\frac{Q\left(u,r\right)\frac{\partial}{\partial
r}Q\left(u,r\right)}{\pi\,r^{4}}$
$\displaystyle+\frac{2P_{t}\left(u,r\right)}{r}+\frac{r\left(\frac{\partial}{\partial
u}P\left(u,r\right)+\frac{\partial}{\partial
u}\rho\left(u,r\right)\right)}{V\left(u,r\right)}$

  * > temp21m0 := collect(simplify(temp21-TOVn[0]), r);

$\mathit{temp21m0:=0}$ (29)

The difference of the two equations brings us back to the conservation
equation of (TOV: Tolman Oppenheimer Volkoff). However, taken separately they
are not proprocionales to $TOV[0]=TOV[3]$.

###  2.10 Einstein field equations

Taking as input the covariant components of the Einstein field equations:

  * >grdef(‘Eins{a b} := G{a b} = 8*Pi*T{a b}‘);

>grcalc(Eins(dn, dn));

> gralter(Eins(dn, dn), simplify, factor, radsimp);

> grdisplay(Eins(dn, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{Eins(dn,dn)}\\\ \mathit{Eins(dn,dn)}\end{array}$
$\displaystyle\mathit{Eins_{uu}}$ $\displaystyle=$
$\displaystyle\left(\frac{r\frac{\partial}{\partial
u}V\left(u,r\right)+2\,\left(V\left(u,r\right)\right)^{2}\frac{\partial}{\partial
r}\beta\left(u,r\right)-2\,rV\left(u,r\right)\frac{\partial}{\partial
u}\beta\left(u,r\right)-V\left(u,r\right)\frac{\partial}{\partial
r}V\left(u,r\right)+V\left(u,r\right)e^{2\,\beta\left(u,r\right)}}{r^{3}}=\right.$
$\displaystyle=$
$\displaystyle\frac{1}{\left(-1+\omega\left(u,r\right)\right)^{2}r^{5}}\left(V\left(u,r\right){\rm
e}^{2\,\beta\left(u,r\right)}\left(8\,r^{4}\pi\,\epsilon\left(u,r\right)\left(\omega\left(u,r\right)\right)^{2}+8\,r^{4}\pi\,\left(\omega\left(u,r\right)\right)^{2}\rho\left(u,r\right)+\right.\right.$
$\displaystyle+8\,r^{4}\pi\,\omega\left(u,r\right)P\left(u,r\right)-16\,r^{4}\pi\,\epsilon\left(u,r\right)\omega\left(u,r\right)-8\,r^{4}\pi\,\omega\left(u,r\right)\rho\left(u,r\right)+8\,r^{4}\pi\,\epsilon\left(u,r\right)+$
$\displaystyle\left.\left.+8\,r^{4}\pi\,\rho\left(u,r\right)+\left(Q\left(u,r\right)\right)^{2}-2\,\left(Q\left(u,r\right)\right)^{2}\omega\left(u,r\right)+\left(Q\left(u,r\right)\right)^{2}\left(\omega\left(u,r\right)\right)^{2}\right)\right)$
$\displaystyle\mathit{Eins_{ru}}$ $\displaystyle=$
$\displaystyle\left(\frac{2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-\frac{\partial}{\partial
r}V\left(u,r\right)+e^{2\,\beta\left(u,r\right)}}{r^{2}}=\frac{e^{2\,\beta\left(u,r\right)}\left(8\,r^{4}\pi\,\rho\left(u,r\right)+\left(Q\left(u,r\right)\right)^{2}\right)}{r^{4}}\right)$
$\displaystyle\mathit{Eins_{ur}}$ $\displaystyle=$
$\displaystyle\left(\frac{2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-\frac{\partial}{\partial
r}V\left(u,r\right)+e^{2\,\beta\left(u,r\right)}}{r^{2}}=\frac{e^{2\,\beta\left(u,r\right)}\left(8\,r^{4}\pi\,\rho\left(u,r\right)+\left(Q\left(u,r\right)\right)^{2}\right)}{r^{4}}\right)$
$\displaystyle\mathit{Eins_{rr}}$ $\displaystyle=$
$\displaystyle\left(4\,\frac{\frac{\partial}{\partial
r}\beta\left(u,r\right)}{r}=8\,\frac{\pi\,r{\rm
e}^{2\,\beta\left(u,r\right)}\left(P\left(u,r\right)+\rho\left(u,r\right)\right)}{V\left(u,r\right)}\right)$
$\displaystyle\mathit{Eins_{\theta\theta}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\,\left(-4\,\left(\frac{\partial^{2}}{\partial
u\partial
r}\beta\left(u,r\right)\right)r^{2}+\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r+2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r+\right.\right.$
$\displaystyle\left.\left.+2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r-2V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)e^{-2\,\beta\left(u,r\right)}=\frac{8\,r^{4}P_{t}\left(u,r\right)\pi+\left(Q\left(u,r\right)\right)^{2}}{r^{2}}\right)$
$\displaystyle\mathit{Eins_{\phi\phi}}$ $\displaystyle=$
$\displaystyle\left(\frac{\sin\left(\theta\right)^{2}}{2}\,\left(-4\,\left(\frac{\partial^{2}}{\partial
u\partial
r}\beta\left(u,r\right)\right)r^{2}+\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r+2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r+\right.\right.$ (30)
$\displaystyle\left.+2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r-2V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)e^{-2\,\beta\left(u,r\right)}=$
$\displaystyle\left.\qquad\frac{\sin\left(\theta\right)^{2}\left(8\,r^{4}P_{t}\left(u,r\right)\pi+\left(Q\left(u,r\right)\right)^{2}\right)}{r^{2}}\right)$

