# COMPLEXITY AND SIMPLICITY OF SELF–GRAVITATING FLUIDS

L. Herrera<EMAIL_ADDRESS>Instituto Universitario de Física Fundamental y
Matemáticas, Universidad de Salamanca, Salamanca 37007, Spain.
<EMAIL_ADDRESS>
(today)

###### Abstract

We review a recently proposed definition of complexity of the structure of
self–gravitating fluids ch1 , and the criterium to define the simplest mode of
their evolution ch2 ; rg5 . We analyze the origin of these concepts and their
possible applications in the study of gravitation collapse. We start by
considering the static spherically symmetric case, extending next the study to
static axially symmetric case cax . Afterward we consider the non–static
spherically symmetric case. Two possible modes of evolution are proposed to be
the simplest one. One is the homologous condition as defined in ch2 , however,
as was shown later on in rg5 , it may be useful to relax this last condition
to enlarge the set of possible solutions, by adopting the so-called quasi-
homologous condition. As another example of symmetry, we consider fluids
endowed with hyperbolical symmetry. Exact solutions for static fluid
distributions satisfying the condition of minimal complexity are presented rg4
. An extension of the complexity factor as defined in ch1 , to the vacuum
solutions of the Einstein equations represented by the Bondi metric is
discussed rg7 . A complexity hierarchy is established in this case, ranging
from the Minkowski spacetime (the simplest one) to gravitationally radiating
systems (the most complex). Finally we propose a list of questions which, we
believe, deserve to be treated in the future

General relativity; Relativistic fluids; Dissipative systems.

##  I Introduction

In recent decades many efforts have been made towards a rigorous definition of
complexity in different branches of science (see 1 ; 2 ; 1bis ; 2bis ; 2bisbis
; 3 ; 3bis ; 3bisbis ; 4 ; 4bis ; 6bis  and references therein). However,
despite all the work done so far, there is not yet a consensus on a precise
definition.

The reason behind such interest stems from the fact that at least at an
intuitive level, complexity, no matter how we define it, is a physical concept
deeply intertwined with fundamental aspects of the system. In other words, we
expect that a suitable definition of complexity of the system could allow us
to infer relevant conclusions about its behavior.

Therefore, it is of utmost relevance to provide a precise definition of an
observable quantity which allows measurement of such an important property of
the system. Thus, when dealing with a situation that intuitively is judged as
“complex”, we have to be able to quantify this complexity by defining an
observable measuring it.

Among the many definitions that have been proposed so far, most of them resort
to concepts such as information and entropy (see for example 3 ; 4 ), and are
based on the intuitive idea that complexity should, somehow, measure a basic
property related to the structural characteristics of the system.

This Chapter is devoted to the review of the concept of complexity introduced
in ch1  for self-gravitating systems, and its applications under variety of
circumstances.

An extension of the definition of complexity based on the work developed by
López-Ruiz and collaborators 3 ; 4  has already been proposed for self-
gravitating systems in 5 ; 6 ; 7 ; 10 .

However, such a poposal suffers from two drawbacks, the most important of
which is the fact that it only involves the energy density, ignoring the role
of stresses. This motivated the introduction of a quite different definition
which was proposed in ch1  for the static spherically symmetric case, and
extended further in ch2  to the general full dynamic case.

The definition given in ch1 , although intuitively associated with the very
concept of “structure” within the fluid distribution, is not related (at least
directly) to information or disequilibrium; rather it stems from the basic
assumption that the simplest system (or at least one of them) is represented
by the homogeneous fluid with isotropic pressure. Having assumed this
conjecture for a vanishing complexity system, the very definition of
complexity emerges in the development of the fundamental theory of self-
gravitating compact objects, in the context of general relativity.

The variable responsible for measuring complexity, which we call the
complexity factor, appears in the orthogonal splitting of the Riemann tensor,
and the justification for such a proposition, roughly speaking, is as follows
(details are given in the next section).

Once the point of view that for a static fluid distribution, the simplest
system is represented by a homogeneous (in the energy density), locally
isotropic fluid (principal stresses equal), then it is reasonable to assign
zero value of the complexity factor to such a distribution. Next, let us
recall the concept of Tolman mass 11 , which may be interpreted as the
“active” gravitational mass, and may be expressed, for an arbitrary
distribution, through its value for the zero-complexity case plus two terms
depending on the energy density inhomogeneity and pressure anisotropy,
respectively. These latter terms in turn may be expressed through a single
scalar function that we call the complexity factor. It obviously vanishes when
the fluid is homogeneous in the energy density, and isotropic in pressure, but
also may vanish when the two terms containing density inhomogeneity and
anisotropic pressure cancel each other out. Thus, as in 3 , vanishing
complexity may correspond to very different systems.

When dealing with time-dependent systems, we face two different problems; on
the one hand, we have to generalize the concept of complexity of the structure
of the fluid distribution to time-dependent dissipative fluids, and on the
other hand we also have to evaluate the complexity of the patterns of
evolution and propose what we consider to be the simplest of them.

In ch2  it was shown that is reasonable to assume that the complexity factor
for the structure of the fluid distribution is the same scalar function as for
the static case, which now includes the dissipative variables. As for the
simplest pattern of evolution, it was shown that the homologous condition
characterizes the simplest possible mode. However, as was shown later on, it
may be useful to relax this last condition to enlarge the set of possible
solutions, by adopting the so-called quasi-homologous condition rg5 .

The axially symmetric static case has been considered in cax , while some
particular cases of cylindrically symmetric fluid distributions have been
studied in ch ; cil . Further applications of the concept of complexity as
defined in ch1  may be found in rg ; rg1 ; rg2 ; rg3 ; rg4 . Always within the
context of general relativity, exact solutions for static fluid distributions
endowed with hyperbolical symmetry and satisfying the condition of minimal
complexity were presented in rg4 ; c13 ; c22 , while dynamic solutions endowed
with hyperbolical symmetry and satisfying the condition of minimal complexity
were obtained in rg6 , such solutions evolve in the so-called quasi-homologous
regime.

The concept of complexity as defined in ch1  has also been extended to other
theories of gravity in ot1 ; ot2 ; ot3 ; ot4 ; n1 ; ot5 ; ot6 ; ot7 ; ot8 ;
ot9 ; ot10 ; ot11 ; ot12 ; ot13 ; ot14 ; ot15 ; ot16 ; ot17 ; ot18 ; ot19 ;
ot20 . More recent developments of this subject within a variety of contexts
may be found in c20 ; c1 ; c6 ; c9 ; c11 ; c15 ; c10 ; c16 ; c5 ; c17 ; c18 ;
c8 ; c12 ; c23 ; c23bis ; c23bc ; Al ; c2 ; c24 ; c14 ; c19 ; c21 ; c21b ; c4
; c3 ; c7 ; c7b ; c7c ; c7d ; rin1 ; rin2 ; ja1  and references therein.

All the above-mentioned cases concern fluid distributions (which eventually
may be charged); however, the vacuum case has barely been treated. The only
known example so far is the extension of the complexity factor as defined in
ch1  to the vacuum solutions of the Einstein equations represented by the
Bondi metric rg7 . A complexity hierarchy may be established in this case,
ranging from the Minkowski spacetime (the simplest one) to gravitationally
radiating systems (the most complex).

##  II Defining complexity for static self–gravitating systems

Let us start by introducing the notion of complexity of a self–gravitating
fluid as represented by a variable (more than one for non–spherical
configurations) aimed to measure the degree of “complexity” of the fluid
distribution. At this level of generality, due to the vagueness of the concept
of “complexity”, the problem appears to be quite difficult to solve,
accordingly let’s consider instead a much simpler question, namely: what is
the simplest fluid configuration we can envisage?

In order to answer to the above question, let us first restrict the discussion
to static and spherically symmetric distributions. Under these conditions, the
fluid is described by the energy density and the pressure. The simplest energy
density distribution corresponds to an incompressible fluid (a constant energy
density) whereas the simplest pressure distribution corresponds to an
isotropic fluid (principal stresses equal). Indeed, it should be noticed that
since for a bounded configuration the pressure vanishes at the boundary
surface, it cannot be homogeneous.

Thus, based on these very simple considerations we shall assume that the
simplest structure (at least one of them) of a static spherically symmetric
fluid corresponds to a homogeneous (in the energy density) and isotropic (in
the pressure) fluid. We shall now see how this assumption leads us to a
reliable definition of complexity.

### The static spherically symmetric case

We consider spherically symmetric distributions of static fluid, which for the
sake of completeness we assume to be locally anisotropic and bounded by a
spherical surface $\Sigma$.

The line element is given in Schwarzschild–like coordinates by

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (1)

where $\nu(r)$ and $\lambda(r)$ are functions of their arguments. We number
the coordinates: $x^{0}=t;\,x^{1}=r;\,x^{2}=\theta;\,x^{3}=\phi$.

The energy–momentum tensor corresponding to our fluid distribution may be
written as

$T^{\mu}_{\nu}=\mu u^{\mu}u_{\nu}-Ph^{\mu}_{\nu}+\Pi^{\mu}_{\nu},$ (2)

with

$\displaystyle\Pi^{\mu}_{\nu}$ $\displaystyle=$
$\displaystyle\Pi(s^{\mu}s_{\nu}+\frac{1}{3}h^{\mu}_{\nu});\quad
P=\frac{P_{r}+2P_{\bot}}{3}$ (3) $\displaystyle\Pi=P_{r}-P_{\bot};\quad
h^{\mu}_{\nu}=\delta^{\mu}_{\nu}-u^{\mu}u_{\nu},$

where $\mu,P_{r},P_{\bot},\Pi_{\mu\nu},u^{\mu}$ denote the energy density, the
radial pressure, the tangential pressure, the anisotropic tensor and the
four–velocity respectively.

The vector $s^{\mu}$ is defined by

$s^{\mu}=(0,e^{-\frac{\lambda}{2}},0,0),$ (4)

whereas the four–velocity vector is given by:

$u^{\mu}=(e^{-\frac{\nu}{2}},0,0,0),$ (5)

with the properties $s^{\mu}u_{\mu}=0$, $s^{\mu}s_{\mu}=-1$.

From (5) we can calculate the four acceleration,
$a^{\alpha}=u^{\alpha}_{;\beta}u^{\beta}$, whose only non–vanishing component
is:

$a_{1}=-\frac{\nu^{\prime}}{2},$ (6)

where prime denotes derivative with respect to $r$.

At the exterior of the fluid distribution the spacetime is described by the
Schwarzschild metric

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (7)

In order to match smoothly the two metrics above on the boundary surface
$r=r_{\Sigma}=constant$, we require the continuity of the first and the second
fundamental forms across that surface, producing

$e^{\nu_{\Sigma}}=1-\frac{2M}{r_{\Sigma}},$ (8)
$e^{-\lambda_{\Sigma}}=1-\frac{2M}{r_{\Sigma}},$ (9)
$\left[P_{r}\right]_{\Sigma}=0,$ (10)

where, from now on, subscript $\Sigma$ indicates that the quantity is
evaluated on the boundary surface $\Sigma$.

Eqs. (8), (9), and (10) are the necessary and sufficient conditions for a
smooth matching of the two metrics (1) and (7) on $\Sigma$.

Next, let us recall that the Tolman mass for a spherically symmetric static
distribution of matter is given by 11

$\displaystyle
m_{T}=4\pi\int^{r_{\Sigma}}_{0}{r^{2}e^{(\nu+\lambda)/2}\left(T^{0}_{0}-T^{1}_{1}-2T^{2}_{2}\right)dr},$
(11)

which we shall extend for any sphere of radius $r$, completely inside
$\Sigma$, as

$\displaystyle
m_{T}=4\pi\int^{r}_{0}{\tilde{r}^{2}e^{(\nu+\lambda)/2}\left(T^{0}_{0}-T^{1}_{1}-2T^{2}_{2}\right)d\tilde{r}}.$

This extension of the global concept of energy to a local level is suggested
by the conspicuous role played by $m_{T}$ as the “active gravitational mass”.

Indeed, as it follows from (6), the gravitational acceleration
($a=-s^{\nu}a_{\nu}$) of a test particle, instantaneously at rest in a static
gravitational field, is given by

$a=\frac{e^{-\lambda/2}\,\nu^{\prime}}{2}=\frac{e^{-\nu/2}m_{T}}{r^{2}}.$ (12)

Another expression for $m_{T}$, which appears to be more suitable for the
discussion is (see 13 ; 14  for details, but notice slight changes in
notation)

$\displaystyle
m_{T}=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}-r^{3}\int^{r_{\Sigma}}_{r}{e^{(\nu+\lambda)/2}\left[\frac{8\pi}{\bar{r}}\left(P_{\bot}-P_{r}\right)+\frac{1}{\bar{r}^{4}}\int^{\bar{r}}_{0}{4\pi\tilde{r}^{3}\mu^{\prime}d\tilde{r}}\right]d\bar{r}},$
(13)

The important point to keep in mind here is that the second integral in (13)
describes the contribution of density inhomogeneity and local anisotropy of
pressure to the Tolman mass.

Next, using the orthogonal splitting of the Riemann tensor first considered by
Bel bel1  (see also GP ), let us introduce the tensor

$Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^{\gamma}u^{\delta},$ (14)

which can be splitted in terms of its trace and the corresponding trace–free
tensor, as (see sc  for details)

$TrY\equiv Y_{T}=4\pi(\rho+3P_{r}-2\Pi),$ (15)

and

$Y_{<\alpha\beta>}=Y_{TF}(s_{\alpha}s_{\beta}+\frac{h_{\alpha\beta}}{3}),$
(16)

with

$Y_{TF}\equiv(4\pi\Pi+E)=8\pi\Pi-\frac{4\pi}{r^{3}}\int^{r}_{0}{\tilde{r}^{3}\mu^{\prime}d\tilde{r}},$
(17)

where $E$ is the scalar defining the electric part of the Weyl tensor
$E_{\alpha\beta}=C_{\alpha\gamma\beta\delta}u^{\gamma}u^{\delta}$, (the
magnetic part vanishes identically), which may be written as

$E_{\alpha\beta}=E(s_{\alpha}s_{\beta}+\frac{1}{3}h_{\alpha\beta}),$ (18)

with

$E=-\frac{e^{-\lambda}}{4}\left[\nu^{\prime\prime}+\frac{{\nu^{\prime}}^{2}-\lambda^{\prime}\nu^{\prime}}{2}-\frac{\nu^{\prime}-\lambda^{\prime}}{r}+\frac{2(1-e^{\lambda})}{r^{2}}\right],$
(19)

satisfying the following properties:

$\displaystyle E^{\alpha}_{\,\,\alpha}=0,\quad
E_{\alpha\gamma}=E_{(\alpha\gamma)},\quad E_{\alpha\gamma}u^{\gamma}=0.$ (20)

Using (17) in (13) we get

$\displaystyle
m_{T}=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}+r^{3}\int^{r_{\Sigma}}_{r}{\frac{e^{(\nu+\lambda)/2}}{\tilde{r}}Y_{TF}d\tilde{r}}.$
(21)

Thus we see that this single scalar function, $Y_{TF}$, encompasses all the
modifications produced by the energy density inhomogeneity and the anisotropy
of the pressure, on the active gravitational (Tolman) mass. More specifically,
it describes how these two factors modify the value of the Tolman mass, with
respect to its value for the homogeneous isotropic fluid.

Then, following our starting conjecture stating that at least one of the
simplest configurations corresponds to an incompressible fluid (a constant
energy density) with isotropic pressure, and noticing that for such a system,
the scalar $Y_{TF}$ vanishes, it appears well justified to identify the
complexity factor with $Y_{TF}$.

The following remarks are in order at this point:

  * •

The scalar $Y_{TF}$ belongs to a family of functions known as structure
scalars, defined in sc  from a detailed analysis of the orthogonal splitting
of the Riemann tensor.

  * •

As is apparent from (12) the Toman mass is a measure of the strength of the
gravitational interaction.

  * •

The complexity factor defined as above, not only vanishes for the homogeneous,
isotropic fluid, where the two terms in (17) vanish identically, but also for
all configurations where the two terms in (17) cancel each other, implying
that there are a wealth of configurations satisfying the vanishing complexity
condition.

  * •

It is worth noticing that whereas the contribution of the pressure anisotropy
to $Y_{TF}$ is local, the contribution of the density energy inhomogeneity is
not.

  * •

For a charged fluid $Y_{TF}$ includes contributions from the electric charge
as well (see eq.(25) in 21 ).

In the next section, we shall present two examples of inhomogeneous and
anisotropic fluid configurations, satisfying the vanishing complexity factor
condition.

###  II.1 Fluid distributions with vanishing complexity factor

The vanishing complexity factor condition, according to (17) reads

$\Pi=\frac{1}{2r^{3}}\int^{r}_{0}{\tilde{r}^{3}\mu^{\prime}d\tilde{r}}.$ (22)

However, since the Einstein equations for a spherically symmetric static,
anisotropic fluid form a system of three ordinary differential equations for
five unknown functions ($\nu,\lambda,\mu,P_{r},P_{\bot}$), the condition
$Y_{TF}=0$ is not enough to close the system and we need still one condition
in order to solve it. Just for the sake of illustration, we shall propose two
examples in the next subsections.

