The Spacetime Geodesy of Perfect Fluid Spheres

Abstract

Herein we shall argue for the utility of “spacetime geodesy”, a point of view where one delays as long as possible worrying about dynamical equations, in favour of the maximal utilization of both symmetries and geometrical features. This closely parallels Weinberg’s distinction between “cosmography” and “cosmology”, wherein maximal utilization of both the symmetries and geometrical features of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes is emphasized. This “spacetime geodesy” point of view is particularly useful in those situations where, for one reason or another, the dynamical equations of motion are either uncertain or completely unknown. Several specific examples are discussed—we shall illustrate what can be done by considering the physics implications of demanding spatially isotropic Ricci tensors as a way of automatically implementing the (isotropic) perfect fluid condition, without committing to a specific equation of state. We also consider the structure of the Weyl tensor in spherical symmetry, with and without the (isotropic) perfect fluid condition, and relate this to the notion of “complexity”. In closing, we indicate some ways in which these considerations might be further generalized to more physically complicated (and technically very much more complicated) situations such as axisymmetric spacetimes.

1. Introduction

When trying to extend and go beyond “known physics”, should you use a dynamical model or a kinematical model? That depends critically on just how much you actually know (or think you know) about the details of the system in question, and how you might wish to modify it. Can you find an actual solution to a reasonably well defined set of equations of motion? If so, great! Otherwise, the next best option is to build a purely kinematic model, using symmetries and general principles. Though careful thought and discretion is highly advised, one should try to minimize the speculative aspects of the model. (A common aphorism in the experimental community is this: “Do not adjust more than one knob on your equipment at a time”. The theoretical community would be well advised to adopt the related maxim: “No more than one miracle at a time”). Extract as much information as possible without committing to a specific choice of dynamics. Sometimes, once you have somehow developed an interesting model, you can even reverse-engineer a suitable Lagrangian. (A potential drawback is that reverse engineering is often fine-tuned and fragile).

1.1. Cosmography Versus Cosmology

A now fully mainstream example of this behaviour is Weinberg’s distinction between “cosmography” and “cosmology” [1]. Cosmography was developed in an attempt to extract as much information as possible from adopting the geometry of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes as a zeroth-order approximation to reality, while eschewing, (as far as possible), any and all arguments regarding the cosmological equation of state. See, for instance, the extensive discussion in references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Some of the implementations of this general idea instead speak in terms of “cosmokinetics” versus “cosmodynamics” [28,29,30,31,32,33]. Work along these lines has by now led to well over 350 published scientific articles. Note that adopting a “cosmographic” point of view implies that one need not make any specific a priori commitment to any specific variant of cosmological inflation, nor, if one wishes to consider any form of “modified gravity”, need one make specific a priori commitment to any specific variant of Horava gravity, or of any specific version of $f\left(R\right)$, $f(R,T)$, $f(R,T,Q)$ gravity or the like. The cosmographic framework is sufficiently flexible to allow one to deal with all of these situations in a theory-agnostic manner.

1.2. Synge’s G-Method Versus Synge’s S-Method

Synge’s G-method of “solving” the Einstein equations amounts to identifying some physically or pedagogically interesting spacetime, and then using the geometry of that spacetime to calculate the Einstein tensor. Thereby, (assuming applicability of the Einstein equations), one can deduce the required stress-energy required to support that spacetime [34]. When used with care and discretion, this process can lead to interesting results. When used without care or discretion the results can, however, be unfortunate. See, for instance, the criticism and discussion in reference [35]. In contrast, Synge’s S-method is much more traditional, and is based on iterating approximate solutions of known equations of motion.

Though not originally phrased in this particular manner, the core of the extensive body of work on “traversable wormholes” [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70], “warp drives” [71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96], and even “tractor beams” [97,98] can be viewed as (by and large successful) applications of Synge’s G-method (see also [99,100,101]). Work along these lines has by now led to well over 600 published scientific articles. As long as one keeps the speculative aspects of the physics under tight control, then useful information can be extracted. We particularly wish to emphasize the need for careful “sanity checking” to make sure one remains compatible with observational reality.

1.3. Immediate Plan of Action

Deferring dynamical considerations, at least until after one has a tolerable grasp on the kinematics, and similarly deferring considerations regarding the equation of state, until after one has a tolerable grasp on the spacetime geometry, can often be an “intermediate” but nevertheless useful route (the “low road”) to significant progress. Below we shall re-assess and re-analyze the venerable century-old problem of the general relativistic perfect fluid sphere (often used as a first approximation to stellar structure) [102]. We shall do so from a purely geometrical perspective, focussing on the spacetime geodesy (rather than the TOV equation and the EOS).

1.4. Long Term Goals

In the longer term, one would certainly be interested in dealing with mathematically more complicated and more physically realistic spacetimes. However, in both cosmography and in spacetime geodesy, there is very definitely a crucial trade-off between generality and symmetry.

For instance, grossly inhomogeneous cosmologies have essentially no symmetries, and there is little that can be said from a cosmographic perspective. On the other hand, rotating stars might plausibly still retain axial symmetry, and so a spacetime geodesy of (horizonless) stationary fluid spheroids might still be practical and useful. We shall leave such considerations for the future.

2. Spacetime Geodesy of Perfect Fluid Spheres: Framework

Terrestrial geodesy, or terrestrial geodetics, is the science of measuring, studying, and representing the geometry, gravity, and spatial orientation of planet Earth. It is called planetary geodesy when studying other astronomical bodies, such as planets or the like. By extension we shall adopt the phrase “spacetime geodesy” to describe the process of probing the geometry, gravity, and shape of some specified spacetime.

A particularly popular spacetime to study, because it is a zeroth order approximation to a stable star, is the static spherically symmetric perfect fluid sphere. The first explicit analysis of such an object was that of Schwarzschild’s constant density star [102]. First, using only the static and spherically symmetric conditions, geometrically this means the spacetime has a line element which in “area coordinates” can be put into the form

$$d{s}^2=-e^{-2\mathrm{\Phi }\left(r\right)}d{t}^2+{\displaystyle \frac{d{r}^2}{1-2m\left(r\right)/r}}+r^{2}d{\Omega }^2.$$

(1)

Area coordinates are often misattributed to Schwarzschild [102,103], though they should more appropriately be attributed to Droste [104,105,106] and Hilbert [107]. Area coordinates have the useful property that the area of a sphere of “coordinate radius” r is simply $A\left(r\right)=4\pi r^2$. This line element, and variations thereof, have been the inspiration for an enormous output of the literature. See, in particular, references [108,109,110,111,112,113,114,115,116,117,118] and, more generally, references [119,120,121,122,123,124,125,126,127,128,129].