Using the simplification routines for the $uu$ component of the Einstein
tensor :$G_{00}=8\pi T_{00}$

  * >lhsEuu := hcollect(lhs(grcomponent(Eins(dn, dn), [u, u])), {r, diff(V(u, r), u),

diff(beta(u, r), u)}, {V(u, r), diff(V(u, r), r), diff(beta(u, r), r)})

$\displaystyle\mathit{lhsEuu}$ $\displaystyle=$
$\displaystyle\frac{-2\,V\left(u,r\right)\frac{\partial}{\partial
u}\beta\left(u,r\right)+\frac{\partial}{\partial u}V\left(u,r\right)}{r^{2}}+$
(31)
$\displaystyle\qquad+\frac{V\left(u,r\right)e^{2\,\beta\left(u,r\right)}+2\,\left(V\left(u,r\right)\right)^{2}\frac{\partial}{\partial
r}\beta\left(u,r\right)-V\left(u,r\right)\frac{\partial}{\partial
r}V\left(u,r\right)}{r^{3}}$

And for the right side of the same component of the Einstein field equation:

  * > rhsEuu := kfactor(hcollect(hcollect(rhs(grcomponent(Eins(dn, dn), [u, u])),

{Q(u, r), V(u, r), exp(2*beta(u, r)), epsilon(u, r)}, {P(u, r), Q(u, r),
rho(u, r),

epsilon(u, r)}), {P(u, r), Q(u, r), rho(u, r), epsilon(u, r)}, {V(u, r),
beta(u, r)}),

8*Pi*exp(2*beta(u, r))*V(u, r)/r);

$\displaystyle\mathit{rhsEuu}$ $\displaystyle:=$
$\displaystyle\frac{8\,\pi\,e^{2\,\beta\left(u,r\right)}V\left(u,r\right)\left(\frac{\omega\left(u,r\right)P\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\frac{1}{8}\,\frac{\left(Q\left(u,r\right)\right)^{2}}{r^{4}\pi}+\frac{\left(\left(\omega\left(u,r\right)\right)^{2}-\omega\left(u,r\right)+1\right)\rho\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\epsilon\left(u,r\right)\right)}{r}$
(32)

Let’s see now its Mixed components:

  * > grcalc(Eins(up, dn));

> gralter(Eins(up, dn), expand, factor);

> grdisplay(Eins(up, dn));

$\begin{array}[t]{c}\mathit{For\,the\,bondi\,spacetime:}\\\
\mathit{Eins(up,dn)}\\\ \mathit{Eins(up,dn)}\end{array}$

$\displaystyle\mathit{Eins_{\>u}^{u}}$ $\displaystyle=$
$\displaystyle\left(\frac{2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-\frac{\partial}{\partial
r}V\left(u,r\right)+\left(e^{\beta\left(u,r\right)}\right)^{2}}{\left(e^{\beta\left(u,r\right)}\right)^{2}r^{2}}=\frac{8\,r^{4}\pi\,\rho\left(u,r\right)+\left(Q\left(u,r\right)\right)^{2}}{r^{4}}\right)$
$\displaystyle\mathit{Eins_{\>r}^{u}}$ $\displaystyle=$
$\displaystyle\left(4\,\frac{\frac{\partial}{\partial
r}\beta\left(u,r\right)}{\left(e^{\beta\left(u,r\right)}\right)^{2}r}=8\,\frac{\pi\,r\left(P\left(u,r\right)+\rho\left(u,r\right)\right)}{V\left(u,r\right)}\right)$
$\displaystyle\mathit{Eins_{\>u}^{r}}$ $\displaystyle=$
$\displaystyle\left(-\frac{2\,V\left(u,r\right)\frac{\partial}{\partial
u}\beta\left(u,r\right)-\frac{\partial}{\partial
u}V\left(u,r\right)}{\left(e^{\beta\left(u,r\right)}\right)^{2}r^{2}}=\right.$
$\displaystyle\left.8\,\frac{\pi\,V\left(u,r\right)\left(\omega\left(u,r\right)P\left(u,r\right)+\epsilon\left(u,r\right)\left(\omega\left(u,r\right)\right)^{2}-2\,\epsilon\left(u,r\right)\omega\left(u,r\right)+\omega\left(u,r\right)\rho\left(u,r\right)+\epsilon\left(u,r\right)\right)}{r\left(-1+\omega\left(u,r\right)\right)^{2}}\right)$
$\displaystyle\mathit{Eins_{\>\theta}^{\theta}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\,\frac{1}{\left(e^{\beta\left(u,r\right)}\right)^{2}r^{2}}\left(4\,\left(\frac{\partial^{2}}{\partial
u\partial
r}\beta\left(u,r\right)\right)r^{2}-\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r-2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r+\right.\right.$
$\displaystyle\left.\left.-2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r+2\,V\left(u,r\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)\right)=-\frac{8\,r^{4}P_{t}\left(u,r\right)\pi+\left(Q\left(u,r\right)\right)^{2}}{r^{4}}\right)$
$\displaystyle\mathit{Eins_{\>\phi}^{\phi}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\,\frac{1}{\left(e^{\beta\left(u,r\right)}\right)^{2}r^{2}}\left(4\,\left(\frac{\partial^{2}}{\partial
u\partial
r}\beta\left(u,r\right)\right)r^{2}-\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r-2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r+\right.\right.$
$\displaystyle\left.\left.-2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r+2\,V\left(u,r\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)\right)=-\frac{8\,r^{4}P_{t}\left(u,r\right)\pi+\left(Q\left(u,r\right)\right)^{2}}{r^{4}}\right)$