####  II.1.1 The Gokhroo and Mehra ansatz

A family of anisotropic spheres has been found in 22 , which leads to
physically satisfactory models for compact objects.

These models are obtained from an ansatz on the form of the metric function
$\lambda$ which reads

$e^{-\lambda}=1-\alpha r^{2}+\frac{3K\alpha r^{4}}{5r_{\Sigma}^{2}},$ (23)

producing,

$\mu=\mu_{0}\left(1-\frac{Kr^{2}}{r_{\Sigma}^{2}}\right),$ (24)

and

$m(r)=\frac{4\pi\mu_{0}r^{3}}{3}\left(1-\frac{3Kr^{2}}{5r_{\Sigma}^{2}}\right),$
(25)

where $K$ is a constant in the range $(0,1)$, $\alpha=\frac{8\pi\mu_{0}}{3}$,
and $m(r)$ is the mass function, defined by

$1-e^{-\lambda}=\frac{2m}{r},$ (26)

or, equivalently

$m=4\pi\int^{r}_{0}\tilde{r}^{2}\mu d\tilde{r}.$ (27)

Then using the field equations it can be shown that the line element becomes
(see ch1  for details)

$ds^{2}=-e^{\int(2z(r)-2/r)dr}dt^{2}+\frac{z^{2}(r)e^{\int(\frac{4}{r^{2}z(r)}+2z(r))dr}}{r^{6}(-2\int\frac{z(r)(1+\frac{\Pi(r)r^{2}}{8\pi})e^{\int(\frac{4}{r^{2}z(r)}+2z(r))dr}}{r^{8}}dr+C)}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2}.$ (28)

where $C$ is a constant of integration, and

$e^{\nu(r)}=e^{\int(2z(r)-2/r)dr}.$ (29)

For the physical variables we have

$4\pi P_{r}=\frac{z(r-2m)+m/r-1}{r^{2}},$ (30)

$4\pi\mu=\frac{m^{\prime}}{r^{2}},$ (31)

and

$4\pi
P_{\bot}=(1-\frac{2m}{r})(z^{\prime}+z^{2}-\frac{z}{r}+\frac{1}{r^{2}})+z(\frac{m}{r^{2}}-\frac{m^{\prime}}{r}).$
(32)

The so obtained solution is regular at the origin, and satisfies the
conditions $\mu>0$, $\mu>P_{r},P_{\bot}$.

Also, to avoid singular behaviour of physical variables on the boundary of the
source ($\Sigma$), the solution should satisfy the Darmois conditions on the
boundary (8), (9), (10).

####  II.1.2 The polytrope with vanishing complexity factor

The polytropic equation of state plays an important role in the study of
self–gravitating systems, both, in Newtonian and general relativistic
astrophysics. In the isotropic pressure case such an equation of state is
sufficient to integrate the field equations. However in case of an anisotropic
fluid additional information is required. The study of polytropes for
anisotropic matter has been considered in detail in 23 ; 24 ; 25 .

Once the polytropic equation of state is assumed, in the case of anisotropic
matter, we still need an additional condition in order to solve the
corresponding system of equations. We propose to assume the vanishing
complexity factor condition as such a complementary information.

Thus our model is characterized by

$P_{r}=K\mu^{\gamma}=K\mu^{1+1/n};\qquad Y_{TF}=0,$ (33)

where constants $K$, $\gamma$, and $n$ are usually called polytropic constant,
polytropic exponent, and polytropic index, respectively.

It is worth mentioning that the generalization of the Newtonian polytrope to
the general relativistic case admits two possibilities. One is the equation
(33), the other is

$P_{r}=K\mu_{b}^{\gamma}=K\mu_{b}^{1+1/n}$ (34)

where $\mu_{b}$ denotes the baryonic (rest) mass density. The treatment of
this latter case has been described in detail in 24 .

From the polytropic equation of state (33) we obtain two equations which read:

$\xi^{2}\frac{d\Psi}{d\xi}\left[\frac{1-2(n+1)\alpha
v/\xi}{1+\alpha\Psi}\right]+v+\alpha\xi^{3}\Psi^{n+1}+\frac{2\Pi\Psi^{-n}\xi}{P_{rc}(n+1)}\left[\frac{1-2\alpha(n+1)v/\xi}{1+\alpha\Psi}\right]=0,$
(35)

and

$\frac{dv}{d\xi}=\xi^{2}\Psi^{n},$ (36)

where

$\alpha=P_{rc}/\mu_{c},\quad r=\xi/A,\quad A^{2}=4\pi\mu_{c}/\alpha(n+1),$
(37)

$\Psi^{n}=\mu/\mu_{c},\quad v(\xi)=m(r)A^{3}/(4\pi\mu_{c}),$ (38)

where subscript $c$ indicates that the quantity is evaluated at the center. At
the boundary surface $r=r_{\Sigma}$ ($\xi=\xi_{\Sigma}$) we have
$\Psi(\xi_{\Sigma})=0$ (see 25  for details).

Equations (35), (36), form a system of two first order ordinary differential
equations for the three unknown functions: $\Psi,v,\Pi$, depending on a duplet
of parameters $n,\alpha$. In order to proceed further with the modeling of a
compact object, we shall assume the vanishing complexity factor condition,
which with the notation above, reads

$\frac{6\Pi}{n\mu_{c}}+\frac{2\xi}{n\mu_{c}}\frac{d\Pi}{d\xi}=\Psi^{n-1}\xi\frac{d\Psi}{d\xi}.$
(39)

Now we have a system of three ordinary differential equations (35), (36), (39)
for the three unknown functions $\Psi,v,\Pi$, which may be integrated for any
arbitrary duplet of values of the parameters $n,\alpha$, only constrained by
the physical conditions (see 24  for details)

$\mu>0,\qquad\alpha\Psi\leq
1,\qquad\frac{3v}{\xi^{3}\Psi^{n}}+\alpha\Psi-1\leq 1.$ (40)

The equations equivalent to (35) and (39), for the equation of state (34) read

$\displaystyle\xi^{2}\frac{d\Psi_{b}}{d\xi}\left[\frac{1-2(n+1)\alpha
v/\xi}{1+\alpha\Psi_{b}}\right]+v+\alpha\xi^{3}\Psi_{b}^{n+1}+\frac{2\Pi\Psi_{b}^{-n}\xi}{P_{rc}(n+1)}\left[\frac{1-2\alpha(n+1)v/\xi}{1+\alpha\Psi_{b}}\right]=0,$
(41)

$\frac{6\Pi}{n\mu_{bc}}+\frac{2\xi}{n\mu_{bc}}\frac{d\Pi}{d\xi}=\Psi_{b}^{n-1}\xi\frac{d\Psi_{b}}{d\xi}\left[1+K(n+1)\mu_{bc}^{1/n}\Psi_{b}\right],$
(42)

with $\Psi_{b}^{n}=\mu_{b}/\mu_{bc}$.

Further research on the application of the concept of complexity in the study
of polytropes may be found in rg2b ; rg2 ; rg3b ; rg4b ; c23 ; c12 .

So far we have been concerned with spherically symmetric systems, we shall
next consider the axially symmetric (static) case.

### The static axially–symmetric case

Let us now extend the concept of complexity as defined for the spherically
symmetric case to the most general axially symmetric static fluid
distributions.

The reason to undertake such a task is based on the well known fact that
(close to the horizon) there is a bifurcation between any finite perturbation
of Schwarzschild spacetime and any Weyl solution, even when the latter is
characterized by parameters arbitrarily close to those corresponding to
spherical symmetry (see i1 ; i5 ; in ; i3 ; i2 ; i4  and references therein
for a discussion on this point). This fact in turn is related to the well
known result, usually referred to as Israel theorem israel , stating that the
only regular static and asymptotically flat vacuum spacetime possessing a
regular horizon is the Schwarzchild solution, while all the others Weyl
exterior solutions exhibit singularities in the curvature invariants (as the
boundary of the source approaches the horizon).

Thus, even though observational evidence suggest that deviations from
spherical symmetry in compact self-gravitating objects (white dwarfs, neutron
stars), are likely to be incidental rather than basic features of these
systems, it is clear that for very compact objects, deviations from spherical
symmetry (no matter how small) should be studied resorting to exact solution
of the Einstein equations, instead as perturbations of spherically symmetric
systems.

As shown in the previous sections, the scalar intended to measure the degree
of complexity for a spherically symmetric fluid distribution (the complexity
factor), is identified as one of the scalar functions (structure scalars)
which appears in the orthogonal splitting of the Riemann tensor. More
specifically, it is related to one of the scalar functions appearing in the
splitting of the electric part of the Riemann tensor.

In spite of the fact that in the axially symmetric case the situation is much
more complicated, and the number of structure scalars is larger than in the
spherically symmetric case, we shall proceed in a similar way to fix the
general criterium to define the variable(s) measuring the complexity of the
fluid distribution.

Thus, we start by asking ourselves the same question as in the previous case,
namely: which is the simplest fluid configuration? As in the spherically
symmetric case we shall assume that such a configuration corresponds to the
incompressible (constant energy density), isotropic (in the pressure)
spheroid. From this simple assumption, we shall see that as the obvious
candidates to measure the degree of complexity of the fluid distribution,
appear three of the eight structure scalars corresponding to the axially
symmetric static fluid distribution. Explicit forms of these structure scalars
as well as some useful differential equations relating the inhomogeneities of
the energy density to some of the structure scalars were already found in as .

As in the spherically symmetric case, the vanishing of the three complexity
factors corresponds not only to the incompressible, isotropic spheroid, but
also to a large family of solutions where the density inhomogeneity terms
cancel the pressure anisotropic terms in the equations relating these
variables to the complexity factors. Some of these solutions will be
exhibited.

Thus, let us consider static and axially symmetric sources. For such a system
the line element may be written in “Weyl spherical coordinates” (please notice
that in this section we are using signature $+2$ instead of $-2$ as in the
previous case, which leads to some changes in the sign of some variables), as

$ds^{2}=-A^{2}dt^{2}+B^{2}\left(dr^{2}+r^{2}d\theta^{2}\right)+D^{2}d\phi^{2},$
(43)

where the coordinates $t$ and $\phi$ are adapted to the two Killing vectors
admitted by our line element, and therefore the metric functions depend only
on $r$ and $\theta$.

We recall that, unlike the vacuum case, the assumption of the Weyl gauge (
$R^{3}_{3}+R^{0}_{0}=0$, where $R^{\alpha}_{\beta}$ denotes the Ricci tensor),
reducing the number of independent metric functions to two, cannot be used
without loss of generality in the interior, which explains why our line
element is described in terms of three independent functions.

In a purely locally Minkowski frame (hereafter referred to as l.M.f.) where
the first derivatives of the metric vanish (locally) Bo , the most general
energy–momentum tensor is given by:

$\widehat{T}_{\alpha\beta}=\left(\begin{array}[]{cccc}\mu&0&0&0\\\
0&P_{xx}&P_{xy}&0\\\ 0&P_{yx}&P_{yy}&0\\\ 0&0&0&P_{zz}\end{array}\right),$
(44)

where $\mu,P_{xy},P_{xx},P_{yy},P_{zz}$ denote the energy density and
different stresses, respectively, as measured by our locally defined
Minkowskian observer.

Also observe that $P_{xy}=P_{yx}$ and, in general $P_{xx}\neq P_{yy}\neq
P_{zz}$.

Then transforming back to our coordinates, we obtain the components of the
energy momentum tensor in terms of the physical variables as defined in the
l.M.f.

$\displaystyle{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\mu+P_{zz})V_{\alpha}V_{\beta}+P_{zz}g_{\alpha\beta}+(P_{xx}-P_{zz})K_{\alpha}K_{\beta}$
(45) $\displaystyle+$
$\displaystyle(P_{yy}-P_{zz})L_{\alpha}L_{\beta}+2P_{xy}K_{(\alpha}L_{\beta)},$

with

$\displaystyle V_{\alpha}=(-A,0,0,0);\quad K_{\alpha}=(0,B,0,0);$
$\displaystyle L_{\alpha}=(0,0,Br,0);\quad S_{\alpha}=(0,0,0,D),$ (46)

where we are considering observers at rest with respect to the fluid
distribution.

Alternatively we may write the energy momentum tensor in the “canonical” form

$\displaystyle{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\mu+P)V_{\alpha}V_{\beta}+Pg_{\alpha\beta}+\Pi_{\alpha\beta},$
(47)

with

$\displaystyle\Pi_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(P_{xx}-P_{zz})\left(K_{\alpha}K_{\beta}-\frac{h_{\alpha\beta}}{3}\right)$
(48) $\displaystyle+$
$\displaystyle(P_{yy}-P_{zz})\left(L_{\alpha}L_{\beta}-\frac{h_{\alpha\beta}}{3}\right)+2P_{xy}K_{(\alpha}L_{\beta)},$
$P=\frac{P_{xx}+P_{yy}+P_{zz}}{3},\quad h_{\mu\nu}=g_{\mu\nu}+V_{\nu}V_{\mu}.$
(49)

The anisotropic tensor may also be written as

$\displaystyle\Pi_{\alpha\beta}=\frac{1}{3}(2\Pi_{I}+\Pi_{II})\left(K_{\alpha}K_{\beta}-\frac{h_{\alpha\beta}}{3}\right)+\frac{1}{3}(2\Pi_{II}+\Pi_{I})\left(L_{\alpha}L_{\beta}-\frac{h_{\alpha\beta}}{3}\right)+\Pi_{KL}\left(K_{\alpha}L_{\beta}+K_{\beta}L_{\alpha}\right),$
(50)

with

$\displaystyle\Pi_{KL}=K^{\alpha}L^{\beta}T_{\alpha\beta},$ (51)

$\Pi_{I}=\left(2K^{\alpha}K^{\beta}-L^{\alpha}L^{\beta}-S^{\alpha}S^{\beta}\right)T_{\alpha\beta},$
(52)
$\Pi_{II}=\left(2L^{\alpha}L^{\beta}-K^{\alpha}K^{\beta}-S^{\alpha}S^{\beta}\right)T_{\alpha\beta}.$
(53)

The relationships between the above scalars and the variables
$P_{xy},P_{xx},P_{yy},P_{zz}$ are (besides (49)),

$\Pi_{2}\equiv\frac{1}{3}(2\Pi_{I}+\Pi_{II})=P_{xx}-P_{zz},$ (54)
$\Pi_{3}\equiv\frac{1}{3}(2\Pi_{II}+\Pi_{I})=P_{yy}-P_{zz},$ (55)
$\Pi_{KL}=P_{xy}.$ (56)

or, inversely:

$P_{zz}=P-\frac{1}{3}(\Pi_{2}+\Pi_{3}),$ (57)
$P_{xx}=P+\frac{1}{3}(2\Pi_{2}-\Pi_{3}),$ (58)
$P_{yy}=P+\frac{1}{3}(2\Pi_{3}-\Pi_{2}).$ (59)

The explicit form of the Einstein equations as well as the conservation
equations, for the line element (43) and the energy–momentum tensor (47), are
given in the Appendix VI

###  II.2 The structure scalars

The structure scalars for our problem were calculated in as . For their
definition we need first to obtain the electric part of the Weyl tensor (the
magnetic part vanishes identically), whose components can be obtained directly
from its definition,

$E_{\mu\nu}=C_{\mu\alpha\nu\beta}\,V^{\alpha}\,V^{\beta},$ (60)

where $C_{\mu\alpha\nu\beta}$ denotes the Weyl tensor.

Equivalently, the electric part of the Weyl tensor may also be written as

$\displaystyle E_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{1}\left(K_{\alpha}L_{\beta}+L_{\alpha}K_{\beta}\right)+\mathcal{E}_{2}\left(K_{\alpha}K_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)$
(61) $\displaystyle+$
$\displaystyle\mathcal{E}_{3}\left(L_{\alpha}L_{\beta}-\frac{1}{3}h_{\alpha\beta}\right),$

where explicit expressions for the three scalars $\mathcal{E}_{1}$,
$\mathcal{E}_{2}$, $\mathcal{E}_{3}$ are given in the Appendix VII.

