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Herrera Complexity and Shadows of Spherically Symmetric Compact Objects

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Pith's one-line read The paper argues that the Herrera complexity factor evaluated at a light ring fixes its stability, with negative complexity marking photon spheres and hence black-hole shadows, and that zero-complexity spherically symmetric objects cannot…

desk verdict Useful relation between complexity and light-ring stability, but the zero-complexity 'no shadow' claim needs an NEC qualifier and Eq. (40) has a numerical slip. read the letter →

arxiv 2501.14282 v2 pith:27MXLQEH submitted 2025-01-24 gr-qc

Subhasis Nalui , Subhra Bhattacharya This is my paper · ORCID
classification gr-qc MSC 83C5783C2083C75 PACS 04.20.cv98.80.-k
keywords complexityfactorphotonsphereblackholeshadowHerrerawormholeeffectivepotentialzerosphericallysymmetriccompactobjects
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper connects a theoretical classification of stellar interiors, the Herrera complexity factor, to an observable feature of compact objects: the photon sphere that produces a black-hole shadow. For spherically symmetric black-hole-like metrics, it derives an exact relation $Y_{TF}(r_{ph}) = (r_{ph}^2/2) V''_{eff}(r_{ph})$ at a light ring, so the sign of the complexity factor at that radius decides whether the ring is an unstable photon sphere (negative complexity) or a stable antiphoton sphere (positive complexity). It then studies zero-complexity geometries, including a one-parameter Finch-Skea family, and argues that no external photon sphere can exist there; the simplest objects would cast no shadow. The wider point is that a purely geometric scalar built from the Riemann tensor can serve as a proxy for whether a compact object will look like a black hole to a distant observer.

What carries the argument

The Herrera complexity factor $Y_{TF} = -\kappa \Pi + E$, equivalently $Y_{TF} = \frac{1}{B^2}\left(\frac{A''}{A} - \frac{A'B'}{AB} - \frac{A'C'}{AC}\right)$ for $C=r$, is the scalar from the orthogonal splitting of the Riemann tensor that measures pressure anisotropy and density inhomogeneity. The identity that carries the argument links $Y_{TF}$ to the effective potential $V_{eff} = A^2/C^2$ for null geodesics; at a light ring, where $V'_{eff}=0$, this reduces to $Y_{TF}(r_{ph}) = \frac{r_{ph}^2}{2} V''_{eff}(r_{ph})$ for black-hole-like metrics, making the sign of the complexity factor equivalent to the stability of the photon sphere.

What would settle it

Find a zero-complexity metric of the form (37) with an external light ring where $V''_{eff}<0$ and $\rho+p_r \ge 0$; a direct numerical scan of $V_{eff} = (1+Rr^2)^{2n+2}/r^2$ over $n\ge -1/2$, $R<0$ for a local maximum outside the throat would settle it. Alternatively, reconstruct $Y_{TF}$ at the photon sphere of a known shadow-casting object from its metric and check whether it is negative, as the rule predicts; a shadow with nonnegative complexity at the light ring would falsify the sign correspondence.

Watch

Extended reading notes

Core claim

The central claim is that the complexity factor $Y_{TF}$, obtained from the orthogonal splitting of the Riemann tensor and normally read as a measure of pressure anisotropy and density inhomogeneity, also encodes the stability of null circular orbits. For metrics with $C(r)=r$ and $A(r)=1/B(r)$, the paper shows $Y_{TF} = \frac{r^2}{2}\left(r V''_{eff} + 3 V'_{eff}\right)$, and at a light ring $V'_{eff}=0$ reduces to $Y_{TF}(r_{ph}) = \frac{r_{ph}^2}{2} V''_{eff}(r_{ph})$. Because unstable light rings (photon spheres, sources of shadows) require $V''_{eff}<0$, a negative complexity factor at $r_{ph}$ signals a photon sphere and a positive one signals an antiphoton sphere. The paper extends the analysis to Morris-Thorne wormholes and to the Damour-Solodukhin family, where the throat and exterior light rings are treated separately, and proposes that zero-complexity spacetimes, exemplified by constant-redshift metrics and the family $B^2 = \alpha^2 (A'/r)^2$, cannot support external photon spheres without violating the radial null energy condition.

Load-bearing premise

The no-shadow conclusion for zero-complexity objects depends on assuming the matter satisfies the radial null energy condition $\rho+p_r \ge 0$; if zero-complexity matter is allowed to violate that condition, as it does at a zero-tidal-force wormhole throat, a photon sphere can coexist with zero complexity.

