Complexity factors for axially symmetric static sources

TL;DR

This paper extends the concept of complexity from spherically symmetric to axially symmetric static fluid sources, defining three complexity factors based on structure scalars, and explores conditions under which these factors vanish, indicating simple configurations.

Contribution

It introduces three new complexity factors for axially symmetric static sources, generalizing previous spherically symmetric results and analyzing their implications for fluid configurations.

Findings

Three complexity factors vanish for simple spheroid configurations.

Exact solutions show complexity factors can vanish in more general cases.

Complexity is not solely determined by symmetry, but also by energy density and pressure anisotropy.

Abstract

A previously found definition of complexity for spherically symmetric fluid distributions [1], is extended to axially symmetric static sources. In this case there are three different complexity factors, defined in terms of three structure scalars obtained from the orthogonal splitting of the Riemann tensor. All these three factors vanish, for what we consider the simplest fluid distribution, i.e a fluid spheroid with isotropic pressure and homogeneous energy density. However, as in the spherically symmetric case, they can also vanish for a variety of configurations, provided the energy density inhomogeneity terms cancel the pressure anisotropic ones in the expressions for the complexity factors. Some exact analytical solutions of this type are found and analyzed. At the light of the obtained results, some conclusions about the correlation (the lack of it) between symmetry and complexity, are put forward.

Equations

$$d{s}^2=-A^{2}d{t}^2+B^2\left(d{r}^2+r^{2}d{\theta }^2\right)+D^{2}d{\varphi }^2,$$

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

$$T_{\alpha \beta }$$

$$V_{\alpha }=(-A,0,0,0);K_{\alpha }=(0,B,0,0);$$

$$L_{\alpha }=(0,0,Br,0);S_{\alpha }=(0,0,0,D),$$

$$T_{\alpha \beta }$$

$${\Pi }_{\alpha \beta }$$

$$P=\frac{P_{xx}+P_{yy}+P_{zz}}{3},h_{\mu \nu }=g_{\mu \nu }+V_{\nu }{V}_{\mu }.$$

$${\Pi }_{\alpha \beta }=\frac{1}{3}(2{\Pi }_I+{\Pi }_{II})\left(K_{\alpha }{K}_{\beta }-\frac{h_{\alpha \beta }}{3}\right)+\frac{1}{3}(2{\Pi }_{II}+{\Pi }_I)\left(L_{\alpha }{L}_{\beta }-\frac{h_{\alpha \beta }}{3}\right)+{\Pi }_{KL}\left(K_{\alpha }{L}_{\beta }+K_{\beta }{L}_{\alpha }\right),$$

$${\Pi }_{KL}=K^{\alpha }{L}^{\beta }{T}_{\alpha \beta },$$

$${\Pi }_I=\left(2K^{\alpha }{K}^{\beta }-L^{\alpha }{L}^{\beta }-S^{\alpha }{S}^{\beta }\right)T_{\alpha \beta },$$

$${\Pi }_{II}=\left(2L^{\alpha }{L}^{\beta }-K^{\alpha }{K}^{\beta }-S^{\alpha }{S}^{\beta }\right)T_{\alpha \beta }.$$

$${\Pi }_2\equiv \frac{1}{3}(2{\Pi }_I+{\Pi }_{II})=P_{xx}-P_{zz},$$

$${\Pi }_3\equiv \frac{1}{3}(2{\Pi }_{II}+{\Pi }_I)=P_{yy}-P_{zz},$$

$${\Pi }_{KL}=P_{xy}.$$

$$P_{zz}=P-\frac{1}{3}({\Pi }_2+{\Pi }_3),$$

$$P_{xx}=P+\frac{1}{3}(2{\Pi }_2-{\Pi }_3),$$

$$P_{yy}=P+\frac{1}{3}(2{\Pi }_3-{\Pi }_2).$$

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

$$E_{\alpha \beta }={\mathcal{E}}_1\left(K_{\alpha }{L}_{\beta }+L_{\alpha }{K}_{\beta }\right)+{\mathcal{E}}_2\left(K_{\alpha }{K}_{\beta }-\frac{1}{3}{h}_{\alpha \beta }\right)+{\mathcal{E}}_3\left(L_{\alpha }{L}_{\beta }-\frac{1}{3}{h}_{\alpha \beta }\right),$$