We shall now seek to abstract the purely geometrical properties of these various and sundry models with a view to developing an appropriate notion of spacetime geodesy. More specifically, and more abstractly, writing the coordinates as $x^a=(t,x^i)$, we quite generally have

$$d{s}^2=-e^{-2\mathrm{\Phi }\left(r\right)}d{t}^2+g_{ij}\left(x^k\right)d{x}^{i}d{x}^j.$$

(2)

Then, for the metric,

$$g_{ab}=\left[\begin{array}{cc}-e^{-2\mathrm{\Phi }\left(r\right)}& 0\\ 0& g_{ij}\left(x^k\right)\end{array}\right].$$

(3)

This is enough to guarantee that the only non-zero parts of the Riemann tensor are $R_{titj}$ and $R_{ijkl}$. In turn, this implies block-diagonalization of the Ricci tensor:

$$R_{ab}=\left[\begin{array}{cc}{R}_{tt}& 0\\ 0& R_{ij}\end{array}\right].$$

(4)

2.1. Generalized Eigenvalues

Useful geometrical quantities are the coordinate independent generalized eigenvalues of the Ricci tensor defined by

$$\Lambda =\left\{\lambda :det\left(R_{ab}-\lambda g_{ab}\right)=0\right\}.$$

(5)

Static spherical symmetry is enough to guarantee $\Lambda =\{{\lambda }_0,{\lambda }_1,{\lambda }_2,{\lambda }_2\}$. The geometrical equivalent of the pressure isotropy condition inherent in demanding a perfect fluid sphere is a condition demanding spatial isotropy of the Ricci tensor—that all three spatial eigenvalues be equal: $\Lambda =\{{\lambda }_0,{\lambda }_1,{\lambda }_1,{\lambda }_1\}$. This in turn implies

$$R_{ij}={\lambda }_1{g}_{ij}={\displaystyle \frac{1}{3}}\left(g^{kl}{R}_{kl}\right)g_{ij}.$$

(6)

This purely geometrical statement now encodes the essential physics of a perfect fluid sphere, without having to resort to detailed investigations of what Einstein referred to as “base matter”. Everything is now encoded in the “marble” of spacetime geometry. For practical computations, it is often preferable to work with the ordinary eigenvalue problem

$$\Lambda =\left\{\lambda :det\left({R^a}_b-\lambda {{\delta }^a}_b\right)=0\right\}.$$

(7)

The only potential disadvantage of the “mixed-index” $T_1^1$ formulation is that, in situations more general than those currently under consideration, there is a risk that ${R^a}_b$, because it no longer need be symmetric, might lead to non-trivial Jordan normal forms [130,131]. Fortunately, this is not an issue in static spherical symmetry.

In this “mixed-index” $T_1^1$ formulation, the spatial Ricci isotropy condition becomes

$${R^i}_j={\lambda }_1{{\delta }^i}_j={\displaystyle \frac{1}{3}}\left({R^k}_k\right){{\delta }^i}_j.$$

(8)

This now completely specifies the purely geometrical spacetime geodesy framework we will more fully investigate below.

2.2. Physical Interpretation of the “Mixed-Index” $T_1^1$ and $T_2^2$ Components

It is also worthwhile to emphasize that, for any diagonal metric, provided the Ricci and Einstein tensors are similarly diagonal, adopting the “mixed-index” $T_1^1$ components for the Ricci ${R^a}_b$ or Einstein ${G^a}_b$ tensors is effectively equivalent to adopting an orthonormal tetrad basis, but without the hassle of actually defining the orthonormal tetrad.

To justify this, note that, when everything is diagonal,

$$R_{ab}={e_a}^{\widehat{a}}{R}_{\widehat{a}\widehat{b}}{e_b}^{\widehat{b}}=\mathrm{diag}\left\{R_{\widehat{a}\widehat{a}}{e_a}^{\widehat{a}}{e_a}^{\widehat{a}}\right\}=\mathrm{diag}\{R_{\widehat{a}\widehat{a}}|g_{aa}\left|\right\}.$$

(9)

Equivalently, when everything is diagonal,

$${R^a}_b={e^a}_{\widehat{a}}{R^{\widehat{a}}}_{\widehat{b}}{e_b}^{\widehat{b}}=\mathrm{diag}\left\{{R^{\widehat{a}}}_{\widehat{a}}{e^a}_{\widehat{a}}{e_a}^{\widehat{a}}\right\}=\mathrm{diag}\left\{{R^{\widehat{a}}}_{\widehat{a}}\right\}={R^{\widehat{a}}}_{\widehat{b}}.$$

(10)

That is, under these circumstances, we have ${R^a}_b={R^{\widehat{a}}}_{\widehat{b}}$. (For this reason, older textbooks sometimes refer to ${R^a}_b$ and ${G^a}_b$ as the “physical” components of the Ricci and Einstein tensors).

This trick also extends to the Riemann and Weyl tensors, provided the Riemann and Weyl tensors are diagonal under the $6\times 6$ Petrov classification, where the “mixed-index” $T_2^2$ components ${R^{ab}}_{cd}$ and ${C^{ab}}_{cd}$ now enjoy all of the benefits of adopting an orthonormal tetrad basis, but without the hassle. To see how this works, view the antisymmetric combination $\left[ab\right]$ or $\left[cd\right]$ as a compound index on six-dimensional space according to the recepie

$$1\leftrightarrow \left[tr\right];2\leftrightarrow \left[t\vartheta \right];3\leftrightarrow \left[t\phi \right];4\leftrightarrow \left[r\vartheta \right];5\leftrightarrow \left[r\phi \right];6\leftrightarrow \left[\vartheta \phi \right].$$

(11)

Then the identification $R_{abcd}⟷R_{AB}$ and $C_{abcd}⟷C_{AB}$ allows you to reinterpret the Riemann and Weyl tensors as $6\times 6$ symmetric matrices. If these $6\times 6$ matrices (and the metric) are diagonal, then, for instance,

$${R^{ab}}_{cd}={e^a}_{\widehat{a}}{e^b}_{\widehat{b}}{R^{\widehat{a}\widehat{b}}}_{\widehat{c}\widehat{d}}{e_c}^{\widehat{c}}{e_d}^{\widehat{d}}=\mathrm{diag}\left\{{R^{\widehat{a}\widehat{b}}}_{\widehat{a}\widehat{b}}\left({e^a}_{\widehat{a}}{e^b}_{\widehat{b}}\right)\left({e_c}^{\widehat{a}}{e_d}^{\widehat{b}}\right)\right\}=\mathrm{diag}\left\{{R^{\widehat{a}\widehat{b}}}_{\widehat{a}\widehat{b}}\right\}={R^{\widehat{a}\widehat{b}}}_{\widehat{c}\widehat{d}}.$$

(12)

That is, under these circumstances, ${R^{ab}}_{cd}={R^{\widehat{a}\widehat{b}}}_{\widehat{c}\widehat{d}}$, and likewise ${C^{ab}}_{cd}={C^{\widehat{a}\widehat{b}}}_{\widehat{c}\widehat{d}}$.