Next, let us calculate the electric part of the Riemann tensor (the magnetic
part vanishes identically), which is defined by

$Y^{\rho}_{\beta}=V^{\alpha}V^{\mu}R^{\rho}_{\alpha\beta\mu}.$ (62)

After some lengthy calculations we find;

$\displaystyle Y_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
Y_{TF_{1}}\left(K_{\alpha}L_{\beta}+K_{\beta}L_{\alpha}\right)+Y_{TF_{2}}\left(K_{\alpha}K_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)$
(63) $\displaystyle+$ $\displaystyle
Y_{TF_{3}}\left(L_{\alpha}L_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)+\frac{1}{3}Y_{T}h_{\alpha\beta},$

where

$Y_{T}=4\pi(\mu+3P),$ (64)

$Y_{TF_{1}}=\mathcal{E}_{1}-4\pi\Pi_{KL},$ (65)
$Y_{TF_{2}}=\mathcal{E}_{2}-4\pi\Pi_{2},$ (66)

$Y_{TF_{3}}=\mathcal{E}_{3}-4\pi\Pi_{3}.$ (67)

Finally, we shall find the tensor associated with the double dual of Riemann
tensor, defined as

$X_{\alpha\beta}=^{*}R^{*}_{\alpha\gamma\beta\delta}V^{\gamma}V^{\delta}=\frac{1}{2}\eta_{\alpha\gamma}^{\quad\epsilon\rho}R^{*}_{\epsilon\rho\beta\delta}V^{\gamma}V^{\delta},$
(68)

with
$R^{*}_{\alpha\beta\gamma\delta}=\frac{1}{2}\eta_{\epsilon\rho\gamma\delta}R_{\alpha\beta}^{\quad\epsilon\rho}$,
where $\eta_{\epsilon\rho\gamma\delta}$ denotes the permutation symbol.

Thus, we find

$\displaystyle X_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
X_{TF_{1}}\left(K_{\alpha}L_{\beta}+K_{\beta}L_{\alpha}\right)+X_{TF_{2}}\left(K_{\alpha}K_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)$
(69) $\displaystyle+$ $\displaystyle
X_{TF_{3}}\left(L_{\alpha}L_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)+\frac{1}{3}X_{T}h_{\alpha\beta},$

where

$X_{T}=8\pi\mu,$ (70)

$X_{TF_{1}}=-(\mathcal{E}_{1}+4\pi\Pi_{KL}),$ (71)
$X_{TF_{2}}=-\left(\mathcal{E}_{2}+4\pi\Pi_{2}\right),$ (72)

$X_{TF_{3}}=-\left(\mathcal{E}_{3}+4\pi\Pi_{3}\right).$ (73)

The scalars $Y_{T}$, $Y_{TF1}$, $Y_{TF2}$,$Y_{TF3}$, $X_{T}$, $X_{TF1}$,
$X_{TF2}$, $X_{TF3}$, are the structure scalars for our system.

Next, we shall need two differential equations which relate the spatial
derivatives of the physical variables and the Weyl tensor, obtained from
Bianchi identities, they have been found before for the spherically symmetric
and the cylindrically symmetric cases (see sc , scc1  and references therein).
For our case they have been calculated in as , and read

$\displaystyle\frac{{\cal E}_{1\theta}}{r}+\frac{1}{3}(2{\cal E}_{2}-{\cal
E}_{3})^{\prime}+\frac{{\cal
E}_{1}}{r}\left(\frac{2B_{\theta}}{B}+\frac{D_{\theta}}{D}\right)$ (74)
$\displaystyle+$ $\displaystyle{\cal
E}_{2}\left(\frac{B^{\prime}}{B}+\frac{D^{\prime}}{D}+\frac{1}{r}\right)-{\cal
E}_{3}\left(\frac{B^{\prime}}{B}+\frac{1}{r}\right)$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}\left(2\mu+3P\right)^{\prime}+4\pi\left[\mu+P+\frac{1}{3}(2\Pi_{2}-\Pi_{3})\right]\frac{A^{\prime}}{A}$
$\displaystyle+$ $\displaystyle 4\pi\Pi_{KL}\frac{A_{\theta}}{Ar},$

$\displaystyle{\cal E}^{\prime}_{1}+\frac{1}{3r}(2{\cal E}_{3}-{\cal
E}_{2})_{\theta}+{\cal
E}_{1}\left(\frac{2B^{\prime}}{B}+\frac{D^{\prime}}{D}+\frac{2}{r}\right)-\frac{{\cal
E}_{2}B_{\theta}}{Br}$ (75) $\displaystyle+$ $\displaystyle\frac{{\cal
E}_{3}}{r}\left(\frac{B_{\theta}}{B}+\frac{D_{\theta}}{D}\right)=\frac{4\pi}{3r}\left(2\mu+3P\right)_{\theta}$
$\displaystyle+$ $\displaystyle
4\pi\left[\mu+P+\frac{1}{3}(2\Pi_{3}-\Pi_{2})\frac{A_{\theta}}{Ar}\right]+4\pi\Pi_{KL}\frac{A^{\prime}}{A},$

which, using (64)-(67) and (70)-73), may be written in terms of structure
scalars, producing

$\displaystyle\frac{8\pi\mu^{\prime}}{3}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left[Y_{TF1\theta}+8\pi\Pi_{KL\theta}+(Y_{TF1}+8\pi\Pi_{KL})(\ln{B^{2}D)}_{\theta}\right]$
(76) $\displaystyle+$
$\displaystyle\left[\frac{2}{3}(Y_{TF2}^{\prime}+8\pi\Pi_{2}^{\prime})+(Y_{TF2}+8\pi\Pi_{2})(\ln{BDr})^{\prime}\right]$
$\displaystyle-$
$\displaystyle\left[\frac{1}{3}(Y_{TF3}^{\prime}+8\pi\Pi_{3}^{\prime})+(Y_{TF3}+8\pi\Pi_{3})(\ln{Br})^{\prime}\right],$

$\displaystyle\frac{8\pi\mu_{\theta}}{3r}$ $\displaystyle=$
$\displaystyle-\frac{1}{r}\left[\frac{1}{3}(Y_{TF2\theta}+8\pi\Pi_{2\theta})+(Y_{TF2}+8\pi\Pi_{2})(\ln
B)_{\theta}\right]$ $\displaystyle+$
$\displaystyle\frac{1}{r}\left[\frac{2}{3}(Y_{TF3\theta}+8\pi\Pi_{3\theta})+(Y_{TF3}+8\pi\Pi_{3})(\ln{BD})_{\theta}\right]$
$\displaystyle+$
$\displaystyle\left[Y_{TF1}^{\prime}+8\pi\Pi_{KL}^{\prime}+(Y_{TF1}+8\pi\Pi_{KL})(\ln{B^{2}Dr^{2}})^{\prime}\right],$

where prime and subscript $\theta$ denote derivatives with respect to $r$ and
$\theta$ respectively.

We have now available all the elements necessary to identify the complexity
factors for the fluid distribution under consideration. For doing so we recall
our basic ansatz consisting in assuming that the simplest possible fluid (or
at least one of them) is the incompressible (constant energy density) fluid
with isotropic pressure.

Now, in as  it has been shown that the necessary and sufficient conditions for
the vanishing of the (invariantly defined) spatial derivatives of the energy
density are $X_{TF1}=X_{TF2}=X_{TF3}=0$. In other words

$X_{TF1}=X_{TF2}=X_{TF3}=0\Leftrightarrow\mu^{\prime}=\mu_{\theta}=0.$ (77)

Therefore the homogeneous energy–density condition implies
$X_{TF1}=X_{TF2}=X_{TF3}=0$, which in turn produces

$Y_{TF1}=-8\pi\Pi_{KL};\quad Y_{TF2}=-8\pi\Pi_{2};\quad Y_{TF3}=-8\pi\Pi_{3}.$
(78)

From the above it follows that the isotropic pressure condition would imply
$Y_{TF1}=Y_{TF2}=Y_{TF3}=0$.

In other words, following the rationale exposed in the spherically symmetric
case, it is reasonable to identify the three structure scalars $Y_{TF}$ (more
precisely, their absolute values) with the complexity factors. As in the
previous case, we notice that they vanish for the incompressible (constant
energy density) fluid with isotropic pressure, but may also vanish for
inhomogeneous, anisotropic fluids, provided these two factors combine in such
a way that they cancel the three complexity factors.

In the next subsections, just to illustrate the way by means of which such
models may be obtained, we shall present two solutions with vanishing
complexity factors.

###  II.3 The incompressible, isotropic spheroid

The first solution we shall present corresponds to the case where the
complexity factors vanish because the energy density is homogeneous and the
pressure is isotropic. This solution was previously obtained and analyzed in
as . Here we just present it without details.

Let us first notice that, from (77), (71)–(73) and $P_{xx}=P_{yy}=P_{zz}=P$,
$P_{xy}=0$, $\mu=\mu_{0}=constant$, it follows that such a solution is also
conformally flat.

Next, for simplicity we shall assume the boundary surface $\Sigma$ to be
defined by the equation:

$r=r_{1}=constant,$ (79)

which is not the most general form of a possible boundary surface.

Then, from the above and (228) and (231) it follows that

$P\stackrel{{\scriptstyle\Sigma}}{{=}}0,$ (80)

where $\stackrel{{\scriptstyle\Sigma}}{{=}}$ means that both sides of the
equation are evaluated on $\Sigma$

Under the conditions above, (233) and (234) can be integrated to obtain:

$P+\mu_{0}=\frac{\zeta}{A},$ (81)

and

$P+\mu_{0}=\frac{\xi(r)}{A},$ (82)

where $\xi$ is an arbitrary function of its argument. Using boundary
conditions (80) in (81) (82) it follows that:

$A(r_{1},\theta)=const.=\frac{\alpha}{\mu_{0}},\qquad\zeta=constant.$ (83)

Finally, the metric for this model can be written as follows

$ds^{2}=\frac{1}{(\gamma r^{2}+\delta+br\cos\theta)^{2}}\left[-(\alpha
r^{2}+\beta+ar\cos\theta)^{2}dt^{2}+dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right],$ (84)

from which, the physical variables can be easily calculated, producing

$8\pi\mu=12\gamma\delta-3b^{2},$ (85)

$8\pi P=(3b^{2}-12\gamma\delta)\left[1-\frac{\alpha r_{1}^{2}+\beta}{\gamma
r_{1}^{2}+\delta}\frac{\gamma r^{2}+\delta+br\cos\theta}{\alpha
r^{2}+\beta+ar\cos\theta}\right],$ (86)

where $b,\gamma,\delta$ are constants, and

$\zeta=\mu_{0}\frac{\alpha r_{1}^{2}+\beta}{\gamma r_{1}^{2}+\delta},\quad
a=\frac{\alpha r_{1}^{2}+\beta}{\gamma r_{1}^{2}+\delta}b,$ (87)

in order to satisfy the junction condition (80).

It is important to stress the fact that this solution cannot be matched to any
Weyl exterior, except in the spherically symmetric case, even though it has a
surface of vanishing pressure (see as  for details). As shown in as  this
result is a consequence of the energy density homogeneity and the pressure
isotropy. So, to find matchable solutions we should relax these two
conditions. Also, it is worth mentioning that the above result is in agreement
with previous works indicating that static, perfect fluid (isotropic in
pressure) sources are spherical (see prueba  and references therein).

###  II.4 Anisotropic inhomogeneous spheroids

In order to obtain a metric smoothly matchable to any Weyl space–time we shall
consider a solution with vanishing complexity factors but inhomogeneous energy
density and anisotropic pressure (see details in cax )

The metric variables of the solution are

$\displaystyle A(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{a_{1}r\sin\theta}{b_{1}r^{2}+b_{2}},$ (88) $\displaystyle
B(r,\theta)$ $\displaystyle=$ $\displaystyle\frac{1}{b_{1}r^{2}+b_{2}},$ (89)
$\displaystyle D(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{b_{1}r^{2}-b_{2}}{b_{1}r^{2}+b_{2}}F\left(\frac{r\cos\theta}{b_{1}r^{2}-b_{2}}\right).$
(90)

It is a simple matter to check that the vanishing complexity factors
conditions (246)–(248) in Appendix VIII, are satisfied for (88)–(90).

From the above, using the Einstein equations (227)–(231), one obtains for the
physical variables.

$\displaystyle 8\pi\mu=12b_{1}b_{2}$
$\displaystyle-\frac{(b_{1}r^{2}+b_{2})^{2}}{(b_{1}r^{2}-b_{2})^{2}}\left[\frac{4b_{1}b_{2}r^{2}cos^{2}\theta}{(b_{1}r^{2}-b_{2})^{2}}+1\right]\frac{F_{zz}}{F},$
(91) $\displaystyle 8\pi P=-12b_{1}b_{2}$
$\displaystyle+\frac{(b_{1}r^{2}+b_{2})^{2}}{3(b_{1}r^{2}-b_{2})^{2}}\left[\frac{4b_{1}b_{2}r^{2}cos^{2}\theta}{(b_{1}r^{2}-b_{2})^{2}}+1\right]\frac{F_{zz}}{F},$
(92) $\displaystyle 8\pi\Pi_{2}\equiv
8\pi(P_{xx}-P_{zz})=\frac{F_{zz}}{4F}\frac{(b_{1}r^{2}+b_{2})^{2}}{(b_{1}r^{2}-b_{2})^{2}}\sin^{2}\theta,$
(93) $\displaystyle 8\pi\Pi_{3}\equiv
8\pi(P_{yy}-P_{zz})=\frac{F_{zz}}{4F}\frac{(b_{1}r^{2}+b_{2})^{4}cos^{2}\theta}{(b_{1}r^{2}-b_{2})^{4}},$
(94) $\displaystyle 8\pi\Pi_{KL}\equiv 8\pi
P_{xy}=-\frac{F_{zz}}{2F}\frac{r(b_{1}r^{2}+b_{2})^{3}}{(b_{1}r^{2}-b_{2})^{3}}\sin
2\theta,$ (95)

where $a_{1},b_{1},b_{2}$ are constant, and

$F(z)\equiv F\left(\frac{r\cos\theta}{b_{1}r^{2}-b_{2}}\right).$ (96)

It is not difficult to find a range of values of the parameters, for which the
physical behavior of physical variables is acceptable and the metric may be
matched smoothly on the boundary surface to a Weyl solution.

### The static hyperbolically symmetric case

Motivated by a new version of the Schwarzschild black hole proposed in 1w ; 2w
, there has been a renewed interest in self–gravitating systems admitting
hyperbolical symmetry.

In this picture, the space–time outside the horizon is represented by the
usual Schwarzschild metric, whereas the region inner to the horizon is
described by the line element

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{2M}{R}-1\right)dt^{2}-\frac{dr^{2}}{\left(\frac{2M}{R}-1\right)}-R^{2}d\Omega^{2},$
$\displaystyle d\Omega^{2}$ $\displaystyle=$ $\displaystyle
d\theta^{2}+\sinh^{2}\theta d\phi^{2},$ (97)

which is a static solution with the $(\theta,\phi)$ space describing a
positive Gaussian curvature, admitting the four Killing vectors

$\displaystyle\mathbf{\chi}_{(\mathbf{0})}=\partial_{\mathbf{t}},\quad{\bf\chi_{(2)}}=-\cos\phi\partial_{\theta}+\coth\theta\sin\phi\partial_{\phi}$
$\displaystyle{\bf\chi_{(1)}}=\partial_{\phi}\quad{\bf\chi_{(3)}}=\sin\phi\partial_{\theta}+\coth\theta\cos\phi\partial_{\phi}.$
(98)

A solution to the Einstein equations of the form given by (97), defined by the
hyperbolic symmetry (98), was first considered by Harrison Ha , and has been
more recently the subject of research in different contexts (see Ga ; Ri ; mim
; Ka ; Ma ; mimII  and references therein).

Our purpose here is to present some exact solutions to Einstein equations
endowed with the symmetry given by (98), satisfying the vanishing complexity
factor condition and which might serve as the source of (97), see rg4  for
details.

Thus, let us consider hyperbolically symmetric distributions of static fluid,
which for the sake of completeness we assume to be locally anisotropic and
which may be (or may be not) bounded from the exterior by a surface
$\Sigma^{e}$ whose equation is $r=r_{\Sigma^{e}}=\rm constant$. On the other
hand as it appears from the study of such fluids (see 2w  for details) the
fluid distribution cannot fill the central region, in which case we may assume
that such a region is represented by an empty vacuole, implying that the fluid
distribution is also bounded from the inside by a surface $\Sigma^{i}$ whose
equation is $r=r_{\Sigma^{i}}=\rm constant$.

The line element is given in polar coordinates by

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}\left(d\theta^{2}+\sinh^{2}\theta
d\phi^{2}\right),$ (99)

where, due to the imposed symmetry, $\nu(r)$ and $\lambda(r)$ are exclusively
functions of $r$. We number the coordinates:
$x^{0}=t;\,x^{1}=r;\,x^{2}=\theta;\,x^{3}=\phi$.