Grouping terms for the right side of the EqnEins $\mathit{Eins(up,dn)[r,u]}$
component has

  * > rhsEru := kfactor(hcollect(rhs(grcomponent(Eins(up, dn), [r, u])), {P(u, r),

epsilon(u, r),omega(u, r)}, {V(u, r)}), 8*V(u, r)*Pi/r);

$\displaystyle\mathit{rhsEru}$ $\displaystyle=$
$\displaystyle\frac{8\,\pi\,V\left(u,r\right)}{r}\left(\frac{\omega\left(u,r\right)P\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}+\epsilon\left(u,r\right)+\frac{\omega\left(u,r\right)\rho\left(u,r\right)}{\left(-1+\omega\left(u,r\right)\right)^{2}}\right)$
(33)

An interesting expression is the one associated with the field equation
$\left(G_{11}=8\pi T_{11}\right)$, which includes the pressure and is very
similar to the one obtained previously $\left(G_{\,1}^{1}=8\pi
T_{\,1}^{1}\right)$, which relates density to the functions of the metric

  * > lhsErr := hcollect(lhs(grcomponent(Eins(up, dn), [r, r])), {P(u, r),

epsilon(u, r), omega(u, r)}, {V(u, r)});

$\mathit{lhsErr}:=\frac{-2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-\frac{\partial}{\partial
r}V\left(u,r\right)+\left(e^{\beta\left(u,r\right)}\right)^{2}}{\left(e^{\beta\left(u,r\right)}\right)^{2}r^{2}}$
(34)

  * > rhsErr := hcollect(rhs(grcomponent(Eins(up, dn), [r, r])), {P(u, r)}, {V(u, r)});

$\mathit{rhsErr}:=-8\,\pi\,P\left(u,r\right)+\frac{\left(Q\left(u,r\right)\right)^{2}}{r^{4}}$
(35)

Both relate the metrical elements – $V(u,r)\,and\,\beta\left(u,r\right)$ –
with the physical quantities – $P(u,r)\,and\,\rho(u,r)$– of pressure and
density.

###  2.11 Trace

Let’s calculate the trace of G

  * > grdef(‘TG := g{^a ^b}*G{a b} ‘);

> grcalcd(TG);

$\begin{array}[t]{c}\mathit{`CPU\,Time`=0.005}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{TG}\end{array}$

$\displaystyle\mathit{TG}$ $\displaystyle:=$
$\displaystyle\frac{1}{e^{2\,\beta\left(u,r\right)}r^{2}}\left(2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-2\,\frac{\partial}{\partial
r}V\left(u,r\right)+2\,e^{2\,\beta\left(u,r\right)}+4\,\left(\frac{\partial^{2}}{\partial
u\partial r}\beta\left(u,r\right)\right)r^{2}\right.$ (36)
$\displaystyle\left.-\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r-2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r-2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r\right)$

Very similar to the equation the left side of the angular component, of the
field equations $G_{2}^{2}=8\pi T_{2}^{2}$. Now the trace of the Stress
tensor:

  * > grdef(‘TT := g{^a ^b}*T{a b} ‘);

> grcalcd(TT);

$\begin{array}[t]{c}\mathit{`CPU\,Time`=0.004}\\\
\mathit{For\,the\,bondi\,spacetime:}\\\ \mathit{TT}\end{array}$
$TT=\rho\left(u,r\right)-P\left(u,r\right)-2P_{t}\left(u,r\right)$ (37)

The contraction of the Field equation $\mathit{Traza=g^{\mu\nu}G_{\mu\nu}=8\pi
g^{\mu\nu}T_{\mu\nu}}$

  * > grdef(‘TrazaTE := g{^a ^b}*G{a b} = 8*Pi* g{^a ^b}*T{a b}‘);

> grcalcd(TrazaTE);

$\displaystyle\mathit{TrazaTE}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{2\,\beta\left(u,r\right)}r^{2}}\left(2\,V\left(u,r\right)\frac{\partial}{\partial
r}\beta\left(u,r\right)-2\,\frac{\partial}{\partial
r}V\left(u,r\right)+2\,e^{2\,\beta\left(u,r\right)}+4\,\left(\frac{\partial^{2}}{\partial
u\partial r}\beta\left(u,r\right)\right)r^{2}\right.$ (38)
$\displaystyle\left.-\left(\frac{\partial^{2}}{\partial
r^{2}}V\left(u,r\right)\right)r-2\,\left(\frac{\partial}{\partial
r}V\left(u,r\right)\right)\left(\frac{\partial}{\partial
r}\beta\left(u,r\right)\right)r-2\,V\left(u,r\right)\left(\frac{\partial^{2}}{\partial
r^{2}}\beta\left(u,r\right)\right)r\right)$ $\displaystyle=$
$\displaystyle-8\,\pi\,\left(-\rho\left(u,r\right)+P\left(u,r\right)+2\,P_{t}\left(u,r\right)\right)$