Unfortunately, this trick is not truly fundamental—it certainly fails whenever the metric is non-diagonal, (e.g., Kerr, Kerr–Newman, and more generically in rotating spacetimes), and can also be problematic in time-dependent situations where both the Einstein and Ricci tensors acquire off-diagonal flux components, and the Riemann and Weyl tensors need no longer be a Petrov diagonal. The “mixed-index” $T_1^1$ and $T_2^2$ still exist, but they no longer have any “nice” straightforward interpretation in terms of an orthonormal tetrad basis. Still, as long as one is aware of the limitations thereof, this trick is a useful way of simplifying explicit calculations.

2.3. Pragmatics

At a purely pragmatic level, we can now proceed simply by writing down the line element for a static spherically symmetric spacetime and solving the spatial isotropy condition for the Ricci tensor:

$${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }.$$

(13)

Of course, spatial isotropy for the Ricci tensor is completely equivalent to spatial isotropy for the Einstein tensor

$${G^r}_r={G^{\vartheta }}_{\vartheta }={G^{\phi }}_{\phi }.$$

(14)

This spatial isotropy condition supplies one ordinary differential equation relating the metric components. Depending on the precise choice of coordinates, and ways of representing the metric components, this ordinary differential equation might be explicitly solvable.

(It should be noted that even symbolic calculation systems such as Maple or Mathematica will often require significant human intervention to obtain a reasonably tractable result). We shall present a number of explicit examples below.

3. Spacetime Geodesy of Perfect Fluid Spheres: Examples

Below we present several specific examples of spacetime geodesy—of various levels of generality and practicality. The primary goal will be to generate purely geometrical models of perfect fluid spheres, focussing on the spacetime geometry and (temporarily) delaying dynamical considerations.

3.1. Example 1: Two Free Functions—Purely Integral Version

We start with a rather non-trivial (certainly not a priori obvious) example of spacetime geodesy. Take two arbitrary integrable functions, $f\left(r\right)$ and $w\left(r\right)$, plus an arbitrary constant $K_0$, and consider this line-element:

$$\begin{array}{ccc}\hfill d{s}^2& =& -exp\left(-2\int f(r)dr\right)d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{f{\left(r\right)}^2{[1+w\left(r\right)]}^{2}exp\left(-2\int f\left(r\right)\frac{[1-w(r\left)\right]}{[1+w(r\left)\right]}dr\right)}{2\int f\left(r\right)[1+w\left(r\right)]exp\left(-2\int \frac{f\left(r\right)[1-2w\left(r\right)-w{\left(r\right)}^2]}{[1+w(r\left)\right]}dr\right)dr-K_0}}\right\}d{r}^2\hfill \\ & & +exp\left(-2\int f(r)w(r)dr\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].\hfill \end{array}$$

(15)

This always has a spatially isotropic Ricci tensor satisfying ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$ (and therefore, assuming the usual Einstein equations, is suitable for describing a perfect fluid sphere, and even if one does not wish to assume the usual Einstein equations, this is nevertheless a geometrically interesting spacetime with locally isotropic spatial curvature).

Verifying this is relatively easy: Just feed the line element into Maple and check that ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$. (Some human intervention is still required). On the other hand, finding this form of the line element is somewhat tedious and requires some inspired guesswork and more than a little direct human intervention. Let us see how to derive this result.

Let us start with the metric in its most general form

$$d{s}^2=g_{tt}\left(r\right)d{t}^2+g_{rr}\left(r\right)d{r}^2+g_{\vartheta \vartheta }\left(r\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(16)

This respects spherical symmetry and time independence but, at this stage, imposes no additional constraint.

Before we do anything else, let us consider the null convergence condition (NCC).

In complete generality,

$${R^t}_t-{R^r}_r=-{\displaystyle \frac{g_{\vartheta \vartheta }^{\prime \prime }}{g_{\vartheta \vartheta }{g}_{rr}}}+{\displaystyle \frac{{\left[g_{\vartheta \vartheta }^{\prime }\right]}^2}{2g_{\vartheta \vartheta }^2{g}_{rr}}}+{\displaystyle \frac{g_{\vartheta \vartheta }^{\prime }}{2g_{rr}{g}_{\vartheta \vartheta }}}\left[{\displaystyle \frac{g_{tt}^{\prime }}{g_{tt}}}+{\displaystyle \frac{g_{rr}^{\prime }}{g_{rr}}}\right].$$

(17)

Thence, at any local extremum of $g_{\vartheta \vartheta }$, we have

$${R^t}_t-{R^r}_r\to -{\displaystyle \frac{g_{\vartheta \vartheta }^{\prime \prime }}{g_{\vartheta \vartheta }{g}_{rr}}}.$$

(18)

This implies that local minima of $g_{\vartheta \vartheta }$ violate the NCC (implying, in standard Einstein gravity, violation of the null energy condition, NEC). Violations of the NCC are not entirely forbidden by quantum physics, but are certainly a significant departure from “usual physics” [132,133,134,135,136]. These considerations are important if you either wish to implement (or prevent) formation of a traversable wormhole throat [36,37].