The metric (99) has to satisfy Einstein field equations

$G^{\nu}_{\mu}=8\pi T^{\nu}_{\mu}.$ (100)

We may write for the energy momentum tensor

$\displaystyle{T}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\mu+P)V_{\alpha}V_{\beta}-Pg_{\alpha\beta}+\Pi_{\alpha\beta},$
(101)

or

${T}_{\alpha\beta}=(\mu+P_{zz})V_{\alpha}V_{\beta}-P_{zz}g_{\alpha\beta}+(P_{xx}-P_{zz})K_{\alpha}K_{\beta}.$
(102)

Since we choose the fluid to be comoving in our coordinates, then

$V^{\alpha}=(e^{-\nu/2},0,0,0);\quad V_{\alpha}=(e^{\nu/2},0,0,0),$ (103)

and

$K_{\alpha}=(0,-e^{\lambda/2},0,0),$ (104)

It would be useful to express the anisotropic tensor in the form

$\displaystyle\Pi_{\alpha\beta}=\Pi\left(K_{\alpha}K_{\beta}+\frac{h_{\alpha\beta}}{3}\right),$
(105)

with $h_{\mu\nu}=g_{\mu\nu}-V_{\nu}V_{\mu}$,

$\displaystyle\Pi=P_{xx}-P_{zz},$ (106)

and

$P=\frac{P_{xx}+2P_{zz}}{3}.$ (107)

Since the Lie derivative and the partial derivative commute, then

$\mathcal{L}_{\chi}G_{\alpha\beta}=8\pi\mathcal{L}_{\chi}T_{\alpha\beta}=0,$
(108)

for any $\chi$ defined by (98), implying that all physical variables only
depend on $r$.

If the fluid is bounded from the exterior by a hypersurface $\Sigma^{e}$
described by the equation $r=r_{\Sigma^{e}}=constant$, then the smooth
matching of (97) and (99) on $\Sigma^{e}$ requires the fulfillment of the
Darmois conditions, imposing the continuity of the first and the second
fundamental forms, which imply

$e^{\nu_{\Sigma^{e}}}=\frac{2M}{r_{\Sigma^{e}}}-1,\qquad
e^{\lambda_{\Sigma^{e}}}=\frac{1}{\frac{2M}{r_{\Sigma^{e}}}-1},\qquad
P_{xx}(r_{\Sigma^{e}})=0,$ (109)

and the continuity of the mass function $m(r)$ defined below. If we assume
that the central region is surrounded by an empty cavity whose delimiting
surface is $r=r_{\Sigma^{i}}=constant$, then the fulfillment of Darmois
conditions on $\Sigma^{i}$ implies

$e^{\nu_{\Sigma^{i}}}=1,\qquad e^{\lambda_{\Sigma^{i}}}=1,\qquad
P_{xx}(r_{\Sigma^{i}})=0,$ (110)

and $m(r_{\Sigma^{i}})=0$.

The non–vanishing components of the Einstein equations for the metric (99) and
the energy momentum tensor (102) are

$\displaystyle 8\pi\mu$ $\displaystyle=$
$\displaystyle-\frac{(e^{-\lambda}+1)}{r^{2}}+\frac{\lambda^{\prime}}{r}e^{-\lambda},$
(111) $\displaystyle 8\pi P_{r}$ $\displaystyle=$
$\displaystyle\frac{(e^{-\lambda}+1)}{r^{2}}+\frac{\nu^{\prime}}{r}e^{-\lambda},$
(112) $\displaystyle 8\pi P_{\bot}$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda}}{2}\left(\nu^{\prime\prime}+\frac{{\nu^{\prime}}^{2}}{2}-\frac{\lambda^{\prime}\nu^{\prime}}{2}+\frac{\nu^{\prime}}{r}-\frac{\lambda^{\prime}}{r}\right),$
(113)

where we have used the standard notation $P_{xx}\equiv P_{r}$ and
$P_{zz}=P_{yy}\equiv P_{\bot}$, and primes denote derivatives with respect to
$r$.

It is worth stressing the differences between these equations and the
corresponding to the spherically symmetric case.

From the equations above or using the conservation laws
$T^{\alpha}_{\beta;\alpha}=0$ we obtain, besides the identity $\dot{\mu}=0$
(where dot denotes derivative with respect to $t$), the corresponding
hydrostatic equilibrium equation (the generalized Tolman–Oppenheimer–Volkoff
equation)

$\displaystyle
P_{r}^{\prime}+(\mu+P_{r})\frac{\nu^{\prime}}{2}+\frac{2}{r}\Pi$
$\displaystyle=$ $\displaystyle 0.$ (114)

Let us now define the mass function $m=m(r)$. For doing so, let us notice that
using (97) we have that outside the fluid distribution (but inside the
horizon)

$M=-\left(\frac{R}{2}\right)R^{3}_{232},$ (115)

where the Riemann tensor component $R^{3}_{232}$, has been calculated with
(97).

Then generalizing the above definition of mass for the interior of the fluid
distribution we may write

$m(r)=-\left(\frac{r}{2}\right)R^{3}_{232}=\frac{r(1+e^{-\lambda})}{2}$ (116)

where now the Riemann tensor component is calculated with (99).

Feeding back (116) into (111) we obtain

$m^{\prime}(r)=-4\pi r^{2}\mu\Rightarrow m=-4\pi\int^{r}_{0}{\mu r^{2}dr}.$
(117)

Since $m$ as defined by (116) is a positive quantity, then $\mu$ should be
negative and therefore the weak energy condition is violated, a result already
obtained in mimII . However it is important to stress that our definition of
mass function differs from the one introduced in mimII . In particular our $m$
is positive defined whereas the expression used in mimII  is negative (for the
hyperbolically symmetric fluid).

The following comments are in order at this point.

  * •

If the energy density is regular everywhere then the mass function must vanish
at the center as $m\sim r^{3}$, this implies (as it follows from (116)) that
the fluid cannot fill the space in the neighborhood of the center, i.e. there
is a cavity around the center which may be, either empty, or filled with a
fluid distribution non endowed with hyperbolical symmetry. Thus the
hyperbolically symmetric fluid spans from a minimal value of the coordinate
$r$ until its external boundary. For the extreme case $\mu=constant$, this
minimal value $r_{min.}$ is defined by $-\frac{8\pi}{3}\mu r^{2}_{min.}>1$.
Obviously, if the energy density is singular in the neighborhood of the
center, then this region must also be excluded by physical reasons.

From the above it follows that, strictly speaking, we should write instead of
(117)

$m=4\pi\int^{r}_{r_{min}}{|\mu|r^{2}dr},$ (118)

where due to the fact that $\mu$ is negative, we have replaced it by $-|\mu|$
(as we shall do from now on).

The situation described above is fully consistent with the results obtained in
2w  where it was shown that test particles cannot reach the center for any
finite value of its energy.

Next, using (112) and (116) we obtain

$\nu^{\prime}=2\frac{4\pi r^{3}P_{r}-m}{r(2m-r)},$ (119)

from which we may write (114) as

$\displaystyle P_{r}^{\prime}+(P_{r}-|\mu|)\frac{4\pi
r^{3}P_{r}-m}{r(2m-r)}+\frac{2}{r}\Pi$ $\displaystyle=$ $\displaystyle 0.$
(120)

This is the hydrostatic equilibrium equation for our fluid. Let us analyze in
some detail the physical meaning of its different terms.

The first term is just the gradient of pressure, which is usually negative and
opposing gravity. The second term describes the gravitational “force” and
contains two different contributions: on the one hand the term $P_{r}-|\mu|$
which we expect to be negative (or zero for the stiff equation of state) and
is usually interpreted as the “passive gravitational mass density” (p.g.m.d.),
and on the other hand the term $4\pi r^{3}P_{r}-m$ that is proportional to the
“active gravitational mass” (a.g.m.), and which is negative if $4\pi
r^{3}P_{r}<m$. Finally the third term describes the effect of the pressure
anisotropy, whose sign depends on the difference between principal stresses.
Two important remarks are in order at this point:

  * •

It is worth stressing that while the self–regenerative pressure effect
(described by the $4\pi r^{3}P_{r}$ term in (120)) has the same sign as in the
spherically symmetric case, the mass function contribution in the second term
has the opposite sign with respect to the latter case. This of course is due
to the fact that the energy density is negative.

  * •

If, both, the p.g.m.d. and the a.g.m. are negative, the final effect of the
gravitational interaction would be as usual, to oppose the negative pressure
gradient. However, because of the equivalence principle, a negative p.g.m.d.
implies a negative inertial mass, which in turn implies that the hydrostatic
force term (the pressure gradient and the anisotropic term), and the
gravitational force term, switch their roles with respect to the positive
energy density case.

In this case the complexity factor $Y_{TF}$ is given by

$Y_{TF}=4\pi\Pi+{\cal E},$ (121)

or

$Y_{TF}=8\pi\Pi+\frac{4\pi}{r^{3}}\int^{r}_{0}{\tilde{r}^{3}|\mu|^{\prime}d\tilde{r}},$
(122)

where ${\cal E}$ is the scalar defining the electric part of the Weyl tensor
(the magnetic part vanishes identically as in the spherically symmetric case).

Also, as in the spherically symetric case $Y_{TF}$ encompasses the influence
of the local anisotropy of pressure and density inhomogeneity on the Tolman
mass. Or, in other words, $Y_{TF}$ describes how these two factors modify the
value of the Tolman mass, with respect to its value for the homogeneous
isotropic fluid.

Indeed, in this case it can be shown that the following expressions for the
Tolman mass can be obtained (see rg4  for details)

$m_{T}=\frac{(cosh\pi-1)}{4}e^{(\nu-\lambda)/2}r^{2}\nu^{\prime},$ (123)

and

$\displaystyle m_{T}$ $\displaystyle=$
$\displaystyle(m_{T})_{\Sigma^{e}}\left(\frac{r}{r_{\Sigma^{e}}}\right)^{3}$
(124) $\displaystyle+$
$\displaystyle\frac{(cosh\pi-1)}{2}r^{3}\int^{r_{\Sigma^{e}}}_{r}{\frac{e^{(\nu+\lambda)/2}}{\tilde{r}}Y_{TF}d\tilde{r}}.$

We shall next present two models endowed with hyperbolical symmetry and
satisfying the vanishing complexity factor condition.

####  II.4.1 A model with vanishing complexity factor and vanishing radial
pressure

Since the vanishing complexity factor is not enough to close the full system
of Einstein equations we have to impose an additional restriction in order to
obtain a specific model. Here we shall assume (besides the vanishing
complexity factor), the condition $P_{r}=0$.

Thus, assuming $P_{r}=0$, we obtain from (112)

$\nu^{\prime}=-\frac{2g}{(2g-1)r},$ (125)

where $g$ is defined by

$e^{-\lambda}=2g-1.$ (126)

Next, imposing $Y_{TF}=0$ in (124) it follows that

$m_{T}=(m_{T})_{\Sigma^{e}}\frac{r^{3}}{r^{3}_{\Sigma}}.$ (127)

The combination of (123), (125), (126) and (127) produces

$e^{\nu}=\frac{4(m^{2}_{T})_{\Sigma^{e}}r^{4}}{r^{6}_{\Sigma}(cosh\pi-1)^{2}}\frac{(2g-1)}{g^{2}}.$
(128)

On the other hand the condition $Y_{TF}=0$ may be written as

$g^{\prime}r(1-g)+g(5g-2)=0,$ (129)

whose solution reads

$C_{2}r^{10}=\frac{g^{5}}{(5g-2)^{3}},$ (130)

where $C_{2}$ is a constant of integration.

Then for the physical variables we obtain

$\displaystyle|\mu|$ $\displaystyle=$ $\displaystyle\frac{3}{4\pi
r^{2}}\frac{g(2g-1)}{(g-1)},$ (131) $\displaystyle P_{\bot}$ $\displaystyle=$
$\displaystyle\frac{3}{8\pi r^{2}}\frac{g^{2}}{(g-1)}.$ (132)

In this case, the fluid distribution is restricted by a minimal value of the
$r$ coordinate, satisfying $g(r_{min})>1$. The specific value of $r_{min}$ is
obtained from (130). For $0<r<r_{min}$ we may assume, as in precedent models,
an empty cavity surrounding the center. Also as in precedent models, the
discontinuity of the mass function across $\Sigma^{i}$ implies that a thin
shell appears on it. Finally, since the radial pressure is assumed to be zero,
both the a.g.m. and the p.g.m.d. are negative.

####  II.4.2 A model with the stiff equation of state and vanishing complexity
factor

Finally we shall consider a solution satisfying the so called stiff equation
of state proposed by Zeldovich zh  and which is thought to be suitable to
describe ultradense matter, and the vanishing complexity factor.

In its original form the stiff equation of state assumes that energy density
equals pressure (in relativistic units). In our case we shall assume

$|\mu|=P_{r}.$ (133)

then (120) becomes

$\displaystyle P_{r}^{\prime}+\frac{2}{r}\Pi$ $\displaystyle=$ $\displaystyle
0.$ (134)

Using this latter condition in (122) with $Y_{TF}=0$, and feeding back the
resulting expression into (134) one obtains

$P^{\prime\prime}_{r}+\frac{3}{r}P^{\prime}_{r}=0,$ (135)

whose solution reads

$P_{r}=\frac{b}{r^{2}}-a,$ (136)

where $a$ and $b$ are two positive constants of integration.

Then from (116) and (117) it follows at once

$m=4\pi r\left(b-\frac{ar^{2}}{3}\right),$ (137)

from which we easily obtain $\lambda$. Finally, feeding back these expressions
into (119) we may obtain $\nu$.

Assuming the fluid distribution to be bounded from the exterior by the surface
$\Sigma^{e}$ described by $r=r_{\Sigma^{e}}=constant$, then we may write

$P_{r}=b\left(\frac{1}{r^{2}}-\frac{1}{{r^{2}_{\Sigma^{e}}}}\right).$ (138)

So far we have only considered static configurations. In the next section we
shall tackle the problem of defining complexity for dynamic dissipative
fluids.

##  III Dynamic spherically symmetric fluids

As mentioned before, when dealing with non–static dissipative fluids we
encounter two additional problems to define complexity. On the one hand we
have to include the dissipative flux in the variable measuring the complexity
of the structure. On the other hand we have also to describe the complexity of
the pattern of evolution.

In other words, in the dynamic case we have to ask ourselves two questions
instead of one, namely, what is the simplest fluid? and, what is the simplest
mode of evolution of the fluid? These two questions, although related in some
way, are completely different. A “simple” fluid may evolve exhibiting a very
complex pattern of evolution, while a fluid with a high degree of complexity
may evolve through a simple pattern of evolution.

This problem was studied in detail in ch2 , here we shall sketch the main
steps leading to the appropriate definition of complexity for dynamic fluids
and discuss about the answer to the question, what is the simplest mode of
evolution of the fluid? Afterwards we shall present some solutions.

We shall restrict the discussion to the spherically symmetric case.

###  III.1 The dynamic spherically symmetric case

So, let us consider a spherically symmetric distribution of collapsing fluid,
which may be bounded by a spherical surface $\Sigma$, or not. The fluid is
assumed to be locally anisotropic (principal stresses unequal) and undergoing
dissipation in the form of heat flow (diffusion approximation).

Choosing comoving coordinates, the general interior metric can be written

$ds^{2}=-A^{2}dt^{2}+B^{2}dr^{2}+R^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$
(139)

where $A$, $B$ and $R$ are functions of $t$ and $r$ and are assumed positive.
We number the coordinates $x^{0}=t$, $x^{1}=r$, $x^{2}=\theta$ and
$x^{3}=\phi$. Observe that $A$ and $B$ are dimensionless, whereas $R$ has the
same dimension as $r$.

The energy–momentum tensor $T_{\alpha\beta}$ of the fluid distribution has the
form

$\displaystyle T_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\mu+P_{\perp})V_{\alpha}V_{\beta}+P_{\perp}g_{\alpha\beta}+(P_{r}-P_{\perp})\chi_{\alpha}\chi_{\beta}$
(140) $\displaystyle+$ $\displaystyle
q_{\alpha}V_{\beta}+V_{\alpha}q_{\beta},$

where $\mu$ is the energy density, $P_{r}$ the radial pressure, $P_{\perp}$
the tangential pressure, $q^{\alpha}$ the heat flux, $V^{\alpha}$ the four
velocity of the fluid, and $\chi^{\alpha}$ a unit four vector along the radial
direction. These quantities satisfy

$\displaystyle
V^{\alpha}V_{\alpha}=-1,\;\;V^{\alpha}q_{\alpha}=0,\;\;\chi^{\alpha}\chi_{\alpha}=1,\;\;\chi^{\alpha}V_{\alpha}=0.$
(141)

Or in the equivalent (canonical) form

$T_{\alpha\beta}={\mu}V_{\alpha}V_{\beta}+Ph_{\alpha\beta}+\Pi_{\alpha\beta}+q\left(V_{\alpha}\chi_{\beta}+\chi_{\alpha}V_{\beta}\right)$
(142)

with

$P=\frac{P_{r}+2P_{\bot}}{3},\qquad
h_{\alpha\beta}=g_{\alpha\beta}+V_{\alpha}V_{\beta},$

$\Pi_{\alpha\beta}=\Pi\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right),\qquad\Pi=P_{r}-P_{\bot},$

where $q$ is a function of $t$ and $r$.

Since we are considering comoving observers, we have

$\displaystyle V^{\alpha}$ $\displaystyle=$ $\displaystyle
A^{-1}\delta_{0}^{\alpha},\;\;q^{\alpha}=qB^{-1}\delta^{\alpha}_{1},\;\;\chi^{\alpha}=B^{-1}\delta^{\alpha}_{1},$
(143)

It is worth noticing that, both, bulk and shear viscosity could be easily
introduced to the system through a redefinition of the radial and tangential
pressures, $P_{r}$ and $P_{\perp}$. Also, dissipation in the free streaming
approximation could be introduced by redefining $\mu,P_{r}$ and $q$.