$$Y_{\beta }^{\rho }=V^{\alpha }{V}^{\mu }{R}_{\alpha \beta \mu }^{\rho }.$$

$$Y_{\alpha \beta }$$

$$Y_T=4\pi (\mu +3P),$$

$$Y_{T{F}_1}={\mathcal{E}}_1-4\pi {\Pi }_{KL},$$

$$Y_{T{F}_2}={\mathcal{E}}_2-4\pi {\Pi }_2,$$

$$Y_{T{F}_3}={\mathcal{E}}_3-4\pi {\Pi }_3.$$

$$X_{\alpha \beta }{=}^{\ast }{R}_{\alpha \gamma \beta \delta }^{\ast }{V}^{\gamma }{V}^{\delta }=\frac{1}{2}{\eta }_{\alpha \gamma }^{ϵ\rho }{R}_{ϵ\rho \beta \delta }^{\ast }{V}^{\gamma }{V}^{\delta },$$

$$X_{\alpha \beta }$$

$$X_T=8\pi \mu ,$$

$$X_{T{F}_1}=-({\mathcal{E}}_1+4\pi {\Pi }_{KL}),$$

$$X_{T{F}_2}=-\left({\mathcal{E}}_2+4\pi {\Pi }_2\right),$$

Full text

Relativistic Fluids, nonspherical sources, interior solutions.

I Introduction

In recent papers a new definition of complexity, in the context of general relativity, has been proposed for spherically symmetric fluid distributions, in the static case css and the dynamical case csd . Applications of this concept to other theories of gravity have been proposed in ot1 ; ot2 , while the charged case has been considered in csd and ch . Also, applications for some particular cases of cylindrically symmetric fluid distributions, may be found in cil . It is our goal in this work to extend this definition of complexity to the most general axially symmetric static fluid distributions.

The motivation for such an endeavour is based on the fact that while it is true that observational evidence seems to 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 (putting aside the evident fact that astrophysical objects are generally endowed with angular momentum), it also true that 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 i2 -in and references therein for a discussion on this point). This fact in turn is related to the well known result that the only regular static and asymptotically flat vacuum spacetime possessing a regular horizon is the Schwarzchild solution israel , while all the others Weyl exterior solutions weyl2 -weyln exhibit singularities in the curvature invariants (as the boundary of the source approaches the horizon).

Sources of different Weyl spacetimes have already been considered by several authors in the past (see for example 1 -sn1 and references therein).

More recently, a renewed interest on this kind of solutions have aroused, particularly in relation to the deviations of spherical symmetry produced by different physical phenomena such as magnetic fields (see for example s3 -s8 and references therein).

In the spherically symmetric case, the complexity factor is a scalar variable intended to measure the degree of complexity of the fluid distribution. For reasons explained in detail in css ; csd , such scalar function may be identified as one of the scalar functions (structure scalars) which appears in the orthogonal splitting of the Riemann tensor 1cil . More specifically, it is related to one of the scalar functions appearing in the splitting of the electric part of the Riemann tensor (see also 2cil ; 3cil ; 4cil ; 5cil ; scc1 ; scc2 ; csp ; hu ; mbi ; ku ; ah and references therein for further discusion on the structure scalar).

In the axially symmetric case the situation is much more complicated, and the number of structure scalars much larger than in the spherically symmetric case. Nevertheless the general criterium to define the variable(s) measuring the complexity of the fluid distribution will be the same, namely: we start by asking ourselves 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 to the complexity factors. Some of these solutions will be exhibited.

Our paper is organized as follows: In the next section we shall review the general framework developped in as to describe the most general non–vacuum, axially symmetric static spacetime. This include relevant variables and equations. Next we define the complexity factors for our system and discuss about their general properties. The two families of solutions found are described in sections IV. A summary of the obtained results as well as a list of some unsolved issues are presented in section V. Finally three Appendices with some useful equations are included.

II The general framework

As we mentioned in the Introduction, a general framework for describing axially symmetric static sources was deployed in as . Here we shall resort (with slight changes in the notation) to such a formalism. However in order to render this manuscript self–consistent we shall provide in this section a brief resume of the approach to be used. The reader may find all the details in as .

II.1 The metric and the source

We shall consider static and axially symmetric sources. For such a system the line element may be written in “Weyl spherical coordinates” as:

[TABLE]

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 $\theta$.

For the sake of generality we shall not assume here the Weyl gauge. In the vacuum case this gauge can be used without loss of generality, and it allows for the reduction of the line element so that only two independent metric functions appear. However, in the interior this is not possible in general, though obviously one may assume it as an additional restriction, which amounts to satisfy $R^{3}_{3}+R^{0}_{0}=0$, where $R^{\alpha}_{\beta}$ denotes the Ricci tensor.