Now consider the ordinary differential equation (ODE)

$${R^r}_r-{R^{\vartheta }}_{\vartheta }=0.$$

(19)

Viewed as an ODE for $g_{rr}\left(r\right)$, this ODE is a Bernoulli ODE, so nominally solvable. There is an art to choosing $g_{tt}\left(r\right)$, $g_{rr}\left(r\right)$, $g_{\vartheta \vartheta }\left(r\right)$ to simplify this as much as possible. Let us now choose

$$d{s}^2=g_{tt}\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{B\left(r\right)}}+g_{\vartheta \vartheta }\left(r\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(20)

The ODE ${R^r}_r-{R^{\vartheta }}_{\vartheta }=0$ is now a first-order linear ODE for $B\left(r\right)$. Write this ODE in the form

$$K_1\left(r\right)B^{\prime }\left(r\right)+K_{2}B\left(r\right)+K_3\left(r\right)=0.$$

(21)

Then one finds

$$\begin{array}{lll}\hfill K_1\left(r\right)& =& {\displaystyle \frac{1}{4}}{\displaystyle \frac{dln\left[g_{tt}\left(r\right)g_{\vartheta \vartheta }\left(r\right)\right]}{dr}};\hfill \end{array}$$

(22)

$$\begin{array}{lll}\hfill K_2\left(r\right)& =& {\displaystyle \frac{1}{2}}{\displaystyle \frac{d^{2}ln\left[g_{tt}\left(r\right)g_{\vartheta \vartheta }\left(r\right)\right]}{d{r}^2}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{dln\left[g_{tt}\left(r\right)\right]}{dr}}{\displaystyle \frac{dln[g_{tt}\left(r\right)/g_{\vartheta \vartheta }\left(r\right)]}{dr}};\hfill \end{array}$$

(23)

$$\begin{array}{lll}\hfill K_3\left(r\right)& =& {\displaystyle \frac{1}{g_{\vartheta \vartheta }\left(r\right)}}.\hfill \end{array}$$

(24)

Looking at $K_1\left(r\right)$ and $K_2\left(r\right)$ suggests that it might be useful to set

$$g_{tt}\left(r\right)=-exp\left(-2\mathit{\int }f(r)dr\right);g_{\vartheta \vartheta }\left(r\right)=exp\left(-2\int h(r)dr\right);$$

(25)

for arbitrary smooth functions $f(r)$ and $h(r)$. Therefore, the line element becomes

$$d{s}^2=-exp\left(-2\int f(r)dr\right)d{t}^2+{\displaystyle \frac{d{r}^2}{B(r)}}+exp\left(-2\int h(r)dr\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(26)

The ODE imposing spatial isotropy, ${R^r}_r-{R^{\vartheta }}_{\vartheta }=0$, when viewed as an ODE for the function $B(r)$, and written as $K_1(r)B^{\prime }(r)+K_{2}B(r)+K_3(r)=0$, now yields

$$\begin{array}{ccc}\hfill K_1(r)& =& -{\displaystyle \frac{1}{2}}[f(r)+h(r)];\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill K_2(r)& =& -[f^{\prime }(r)+h^{\prime }(r)]+f(r)[f(r)-h(r)];\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill K_3(r)& =& exp\left(+2\int h(r)dr\right).\hfill \end{array}$$

(29)

We know that this ODE has the explicit solution

$$B(r)=-\left(\int {\displaystyle \frac{K_3\left(r\right)}{K_1\left(r\right)}}exp\left(\int {\displaystyle \frac{K_2\left(r\right)}{K_1\left(r\right)}}dr\right)dr+K_0\right)exp\left(-\int {\displaystyle \frac{K_2\left(r\right)}{K_1\left(r\right)}}dr\right).$$

(30)

Note the following:

$$\begin{array}{ccc}\hfill \int {\displaystyle \frac{K_2\left(r\right)}{K_1\left(r\right)}}dr& =& \int {\displaystyle \frac{[f^{\prime }\left(r\right)+h^{\prime }\left(r\right)]-f\left(r\right)[f\left(r\right)-h\left(r\right)]}{\frac{1}{2}[f\left(r\right)+h\left(r\right)]}}dr\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& =& 2ln[f\left(r\right)+h\left(r\right)]-2\int {\displaystyle \frac{f\left(r\right)\left[f\right(r)-h(r\left)\right]}{\left[f\right(r)+h(r\left)\right]}}dr.\hfill \end{array}$$

(32)

Thence,

$$exp\left(\int {\displaystyle \frac{K_2\left(r\right)}{K_1\left(r\right)}}dr\right)={[f\left(r\right)+h\left(r\right)]}^{2}exp\left(-2\int {\displaystyle \frac{f\left(r\right)\left[f\right(r)-h(r\left)\right]}{\left[f\right(r)+h(r\left)\right]}}dr\right).$$

(33)

Furthermore,

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{K_3\left(r\right)}{K_1\left(r\right)}}exp\left(\int {\displaystyle \frac{K_2\left(r\right)}{K_1\left(r\right)}}dr\right)\hfill \\ \hfill & & =-2exp\left(+2\int h\left(r\right)dr\right)[f\left(r\right)+h\left(r\right)]exp\left(-2\int {\displaystyle \frac{f\left(r\right)\left[f\right(r)-h(r\left)\right]}{\left[f\right(r)+h(r\left)\right]}}dr\right)\hfill \\ \hfill & & =-2[f\left(r\right)+h\left(r\right)]exp\left(-2\int {\displaystyle \frac{[f{\left(r\right)}^2-2f\left(r\right)h\left(r\right)-h{\left(r\right)}^2]}{\left[f\right(r)+h(r\left)\right]}}dr\right).\hfill \end{array}$$

(34)

Thence,

$$g_{rr}(r)={\displaystyle \frac{1}{B\left(r\right)}}={\displaystyle \frac{{[f\left(r\right)+h\left(r\right)]}^{2}exp\left(-2\int \frac{f\left(r\right)\left[f\right(r)-h(r\left)\right]}{\left[f\right(r)+h(r\left)\right]}dr\right)}{2\int [f\left(r\right)+h\left(r\right)]exp\left(-2\int \frac{[f{\left(r\right)}^2-2f\left(r\right)h\left(r\right)-h{\left(r\right)}^2]}{\left[f\right(r)+h(r\left)\right]}dr\right)dr-K_0}}.$$

(35)

Therefore, we have the line element

$$\begin{array}{ccc}\hfill d{s}^2& =& -exp\left(-2\int f\left(r\right)dr\right)d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{{[f\left(r\right)+h\left(r\right)]}^{2}exp\left(-2\int \frac{f\left(r\right)\left[f\right(r)-h(r\left)\right]}{\left[f\right(r)+h(r\left)\right]}dr\right)}{2\int [f\left(r\right)+h\left(r\right)]exp\left(-2\int \frac{[f{\left(r\right)}^2-2f\left(r\right)h\left(r\right)-h{\left(r\right)}^2]}{\left[f\right(r)+h(r\left)\right]}dr\right)dr-K_0}}\right\}d{r}^2\hfill \\ & & +exp\left(-2\int h\left(r\right)dr\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].\hfill \end{array}$$

(36)

Finally, define $w(r)$ by setting $h(r)=f(r)w(r)$ to obtain the result previously announced in Equation (15). The good news is that we have a fully explicit line element always guaranteed to generate a spatially isotropic Ricci tensor (ditto a spatially isotropic Einstein tensor). The not so good news is that considerable brute force was required to obtain this result, and that the resulting curvature tensors are extremely clumsy and difficult to deal with. Still, with 2 free functions at one’s disposal, either $\{f(r),w(r)\}$ in Equation (15) or $\{f(r),h(r)\}$ in Equation (36), one has a very powerful algorithm for generating bespoke perfect fluid spheres.