The Einstein equations for (139) and (142), are explicitly written in Appendix
IX.

The acceleration $a_{\alpha}$ and the expansion $\Theta$ of the fluid are
given by

$a_{\alpha}=V_{\alpha;\beta}V^{\beta},\;\;\Theta={V^{\alpha}}_{;\alpha}.$
(144)

and its shear $\sigma_{\alpha\beta}$ by

$\sigma_{\alpha\beta}=V_{(\alpha;\beta)}+a_{(\alpha}V_{\beta)}-\frac{1}{3}\Theta
h_{\alpha\beta},$ (145)

from which we easily obtain

$a_{1}=\frac{A^{\prime}}{A},\;\;a=\sqrt{a^{\alpha}a_{\alpha}}=\frac{A^{\prime}}{AB},$
(146)

$\Theta=\frac{1}{A}\left(\frac{\dot{B}}{B}+2\frac{\dot{R}}{R}\right),$ (147)

$\sigma_{11}=\frac{2}{3}B^{2}\sigma,\;\;\sigma_{22}=\frac{\sigma_{33}}{\sin^{2}\theta}=-\frac{1}{3}R^{2}\sigma,$
(148)

where

$\sigma^{\alpha\beta}\sigma_{\alpha\beta}=\frac{2}{3}\sigma^{2},$ (149)

with

$\sigma=\frac{1}{A}\left(\frac{\dot{B}}{B}-\frac{\dot{R}}{R}\right),$ (150)

where the prime stands for $r$ differentiation and the dot stands for
differentiation with respect to $t$.

Next, the mass function $m(t,r)$ introduced by Misner and Sharp Misner  reads

$m=\frac{R^{3}}{2}{R_{23}}^{23}=\frac{R}{2}\left[\left(\frac{\dot{R}}{A}\right)^{2}-\left(\frac{R^{\prime}}{B}\right)^{2}+1\right],$
(151)

and introducing the proper time derivative $D_{T}$ given by

$D_{T}=\frac{1}{A}\frac{\partial}{\partial t},$ (152)

we can define the velocity $U$ of the collapsing fluid as the variation of the
areal radius with respect to proper time, i.e.

$U=D_{T}R,$ (153)

where $R$ defines the areal radius of a spherical surface inside the fluid
distribution (as measured from its area).

Then (151) can be rewritten as

$E\equiv\frac{R^{\prime}}{B}=\left(1+U^{2}-\frac{2m}{R}\right)^{1/2}.$ (154)

Using (154) we can express (254) as

$4\pi q=E\left[\frac{1}{3}D_{R}(\Theta-\sigma)-\frac{\sigma}{R}\right],$ (155)

where $D_{R}$ denotes the proper radial derivative,

$D_{R}=\frac{1}{R^{\prime}}\frac{\partial}{\partial r}.$ (156)

Using (250)-(252) with (156) we obtain from (151)

$\displaystyle D_{R}m=4\pi\left(\mu+q\frac{U}{E}\right)R^{2},$ (157)

which implies

$m=4\pi\int^{r}_{0}\left(\mu+q\frac{U}{E}\right)R^{2}R^{\prime}dr,$ (158)

satisfying the regular condition $m(t,0)=0$.

Integrating (158) we find

$\frac{3m}{R^{3}}=4\pi{\mu}-\frac{4\pi}{R^{3}}\int^{r}_{0}{R^{3}\left(D_{R}{\mu}-3q\frac{U}{RE}\right)R^{\prime}dr}.$
(159)

###  III.2 Defining complexity for the dynamic fluid

As we have already mentioned, in the dynamic case the definition of a quantity
measuring the complexity of the system poses two additional problems with
respect to the static case.

On the one hand, the definition of the complexity of the structure of the
fluid, which in this case also involves dissipative variables, and on the
other hand the problem of defining the complexity of the pattern of evolution
of the system.

For the static fluid distribution it was assumed that the scalar function
$Y_{TF}$ is an appropriate measure of the complexity of the fluid, and
therefore was identified as the complexity factor.

We shall assume in the dynamic case that $Y_{TF}$ still measures the
complexity of the system, in what corresponds to the structure of the object,
and we shall adopt initially an assumption about the simplest possible pattern
of evolution. Specifically, we shall assume that the simplest evolution
pattern (one of them at least) is described by the homologous evolution.
However, as we shall see below, this last condition might be too stringent,
ruling out many interesting scenarios from the astrophysical point of view and
therefore we shall consider also other possible (less restrictive) mode of
evolution which also could be used to describe the simplest mode of evolution,
and which we call quasi–homologous.

In order to provide the necessary mathematical expressions for carrying out
our task, let us start by finding the expression for the Weyl tensor.

As is well known, in the spherically symmetric case the Weyl tensor
($C^{\rho}_{\alpha\beta\mu}$) is defined by its “electric” part
$E_{\gamma\nu}$ alone, since its “magnetic” part vanishes, with

$E_{\alpha\beta}=C_{\alpha\mu\beta\nu}V^{\mu}V^{\nu},$ (160)

where the electric part of Weyl tensor may also be written as

$E_{\alpha\beta}={\cal
E}(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}).$ (161)

with

$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\frac{1}{2A^{2}}\left[\frac{\ddot{R}}{R}-\frac{\ddot{B}}{B}-\left(\frac{\dot{R}}{R}-\frac{\dot{B}}{B}\right)\left(\frac{\dot{A}}{A}+\frac{\dot{R}}{R}\right)\right]+\frac{1}{2B^{2}}\left[\frac{A^{\prime\prime}}{A}-\frac{R^{\prime\prime}}{R}+\left(\frac{B^{\prime}}{B}+\frac{R^{\prime}}{R}\right)\left(\frac{R^{\prime}}{R}-\frac{A^{\prime}}{A}\right)\right]-\frac{1}{2R^{2}}.$
(162)

Then, proceeding as in the previous cases (see ch2  for details) we obtain

$\displaystyle Y_{T}=4\pi(\mu+3P_{r}-2\Pi),\qquad Y_{TF}={\cal E}-4\pi\Pi.$
(163)

Next, using (250), (252), (253) with (151) and (162) we obtain

$\frac{3m}{R^{3}}=4\pi\left({\mu}-\Pi\right)-\cal{E},$ (164)

which combined with (159) and (163) produces

$Y_{TF}=-8\pi\Pi+\frac{4\pi}{R^{3}}\int^{r}_{0}{R^{3}\left(D_{R}{\mu}-3{q}\frac{U}{RE}\right)R^{\prime}dr}.$
(165)

Again, we notice that due to a different signature, the sign of $Y_{TF}$ in
the above equation differs from the sign of the $Y_{TF}$ used in ch1  for the
static case.

Thus the scalar $Y_{TF}$ may be expressed through the Weyl tensor and the
anisotropy of pressure or in terms of the anisotropy of pressure, the density
inhomogeneity and the dissipative variables.

Once the complexity factor for the structure of the fluid distribution has
been established, it remains to elucidate what is the simplest pattern of
evolution. Based on purely intuitive thoughts we shall first identify the
homologous evolution as the simplest mode of evolution.

In order to obtain a mathematical description of the homologous evolution, we
shall proceed as follows

First of all observe that we can write (155) as

$D_{R}\left(\frac{U}{R}\right)=\frac{4\pi}{E}q+\frac{\sigma}{R},$ (166)

which after integration becomes

$U=\tilde{a}(t)R+R\int^{r}_{0}\left(\frac{4\pi}{E}q+\frac{\sigma}{R}\right)R^{\prime}dr,$
(167)

where $\tilde{a}$ is an integration function, or,

$U=\frac{U_{\Sigma}}{R_{\Sigma}}R-R\int^{r_{\Sigma}}_{r}\left(\frac{4\pi}{E}q+\frac{\sigma}{R}\right)R^{\prime}dr.$
(168)

If the integrand in the above equations vanishes we have from (167) or (168)

$U=\tilde{a}(t)R.$ (169)

This relationship is characteristic of the homologous evolution in Newtonian
hydrodynamics 22s ; 21s ; 20s . In our case this may occur if the fluid is
shear–free and non dissipative, or if the two terms in the integral cancel
each other.

In ch2 , the term “homologous evolution” was used to characterize relativistic
systems satisfying, besides (169), the condition

$\frac{R_{I}}{R_{II}}=\mbox{constant},$ (170)

where $R_{I}$ and $R_{II}$ denote the areal radii of two concentric shells
($I,II$) described by $r=r_{I}={\rm constant}$, and $r=r_{II}={\rm constant}$,
respectively.

Now, it is very important to be aware of the fact that conditions (169) and
(170, in the general relativistic case, are different and more specifically,
that (169) does not imply (170).

Indeed, (169) implies that for the two shells of fluids $I,II$ we have

$\frac{U_{I}}{U_{II}}=\frac{A_{II}\dot{R}_{I}}{A_{I}\dot{R}_{II}}=\frac{R_{I}}{R_{II}},$
(171)

that implies (170) only if $A=A(t)$, which by a simple coordinate
transformation becomes $A={\rm constant}$. Thus in the non–relativistic
regime, (170) always follows from the condition that the radial velocity is
proportional to the radial distance, whereas in the relativistic regime the
condition (169) implies (170), only if the fluid is geodesic.

We shall define quasi–homologous evolution as that restricted only by
condition (169), implying

$\frac{4\pi}{R^{\prime}}Bq+\frac{\sigma}{R}=0.$ (172)

Let us first consider the homologous evolution (both (169) and (170) are
satisfied).

From the equation (170) it follows that $R$ is a separable function, i.e. we
can write

$R=R_{1}(t)R_{2}(r).$ (173)

To summarize, the homologous condition implies (173), and

$\frac{4\pi}{R^{\prime}}Bq+\frac{\sigma}{R}=0.$ (174)

Feeding back this last expression into (254), we obtain

$(\Theta-\sigma)^{\prime}=0,$ (175)

whereas, using (147) and (150) we get

$\left(\Theta-\sigma\right)^{\prime}=\left(\frac{3}{A}\frac{\dot{R}}{R}\right)^{\prime}=0.$
(176)

Then using (173) it follows at once that

$A^{\prime}=0,$ (177)

implying that the fluid is geodesic, as it follows from (146). Also, by
reparametrizing the coordinate $r$, we may put, without loss of generality,
$A=1$.

The inverse is in general untrue, e.g. the Lemaitre–Tolman– Bondi case, unless
we assume that $(\Theta-\sigma)^{\prime}$ is of class $C^{\omega}$.

In the non–dissipative case, the homologous condition not only implies that
the fluid is geodesic, but also that it is shear–free, as it follows at once
from (174).

An important point to mention here is that as it has been shown in 26 , an
initially shear–free geodesic fluid remains shear–free during the evolution
iff $Y_{TF}=0$. This implies that a system which starts its evolution from the
rest ($\sigma=0$), will remain shear–free if the fluid is geodesic (or
equivalently, homologous) and $Y_{TF}=0$. This is an additional argument
supporting our choice of $Y_{TF}$ as the complexity factor.

If we impose the homologous condition, the equation (259) in Appendix X
becomes

$D_{T}U=-\frac{m}{R^{2}}-4\pi P_{r}R,$ (178)

or in terms of $Y_{TF}$

$\frac{3D_{T}U}{R}=-4\pi\left(\mu+3P_{r}-2\Pi\right)+Y_{TF},$ (179)

where (163) has been used.

Next from the field equations we obtain

$4\pi\left(\mu+3P_{r}-2\Pi\right)=-\frac{2\ddot{R}}{R}-\frac{\ddot{B}}{B},$
(180)

and from the definition of $U$

$\frac{3D_{T}U}{R}=\frac{3\ddot{R}}{R},$ (181)

feeding back the two equations above into (179), it follows that

$\frac{\ddot{R}}{R}-\frac{\ddot{B}}{B}=Y_{TF}.$ (182)

Now, the vanishing complexity factor condition ($Y_{TF}=0$), produces after
the integration of (182)

$B=R_{1}(t)\left(b_{1}(r)\int{\frac{dt}{R_{1}(t)^{2}}+b_{2}(r)}\right),$ (183)

where $b_{1}(r)$ and $b_{2}(r)$ are two functions of integration, or

$B=R_{1}(t)R^{\prime}_{2}(r)\left(\tilde{b}_{1}(r)\int{\frac{dt}{R_{1}(t)^{2}}+\tilde{b}_{2}(r)}\right),$
(184)

with $b_{1}(r)=\tilde{b}_{1}(r)R_{2}^{\prime}$ and
$b_{2}(r)=\tilde{b}_{2}(r)R_{2}^{\prime}$.

Then introducing the variable

$Z=\tilde{b}_{1}(r)\int{\frac{dt}{R_{1}(t)^{2}}+\tilde{b}_{2}(r)},$ (185)

we may write

$B=ZR^{\prime}.$ (186)

Let us now use the expressions above to analyze first the non–dissipative
case.

Thus, if we further assume the fluid to be non–dissipative, recalling that in
this case the homologous condition implies the vanishing of the shear, we
obtain because of (150)

$\frac{\ddot{R}}{R}-\frac{\ddot{B}}{B}=0\quad\Rightarrow Y_{TF}=0.$ (187)

In other words, in this particular case, the homologous condition already
implies the vanishing complexity factor condition.

More so, since the fluid is shear–free, we have because of (150) and (183)

$b_{1}(r)=0\Rightarrow
B=b_{2}(r)R_{1}(t)=\tilde{b}_{2}(r)R_{1}(t)R_{2}^{\prime}.$ (188)

Then reparametrising $r$ as $\tilde{b}_{2}(r)dr\Rightarrow dr$, we may put
without loss of generality $B=R_{1}(t)R_{2}(r)^{\prime}$, or equivalently
$Z=1$, implying that all non–dissipative configurations evolving homologously
(and thereby satisfying $Y_{TF}=0$), belong to what are known as “Euclidean
stars” euc , characterized by the condition $Z=1\Rightarrow B=R^{\prime}$.
However among all possible solutions satisfying the “Euclidean condition” ,
only one evolves homologously and satisfies the condition $Y_{TF}=0$.

Indeed, from the field equations (252) and (253) we may write

$8\pi(P_{r}-P_{\bot})=\frac{\dot{Z}\dot{R}}{ZR}+\frac{1}{Z^{2}R^{2}}\left(\frac{Z^{\prime}R}{ZR^{\prime}}+1-Z^{2}\right).$
(189)

Since in this case we have $Z=1$ then $\Pi=P_{r}-P_{\bot}=0$ which implies
because of the $Y_{TF}=0$ condition, that $\mu^{\prime}=0$.

However it is known that a shear–free, geodesic (non–dissipative) fluid with
isotropic pressure is necessarily dust with homogeneous energy density and
vanishing Weyl tensor (see gen ; sc ). It goes without saying that this kind
of system represents the simplest conceivable configuration
(Friedman–Robertson–Walker).

Thus for the non–dissipative case, the homologous condition implies $Y_{TF}=0$
and produces the simplest configuration. This configuration is the only one
evolving homologously and satisfying $Y_{TF}=0$.

Of course, solutions satisfying $Y_{TF}=0$ but not evolving homologously do
exist. They only require
$8\pi\Pi=\frac{4\pi}{R^{3}}\int^{r}_{0}{R^{3}\mu^{\prime}dr}$. In such a case
the solutions are shearing, and neither conformally flat nor geodesic.

Based on all the precedent comments, it seems reasonable to consider the
homologous condition as a good candidate to describe the simplest mode of
evolution.

###  III.3 The dissipative case

In the dissipative case, we may obtain from (150) and (187),

$\dot{\sigma}=-Y_{TF}+\left(\frac{\dot{R}}{R}\right)^{2}-\left(\frac{\dot{B}}{B}\right)^{2}.$
(190)

Then, taking the $t$-derivative of (174) and using (190) we obtain

$Y_{TF}\frac{R^{\prime}}{R}=4\pi
Bq\left(\frac{\dot{q}}{q}+2\frac{\dot{B}}{B}+\frac{\dot{R}}{R}\right).$ (191)

If we assume $Y_{TF}=0$, then we obtain

$q=\frac{f(r)}{B^{2}R},$ (192)

implying

$\dot{q}=-q(\Theta+\sigma),$ (193)

where $f$ is an arbitrary integration function. Solutions of this kind might
be found by using the general methods presented in TM ; TMb ; ivanov0 ;
ivanov1 ; ivanov .

In the dissipative case we need to provide a transport equation to describe
the evolution and distribution of temperature. Assuming a causal dissipative
theory (e.g. the Israel– Stewart theory 19nt ; 20nt ; 21nt  ), the transport
equation for the heat flux reads

$\tau h^{\alpha\beta}V^{\gamma}q_{\beta;\gamma}+q^{\alpha}=-\kappa
h^{\alpha\beta}\left(T_{,\beta}+Ta_{\beta}\right)-\frac{1}{2}\kappa
T^{2}\left(\frac{\tau V^{\beta}}{\kappa T^{2}}\right)_{;\beta}q^{\alpha},$
(194)

where $\kappa$ denotes the thermal conductivity, and $T$ and $\tau$ denote
temperature and relaxation time, respectively.