Let us now provide a full description of the source. In order to give physical significance to the components of the energy momentum tensor, we shall apply the Bondi approach Bo , which consists in defining the physical variables in a purely locally Minkowski frame ($\tau,x,y,z$) (hereafter referred to as l.M.f.) where the first derivatives of the metric vanish (locally), or, equivalently, consider a tetrad field attached to such l.M.f.

For the system under consideration, the most general energy–momentum tensor in such locally defined coordinate system is given by:

[TABLE]

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.

[TABLE]

where

[TABLE]

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:

[TABLE]

with

[TABLE]

and

[TABLE]

and:

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

The relationships between the above scalars and the variables $P_{xy},P_{xx},P_{yy},P_{zz}$ are, (besides (7)), as follows:

[TABLE]

[TABLE]

[TABLE]

or, inversely:

[TABLE]

[TABLE]

[TABLE]

The explicit form of the Einstein equations as well as the conservation equations, for the line element (1) and the energy–momentum tensor (3), are given in the Appendix A and B respectively.

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,

[TABLE]

where $C_{\mu\alpha\nu\beta}$ denotes the Weyl tensor. These are exhibited in the Appendix B.

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

[TABLE]

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

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

[TABLE]

After some lengthy calculations we find;

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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

II.3 Some differential equations for the structure scalars

Two differential equations which relate the spatial derivatives of the physical variables and the Weyl tensor may be obtained using Bianchi identities, they have been found before for the spherically symmetric and the cylindrically symmetric cases (see 1cil , scc1 and references therein). For our case they have been calculated in as :

[TABLE]

[TABLE]

which, using (22)-(25) and (28)-31), may be written in terms of structure scalars:

[TABLE]

[TABLE]

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

III The complexity factors

We have now available all the elements necessary to define the complexity factors for the fluid distribution under consideration. For doing so we have first to establish what we consider is the simplest possible fluid (or at least one of them). From elementary considerations, as we did in css ; csd , we assume that the incompressible (constant energy density) fluid with isotropic pressure is one of the simplest fluid distributions. 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

[TABLE]

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

[TABLE]

Obviously, the isotropic pressure condition would imply $Y_{TF1}=Y_{TF2}=Y_{TF3}=0$.

Thus from the above considerations, and following the rationale exposed in the spherically symmetric case, we shall identify the three structure scalars $Y_{TF}$ (more precisely, their absolute values) as the complexity factors. They vanish for the 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.

We shall next find some explicit analytical solutions.

IV Solutions satisfying the vanishing complexity factors condition

As was mentioned above, the fluid distribution with homogeneous energy density and isotropic pressure satisfies the vanishing complexity factors condition, but it is not the only one. These conditions may also be satisfied if the terms describing the energy density inhomogeneity cancel the anisotropic terms in (34), (35).

In this section we shall present some solutions of this kind. It should be kept in mind that our purpose here is not to present solutions representing specific physically meaningful compact object, but just to illustrate the way by means of which such models might be obtained.

IV.1 The incompressible, isotropic spheroid

As we have already seen, the incompressible isotropic spheroid represents a fluid distribution for which the three complexity factors vanish. This solution was obtained and analyzed in as . Here we just reproduce it without details. Thus from (36), (29)–(31) and $P_{xx}=P_{yy}=P_{zz}=P$, $P_{xy}=0$, $\mu=\mu_{0}=constant$, it is evident that such a solution is also conformally flat.

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

[TABLE]

From the above and (57) and (60) it follows that

[TABLE]

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

Under the conditions above (62) and (63) can be integrated to obtain:

[TABLE]

and

[TABLE]

where $\xi$ is an arbitrary function of its argument. Using boundary conditions (39) in (40) (41) it follows that:

[TABLE]

Finally, the metric of incompressible conformally flat isotropic fluids can be written as follows.

[TABLE]

Next, the physical variables can be easily calculated. Thus, using (43) into (56) the energy density reads:

[TABLE]

To obtain the pressure we shall use (40) and (42), which produce

[TABLE]

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

[TABLE]

in order to satisfy the junction condition (39).

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 csd for details). This result is in agreement with theorems indicating that static, perfect fluid (isotropic in pressure) sources are spherical (see prueba and references therein).

IV.2 Anisotropic inhomogeneous spheroids

Although the inhomogeneous anisotropic spheroids exhibited in as do not satisfy the vanishing complexity factors conditions, solutions with vanishing complexity factors but inhomogeneous energy density and anisotropic pressure do exist, as we shall show in this subsection.

The metric variables for the solution are:

[TABLE]

It is a simple matter to check that conditions (75)–(77) are satisfied for (47)–(49).