3.2. Example 2: Two Free Functions—Integro-Differential Version

A perhaps slightly more tractable version of the above is to simply set

$$F(r)=\int f(r)dr;H(r)=\int h(r)dr,$$

(37)

so that the line element becomes

$$d{s}^2=-exp\left(-2F(r)\right)d{t}^2+{\displaystyle \frac{d{r}^2}{B\left(r\right)}}+exp\left(-2H(r)\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(38)

Then, working directly from Equation (36), we obtain

$$\begin{array}{ccc}\hfill d{s}^2& =& -exp\left(-2F\left(r\right)\right)d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{{[F^{\prime }\left(r\right)+H^{\prime }\left(r\right)]}^{2}exp\left(-2\int F^{\prime }\left(r\right)\frac{[F^{\prime }\left(r\right)-H^{\prime }\left(r\right)]}{[F^{\prime }\left(r\right)+H^{\prime }\left(r\right)]}dr\right)}{2\int [F^{\prime }\left(r\right)+H^{\prime }\left(r\right)]exp\left(2H\left(r\right)-2\int F^{\prime }\left(r\right)\frac{[F^{\prime }\left(r\right)-H^{\prime }\left(r\right)]}{[F^{\prime }\left(r\right)+H^{\prime }\left(r\right)]}dr\right)dr-K_0}}\right\}d{r}^2\hfill \\ & & +exp\left(-2H\left(r\right)\right)[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].\hfill \end{array}$$

(39)

This again has a spatially isotropic Ricci tensor satisfying ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$. Verifying this is now somewhat easier—Maple requires significantly less human intervention. Unfortunately, the resulting curvature tensors are still extremely clumsy. On the other hand, we still have two free functions available, so we still have room for using coordinate freedom to further simplify the situation.

Still, with 2 free functions at one’s disposal, $\{F(r),H(r)\}$ in Equation (39), one again has a very powerful algorithm for generating bespoke perfect fluid spheres.

3.3. Example 3: One Free Function—Area Coordinates

The Hilbert–Droste area coordinates [104,105,106,107] amount to setting $g_{\vartheta \vartheta }=r^2$ so that the area of the sphere of coordinate radius r is simply $A(r)=4\pi r^2$. There is a mild technical constraint involved here—if $A(r)$ is not monotone increasing as one moves outwards, then one needs multiple coordinate patches of this type, which overlap at wormhole throats or anti-throats. (In particular, as already noted, local minima of $g_{rr}(r)$, and hence local minima of $A(r)=4\pi g_{\vartheta \vartheta }(r)=4\pi r^2$ correspond to violations of the null convergence condition (NCC). Consequently, globally enforcing the existence of a single patch of area coordinates automatically precludes the existence of traversable wormholes).

In normal situations, this is not an issue and one globally asserts the following.

$$d{s}^2=-exp\left(-2F(r)\right)d{t}^2+{\displaystyle \frac{d{r}^2}{B\left(r\right)}}+r^2[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(40)

(If one does want to allow the possibility of traversable wormholes, then area coordinates should be used with care and discretion—with wormhole throats and anti-throats being located right on the edge of these coordinate patches, it is easy to become confused. See references [137,138,139] for some examples of potential pitfalls.)

Then, working directly from Equation (39), we obtain

$$\begin{array}{ccc}\hfill d{s}^2& =& -exp\left(-2F\left(r\right)\right)d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{{[1-r{F}^{\prime }\left(r\right)]}^2{r}^{-2}exp\left(-2\int F^{\prime }\left(r\right)\frac{[1+r{F}^{\prime }\left(r\right)]}{[1-r{F}^{\prime }\left(r\right)]}dr\right)}{-2\int [1-r{F}^{\prime }\left(r\right)]r^{-3}exp\left(-2\int F^{\prime }\left(r\right)\frac{[1+r{F}^{\prime }\left(r\right)]}{[1-r{F}^{\prime }\left(r\right)]}dr\right)dr+K_0}}\right\}d{r}^2\hfill \\ & & +r^2[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].\hfill \end{array}$$

(41)

This again has a spatially isotropic Ricci tensor satisfying ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$. Verifying this is now considerably easier—in this situation, Maple requires essentially no direct human intervention. This result is essentially a repackaging of the Boonserm–Visser–Weinfurtner algorithm [121,122] (see also [123,124,125]), which we now see is a special case of a considerably more general analysis.

3.4. Example 4: One Free Function—Redshift Coordinates

Redshift coordinates are somewhat unusual. First, re-label $r\to z$, that is just a name change, but now additionally use coordinate freedom to choose $g_{tt}\left(z\right)=-1/{(1+z)}^2$ so that

$$d{s}^2=-{\displaystyle \frac{d{t}^2}{{(1+z)}^2}}+{\displaystyle \frac{d{z}^2}{B\left(z\right)}}+exp(-2H\left(z\right))[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(42)

The proper time measured by a clock at position z is then $\Delta \tau ={\displaystyle \frac{\Delta t}{1+z}}$, which means that signals emitted from the location z will be redshifted by exactly an amount z by the time they reach spatial infinity ($T_{emitted}=T_{\infty }/(1+z)$). The range of the z coordinate will be $z\in [0,z_0]$, where $z_0$ is the “central redshift” of a signal emitted from the centre of the spacetime. Coordinates of this type do require the mild technical restriction that $g_{tt}\left(z\right)$ be a monotone as one moves outwards, corresponding to the gravitational field always being attractive.

Then, working directly from Equation (39), we now obtain

$$\begin{array}{ccc}\hfill d{s}^2& =& -{(1+z)}^{-2}d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{{[1-(1+z)H^{\prime }\left(z\right)]}^2{[1+z]}^{-2}exp\left(+2\int \frac{[1+(1+z)H^{\prime }\left(z\right)]}{[1-(1+z)H^{\prime }\left(z\right)]}\frac{dz}{1+z}\right)}{-2\int [1-(1+z)H^{\prime }\left(z\right)]exp\left(2H\left(z\right)+2\int \frac{[1+(1+z)H^{\prime }\left(z\right)]}{[1-(1+z)H^{\prime }\left(z\right)]}\frac{dz}{1+z}\right)\frac{dz}{1+z}+K_0}}\right\}d{z}^2\hfill \\ & & +exp(-2H\left(z\right))[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2]\hfill \end{array}$$

(43)

This again has a spatially isotropic Ricci tensor satisfying ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$. Verifying this is again relatively easy—Maple requires essentially no direct human intervention.