This equation of the trace is important since it can be shown that it is
equivalent to the equation $\left(T_{\quad;\mu}^{\mu 0}=0\right)$, the Tolman-
Oppenheimer-Volkoff equation. When comparing the left side of the trace
equation with the left side of the angular component $G_{\,2}^{2}=8\pi
T_{\>2}^{2}$, of the equations from field, we get

  * >difG22TG := simplify((1/2)*grcomponent(TG)-lhs(grcomponent(Eins(up, dn), [theta, theta])))

$difG22TG:=\frac{e^{-2\,\beta\left(u,r\right)}\left(-\frac{\partial}{\partial
r}V\left(u,r\right)+e^{2\,\beta\left(u,r\right)}\right)}{r^{2}}$ (39)

##  3 Results and Comments

This section is dedicated to presenting some results obtained when executing
the established and described commands _(MAPLE17 + GRTensorIII)_ of the
previous section. We can summarize that the spreadsheet allows us to calculate
the different tensor terms for a radiative coordinate system from the Einstein
Field equations and their Conservation equations. Throughout the spreadsheet
development process, the importance of the expression to determine the
components of the Stress tensor from the Minkowski metric to its connection
with the Bondi radiative coordinate metric of the subsection (2.4) should be
taken into account (9). You can see that it is necessary to specify the scope
of each component of the metric (9), to make the calculation of the entire
sheet flow without interruption ([74]). This procedure has been used to
perform the calculations for ([67, 68]). We prove that this calculation
process fully coincides with the result obtained manually. In these articles
the equations allow us to carry out the study and behavior of the physical
variables calculated in the articles cited above. For example, the
Conservation equations$\left(T_{\nu;\mu}^{\mu}=0\right)$ for the charged case
are:

$\displaystyle T_{0;\lambda}^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{V}{r^{2}}\left(1+\frac{rV_{,1}}{V}\right)\left[\varepsilon+\frac{\omega\left(\rho+P\right)}{\left(1-\omega\right)^{2}}\right]+$
(40) $\displaystyle+\frac{V}{r}\frac{\partial}{\partial
r}\left[\varepsilon+\frac{\omega\left(\rho+P\right)}{\left(1-\omega\right)^{2}}\right]+\frac{\partial}{\partial
u}\left(\rho+\frac{Q^{2}}{8\pi r^{4}}\right)=0$ $\displaystyle
T_{1;\lambda}^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{e^{-2\beta}}{2\pi r}\beta_{,10}-\frac{\partial}{\partial
r}\left(P-\frac{Q^{2}}{8\pi
r^{4}}\right)-\frac{1}{2}\left(2\beta_{,1}+\frac{V_{,1}}{V}-\frac{1}{r}\right)\left(\rho+P\right)+$
(41) $\displaystyle-\frac{2}{r}\left[\left(P-\frac{Q^{2}}{8\pi
r^{4}}\right)-\left(P_{t}+\frac{Q^{2}}{8\pi r^{4}}\right)\right]=0$
$\displaystyle T_{\,;\lambda}^{\lambda 0}$ $\displaystyle=$ $\displaystyle
e^{-2\beta}T_{1;\lambda}^{\lambda}=0$ (42) $\displaystyle
e^{2\beta}T_{\,;\lambda}^{\lambda 1}$ $\displaystyle=$
$\displaystyle\frac{V}{r^{2}}\left(1+4r\beta_{,1}+\frac{rV_{,1}}{V}\right)\left[\varepsilon+\frac{\omega\left(\rho+P\right)}{\left(1-\omega\right)^{2}}\right]+\frac{V}{2r}\left(2\beta_{,1}+\frac{V_{,1}}{V}-\frac{1}{r}\right)\left(\rho+P\right)+$
(43) $\displaystyle+\frac{2V}{r^{2}}\left[\left(P-\frac{Q^{2}}{8\pi
r^{4}}\right)-\left(P_{t}+\frac{Q^{2}}{8\pi r^{4}}\right)\right]+$
$\displaystyle\frac{V}{r}\frac{\partial}{\partial r}\left(P-\frac{Q^{2}}{8\pi
r^{4}}\right)+\frac{V}{r}\frac{\partial}{\partial
r}\left[\varepsilon+\frac{\omega\left(\rho+P\right)}{\left(1-\omega\right)^{2}}\right]-\frac{\partial}{\partial
u}\left(P-\frac{Q^{2}}{8\pi r^{4}}\right)=0$

As can be seen in (2.9) and the concordance of these results with those
obtained manually, it can allow us to trust that this calculation to be
carried out in another coordinate system should have a similar performance. Of
course, it is necessary to point out that it is still pending to use the
complete equation $\left(31\right)$ shown in the article ([68]). This
comparative study between the results obtained so far and solving the complete
equation with the terms $\frac{\partial\varepsilon}{\partial
r}\>and\>\frac{\partial\omega}{\partial r}$ will be carried out in a future
work to publish.