Next, using the Einstein equations (56)–(60), the metric above produces the following expressions for the physical variables.

[TABLE]

with

[TABLE]

and where $a_{1},b_{1},b_{2}$ are constants. For a range of values of these parameters, the physical behaviour of physical variables is acceptable and the metric may be matched smoothly on the boundary surface to a Weyl solution. However, our only purpose in this section is to illustrate the existence of solutions admitting the vanishing complexity factors condition, and not to model specific astrophysical objects.

V conclusions

We have extended a previously proposed definition of complexity for spherically symmetric fluid distributions, to the axially symmetric static case. We have considered the most general fluid distribution compatible with this latter symmetry. Unlike the spherically symmetric case, the complexity is now defined in terms of three scalar functions (complexity factors). This fact opens the possibility to establish a more elaborated hierarchy of models, which runs from the simplest case (the vanishing of the three complexity factors) through semi–simple (semi–complex) models with only one or two vanishing complexity factors, until the more complex (the less simple) models with all the three complexity factors different from zero. Also, it is worth noticing that the three scalars $Y_{TF,1,2,3}$ may be positive or negative (if they are non–vanishing), depending on the interplay between energy density inhomogeneity and pressure anisotropy. Accordingly it is evident from purely physical considerations that we have to choose the absolute values of these scalars as the measure of the complexity of any fluid distribution.

As it happens in the spherically symmetric case, there are more than one model compatible with the vanishing of all the complexity factors. It remains as a pending task to find out what all these models have in common (besides the fact that the complexity factors vanish). In other words, what are the physical consequences derived from the vanishing of the complexity factors? In the spherically symmetric case the consequence derived from the vanishing of the complexity factor is very simple: the distribution of the Tolman (active gravitational) mass is the same for all these configurations. We don’t know if something similar appears in the axially symmetric case.

In relation to the comment above, we would like to stress one point which deserves to be explored in some detail: we refer to the study of the possible relationship between complexity (as defined here) and the stability of the fluid distribution. Such relationship is apparent in the spherically symmetric case through the influence of the complexity factor in the value of the active gravitational mass (Tolman) within the fluid distribution.

Finally we would like to call the attention to an issue which may be relevant in the discussion about the definition of complexity. We have in mind here the possible link between symmetry (expressed through the admittance of Killing vectors), and complexity. Indeed, even though, at purely intuitive level, one might expect these two concepts to be closely intertwined, the fact is that our results in this work as well as in css ; csd point in the opposite direction.

In the spherically symmetric case, both in the static and in the dynamic case, there are three Killing vectors which are compatible with a broad hierarchy in the degree of complexity. The situation analyzed in this manuscript reinforces further this picture, by admitting a wider hierarchy of complexity, for a lesser degree of symmetry.

Furthermore, there is an example that illustrates the lack of correlation between symmetry and complexity, in a particularly sharp and forceful way. Such an example is provided by the Szekeres spacetime 1sz ; 2sz . These are time dependent metrics sourced by pure dust, which in general do not admit a single Killing vector 3sz . However, in spite of the absence of symmetry, the electric part of the Weyl tensor is defined through a single scalar function 4sz . Then, since there are no pressure terms, if we restrict ourselves to the class of axially symmetric Szekeres metrics, we conclude that there is only one complexity factor (see eqs.(43-44) in 5sz ), as in the spherically symmetric case, revealing thereby that its degree of complexity is low, in spite of the fact that there is only one Killing vector (in the axially symmetric case).

Thus the qualification of “quasispherical” assigned by Szekeres himself to his solution appears to be well justified, due to the similar degree of complexity of both spacetimes. In other words, the concept of complexity adopted here, seems to represent better than its symmetry, some deeper aspects of the system.

Acknowledgements.

This work was partially supported by the Spanish Ministry of Science and Innovation (grant FIS2010-15492). J.O. acknowledges financial support from the Spanish Ministry of Science and Innovation (grant FIS2009-07238).

Appendix A The Einstein and the conservation equations

For the line element (1) and the energy momentum (3), the Einstein equations read:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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

[TABLE]

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

[TABLE]

[TABLE]

Appendix B Expression for the components of the electric Weyl tensor

There are four nonvanishing components as calculated from (18), however they are not independent since they satisfy the relationship:

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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

[TABLE]

[TABLE]

[TABLE]

Or, using Einstein equations we may also write:

[TABLE]

[TABLE]

[TABLE]

Appendix C Vanishing complexity conditions

[TABLE]

[TABLE]

[TABLE]