3.5. Example 5: One Free Function—Exponential Redshift Coordinates

It is also worthwhile to consider an “exponential” version of redshift coordinates with $g_{tt}\left(z\right)=-e^{-2z}$ and $T_{emitted}=e^{-z}{T}_{\infty }$. This is not precisely the standard definition of redshift, but ultimately it carries the same physical information and is somewhat easier to deal with mathematically. We start from

$$d{s}^2=-e^{-2z}d{t}^2+{\displaystyle \frac{d{z}^2}{B\left(z\right)}}+exp(-2H\left(z\right))[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

Then, working directly from Equation (39), we now obtain

$$\begin{array}{ccc}\hfill d{s}^2& =& -e^{-2z}d{t}^2\hfill \\ & & +\left\{{\displaystyle \frac{{[1+H^{\prime }\left(z\right)]}^{2}exp\left(-2\int \frac{[1-H^{\prime }\left(z\right)]}{[1+H^{\prime }\left(z\right)]}dz\right)}{2\int [1+H^{\prime }\left(z\right)]exp\left(2H\left(z\right)-2\int \frac{[1-H^{\prime }\left(z\right)]}{[1+H^{\prime }\left(z\right)]}dz\right)dz+K_0}}\right\}d{z}^2\hfill \\ & & +exp(-2H\left(z\right))[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2]\hfill \end{array}$$

(44)

This again has a spatially isotropic Ricci tensor satisfying ${R^r}_r={R^{\vartheta }}_{\vartheta }={R^{\phi }}_{\phi }$. Verifying this is again relatively easy—Maple requires essentially no direct human intervention.

3.6. Discussion of These Perfect Fluid Spheres

In this section, we have developed a general algorithm guaranteed to always generate spacetime geometries describing perfect fluid spheres—this would only be a first step in investigating the physical properties of such spacetimes [109]. The isotropy condition, while central to the effort, is only part of what makes perfect fluid spheres physically acceptable [109,110,111,112,113,114,115,116,117,118]. Overall, this spacetime geodesy approach is a very practical and pragmatic first step in any such analysis.

4. Weyl Tensor and Weyl Scalar in Spherical Symmetry

A quite remarkable feature of four-dimensional spacetime geodesy is that, for spherical symmetry, the Weyl tensor is particularly and perhaps unexpectedly simple.

4.1. Generalities

Even allowing for the possibility of time dependence in the spacetime geometry, the Weyl tensor in four-dimensional spherical symmetry always takes the very simple form

$${C^{ab}}_{cd}={Z^{ab}}_{cd}{C}_0(r,t),$$

(45)

where the $T_2^2$ tensor ${Z^{ab}}_{cd}$ has only fixed integer-valued components, whereas the common scalar $C_0(r,t)$ depends on the specific details of the spacetime. ($C_0(r,t)$ will soon be seen to be particularly simple for static fluid spheres).

Furthermore, since the single quantity $C_0(r,t)$ completely characterizes the deviation from conformal flatness, it also completely characterizes the lensing properties of any spherically symmetric spacetime. Specifically, it is convenient for this section to set

$$d{s}^2=-exp(-2\mathrm{\Phi }(r,t))\left(1-{\displaystyle \frac{2m(r,t)}{r}}\right)d{t}^2+{\displaystyle \frac{d{r}^2}{1-2m(r,t)/r}}+r^2[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2].$$

(46)

Now explicitly calculate the $T_2^2$ Weyl tensor ${C^{ab}}_{cd}$ and (by inspection) note that it is always of the form ${C^{ab}}_{cd}={Z^{ab}}_{cd}{C}_0(r,t)$, with

$${Z^{tr}}_{tr}={Z^{\vartheta \phi }}_{\vartheta \phi }=2;$$

(47)

$${Z^{t\vartheta }}_{t\vartheta }={Z^{t\phi }}_{t\phi }={Z^{r\vartheta }}_{r\vartheta }={Z^{r\phi }}_{r\phi }=-1;$$

(48)

and

$$C_0(r,t)={\displaystyle \frac{1}{6}}\left({G^t}_t+{G^r}_r-{G^{\vartheta }}_{\vartheta }\right)+{\displaystyle \frac{m(r,t)}{r^3}}.$$

(49)

(All other components, not related by the standard Weyl symmetries, vanish.) Note that this simple formula for $C_0(r,t)$ holds even though the time-dependent Einstein tensor ${G^a}_b$ has two non-zero off-diagonal components ${G^t}_r$ and ${G^r}_t$. Note that, whereas the Einstein tensor components ${G^t}_t$, ${G^r}_r$, and ${G^{\vartheta }}_{\vartheta }$ by construction depend only on local physics, the quantity $C_0(r,t)$ also explicitly depends on one quasi-local concept—the Misner–Sharp quasi-local mass $m(r,t)$.

A side-effect of these results is that, in four-dimensional spherical symmetry, the Weyl scalar is always a perfect square

$$W=C_{abcd}{C}^{abcd}=\left[Z_{abcd}{Z}^{abcd}\right]{\left[C_0(r,t)\right]}^2=48{\left[C_0(r,t)\right]}^2.$$

(50)

For the dual Weyl tensor ${}^{*}{C}^{ab}_{cd}={\displaystyle \frac{1}{2}}{ϵ}^{ab}{}_{ef}{C}^{ef}_{cd}={}^{*}{Z}^{ab}_{cd}{C}_0(r,t)$, we have

$${}^{*}{Z}^{tr}_{\vartheta \phi }=2;{}^{*}{Z}^{r\vartheta }_{\phi r}={}^{*}{Z}^{t\phi }_{r\vartheta }=-1;$$

(51)

(All other components, not related by the standard dual-Weyl symmetries, vanish.)

Defining V to be a unit vector in the t direction, the “electric” part of the Weyl tensor is defined to be ${E^a}_b={C^{ac}}_{bd}{V}_c{V}^d$. Consequently, ${E^a}_b={{\left[Z_E\right]}^a}_b{C}_0$, where the only non-zero components of ${{\left[Z_E\right]}^a}_b$ are

$${{\left[Z_E\right]}^r}_r=-2;{{\left[Z_E\right]}^{\vartheta }}_{\vartheta }={{\left[Z_E\right]}^{\phi }}_{\phi }=1.$$

(52)

Thence,

$${{\left[Z_E\right]}^a}_b=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -2& 0& 0\\ 0& 0& +1& 0\\ 0& 0& 0& +1\end{array}\right].$$

(53)

(Note the tensor ${{\left[Z_E\right]}^a}_b$ is traceless.) In contrast, it is easy to see that, in spherical symmetry, the “magnetic” part of the Weyl tensor ${H^a}_b={}^{*}{C}^{ab}_{bd}{V}_c{V}^d$ vanishes identically. All in all, spherical symmetry very tightly constrains the Weyl tensor.