##  4 Conclusions

In this article we describe a series of computer procedures used in GR relying
on the facilities of an integrated platform such as the _Maple_ package and
_GRTensorIII._ These procedures of computational algebra, numerical and
graphic computation can facilitate algebraic calculation both for research
purposes and for teaching GR at different levels.

## References

  * Nolan [1953] J. F. Nolan.  Analytical differentiation on a digital computer.  mathesis, Massachussettes Institute of Tecnology, may 1953.
  * Kahrimanian [1953] H. G. Kahrimanian.  Analytical differentiation by a digital computer.  Master’s thesis, Temple University, May 1953.
  * McCarthy [1960] John McCarthy.  Recursive functions symbolic expressions and their computation by machine, part i.  _Communications of the ACM_ , 3, 1960.  10.1145/367177.367199.  URL http://gen.lib.rus.ec/scimag/index.php?s=10.1145/367177.367199.
  * Bond et al. [1964] E. Bond, M. Auslander, S. Grisoff, R. Kenney, M. Myszewski, J. Sammet, R. Tobey, and S. Zilles.  Formac an experimental formula manipulation compiler.  In _Proceedings of the 1964 19th ACM National Conference_ , ACM ’64, pages 112.101–112.1019, New York, NY, USA, 1964. ACM.  10.1145/800257.808916.  URL http://doi.acm.org/10.1145/800257.808916.
  * P.W. and E. [2017] Hawkes P.W. and Kasper E.  Chapter 34 - numerical calculation of trajectories, paraxial properties and aberrations.  In Hawkes P.W. and Kasper E., editors, _Principles of Electron Optics (Second Edition)_ , volume 1 of _-_ , pages 585–626. Elsevier, second edition edition, oct 2017.  ISBN 978-0-08-102256-6.  https://doi.org/10.1016/B978-0-08-102256-6.00034-1.  URL https://www.sciencedirect.com/science/article/pii/B9780081022566000341.
  * Laskar and Gastineau [2012] J. Laskar and M. Gastineau.  Trip: General computer algebra system for celestial mechanics, oct 2012\.
  * Hearn [1966] A. C. Hearn.  Computation of algebraic properties of elementary particle reactions using a digital computer.  _Communications of the ACM_ , 9(8):573–577, aug 1966.  ISSN 0001-0782.  10.1145/365758.365766.  URL http://doi.acm.org/10.1145/365758.365766.
  * Hearn [1971] Anthony C. Hearn.  Applications of symbol manipulation in theoretical physics.  _Commun. ACM_ , 14(8):511–516, August 1971.  ISSN 0001-0782.  10.1145/362637.362641.  URL http://doi.acm.org/10.1145/362637.362641.
  * Brown and Hearn [1979] W.S. Brown and A.C. Hearn.  Applications of symbolic algebraic computation.  _Computer Physics Communications_ , 17(1):207 – 215, 1979.  ISSN 0010-4655.  https://doi.org/10.1016/0010-4655(79)90082-1.  URL http://www.sciencedirect.com/science/article/pii/0010465579900821.
  * Laporta and Remiddi [1996] S. Laporta and E. Remiddi.  The analytic value of (ge-2) at three loops in QED.  _Nuclear Physics B Proceedings Supplements, Vol.51_ , 51:142–147, December 1996.  10.1016/S0920-5632(96)90018-3.
  * Grozin [2005] A.G. Grozin.  _Using REDUCE in High Energy Physics_.  Cambridge University Press, 2005, sep 2005.
  * Friedrich Jegerlehner [auth.] Friedrich Jegerlehner (eds.) Friedrich Jegerlehner (auth.).  _The Anomalous Magnetic Moment of the Muon_ , volume 274 of _Springer Tracts in Modern Physics 274_.  Springer-Verlag Berlin Heidelberg, 2 edition, 2017.  ISBN 9783319635750 9783319635774.
  * Fitch and Cohen [1979] J. P. Fitch and H. I. Cohen.  Using camal for algebraic computations in general relativity.  _General Relativity and Gravitation_ , 11(6):411–418, dec 1979.  10.1007/BF00759304.
  * Campbell and Wainwright [1977] S. J. Campbell and J. Wainwright.  Algebraic computing and the newman-penrose formalism in general relativity.  _General Relativity and Gravitation_ , 8(12):987–1001, dec 1977.  10.1007/BF00759742.
  * Wainwright [1977] J. Wainwright.  Algebraic computing in general relativity.  In _General Relativity and Gravitation 1977_ , page 19, aug 1977.
  * Krasiński and Perkowski [1981] Andrzej Krasiński and Marek Perkowski.  Ortocartan a new computer program for analytic calculations in general relativity.  _General Relativity and Gravitation_ , 13(1):67–77, jan 1981.  10.1007/BF00766299.
  * Fletcher et al. [1967] John G. Fletcher, Reg Clemens, Richard Matzner, Kip S. Thorne, and Barbara A. Zimmerman.  Computer programs for calculating general-relativistic curvature tensors.  _Astrophysical Journal, Letters_ , 148:L91, May 1967.  10.1086/180022.