In the spherically symmetric case under consideration, the transport equation
has only one independent component, which may be obtained from (194) by
contracting with the unit spacelike vector $K^{\alpha}$, producing

$\tau
V^{\alpha}q_{,\alpha}+q=-\kappa\left(K^{\alpha}T_{,\alpha}+Ta\right)-\frac{1}{2}\kappa
T^{2}\left(\frac{\tau V^{\alpha}}{\kappa T^{2}}\right)_{;\alpha}q.$ (195)

Sometimes, it is possible to simplify the equation above, assuming the so-
called truncated transport equation when the last term in (194) may be
neglected t8 , producing

$\tau V^{\alpha}q_{,\alpha}+q=-\kappa\left(K^{\alpha}T_{,\alpha}+Ta\right).$
(196)

Now, in a dissipative process, it appears reasonable to consider the
stationary state, prevailing once the system has relaxed and transient
phenomena have vanished, as an example of the simplest dissipative regime.
Thus, if we assume the stationary state (neglecting the relaxation time), then
the transport equation (194) reads

$q=-\frac{\kappa T^{\prime}}{B}.$ (197)

Combining the above equation with (192) we obtain

$T^{\prime}=-\frac{f(r)}{\kappa BR}.$ (198)

At this point, however, neither can we provide solid arguments to support
further the assumption about the vanishing of the relaxation time as an
indicator of minimum complexity of the dissipative regime, nor can we prove
that exact solutions of this kind exist.

So far we have assumed the homologous condition in order to describe the
simplest mode of evolution, however as indicated in the lines above, the
resulting models are perhaps too restrictive, and it could be wise to consider
less stringent conditions. One possible example could be the quasi–homologous
condition (174). In rg5 ; rg6n ; hcm  the reader may find a long list of exact
solutions satisfying the vanishing complexity factor and the quasi–homologous
condition for spherically symmetric and hyperbolically symmetric dissipative
fluids.

Finally we shall tackle the problem of defining complexity of vacuum solutions
to Einstein equations. We shall restrict our discussion to the Bondi metric.

##  IV Complexity of the Bondi space–time

Let us now consider the extension of our concept of complexity to vacuum
spacetimes. More specifically we shall consider the Bondi metric Boal , which
encompasses a vast numbers of spacetimes, including the Minkowski spacetime,
the static Weyl metrics, non–radiative non–static metrics and gravitationally
radiating metrics. Furthermore, the Bondi approach has the virtue of providing
a clear and precise criterion for the existence of gravitational radiation.
Namely, if the news function is zero over a time interval, then there is no
radiation over that interval.

As we have seen, in the case of fluid distributions, the variable(s) measuring
the complexity of the fluid (the complexity factor(s)) appear in the trace
free part of the orthogonal splitting of the electric Riemann tensor. In
vacuum the Riemann tensor and the Weyl tensor are the same, so if we
extrapolate to the vacuum case the same definition of complexity as for the
fluid self–gravitating system, we shall need the scalar functions defining the
electric part of the Weyl tensor for the Bondi metric.

The general form of an axially and reflection symmetric asymptotically flat
metric given by Bondi Boal  is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{V}{r}e^{2\beta}-U^{2}r^{2}e^{2\gamma}\right)du^{2}+2e^{2\beta}dudr+2Ur^{2}e^{2\gamma}dud\theta$
(199) $\displaystyle-$ $\displaystyle
r^{2}\left(e^{2\gamma}d\theta^{2}+e^{-2\gamma}\sin^{2}{\theta}d\phi^{2}\right),$

where $V,\beta,U$ and $\gamma$ are functions of $u,r$ and $\theta$.

We number the coordinates $x^{0,1,2,3}=u,r,\theta,\phi$ respectively. $u$ is a
timelike coordinate ($g_{uu}>0$ ) converging to the retarded time as
$r\rightarrow\infty$. The hypersurfaces $u=constant$ define null surfaces
(their normal vectors are null vectors), which at null infinity
($r\rightarrow\infty$) coincides with the Minkowski null light cone open to
the future. $r$ is a null coordinate ($g_{rr}=0$) and $\theta$ and $\phi$ are
two angle coordinates (see Boal  for details).

Regularity conditions in the neighborhood of the polar axis
($\sin{\theta}=0$), imply that as $\sin{\theta}\rightarrow 0$

$V,\beta,U/\sin{\theta},\gamma/\sin^{2}{\theta},$ (200)

each equals a function of $\cos{\theta}$ regular on the polar axis.

The four metric functions are assumed to be expanded in series of $1/r$, then
using the field equations, Bondi gets

$\gamma=cr^{-1}+\left(C-\frac{1}{6}c^{3}\right)r^{-3}+...,$ (201)
$U=-\left(c_{\theta}+2c\cot{\theta}\right)r^{-2}+\left[2N+3cc_{\theta}+4c^{2}\cot{\theta}\right]r^{-3}...,$
(202)

$\displaystyle
V=r-2M-\left(N_{\theta}+N\cot{\theta}-c_{\theta}^{2}-4cc_{\theta}\cot{\theta}-\right.$
(203)
$\displaystyle\left.\frac{1}{2}c^{2}(1+8\cot^{2}{\theta})\right)r^{-1}...,$
(204) $\beta=-\frac{1}{4}c^{2}r^{-2}+...,$ (205)

where $c$, $C$, $N$ and $M$ are functions of $u$ and $\theta$ satisfying the
constraint

$4C_{u}=2c^{2}c_{u}+2cM+N\cot{\theta}-N_{\theta},$ (206)

and letters as subscripts denote derivatives. The three functions $c,M$ and
$N$ are further related by the supplementary conditions

$M_{u}=-c_{u}^{2}+\frac{1}{2}\left(c_{\theta\theta}+3c_{\theta}\cot{\theta}-2c\right)_{u},$
(207) $-3N_{u}=M_{\theta}+3cc_{u\theta}+4cc_{u}\cot{\theta}+c_{u}c_{\theta}.$
(208)

In the static case $M$ equals the mass of the system and is called by Bondi
the “mass aspect”, whereas $N$ and $C$ are closely related to the dipole and
quadrupole moments respectively.

Next, Bondi defines the mass $m(u)$ of the system as

$m(u)=\frac{1}{2}\int_{0}^{\pi}{M\sin{\theta}d\theta},$ (209)

which by virtue of (207) and (200) yields

$m_{u}=-\frac{1}{2}\int_{0}^{\pi}{c_{u}^{2}\sin{\theta}d\theta}.$ (210)

Arriving at this point let us summarize the main conclusions emerging from
Bondi’s approach.

  1. 1.

If $\gamma,M$ and $N$ are known for some $u=a$(constant) and $c_{u}$ (the news
function) is known for all $u$ in the interval $a\leq u\leq b$, then the
system is fully determined in that interval. In other words, whatever happens
at the source, leading to changes in the field, it can only do so by affecting
$c_{u}$ and vice versa. In the light of this comment the relationship between
news function and the occurrence of radiation becomes clear.

  2. 2.

As it follows from (210), the mass of a system is constant if and only if
there is no news.

Now, for an observer at rest in the frame of (199), the four-velocity vector
has components

$V^{\alpha}=\left(\frac{1}{A},0,0,0\right),$ (211)

with

$A\equiv\left(\frac{V}{r}e^{2\beta}-U^{2}r^{2}e^{2\gamma}\right)^{1/2}.$ (212)

Next, let us introduce the unit, spacelike vectors ${\bf K}$, ${\bf L}$, ${\bf
S}$, with components

$K^{\alpha}=\left(\frac{1}{A},-e^{-2\beta}A,0,0\right)\quad
L^{\alpha}=\left(0,Ure^{\gamma}e^{-2\beta},-\frac{e^{-\gamma}}{r},0\right)$
(213) $S^{\alpha}=\left(0,0,0,-\frac{e^{\gamma}}{r\sin\theta}\right),$ (214)

For the observer defined by (211) the vorticity vector may be written as (see
HSC  for details)

$\omega^{\alpha}=\left(0,0,0,\omega^{\phi}\right).$ (215)

The explicit expressions for $\omega^{\phi}$ and its absolute value
$\Omega\equiv\left(-\omega_{\alpha}\omega^{\alpha}\right)^{1/2}$ are given in
the Appendix XIII.

The electric and magnetic parts of Weyl tensor, $E_{\alpha\beta}$ and
$H_{\alpha\beta}$, respectively, are formed from the Weyl tensor
$C_{\alpha\beta\gamma\delta}$ and its dual
$\tilde{C}_{\alpha\beta\gamma\delta}$ by contraction with the four velocity
vector given by (211)

$E_{\alpha\beta}=C_{\alpha\gamma\beta\delta}V^{\gamma}V^{\delta},$ (216)
$\displaystyle H_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\tilde{C}_{\alpha\gamma\beta\delta}V^{\gamma}V^{\delta}=\frac{1}{2}\epsilon_{\alpha\gamma\epsilon\delta}C^{\epsilon\delta}_{\quad\beta\rho}V^{\gamma}V^{\rho},$
(217)
$\displaystyle\epsilon_{\alpha\beta\gamma\delta}\equiv\sqrt{-g}\;\;\eta_{\alpha\beta\gamma\delta},$

where $\eta_{\alpha\beta\gamma\delta}$ is the permutation symbol.

The electric part of the Weyl tensor has only three independent non-vanishing
components, whereas only two components define the magnetic part. Thus we may
write

$\displaystyle E_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{1}\left(K_{\alpha}L_{\beta}+L_{\alpha}K_{\beta}\right)+\mathcal{E}_{2}\left(K_{\alpha}K_{\beta}+\frac{1}{3}h_{\alpha\beta}\right)$
(218) $\displaystyle+$
$\displaystyle\mathcal{E}_{3}\left(L_{\alpha}L_{\beta}+\frac{1}{3}h_{\alpha\beta}\right),$

and

$H_{\alpha\beta}=H_{1}(S_{\alpha}K_{\beta}+S_{\beta}K_{\alpha})+H_{2}(S_{\alpha}L_{\beta}+S_{\beta}L_{\alpha}).$
(219)

with $h_{\mu\nu}=g_{\mu\nu}-V_{\nu}V_{\mu}$, and

$\mathcal{E}_{1}=L^{\alpha}K^{\beta}E_{\alpha\beta},$ (220)
$\mathcal{E}_{2}=(2K^{\alpha}K^{\beta}+L^{\alpha}L^{\beta})E_{\alpha\beta},$
(221)
$\mathcal{E}_{3}=(2L^{\alpha}L^{\beta}+K^{\alpha}K^{\beta})E_{\alpha\beta},$
(222)

these three scalars represent the complexity factors of our solutions.

For the magnetic part we have

$H_{2}=S^{\alpha}L^{\beta}H_{\alpha\beta},$ (223)
$H_{1}=S^{\alpha}K^{\beta}H_{\alpha\beta}.$ (224)

Explicit expressions for these scalars are given in the Appendixes XI and XII.

In HSC  it was obtained that if we put $H^{\alpha}_{\beta}=0$ then the field
is non–radiative and up to order $1/r^{3}$ in $\gamma$, the metric is static,
and the mass, the “dipole” ($N$) and the “quadrupole” ($C$) moments correspond
to a static situation. However, the time dependence might enter through
coefficients of higher order in $\gamma$, giving rise to what Bondi calls
“non–natural–non–radiative moving system” (NNNRS). In this latter case, the
system keeps the first three moments independent of time, but allows for time
dependence of higher moments. This class of solutions is characterized by
$M_{\theta}=0$.

A second family of time dependent non–radiative solutions exists for which
$M_{\theta}\neq 0$. These are called natural non–radiative moving system”
(NNRS), and their magnetic Weyl tensor is non–vanishing.

Let us now discuss the hierarchy of different spacetimes belonging to the
Bondi family, according to their complexity.

The simplest spacetime corresponds to the vanishing of the three complexity
factors, and this is just Minkowski.

Indeed, as it was shown in HSC , if we assume $E^{\alpha}_{\beta}=0$ and use
regularity conditions, we find that the spacetime must be Minkowski, giving
further support to the conjecture that there are no purely magnetic vacuum
space–times Bonnor .

At the other end (maximal complexity) we have a gravitationally radiating
system which requires all three complexity factors to be different from zero.

Indeed, if we assume that $\mathcal{E}_{1}=0$, then it follows at once from
(261) that $c_{u}=0$ (otherwise $c_{u}$ would be a non–regular function of
$\theta$ on the symmetry axis). Thus $\mathcal{E}_{1}=0$ implies that the
system is non–radiative.

If instead we assume that $\mathcal{E}_{2}=0$, then from the first order in
(262) we obtain that $c_{uu}=0$, this implies that either $c_{u}=0$ or
$c_{u}\sim u$. Bondi refers to this latter case as “mass loss without
radiative Riemann tensor” and dismisses it as being of little physical
significance. As a matter of fact, in this latter case the system would be
radiating “forever”, which according to (210) requires an unbounded source,
incompatible with an asymptotically flat spacetime. Thus in this case too, we
have $c_{u}=0$, and the system is non–radiative.

Finally, if we assume $\mathcal{E}_{3}=0$ it follows at once from the first
order in (263), that $c_{uu}=0$, leading to $c_{u}=0$, according to the
argument above.

Thus, a radiative system requires all three complexity factors to be
nonvanishing, implying a maximal complexity.

In the middle of the two extreme cases we have, on the one hand the
spherically symmetric spacetime (Schwarzschild), characterized by a single
complexity factor (the same applies for any static metric),
$\mathcal{E}_{1}=\mathcal{E}_{3}=0$, and $\mathcal{E}_{2}=\frac{3M}{r^{3}}$.
On the other hand, we have the non–static non–radiative case.

Let us now analyze in detail this latter case. This group of solutions
encompasses two subclasses, which using Bondi notation are:

  1. 1.

Natural–non–radiative systems (NNRS) characterized by $M_{\theta}\neq 0$.

  2. 2.

Non–natural–non–radiative systems (NNNRS) characterized by $M_{\theta}=0$.

Let us first consider the NNNRS subcase. Using (261) we obtain ${\cal
E}_{1}=0$, (up to order $1/r^{3}$), while the first non–vanishing terms in
${\cal E}_{2}$ and ${\cal E}_{3}$ are respectively, $3M$ and $0$, where
(VIII), (207), (208), (262) and (263) have been used.

Thus, the NNNRS are characterized by only one non–vanishing complexity factor
(${\cal E}_{2}$). Furthermore, as it follows from (267) the vorticity of the
congruence of observers at rest with respect to the frame of (199) vanishes,
and the field is purely electric. However as mentioned before we cannot
conclude that the field is static, since the $u$ dependence might appear
through coefficients of higher order in $\gamma$.

Let us now consider the “natural–non–radiative system” (NNRS). In this
subcase, using (261) we obtain ${\cal E}_{1}=0$, (up to order $1/r^{3}$) as
for the NNNRS subcase, while the first non–vanishing term in ${\cal E}_{2}$
and ${\cal E}_{3}$ (up to order $1/r^{3}$) are respectively
$3M+\frac{M_{\theta\theta}}{4}-\frac{M_{\theta}\cot\theta}{4}$ and
$\frac{M_{\theta\theta}}{2}-\frac{M_{\theta}\cot\theta}{2}$.

Also, up to the same order, it follows from (264) and (265) that $H_{1}=0$ for
both subcases, while the corresponding term in $H_{2}$ is (for NNRS)

$-\frac{1}{4}(M_{\theta\theta}-M_{\theta}\cot\theta),$ (225)

which of course vanishes for the NNNRS subcase.

It should be observed that if we assume ${\cal E}_{3}=0$ or $H_{2}=0$ then it
follows at once from the above that

$M_{\theta\theta}-M_{\theta}\cot\theta=0\Rightarrow M=a\cos\theta,\quad
a=constant.$ (226)

But this implies because of (209) that the Bondi mass function of the system
vanishes. Therefore, the only physically meaningful NNRS requires ${\cal
E}_{3}\neq 0$, $\Omega\neq 0$ and $H_{2}\neq 0$ implying that the complexity
is characterized by two complexity factors (${\cal E}_{2}$, ${\cal E}_{3}$).

Thus a hierarchy of spacetimes according to their complexity has been
established. This allows us to discriminate between two classes of spacetimes
that depend on time but are not radiative (vanishing of the news function).
These two classes were called by Bondi Boal  natural and non–natural
non–radiative moving systems, and are characterized by different forms of the
mass aspect. They exhibit different degree of complexity.

Unfortunately, though, up to the leading order of the complexity factors
analyzed here, it is impossible to discriminate between different radiative
systems according to their complexity. Higher order terms would be necessary
for that purpose, although it is not clear at this point if it is possible to
establish such a hierarchy of radiative systems after all.