Editorial extensions

If this is right

  • For black-hole-like spacetimes, a negative complexity factor at a light ring implies a photon sphere and hence a shadow, while a positive one implies a stable antiphoton sphere and no shadow.
  • Zero-complexity metrics of the form $ds^2 = -A^2 dt^2 + \alpha^2(A'/r)^2 dr^2 + r^2 d\Omega^2$ cannot be black holes and admit no external photon spheres as long as the matter satisfies $\rho+p_r \ge 0$.
  • The zero-tidal-force Morris-Thorne wormhole is zero-complexity but still has a photon sphere at its throat, so the no-shadow conclusion applies to exterior light rings $r>r_0$, not necessarily to the throat itself.
  • In wormhole throats, an antiphoton sphere implies positive complexity, and negative complexity implies a photon sphere, although a positive complexity can accompany either a photon sphere or an antiphoton sphere depending on the metric functions.
  • For the Damour-Solodukhin family, the paper's table shows negative $Y_{TF}$ at the external photon spheres of black holes and naked singularities with shadows, while the throat photon sphere of the $\lambda>1/2$ wormhole carries positive complexity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the sign rule is general, then the observed shadows of M87* and Sgr A* imply that the effective geometries have negative $Y_{TF}$ at their light rings, so future metric reconstructions from shadow data could be checked against this complexity condition.
  • The no-shadow claim for zero-complexity objects is really 'zero complexity plus $\rho+p_r\ge 0$ implies no external photon sphere'; allowing NEC-violating zero-complexity matter, as at a zero-tidal-force wormhole throat, can bring back a photon sphere.
  • A natural test of the extension to rotation is whether $Y_{TF}$ evaluated on equatorial light rings of rotating metrics still predicts their stability; the paper leaves rotation open, and a failure there would delimit the rule to spherical symmetry.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper investigates the relation between Herrera's complexity factor Y_TF, obtained from the orthogonal splitting of the Riemann tensor, and the existence and stability of photon spheres in static, spherically symmetric spacetimes. The authors express Y_TF in terms of the null effective potential Veff (Eq. (27)), specialize to Schwarzschild-like metrics (Section 5), where they find Y_TF = (r^2/2) V''_eff at a light ring, and conclude that negative Y_TF marks an unstable photon sphere while positive Y_TF marks an antiphoton sphere. For Morris-Thorne wormholes (Section 6) they derive a more involved relation and discuss throat versus external light rings. Section 7 analyzes zero-complexity metrics with B^2 = α^2(A'/r)^2 and claims that such spacetimes cannot have photon spheres. Section 8 provides a generalized Damour-Solodukhin example with a table of photon-sphere radii, complexity values, and shadow radii.

Significance. If the main claims held, the paper would establish a bridge between a purely theoretical classification of compact objects (complexity) and observable shadows, providing a new diagnostic for black hole mimickers. The derivation in Section 5 is clean, self-contained, and correctly reproduces the Schwarzschild limit Y_TF = -1/(9M^2) at r=3M; the sign correspondence (31) is a genuinely useful result. The authors also make a good choice to work directly with the metric components rather than matter variables. However, the generalized formula in Section 8 is algebraically wrong, and the zero-complexity no-shadow conclusion in Section 7 relies on an unstated NEC assumption that the paper's own wormhole example violates. These issues are substantive but local; the clean part of the paper is worth salvaging.