  * d’Inverno [1969] R. A. d’Inverno.  Alam - atlas lisp algebraic manipulator.  _The Computer Journal_ , 12(2):124–127, 01 1969\.  ISSN 0010-4620.  10.1093/comjnl/12.2.124.  URL https://doi.org/10.1093/comjnl/12.2.124.
  * D’Inverno [1975] Raymon A. D’Inverno.  Algebraic computing in general relativity.  _General Relativity and Gravitation_ , 6(6):567–593, dec 1975.  10.1007/BF00761964.
  * Gourgoulhon and Mancini [2018] Éric Gourgoulhon and Marco Mancini.  Symbolic tensor calculus on manifolds: a SageMath implementation.  _Les cours du CIRM_ , 6(1):1–54, 2018.  10.5802/ccirm.26.  URL https://doi.org/10.5802%2Fccirm.26.
  * MacCallum [2018] M.A.H. MacCallum.  Computer algebra in gravity research.  _Living Rev Relativ_ , 21:6, aug 2018.  10.1007/s41114-018-0015-6.
  * Bolotin and Poslavsky [2013] D.A. Bolotin and S.V. Poslavsky.  Introduction to Redberry: a Computer Algebra System designed for tensor manipulation.  _arXiv e-prints_ , feb 2013.
  * Poslavsky and Bolotin [2015] S. Poslavsky and D. Bolotin.  Redberry: a Computer Algebra System designed for tensor manipulation.  In _Journal of Physics Conference Series_ , volume 608 of _Journal of Physics Conference Series_ , page 012060, may 2015.  10.1088/1742-6596/608/1/012060.
  * G. [2018] Zimmermann P.; Casamayou A.; Cohen N.; Connan G.  _Computational Mathematics with SageMath —— Polynomials_ , volume 10.1137/1.9781611975468, chapter Chapter 7, pages 125–151.  SIAM, 2018.  ISBN 978-1-61197-545-1,978-1-61197-546-8.  10.1137/1.9781611975468.ch7.  URL http://gen.lib.rus.ec/scimag/index.php?s=10.1137/1.9781611975468.ch7.
  * Peeters [2007] Kasper Peeters.  Cadabra: a field-theory motivated symbolic computer algebra system.  _Computer Physics Communications_ , 176(8):550 – 558, 2007.  ISSN 0010-4655.  https://doi.org/10.1016/j.cpc.2007.01.003.  URL http://www.sciencedirect.com/science/article/pii/S0010465507000318.
  * Brewin [2010] L. Brewin.  A brief introduction to Cadabra: A Tool for Tensor Computations in General Relativity.  _Computer Physics Communications_ , 181:489–498, mar 2010\.  10.1016/j.cpc.2009.10.020.
  * Gerdt [1980] V.P. Gerdt.  Analytical calculations in high energy physics by computer.  _Computer Physics Communications_ , 20:85–90, sep 1980\.  10.1016/0010-4655(80)90113-7.
  * Gerdt et al. [1980] V.P. Gerdt, O.V. Tarasov, and D.V. Shirkov.  Reviews of topical problems: Analytic calculations on digital computers for applications in physics and mathematics.  _Soviet Physics Uspekhi_ , 23:59–77, jan 1980.  10.1070/PU1980v023n01ABEH004898.
  * Strubbe [1979] H. Strubbe.  Development of the schoonship program.  _Computer Physics Communications_ , 18:1–5, sep 1979.  10.1016/0010-4655(79)90019-5.
  * Martinus J. G. Veltman and David N. Williams [1993] Martinus J. G. Veltman and David N. Williams.  Schoonschip ’91.  _The University of Michigan Ann Arbor, MI 48109-1120_ , 8, 06 1993\.  URL http://arxiv.org/abs/hep-ph/9306228v2.
  * Levine [1967] M. J. Levine.  Dirac matrix and tensor algebra on a digital computer.  pages 454–455, 1967.
  * Levine and Roskies [1974] M.J. Levine and R. Roskies.  Hyperspherical approach to quantum electrodynamics: sixth-order magnetic moment.  _Physical Review D_ , 9(2):421–429, January 1974\.  10.1103/PhysRevD.9.421.
  * Levine and Roskies [1976] M. J. Levine and R. Roskies.  Ashmedai and a large algebraic problem.  In _Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation_ , SYMSAC ’76, pages 359–364, New York, NY, USA, 1976. ACM.  10.1145/800205.806357.  URL http://doi.acm.org/10.1145/800205.806357.
  * Lewis et al. [1979] V.E. Lewis, Massachusetts Institute of Technology, Massachusetts Institute of Technology. Laboratory for Computer Science, United States. Army Research Office, and United States. Office of Naval Research.  _Proceedings of the 1979 MACSYMA Users’ Conference: Washington, D.C., June 20-22, 1979_.  MIT Laboratory for Computer Science, 1979.  URL https://books.google.co.ve/books?id=NNY-AAAAIAAJ.
  * Pavelle and Wang [1985] Richard Pavelle and Paul S. Wang.  Macsyma from f to g.  _Journal of Symbolic Computation_ , 1(1):69 – 100, 1985.  ISSN 0747-7171.  https://doi.org/10.1016/S0747-7171(85)80030-4.  URL http://www.sciencedirect.com/science/article/pii/S0747717185800304.