The simplest system (Minkowski) is characterized by the vanishing of all the
complexity factors. Next, the static case (including Schwarzchild) is
described by a single complexity factor.

The time dependent non–radiative solutions split in two subgroups depending on
the form of the mass aspect $M$. If $M_{\theta}=0$ which corresponds to the
NNNRS the complexity is similar to the static case. Also, in this case, as in
the static situation, the vorticity vanishes and the field is purely electric.
This result could suggest that in fact NNNRS are just static, and no time
dependence appears in the coefficients of higher order in $\gamma$. On the
contrary for the NNRS there are two complexity factors, the vorticity is
non–vanishing and the field is not purely electric.

All these results are summarized in Tables I and II. Thus, NNNRS and NNRS are
clearly differentiated through their degree of complexity, as measured by the
complexity factors considered here.

The fact that radiative systems necessarily decay into NNRS, NNNRS or static
systems, since the Bondi mass function must be finite, suggests that higher
degrees of complexity might be associated with stronger stability. Of course a
proof of this conjecture requires a much more detailed analysis.

It is also worth mentioning the conspicuous link between vorticity and
complexity factors. Indeed vorticity appears only in NNRS and radiative
systems, which are the most complex systems, while it is absent in the
simplest systems (Minkowski, static, NNNRS). In the radiative case there are
contributions at order $\mathcal{O}(r^{-1})$ related to the news function, and
at order $\mathcal{O}(r^{-2})$, while for the NNRS there are only
contributions at order $\mathcal{O}(r^{-2})$, these describe the effect of the
tail of the wave.

Table 1: Complexity factors for different spacetimes of the Bondi metric

COMPLEXITY HIERARCHY

$complex.fac.\diagdown spacetimes$ |  | Minkowski |  | Static |  | NNNRS |  | NNRS |  | Radiative |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
${\cal E}_{1}$ |  | 0 |  | 0 |  | 0 |  | 0 |  |  ${\cal E}^{(n)}_{1}\neq 0$, $n\geq 1$ |  |
${\cal E}_{2}$ |  | 0 |  | ${\cal E}^{(3)}_{2}=3M$ |  | ${\cal E}^{(3)}_{2}=3M$ |  | ${\cal E}^{(3)}_{2}=3M+\frac{M_{\theta\theta}}{4}-\frac{M_{\theta}\cot\theta}{4}$ |  |  ${\cal E}^{(n)}_{2}\neq 0$, $n\geq 1$ |  |
${\cal E}_{3}$ |  | 0 |  | 0 |  | 0 |  | ${\cal E}^{(3)}_{3}=\frac{1}{2}(M_{\theta\theta}-M_{\theta}\cot\theta)$ |  |  ${\cal E}^{(n)}_{3}\neq 0$, $n\geq 1$ |  |

where ${\cal E}^{(n)}_{1,2,3}$ are de coefficients of order
$\mathcal{O}(r^{-n})$.

Table 2: The magnetic parts of the Weyl tensor and the vorticity for different
spacetimes of the Bondi metric

Magnetic parts and vorticity

$Mag.Weyl;\Omega.\diagdown spacetimes$ |  | Minkowski |  | Static |  | NNNRS |  | NNRS |  | Radiative |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
$H_{1}$ |  | 0 |  | 0 |  | 0 |  | 0 |  |  $H^{(n)}_{1}\neq 0$, $n\geq 1$ |  |
$H_{2}$ |  | 0 |  | $0$ |  | $0$ |  | $H^{(3)}_{2}=-\frac{1}{4}(M_{\theta\theta}-M_{\theta}\cot\theta)$ |  |  $H^{(n)}_{2}\neq 0$, $n\geq 1$ |  |
$\Omega$ |  | 0 |  | 0 |  | 0 |  | $\Omega^{(2)}=M_{\theta}$ |  |  $\Omega^{(n)}\neq 0$, $n\geq 1$ |  |

##  V Conclusions

We have discussed about the concept of complexity of self–gravitating
relativistic fluid distributions. For the static case we were concerned
exclusively with the notion of complexity of the structure of the fluid.
However in the dynamic case we have also tackled the question about the
complexity of the pattern of evolution. These two questions, although related,
refer to different aspects of the definition of complexity. However, it is
remarkable that in the non–dissipative case the homologous condition implies
the vanishing complexity factor.

As a measure of complexity of the structure of the fluid (the complexity
factor) we have chosen the scalar function(s) $Y_{TF}$ defining the trace–free
part of the electric Riemann tensor.

Next, we discussed about the complexity of the pattern of evolution. Two
possibilities appear as the more obvious candidates: the homologous condition
and the quasi–homologous condition. The latter, being less restricted than the
former allows to consider larger number of models.

All this having been said, we are well aware of the fact that the above
mentioned candidate for measuring the complexity of a self-gravitating system
is by no means unique and many different alternatives may be proposed. In the
same line of arguments, the simplest patterns of evolution assumed so far are
the homologous and the quasi-homologous regimes. But, it is not clear whether
or not other patterns of evolution could also fit the role of the simplest
pattern of evolution.

Additionally, we believe that alternative definitions of complexity for vacuum
space–time is worth considering. Finally, new exact solutions to the field
equations in the context of the Einstein theory or any alternative one, would
serve as a test-bed for the definition of complexity.

These remarks suggest a list of questions and open issues which we believe
that deserve to be considered further.

  * •

Are there alternative definitions of complexity, different from the one
proposed in ch1 ?

  * •

Are there other ways to extend the definition of complexity for vacuum
space–time?

  * •

Besides the homologous and the quasi-homologous regime, could we define
another pattern of evolution that could qualify as the simplest one?

  * •

Do physically meaningful dissipative models satisfying (192) exist?

  * •

If the answer to the above question is positive then, is there a unique
solution or are there a large number of them?

  * •

What is the physical meaning of such solution(s)?

  * •

Is it physically reasonable to neglect transient effects when considering the
simplest dissipative system, and assume that the relaxation time vanishes?

  * •

To summarize the four points above: is there a specific dissipative regime
that could be considered as the simplest one?

  * •

Can we relate the complexity factor(s) in the non-spherically symmetric case
to the active gravitational mass, as in the spherically symmetric case?

  * •

Can we single out a specific family of exact axially symmetric static
solutions satisfying the vanishing complexity factor(s) condition?

  * •

Can any of the above solutions be matched smoothly to any vacuum Weyl
solution?

  * •

The definition of complexity proposed in ch1  is not directly related to
entropy or disequilibrium, although it is possible that such a link might
exist after all. If so, how could such relationship be brought out?

  * •

What is the simplest mode of evolution of axially–symmetric fluids? How is
such a mode related to the emission (or not) of gravitational radiation?

  * •

Could it be possible to provide a definition of the arrow of time in terms of
the complexity factor?

  * •

How is the complexity factor related to physical relevant properties of the
source, in terms of stability or maximal degree of compactness?

  * •

How does the complexity factor evolve? Do physically meaningful systems prefer
vanishing complexity factors?

  * •

As we have seen the FRW model satisfies the vanishing complexity factor and
evolves homologously. Should any other, physically sound, cosmological model
have a vanishing complexity factor? Should it evolve in the homologous or
quasi-homologous regime?

  * •

The complexity factor for a charged fluid is known, but what is the complexity
factor for a different type of field (e.g., scalar field?).

  * •

How should we define the complexity factor in the context of other alternative
theories of gravity that have not been considered so far?

  * •

How can we find new solutions satisfying the vanishing complexity factor?
Could we use the general methods described in TM ; TMb ; ivanov ; ivanov0 ;
ivanov1  to obtain such solutions?

  * •

What relevant physical features share solutions satisfying the vanishing
complexity factor?

  * •

Is there a link between the concept of complexity and some kind of symmetry
(e.g. motions, conformal motions, affine collineations, curvature
collineations, matter collineations, etc.)?

  * •

We have extended the concept of complexity, adopted for fluids, to the vacuum
case for the Bondi metric, the three complexity factors corresponding to the
three scalars defining the electric part of the Weyl tensor. Could it be
possible to further refine the scheme proposed here so as to discriminate
between different radiative systems according to their complexity? Or, in
other words, among radiative systems is there a simplest one ?

  * •

Also in the vacuum case, could it be possible to discriminate between
different static spacetimes of the Weyl family?

## Acknowledgments

This work was partially supported by Grant PID2021-122938NB-I00 funded by
MCIN/AEI/ 10.13039/501100011033 and by ERDF A way of making Europe. I also
wish to thank Universitat de les Illes Balears for financial support and
hospitality.

## Appendix

##  VI The Einstein and the conservation equations for the axially symmetric
static case

For the line element (43) and the energy momentum (47), the Einstein equations
read:

$\displaystyle
8\pi\mu=-\frac{1}{B^{2}}\left\\{\frac{B^{\prime\prime}}{B}+\frac{D^{\prime\prime}}{D}+\frac{1}{r}\left(\frac{B^{\prime}}{B}+\frac{D^{\prime}}{D}\right)-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{1}{r^{2}}\left[\frac{B_{\theta\theta}}{B}+\frac{D_{\theta\theta}}{D}-\left(\frac{B_{\theta}}{B}\right)^{2}\right]\right\\},$
(227)

$\displaystyle 8\pi
P_{xx}=\frac{1}{B^{2}}\left[\frac{A^{\prime}B^{\prime}}{AB}+\frac{A^{\prime}D^{\prime}}{AD}+\frac{B^{\prime}D^{\prime}}{BD}+\frac{1}{r}\left(\frac{A^{\prime}}{A}+\frac{D^{\prime}}{D}\right)+\frac{1}{r^{2}}\left(\frac{A_{\theta\theta}}{A}+\frac{D_{\theta\theta}}{D}-\frac{A_{\theta}B_{\theta}}{AB}+\frac{A_{\theta}D_{\theta}}{AD}-\frac{B_{\theta}D_{\theta}}{BD}\right)\right],$
(228) $\displaystyle 8\pi
P_{yy}=\frac{1}{B^{2}}\left[\frac{A^{\prime\prime}}{A}+\frac{D^{\prime\prime}}{D}-\frac{A^{\prime}B^{\prime}}{AB}+\frac{A^{\prime}D^{\prime}}{AD}-\frac{B^{\prime}D^{\prime}}{BD}+\frac{1}{r^{2}}\left(\frac{A_{\theta}B_{\theta}}{AB}+\frac{A_{\theta}D_{\theta}}{AD}+\frac{B_{\theta}D_{\theta}}{BD}\right)\right],$
(229)

$\displaystyle 8\pi
P_{zz}=\frac{1}{B^{2}}\left\\{\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{1}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)+\frac{1}{r^{2}}\left[\frac{A_{\theta\theta}}{A}+\frac{B_{\theta\theta}}{B}-\left(\frac{B_{\theta}}{B}\right)^{2}\right]\right\\},$
(230)

$\displaystyle 8\pi
P_{xy}=\frac{1}{B^{2}}\left\\{\frac{1}{r}\left[-\frac{A^{\prime}_{\theta}}{A}-\frac{D^{\prime}_{\theta}}{D}+\frac{B_{\theta}}{B}\left(\frac{A^{\prime}}{A}+\frac{D^{\prime}}{D}\right)+\frac{B^{\prime}}{B}\frac{A_{\theta}}{A}+\frac{B^{\prime}}{B}\frac{D_{\theta}}{D}\right]+\frac{1}{r^{2}}\left(\frac{A_{\theta}}{A}+\frac{D_{\theta}}{D}\right)\right\\}.$
(231)

The nonvanishing components of the conservation equations
$T^{\alpha\beta}_{;\beta}=0$ yield: the trivial equation

$\dot{\mu}=0,$ (232)

where the overdot denotes derivative with respect to $t$, and the two
hydrostatic equilibrium equations

$\displaystyle\left[P+\frac{1}{3}(2\Pi_{2}-\Pi_{3})\right]^{\prime}+\frac{A^{\prime}}{A}\left[\mu+P+\frac{1}{3}(2\Pi_{2}-\Pi_{3})\right]+\frac{B^{\prime}}{B}(\Pi_{2}-\Pi_{3})+\frac{D^{\prime}}{D}\Pi_{2}$
$\displaystyle+\frac{1}{r}\left[\left(\frac{A_{\theta}}{A}+2\frac{B_{\theta}}{B}+\frac{D_{\theta}}{D}\right)\Pi_{KL}+\Pi_{Kl\theta}+\Pi_{2}-\Pi_{3}\right]=0,$
(233)
$\displaystyle\left[P+\frac{1}{3}(2\Pi_{3}-\Pi_{2})\right]_{\theta}+\frac{A_{\theta}}{A}\left[\mu+P+\frac{1}{3}(2\Pi_{3}-\Pi_{2})\right]+\frac{B_{\theta}}{B}(\Pi_{3}-\Pi_{2})$
$\displaystyle+\frac{D_{\theta}}{D}\Pi_{3}+r\left[\left(\frac{A^{\prime}}{A}+2\frac{B^{\prime}}{B}+\frac{D^{\prime}}{D}\right)\Pi_{KL}+\Pi^{\prime}_{KL}\right]+2\Pi_{KL}=0.$
(234)

##  VII Expression for the components of the electric Weyl tensor

There are four nonvanishing components as calculated from (60), however they
are not independent since they satisfy the relationship:

$E_{11}+\frac{1}{r^{2}}E_{22}+\frac{B^{2}}{D^{2}}E_{33}=0,$ (235)

implying that the Weyl tensor may be expressed through three independent
scalar functions $\mathcal{E}_{1},\mathcal{E}_{2},\mathcal{E}_{3}$.

These four components are

$\displaystyle E_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[\frac{2A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}-\frac{D^{\prime\prime}}{D}-\frac{3A^{\prime}B^{\prime}}{AB}-\frac{A^{\prime}D^{\prime}}{AD}+\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{3B^{\prime}D^{\prime}}{BD}+\frac{1}{r}\left(2\frac{D^{\prime}}{D}-\frac{B^{\prime}}{B}-\frac{A^{\prime}}{A}\right)\right]$
(236) $\displaystyle+$
$\displaystyle\frac{1}{6r^{2}}\left[-\frac{A_{\theta\theta}}{A}-\frac{B_{\theta\theta}}{B}+\frac{2D_{\theta\theta}}{D}+\frac{3A_{\theta}B_{\theta}}{AB}-\frac{A_{\theta}D_{\theta}}{AD}+\left(\frac{B_{\theta}}{B}\right)^{2}-\frac{3B_{\theta}D_{\theta}}{BD}\right],$

$\displaystyle E_{22}$ $\displaystyle=$
$\displaystyle-\frac{r^{2}}{6}\left[\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}-\frac{2D^{\prime\prime}}{D}-\frac{3A^{\prime}B^{\prime}}{AB}+\frac{A^{\prime}D^{\prime}}{AD}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{3B^{\prime}D^{\prime}}{BD}+\frac{1}{r}\left(\frac{D^{\prime}}{D}+\frac{B^{\prime}}{B}-\frac{2A^{\prime}}{A}\right)\right]$
(237) $\displaystyle-$
$\displaystyle\frac{1}{6}\left[-\frac{2A_{\theta\theta}}{A}+\frac{B_{\theta\theta}}{B}+\frac{D_{\theta\theta}}{D}+\frac{3A_{\theta}B_{\theta}}{AB}+\frac{A_{\theta}D_{\theta}}{AD}-\left(\frac{B_{\theta}}{B}\right)^{2}-\frac{3B_{\theta}D_{\theta}}{BD}\right],$

$\displaystyle E_{33}$ $\displaystyle=$
$\displaystyle-\frac{D^{2}}{6B^{2}}\left[\frac{A^{\prime\prime}}{A}-\frac{2B^{\prime\prime}}{B}+\frac{D^{\prime\prime}}{D}-\frac{2A^{\prime}D^{\prime}}{AD}+2\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{1}{r}\left(\frac{D^{\prime}}{D}-\frac{2B^{\prime}}{B}+\frac{A^{\prime}}{A}\right)\right]$
(238) $\displaystyle-$
$\displaystyle\frac{D^{2}}{6B^{2}r^{2}}\left[\frac{A_{\theta\theta}}{A}-\frac{2B_{\theta\theta}}{B}+\frac{D_{\theta\theta}}{D}-\frac{2A_{\theta}D_{\theta}}{AD}+2\left(\frac{B_{\theta}}{B}\right)^{2}\right],$

$\displaystyle
E_{12}=\frac{1}{2}\left[\frac{A^{\prime}_{\theta}}{A}-\frac{D^{\prime}_{\theta}}{D}+\frac{B_{\theta}}{B}\frac{D^{\prime}}{D}-\frac{A^{\prime}B_{\theta}}{AB}-\frac{B^{\prime}A_{\theta}}{AB}+\frac{D_{\theta}}{D}\frac{B^{\prime}}{B}-\frac{1}{r}\left(\frac{A_{\theta}}{A}-\frac{D_{\theta}}{D}\right)\right].$
(239)

For the three scalars $\mathcal{E}_{1}$, $\mathcal{E}_{2}$, $\mathcal{E}_{3}$
we obtain