We could also work with the $6\times 6$ Petrov embedding of ${Z^{ab}}_{cd}⟷{Z^A}_B$ in which case

$${Z^A}_B=\left[\begin{array}{cccccc}+2& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& +2\end{array}\right]$$

(54)

Finally, another way of characterizing the tensor structure is as follows: Ordering the coordinates as $\{t,r,\vartheta ,\phi \}$, define the $T_2^2$ tensor:

$${{\Delta }^a}_b=\left[\begin{array}{cccc}\hfill +1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill +1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1\end{array}\right].$$

(55)

This tensor squares to the ordinary Kronecker delta, ${{\Delta }^a}_b{{\Delta }^b}_c={{\delta }^a}_c$, but its trace is zero ${{\Delta }^a}_a=0$. Then it is a easy exercise to check:

$${Z^{ab}}_{cd}={\displaystyle \frac{3}{2}}({{\Delta }^a}_c{{\Delta }^b}_d-{{\Delta }^a}_d{{\Delta }^b}_c)+{\displaystyle \frac{1}{2}}({{\delta }^a}_c{{\delta }^b}_d-{{\delta }^a}_d{{\delta }^b}_c).$$

(56)

First, check that this reproduces

$${Z^{tr}}_{tr}={Z^{\vartheta \phi }}_{\vartheta \phi }=2,$$

(57)

and then check that this also reproduces

$${Z^{t\vartheta }}_{t\vartheta }={Z^{t\phi }}_{t\phi }={Z^{r\vartheta }}_{r\vartheta }={Z^{r\phi }}_{r\phi }=-1.$$

(58)

As a final consistency check, note that, when one takes the trace,

$${Z^{ab}}_{cb}=-{\displaystyle \frac{3}{2}}{{\delta }^a}_c+{\displaystyle \frac{1}{2}}\left([4-1]{{\delta }^a}_c\right)=0.$$

(59)

Thus, all the Weyl symmetries, and the trace-free properties, are respected.

All in all, in spherical symmetry in (3 + 1) dimensions, we have

$${C^{\mu \nu }}_{\alpha \beta }=\left\{{\displaystyle \frac{3}{2}}({{\Delta }^{\mu }}_{\alpha }{{\Delta }^{\nu }}_{\beta }-{{\Delta }^{\mu }}_{\beta }{{\Delta }^{\nu }}_{\alpha })+{\displaystyle \frac{1}{2}}({{\delta }^{\mu }}_{\alpha }{{\delta }^{\nu }}_{\beta }-{{\delta }^{\mu }}_{\beta }{{\delta }^{\nu }}_{\alpha })\right\}{C}_0.$$

(60)

4.2. Weyl Tensor for Static Perfect Fluid Spheres

If we now consider static spacetimes, then the off-diagonal part of the Einstein (and Ricci) tensors vanish. Furthermore, if (with perfect fluids in mind) we demand spatial isotropy for the Einstein tensor

$${G^r}_r={G^{\vartheta }}_{\vartheta }={G^{\phi }}_{\phi },$$

(61)

then the common factor $C_0(r,t)$ simplifies considerably:

$$C_0(r,t)\to {\displaystyle \frac{1}{6}}{G^t}_t+{\displaystyle \frac{m\left(r\right)}{r^3}}=-{\displaystyle \frac{m^{\prime }\left(r\right)}{3r^2}}+{\displaystyle \frac{m\left(r\right)}{r^3}}=-{\displaystyle \frac{r{(m\left(r\right)/r^3)}^{\prime }}{3}}.$$

(62)

This is a very simple function of the Misner–Sharp quasi-local mass. (Indeed $m(r)/r^3$ can be thought of as the “average density” inside coordinate radius r). It is notable that the metric function $\mathrm{\Phi }(r)$ has completely dropped out of the analysis. Furthermore, for a constant density perfect fluid sphere (Schwarzschild’s constant density star, Schwarzschild’s “interior” solution [102]), the quantity $C_0\left(t\right)$, and hence the Weyl tensor itself, vanishes identically.

One could even reverse-engineer $m(r)$ to set up a chosen profile $C_0(r)$ for the Weyl tensor. Merely set

$$m(r)={\displaystyle \frac{\left(m_0+3\int C_0\left(r\right)r^{5}dr\right)}{r^3}}.$$

(63)

Unfortunately, for this choice of mass profile, the perfect fluid condition ${G^r}_r={G^{\vartheta }}_{\vartheta }$ turns into a quite nasty nonlinear ODE for $\mathrm{\Phi }(r)$.

4.3. PFDM—Not a Perfect Fluid

The most critical thing one really needs to know about PFDM (so-called “perfect fluid dark matter”) is that it is not a perfect fluid [140,141]. Instead, PFDM is (at best) interpretable as an anisotropic fluid—indeed one should really call it a “Kiselev fluid”, and refer to KFDM. The terminology originated in some (quickly corrected) awkward phrasing in reference [142], but some segments of the community continue to abuse terminology in this particular way.

To drive the message home, the eigenvalues $\{\lambda :det({T^a}_b-\lambda {{\delta }^a}_b)=0\}$ of the stress-enegy tensor for a perfect fluid are always of the form $\{{\lambda }_0,{\lambda }_1,{\lambda }_1,{\lambda }_1\}$, whereas the eigenvalues of Kiselev’s stress-energy tensor are of the form $\{{\lambda }_0,{\lambda }_0,{\lambda }_1,{\lambda }_1\}$. These eigenvalue patterns are just not the same (except in the trivial case ${\lambda }_1={\lambda }_0$, corresponding to “vacuum energy”/cosmological constant $\{{\lambda }_0,{\lambda }_0,{\lambda }_0,{\lambda }_0\}$).