  * Moses [2012] Joel Moses.  Macsyma: A personal history.  _Journal of Symbolic Computation_ , 47(2):123 – 130, 2012.  ISSN 0747-7171.  https://doi.org/10.1016/j.jsc.2010.08.018.  URL http://www.sciencedirect.com/science/article/pii/S0747717110001483.
  * Steinhauser et al. [2015] Matthias Steinhauser, Takahiro Ueda, and Jos A.M. Vermaseren.  Parallel versions of form and more.  _Nuclear and Particle Physics Proceedings_ , 261-262:45–57, apr 2015.  10.1016/j.nuclphysbps.2015.03.006.
  * Ruijl et al. [2017] B. Ruijl, T. Ueda, and J. Vermaseren.  Form version 4.2.  _arXiv e-prints_ , July 2017.
  * Vermaseren [2017] J. Vermaseren.  New features of form and future plans.  In _13th International Symposium on Radiative Corrections; Applications of Quantum Field Theory to Phenomenology. 25-29 September, 2017 . St. Gilgen, Austria. Online at ¡A href=“https://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=290”¿https://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=290¡/A¿, id.22_ , page 22, January 2017.
  * Wolfram [1980] Stephen Wolfram.  _Some Topics in Theoretical High-Energy Physics._ PhD thesis, CALIFORNIA INSTITUTE OF TECHNOLOGY., jan 1980.
  * Wolfram [1985] Stephen Wolfram.  Undecidability and intractability in theoretical physics.  _prl_ , 54(8):735–738, February 1985.  10.1103/PhysRevLett.54.735.
  * Wolfram [2019] Stephen Wolfram.  The practical and theoretical future of computation in physics.  In _APS March Meeting Abstracts_ , volume 2019 of _APS Meeting Abstracts_ , page F22.001, January 2019.
  * Wolfram [2020] Stephen Wolfram.  A class of models with the potential to represent fundamental physics.  _arXiv e-prints_ , art. arXiv:2004.08210, April 2020.
  * Char et al. [1983] Bruce W. Char, Keith O. Geddes, W. Morven Gentleman, and Gaston H. Gonnet.  The design of maple: A compact, portable and powerful computer algebra system.  In _In EUROCAL ’83, Proceedings of the International Symposium on Symbolic and Algebraic Computation_ , EUROCAL ’83, pages 101–115, London, UK, UK, apr 1983. Springer-Verlag.  ISBN 3-540-12868-9.  URL http://dl.acm.org/citation.cfm?id=646657.700400.
  * R. and S.L. [1997] Portugal R. and Sautú S.L.  Applications of maple to general relativity.  _Computer Physics Communications_ , 105, 1997.  URL http://www.sciencedirect.com/science/article/pii/S0010465597000787.
  * Vulcanov and Ciobanu [2000] Dumitru N. Vulcanov and Gabriela Ciobanu.  Maple procedures in teaching the canonical formalism of general relativity.  _arXiv e-prints_ , art. physics/0010053, oct 2000.
  * Kalashnikov [2001] V.L. Kalashnikov.  Introduction to Relativistic Astrophysics and Cosmology through Maple.  _arXiv General Relativity and Quantum Cosmology e-prints_ , mar 2001\.
  * Vulcanov [2002] D.N. Vulcanov.  Gravity, torsion, dirac field and computer algebra using maple and reduce.  _arXiv General Relativity and Quantum Cosmology e-prints_ , September 2002.
  * Schreiner et al. [2003] Wolfgang Schreiner, Christian Mittermaier, and Karoly Bosa.  Distributed maple: parallel computer algebra in networked environments.  _Journal of Symbolic Computation_ , 35(3):305 – 347, 2003.  ISSN 0747-7171.  https://doi.org/10.1016/S0747-7171(02)00137-2.  URL http://www.sciencedirect.com/science/article/pii/S0747717102001372.
  * Vulcanov and Vulcanov [2004] D.N. Vulcanov and V.D. Vulcanov.  Maple libraries for cosmology.  _eprint arXiv:cs/0409006_ , sep 2004.
  * Wang [2006] Frank Y. Wang.  _Physics with MAPLE: The Computer Algebra Resource for Mathematical Methods in Physics_.  Physics Textbook. Wiley-VCH, 1 edition, may 2006.  ISBN 3-527-40640-9,9783527406401.  URL http://gen.lib.rus.ec/book/index.php?md5=1BFCDB89E13646633CC45CE9922112B6.
  * Sporea and Vulcanov [2016] C.A. Sporea and D.N. Vulcanov.  Using maple + grtensorii in teaching basics of general relativity and cosmology.  _Rom. Rep. Phys._ , (68):29–40, 2016.
  * SWMATH [2021] SWMATH.  Information service for mathematical software.  SWMATH, 2021.  https://www.swmath.org/.
  * Bulletin [2021] SIGSAM Bulletin.  Special interest group on symbolic & algebraic manipulation.  Dictionary Of Computating, 2021.  http://www.sigsam.org/Resources/Software.html.