$\displaystyle\mathcal{E}_{1}=\frac{1}{2B^{2}}\left[\frac{1}{r}\left(\frac{A^{\prime}_{\theta}}{A}-\frac{D^{\prime}_{\theta}}{D}-\frac{B_{\theta}}{B}\frac{A^{\prime}}{A}+\frac{D^{\prime}}{D}\frac{B_{\theta}}{B}-\frac{B^{\prime}}{B}\frac{A_{\theta}}{A}+\frac{D_{\theta}}{D}\frac{B^{\prime}}{B}\right)+\frac{1}{r^{2}}\left(\frac{D_{\theta}}{D}-\frac{A_{\theta}}{A}\right)\right],$
(240)

$\displaystyle\mathcal{E}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2B^{2}}\left[-\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}+\frac{A^{\prime}B^{\prime}}{AB}+\frac{A^{\prime}D^{\prime}}{AD}-\left(\frac{B^{\prime}}{B}\right)^{2}-\frac{B^{\prime}D^{\prime}}{BD}+\frac{1}{r}\left(\frac{B^{\prime}}{B}-\frac{D^{\prime}}{D}\right)\right]$
(241) $\displaystyle-$
$\displaystyle\frac{1}{2B^{2}r^{2}}\left[\frac{B_{\theta\theta}}{B}-\frac{D_{\theta\theta}}{D}-\frac{A_{\theta}B_{\theta}}{AB}+\frac{A_{\theta}D_{\theta}}{AD}-\left(\frac{B_{\theta}}{B}\right)^{2}+\frac{B_{\theta}D_{\theta}}{BD}\right],$

$\displaystyle\mathcal{E}_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2B^{2}}\left[\frac{B^{\prime\prime}}{B}-\frac{D^{\prime\prime}}{D}-\frac{A^{\prime}B^{\prime}}{AB}+\frac{A^{\prime}D^{\prime}}{AD}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{B^{\prime}D^{\prime}}{BD}+\frac{1}{r}\left(\frac{B^{\prime}}{B}-\frac{A^{\prime}}{A}\right)\right]$
(242) $\displaystyle-$
$\displaystyle\frac{1}{2B^{2}r^{2}}\left[\frac{B_{\theta\theta}}{B}-\frac{A_{\theta\theta}}{A}+\frac{A_{\theta}B_{\theta}}{AB}+\frac{A_{\theta}D_{\theta}}{AD}-\left(\frac{B_{\theta}}{B}\right)^{2}-\frac{B_{\theta}D_{\theta}}{BD}\right].$

or using Einstein equations

$\displaystyle\mathcal{E}_{1}=\frac{E_{12}}{B^{2}r}=4\pi\Pi_{KL}+\frac{1}{B^{2}r}\left[\frac{A^{\prime}_{\theta}}{A}-\frac{A^{\prime}B_{\theta}}{AB}-\frac{A_{\theta}}{A}\left(\frac{B^{\prime}}{B}+\frac{1}{r}\right)\right],$
(243)

$\displaystyle\mathcal{E}_{2}$ $\displaystyle=$
$\displaystyle-\frac{2E_{33}}{D^{2}}-\frac{E_{22}}{B^{2}r^{2}}={4\pi}(\mu+3P+\Pi_{2})-\frac{A^{\prime}}{B^{2}A}\left(\frac{2D^{\prime}}{D}+\frac{B^{\prime}}{B}+\frac{1}{r}\right)$
(244) $\displaystyle+$
$\displaystyle\frac{A_{\theta}}{AB^{2}r^{2}}\left(\frac{B_{\theta}}{B}-\frac{2D_{\theta}}{D}\right)-\frac{1}{B^{2}r^{2}}\frac{A_{\theta\theta}}{A},$

$\displaystyle\mathcal{E}_{3}=-\frac{E_{33}}{D^{2}}+\frac{E_{22}}{B^{2}r^{2}}=4\pi\Pi_{3}-\frac{A^{\prime}}{B^{2}A}\left(\frac{D^{\prime}}{D}-\frac{B^{\prime}}{B}-\frac{1}{r}\right)-\frac{A_{\theta}}{AB^{2}r^{2}}\left(\frac{D_{\theta}}{D}+\frac{B_{\theta}}{B}\right)+\frac{1}{B^{2}r^{2}}\frac{A_{\theta\theta}}{A}.$
(245)

##  VIII Vanishing complexity factor conditions

$\displaystyle Y_{TF_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{B^{2}r}\left[\frac{A^{\prime}_{\theta}}{A}-\frac{A^{\prime}B_{\theta}}{AB}-\frac{A_{\theta}}{A}\left(\frac{B^{\prime}}{B}+\frac{1}{r}\right)\right]=0,$
(246)

$\displaystyle Y_{TF_{2}}$ $\displaystyle=$
$\displaystyle\frac{A^{\prime\prime}}{B^{2}A}-\frac{A^{\prime}}{B^{2}A}\left(\frac{D^{\prime}}{D}+\frac{B^{\prime}}{B}\right)+\frac{A_{\theta}}{AB^{2}r^{2}}\left(\frac{B_{\theta}}{B}-\frac{D_{\theta}}{D}\right)=0,$
(247)

$\displaystyle Y_{TF_{3}}$ $\displaystyle=$
$\displaystyle-\frac{A^{\prime}}{B^{2}A}\left(\frac{D^{\prime}}{D}-\frac{B^{\prime}}{B}-\frac{1}{r}\right)-\frac{A_{\theta}}{AB^{2}r^{2}}\left(\frac{D_{\theta}}{D}+\frac{B_{\theta}}{B}\right)+\frac{1}{B^{2}r^{2}}\frac{A_{\theta\theta}}{A}$
(248) $\displaystyle=$ $\displaystyle 0.$

##  IX Einstein equations for the dynamical spherically symmetric case

Einstein’s field equations

$G_{\alpha\beta}=8\pi T_{\alpha\beta},$ (249)

for the interior spacetime (139) read

$\displaystyle 8\pi T_{00}=8\pi\mu
A^{2}=\left(2\frac{\dot{B}}{B}+\frac{\dot{R}}{R}\right)\frac{\dot{R}}{R}-\left(\frac{A}{B}\right)^{2}\left[2\frac{R^{\prime\prime}}{R}+\left(\frac{R^{\prime}}{R}\right)^{2}-2\frac{B^{\prime}}{B}\frac{R^{\prime}}{R}-\left(\frac{B}{R}\right)^{2}\right],$
(250) $8\pi T_{01}=-8\pi
qAB=-2\left(\frac{{\dot{R}}^{\prime}}{R}-\frac{\dot{B}}{B}\frac{R^{\prime}}{R}-\frac{\dot{R}}{R}\frac{A^{\prime}}{A}\right),$
(251) $\displaystyle 8\pi T_{11}=8\pi
P_{r}B^{2}=-\left(\frac{B}{A}\right)^{2}\left[2\frac{\ddot{R}}{R}-\left(2\frac{\dot{A}}{A}-\frac{\dot{R}}{R}\right)\frac{\dot{R}}{R}\right]+\left(2\frac{A^{\prime}}{A}+\frac{R^{\prime}}{R}\right)\frac{R^{\prime}}{R}-\left(\frac{B}{R}\right)^{2},$
(252)

$\displaystyle 8\pi T_{22}$ $\displaystyle=$
$\displaystyle\frac{8\pi}{\sin^{2}\theta}T_{33}=8\pi
P_{\perp}R^{2}=-\left(\frac{R}{A}\right)^{2}\left[\frac{\ddot{B}}{B}+\frac{\ddot{R}}{R}-\frac{\dot{A}}{A}\left(\frac{\dot{B}}{B}+\frac{\dot{R}}{R}\right)+\frac{\dot{B}}{B}\frac{\dot{R}}{R}\right]$
(253) $\displaystyle+$
$\displaystyle\left(\frac{R}{B}\right)^{2}\left[\frac{A^{\prime\prime}}{A}+\frac{R^{\prime\prime}}{R}-\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}+\left(\frac{A^{\prime}}{A}-\frac{B^{\prime}}{B}\right)\frac{R^{\prime}}{R}\right].$

The component (251) can be rewritten with (147) and (150) as

$4\pi qB=\frac{1}{3}(\Theta-\sigma)^{\prime}-\sigma\frac{R^{\prime}}{R}.$
(254)

##  X Dynamical equations

The non trivial components of the Bianchi identities,
$T^{\alpha\beta}_{;\beta}=0$, from (249) yield

$\displaystyle T^{\alpha\beta}_{;\beta}V_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{A}\left[\dot{\mu}+\left(\mu+P_{r}\right)\frac{\dot{B}}{B}+2\left(\mu+P_{\perp}\right)\frac{\dot{R}}{R}\right]$
(255) $\displaystyle-$
$\displaystyle\frac{1}{B}\left[q^{\prime}+2q\frac{(AR)^{\prime}}{AR}\right]=0,$
$\displaystyle
T^{\alpha\beta}_{;\beta}\chi_{\alpha}=\frac{1}{A}\left[\dot{q}+2q\left(\frac{\dot{B}}{B}+\frac{\dot{R}}{R}\right)\right]$
$\displaystyle+\frac{1}{B}\left[P_{r}^{\prime}+\left(\mu+P_{r}\right)\frac{A^{\prime}}{A}+2(P_{r}-P_{\perp})\frac{R^{\prime}}{R}\right]=0,$
(256)

or, by using (146), (147), (156) and (154), they become, respectively,

$\displaystyle
D_{T}\mu+\frac{1}{3}\left(3\mu+P_{r}+2P_{\perp}\right)\Theta+\frac{2}{3}(P_{r}-P_{\perp})\sigma+ED_{R}q+2q\left(a+\frac{E}{R}\right)=0,$
(257) $\displaystyle
D_{T}q+\frac{2}{3}q(2\Theta+\sigma)+ED_{R}P_{r}+\left(\mu+P_{r}\right)a+2(P_{r}-P_{\perp})\frac{E}{R}=0.$
(258)

This last equation may be further tranformed as follows, the acceleration
$D_{T}U$ of an infalling particle can be obtained by using (146), (252), (151)
and (154), producing

$D_{T}U=-\frac{m}{R^{2}}-4\pi P_{r}R+Ea,$ (259)

and then, substituting $a$ from (259) into (258), we obtain

$\displaystyle\left(\mu+P_{r}\right)D_{T}U$ $\displaystyle=$
$\displaystyle-\left(\mu+P_{r}\right)\left[\frac{m}{R^{2}}+4\pi
P_{r}R\right]-E^{2}\left[D_{R}P_{r}+2(P_{r}-P_{\perp})\frac{1}{R}\right]$
(260) $\displaystyle-$ $\displaystyle
E\left[D_{T}q+2q\left(2\frac{U}{R}+\sigma\right)\right].$

##  XI The complexity factors for the Bondi metric

$\displaystyle{\cal E}_{1}=\frac{1}{r^{2}}\left(2c_{u}\cot{\theta}+c_{\theta
u}\right)+\mathcal{O}(r^{-n}),\quad n\geq 4,$ (261) $\displaystyle{\cal
E}_{2}=\frac{1}{r}c_{uu}-\frac{1}{2r^{2}}\left(c_{\theta\theta
u}-4Mc_{uu}+2c_{u}+c_{\theta
u}\cot{\theta}-\frac{4c_{u}}{\sin^{2}{\theta}}\right)$
$\displaystyle+\frac{1}{r^{3}}\left[cc_{u}+2c_{\theta}c_{\theta
u}+3M+\frac{\cot{\theta}}{2}\left(3c_{u}c_{\theta}+5cc_{\theta
u}\right)-M_{u}c+\frac{1}{2}M_{\theta\theta}+N_{\theta u}+P_{uu}\right.$
$\displaystyle\left.-\cot{\theta}\left(Mc_{\theta
u}+\frac{1}{2}M_{\theta}+N_{u}-Nc_{uu}\right)-Mc_{u}\left(1-\frac{4}{\sin^{2}{\theta}}\right)+c_{u}\left(cc_{u}+\frac{1}{2}c_{\theta\theta}\right)\right.$
$\displaystyle\left.+c_{uu}\left(4M^{2}+N_{\theta}\right)-c_{\theta\theta
u}\left(M-\frac{3}{2}c\right)\right]$ $\displaystyle+\mathcal{O}(r^{-n}),$
(262) $\displaystyle{\cal
E}_{3}=\frac{2}{r}c_{uu}-\frac{1}{r^{2}}\left(c_{\theta\theta
u}-4Mc_{uu}+2c_{u}+c_{\theta
u}\cot{\theta}-\frac{4c_{u}}{\sin^{2}{\theta}}\right)$
$\displaystyle+\frac{1}{r^{3}}\left[-4cc_{u}+4c_{\theta}c_{\theta
u}+\cot{\theta}\left(3c_{u}c_{\theta}+5cc_{\theta
u}\right)-2M_{u}c+M_{\theta\theta}+2N_{\theta u}+2P_{uu}\right.$
$\displaystyle\left.-\cot{\theta}\left(2Mc_{\theta
u}+M_{\theta}+2N_{u}-2Nc_{uu}\right)-2Mc_{u}\left(1-\frac{4}{\sin^{2}{\theta}}\right)+c_{u}\left(2cc_{u}+c_{\theta\theta}\right)\right.$
$\displaystyle\left.+2c_{uu}\left(4M^{2}+N_{\theta}\right)-c_{\theta\theta
u}\left(2M-3c\right)\right]$ $\displaystyle+\mathcal{O}(r^{-n}).$ (263)

##  XII The magnetic part of the Weyl tensor

$\displaystyle H_{1}=-\frac{1}{r^{2}}\left(2c_{u}\cot{\theta}+c_{\theta
u}\right)+\mathcal{O}(r^{-n}),\quad n\geq 4,$ (264)
$\displaystyle
H_{2}=-\frac{1}{r}c_{uu}-\frac{1}{r^{2}}\left[-c_{u}\left(1-\frac{2}{\sin^{2}{\theta}}\right)-\frac{\cot{\theta}}{2}c_{\theta
u}+2c_{uu}(M-c)-\frac{1}{2}c_{\theta\theta u}\right]$
$\displaystyle-\frac{1}{r^{3}}\left\\{-Mc_{u}\left(1-\frac{4}{\sin^{2}{\theta}}\right)-\frac{4cc_{u}}{\sin^{2}{\theta}}+\cot{\theta}\left[\frac{3}{2}c_{u}c_{\theta}-N_{u}-\frac{1}{2}M_{\theta}+Nc_{uu}+\left(\frac{7}{2}c-M\right)c_{\theta
u}\right]+\left(\frac{5}{2}c-M\right)c_{\theta\theta u}\right.$
$\displaystyle\left.+\frac{1}{2}c_{\theta\theta}c_{u}+2c_{\theta}c_{\theta
u}+cc_{u}^{2}+\frac{1}{2}M_{\theta\theta}-cM_{u}+N_{\theta
u}+P_{uu}+c_{uu}\left(4c^{2}+4M^{2}-4Mc+N_{\theta}\right)\right\\}+\mathcal{O}(r^{-n}),\quad
n\geq 4$ (265)

where $P=C-\frac{c^{3}}{6}$.

##  XIII The vorticity

$\displaystyle\omega^{\phi}=-\frac{e^{-2\beta}}{2r^{2}\sin{\theta}}\left[2\beta_{\theta}e^{2\beta}-\frac{2e^{2\beta}A_{\theta}}{A}-\left(Ur^{2}e^{2\gamma}\right)_{r}+\frac{2Ur^{2}e^{2\gamma}}{A}A_{r}+\frac{e^{2\beta}\left(Ur^{2}e^{2\gamma}\right)_{u}}{A^{2}}-\frac{Ur^{2}e^{2\gamma}}{A^{2}}2\beta_{u}e^{2\beta}\frac{}{}\right],$
(266)

and for the absolute value of $\omega^{\alpha}$ we get

$\displaystyle\Omega\equiv\left(-\omega_{\alpha}\omega^{\alpha}\right)^{1/2}=\frac{e^{-2\beta-\gamma}}{2r}\left[2\beta_{\theta}e^{2\beta}-2e^{2\beta}\frac{A_{\theta}}{A}-\left(Ur^{2}e^{2\gamma}\right)_{r}+2Ur^{2}e^{2\gamma}\frac{A_{r}}{A}+\frac{e^{2\beta}}{A^{2}}\left(Ur^{2}e^{2\gamma}\right)_{u}-2\beta_{u}\frac{e^{2\beta}}{A^{2}}Ur^{2}e^{2\gamma}\right]$
(267)

Feeding back (201–205) into (267) and keeping only the two leading terms, we
obtain

$\displaystyle\Omega=-\frac{1}{2r}(c_{u\theta}+2c_{u}\cot\theta)+\frac{1}{r^{2}}\left[M_{\theta}-M(c_{u\theta}+2c_{u}\cot\theta)-cc_{u\theta}+6cc_{u}\cot\theta+2c_{u}c_{\theta}\right].$
(268)

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