The Kiselev geometry (at its most general) corresponds to the metric ansatz

$$d{s}^2=-\left(1-{\displaystyle \frac{2m\left(r\right)}{r}}\right)d{t}^2+{\displaystyle \frac{d{r}^2}{1-2m\left(r\right)/r}}+r^2[d{\vartheta }^2+{sin}^2\vartheta d{\phi }^2],$$

(64)

for which

$${G^t}_t={G^r}_r=-{\displaystyle \frac{2m^{\prime }\left(r\right)}{r^2}};{G^{\vartheta }}_{\vartheta }={G^{\phi }}_{\phi }=-{\displaystyle \frac{m^{″}\left(r\right)}{r}}.$$

(65)

Thence, for a Kiselev fluid,

$$C_0(r)={\displaystyle \frac{m^{″}\left(r\right)}{6r}}-{\displaystyle \frac{2m^{\prime }\left(r\right)}{3r^2}}+{\displaystyle \frac{m\left(r\right)}{r^3}}={\displaystyle \frac{r^2{m}^{″}\left(r\right)-4r{m}^{\prime }\left(r\right)+6m\left(r\right)}{6r^3}}={\displaystyle \frac{r{(m\left(r\right)/r^2)}^{″}}{6}},$$

(66)

which generically is simply different from that of a perfect fluid.

A much more specific form of Kiselev’s spacetime is to take [142]

$$m(r)=m_0\left(1+{\displaystyle \frac{k_{-1}}{r}}-k_2{r}^2+{\displaystyle \frac{k_{-2w}}{r^{2w}}}\right),$$

(67)

with non-constant contributions to $m(r)$ coming from electrical charge, cosmological constant, and a specific version of the Kiselev fluid, respectively. In this situation, one obtains

$$C_0(r)={\displaystyle \frac{m_0}{r^3}}\left(1+{\displaystyle \frac{2k_{-1}}{r}}+{\displaystyle \frac{(1+w)(1+2w/3)k_{-2w}}{r^{2w}}}\right).$$

(68)

This generically is simply different from what happens for a perfect fluid (except in the very special case $k_{-1}=0=k_{-2w}$, which corresponds to a pure cosmological constant).

4.4. Complexity Factor

The Weyl component $C_0(r)$ is almost identical to Herrara’s “complexity factor”. Indeed, assuming the usual Einstein equations, Herrara defines $Y_{TF}$ as the equivalent of [143,144,145,146,147,148,149,150,151,152,153,154]:

$$Y_{TF}=4\pi (p_r-p_t)+C_0(r).$$

(69)

(See also various applications of related ideas in references [155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173].) Certainly for any perfect fluid Herrara’s complexity factor is identical to the Weyl component $C_0(r)$. Furthermore, from a purely geometrical perspective,

$$Y_{TF}={\displaystyle \frac{1}{2}}({G_r}^r-{G_{\vartheta }}^{\vartheta })+C_0(r)={\displaystyle \frac{1}{6}}{G^t}_t+{\displaystyle \frac{2}{3}}\left({G^r}_r-{G^{\vartheta }}_{\vartheta }\right)+{\displaystyle \frac{m\left(r\right)}{r^3}}.$$

(70)

We note that both Herrara’s “complexity factor” and the Weyl component $C_0(r,t)$ vanish for Schwarzschild’s constant density perfect fluid star. Therefore, both Herrara’s “complexity factor” and the Weyl component $C_0(r,t)$ characterize deviations from Schwarzschild’s constant density perfect fluid star. Furthermore, note that any constant coefficient linear combination of $({G_r}^r-{G_{\vartheta }}^{\vartheta })$ and $C_0(r)$ also satisfies this property. Whether one prefers to use Herrara’s “complexity factor” or the Weyl component $C_0(r)$ seems largely a matter of taste—though we would argue that the Weyl component $C_0(r)$ has a considerably clearer and direct geometrical interpretation in terms of the entire Weyl tensor vanishing identically.

A more subtle approach would be to introduce a quadratic notion of complexity:

$$Q=\sqrt{{({G_r}^r-{G_{\vartheta }}^{\vartheta })}^2+C_0{\left(r\right)}^2}.$$

(71)

This quadratic complexity vanishes if and only if both ${G_r}^r={G_{\vartheta }}^{\vartheta }$ (corresponding to a perfect fluid), and then also ${(m\left(r\right)/r^3)}^{\prime }=0$ (corresponding to constant density). Therefore, this quadratic complexity vanishes if and only if one is dealing with Schwarzschild’s constant density star, and we can interpret this quadratic complexity Q as a “distance” between the geometry of interest and Schwarzschild’s constant density star.

4.5. Discussion of the Weyl Tensor Analysis

In this section, we have used a spacetime geodesy perspective to see that that, in spherical symmetry, the Weyl tensor takes on a very simple form—effectively with only a single independent tensor component. We have seen that the Weyl tensor simplifies even further for perfect fluid spheres. Subsequently, we have considered the example of so-called PFDM (perfect fluid dark matter), verifying that it is not a perfect fluid and suggesting it be renamed KFDM (Kiselev fluid dark matter). Finally we have related these considerations to the notion of “complexity”.

5. Spacetime Geodesy of Anisotropic Fluid Spheres

Finally, let us consider the spacetime geodesy of generic anisotropic fluid spheres [174,175,176]. In this situation, spacetime geodesy is relatively uninteresting, because once you allow for generic anisotropic fluids, in principle, any spherically symmetric spacetime is compatible with representing an anisotropic fluid sphere [177].

That is, for a generic anisotropic fluid sphere, because there is no longer any geometrical constraint on the Einstein or Ricci tensors, the spacetime geodesy approach is not particularly useful. For some specific anisotropic fluid spheres, such as the Kiselev fluid considered above, one does have some geometrical constraint coming from an eigenvalue degeneracy in the Ricci tensor. However, for a generic anisotropic fluid sphere, free of any such constraint, almost nothing can be said.

6. Conclusions

In this article, we have advocated for the utility of a spacetime geodesy point of view. Delay, at least temporarily, concerns regarding equations of state and/or energy conditions, until one at least has a clear geometrical picture of the spacetime of interest. We have illustrated the discussion with a number of examples, including both perfect fluid spheres and imperfect fluid spheres, and have placed rather remarkably tight constraints on the Weyl tensor in spherical symmetry.

We again emphasize the theory-agnostic nature of the cosmographic/geodesy framework. Cosmography is indifferent to the particular implementation of cosmological inflation one wishes to impose, and is indifferent to specific choices of “modified gravity”. Cosmography provides a general framework for studying FLRW cosmologies without necessitating a specific choice of dynamics. Similarly, spacetime geodesy provides a general framework for studying the general features of localized clumps of gravitating matter without necessitating a specific choice of dynamics.

As regards future plans, perhaps the most plausibly fruitful extension of these current ideas would be to developing a spacetime geodesy for the rotating axisymmetric spacetimes appropriate for modelling rotating stars. Given what we already know regarding rotating spacetimes, it is clear that such a project would be technically challenging, but relatively straightforward, and with high scientific impact. We hope to address such issues in the future.