  * System [2021] Maxima A Computer Algebra System.  Other free computer algebra systems.  MAXIMA, 2021.  https://maxima.sourceforge.io/compalg.html.
  * for Symbolic Computation [2021] RISC Research Institute for Symbolic Computation.  Research institute for symbolic computation.  Research Institute for Symbolic Computation, 2021.  https://risc.jku.at/software/.
  * FOLDOC [2021] FOLDOC.  Free on line dictionary of computing.  Dictionary Of Computating, 2021.  https://foldoc.org/symbolic+mathematics.
  * Netherlands [2021] Computer Algebra Netherlands.  Foundation can (computer algebra netherlands).  Computer Algebra Netherlands, 2021.  https://computeralgebra.nl/.
  * Brown [1966] W. S. Brown.  Proceedings of the first acm symposium on symbolic and algebraic manipulation.  In SYSMAC ’66, editor, _A language and system for symbolic algebra on a digital computer ALTRAN_ , number 1 in SAAC, pages 501–540, 1966\.  10.1145/800005.807955.
  * Hearn [1973] A. C. Hearn.  The reduce program for computer algebra.  In A. Visconti, editor, _3th Colloquium on Advanced Computing Methods in Theoretical Physics_ , pages AV/1–AV/19, Marseilles, jun 1973. Centre de Physique Théorique CNRS.  URL https://inis.iaea.org/collection/NCLCollectionStore/_Public/07/229/7229583.pdf?r=1&r=1.  CNRS technical report 73/P.550.
  * Monagan and Pearce [2013] Michael Monagan and Roman Pearce.  Poly: A new polynomial data structure for maple 17.  _ACM Commun. Comput. Algebra_ , 46(3/4):164 – 167, jan 2013.  ISSN 1932-2240.  10.1145/2429135.2429173.  URL https://doi.org/10.1145/2429135.2429173.
  * W. et al. [1991] Char Bruce W., Geddes Keith O., Gonnet Gaston H., Leong Benton, Monagan Michael B., and Watt Stephen M.  _Maple V Language Reference Manual_.  Springer US, 1 edition, 1991.  ISBN 978-0-387-94124-0,978-1-4615-7386-9.  URL http://gen.lib.rus.ec/book/index.php?md5=33976bde4f045b1ce2e97f0abe593d0e.
  * Geddes Keith O. [1992] Labahn George Geddes Keith O., Czapor Stephen R.  _Algorithms for Computer Algebra_.  Springer, 1992 edition, 1992.  ISBN 0792392590,9780792392590.  URL http://gen.lib.rus.ec/book/index.php?md5=8dc5ec01fc92cffddeeb422cc6149edb.
  * Maplesoft [2021] Maplesoft.  Index of descriptions for packages of library functions.  Maple, 7 2021.  https://www.maplesoft.com/support/help/Maple/view.aspx?path=index/package.
  * Musgrave P. [2021] Lake K. Musgrave P., Pollney D.  Grtensoriii package for maple.  GRTensorIII, 2021.  https://github.com/grtensor/grtensor.
  * Musgrave and Lake [1998] P. Musgrave and K. Lake.  Engineering applications of grtensorii 1:elasticity-finite deformations.  _GRTensorII_ , 1(1):22, aug 1998.
  * Medina and Falcon [2018] V. Medina and N. Falcon .  Relativistic model for radiating spherical collapse.  _Progress in Physics_ , 14(1):46–53, jan 2018\.
  * Medina [2022] Víctor Medina.  A self-similar solution of a fluid with spherical distribution in general relativity.  _Indian Journal of Physics_ , 96(1):317–328, jan 2022.  10.1007/s12648-020-01959-1.
  * G. [1998] Davies G.  pertutils.mpl by george davies written for grtensorii.  PertUtils, 1998.  http://grtensor.phy.queensu.ca/NewDemo/Blackholes/Utils/pertutils.mpl.
  * Davies [1998] G. Davies.  Second-order black hole perturbations: A computer algebra approach, i \- the schwarzschild spacetime.  _arXiv General Relativity and Quantum Cosmology e-prints_ , October 1998.
  * Herrera et al. [1980] L. Herrera, J. Jiménez, and G. J. Ruggeri.  Evolution of radiating fluid spheres in general relativity.  _Phys. Rev. D_ , 22:2305–2316, nov 1980.  10.1103/PhysRevD.22.2305.
  * Medina et al. [1988] V. Medina, L. Núñez, H. Rago, and A. Patiño.  Evolution of radiating charged spheres in general relativity.  _Canadian Journal of Physics_ , 66:981–986, nov 1988.  10.1139/p88-158.  URL http://gen.lib.rus.ec/scimag/index.php?s=10.1139/p88-158.
  * Bondi [1964] H. Bondi.  The Contraction of Gravitating Spheres.  _Proceedings of the Royal Society of London Series A_ , 281:39–48, aug 1964.  10.1098/rspa.1964.0167.
  * Musgrave and Lake [1996] P. Musgrave and K. Lake.  Junctions and thin shells in general relativity using computer algebra: I. the darmois - israel formalism.  _Classical and Quantum Gravity_ , 13:1885–1899, July 1996\.  10.1088/0264-9381/13/7/018.