major comments (2)
  1. [Section 8, Eq. (40)] Equation (40) is inconsistent with the paper's own Schwarzschild limit and Table I. Substituting the Schwarzschild parameters q=0, λ=0 and r_ph=3M into Eq. (40) gives -13/(9M^2), not -1/(9M^2) as in Table I and as obtained from Eq. (23). For q=1, λ=0, r_ph=4M, Eq. (40) gives -7/(32M^2), whereas Table I and a direct calculation from Eq. (23) give -1/(32M^2); for the λ=1/4 wormhole at r_ph=12M/5, Eq. (40) gives -1525/(576M^2) versus the table's -25/(576M^2). The formula must be re-derived, and the table and figures recomputed, before the quantitative conclusions can be accepted.
  2. [Section 7] The conclusion 'photon spheres cannot exist for any configuration where Y_TF=0' does not follow from the displayed calculation. The authors show that for the metric family (37) a would-be photon sphere has rho+p_r = V''_eff/(r_ph^2 α^2 (A'(r_ph))^4) < 0, i.e., it requires NEC-violating matter. Since the paper does not impose the NEC on zero-complexity configurations, and since it explicitly discusses zero-tidal-force wormholes (which violate NEC at the throat, Section 6) and dark-energy stars (Section 7), the valid conclusion is only that a photon sphere for this family would require NEC-violating matter. To claim impossibility, the authors must either assume the NEC as a physical-viability criterion for zero-complexity objects or exclude all NEC-violating zero-complexity geometries; neither is stated. The parenthetical exclusion of the throat photon sphere in the zero-tidal-force wormhole further shows that the unqualified statement overreaches.
minor comments (6)
  1. [Section 1, reference [5]] 'Page and Thorn' should be 'Page and Thorne'.
  2. [Throughout] Equation cross-references such as '(1equation.2.1)' and '(12equation.2.11)' appear to be formatting artifacts; they should be cleaned.
  3. [Section 6] The bullet points mix necessary and sufficient conditions; for example, 'if YTF < 0 at the throat then a photon sphere will sufficiently exist at the throat but not necessarily' is internally contradictory and should be rewritten.
  4. [Section 7] The relation 'grr = gtt/r^4' should be written explicitly including the α^2 factor and signs, and its consequence 'this cannot be a black hole' should be justified by a short argument.
  5. [Figures 2 and 3] The captions contain the typo 'Critiical Radius', and Figure 3(c) refers to a naked singularity although the panel corresponds to a wormhole.
  6. [Table I] For the q=-3, M<0 row, the entry 'shadow radius 2M' without a photon sphere is surprising and should be explained, since the paper elsewhere associates shadows with photon spheres.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the complexity-factor/photon-sphere relation is a derived algebraic identity, and the zero-complexity conclusion, while carrying an implicit NEC caveat, is not obtained by fitting or by a self-citation chain.

full rationale

The paper's main derivation is self-contained. Equation (23) expresses the Herrera complexity factor Y_TF directly in terms of the metric functions, and equation (27) is obtained by substituting A^2 = r^2 V_eff into that expression; at a light ring, V'_eff = 0 reduces it to Y_TF(r_ph) = (r_ph^2/2) V''_eff(r_ph) in the Schwarzschild-like case. This is an algebraic consequence of the definitions, not a definition of Y_TF in terms of V''_eff, and no fitted parameter is renamed as a prediction. The wormhole relation (35)-(36) is likewise a direct coordinate transform of the same identity. Citations to Herrera and to the authors' earlier work appear, but they provide background or the original complexity framework; the central stability relation is re-derived from the metric in this paper and does not rely on those citations as load-bearing evidence. The one caveat worth flagging is in Section 7: the claim that a zero-complexity configuration cannot have a photon sphere is valid only if the null energy condition is imposed, because the derivation itself shows that a would-be photon sphere would require rho + p_r < 0. Since the paper elsewhere considers wormholes and dark-energy stars that violate NEC, the unqualified sentence "Clearly photon spheres cannot exist for any configuration where Y_TF = 0" is an overgeneralization. That is a correctness or assumption issue, however, not a circularity: the zero-complexity condition is not defined through the photon-sphere condition, and the argument does not reduce to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard GR, Herrera's complexity factor as an accepted scalar (Eq. 23), and the metric ansatz. The zero-complexity no-photon-sphere conclusion additionally assumes the NEC as a viability condition, which is the main added assumption. No new particles or forces are postulated.

free parameters (3)
  • alpha = arbitrary constant in Eq. (37)
    Integration constant of the zero-complexity family B = alpha * A'/r; not fitted to data, conclusions hold for any value.
  • R = free constant in Finch-Skea ansatz (Eq. 38)
    Scales the radial coordinate; examples use R<0 for gravastars and R>0 for dark energy star, but not fitted.
  • n = real number (examples 1, 1/2, -1/2)
    Exponent in the Finch-Skea ansatz; the paper notes n=1 for Finch-Skea star, n=1/2 anisotropic analogue, n=-1/2 dark energy star.
assumptions (5)
  • standard math Einstein field equations G_alpha_beta = kappa T_alpha_beta with kappa = 8*pi*G
    Used in Section 2 to relate metric to matter (Eqs. 3-5).
  • domain assumption Orthogonal splitting of the Riemann tensor and Herrera's definition of the complexity factor Y_TF (Eqs. 13-23)
    The paper adopts Herrera's complexity factor as the measure of complexity; this is a physical definition from the literature [1,2].
  • domain assumption Static spherically symmetric line element ds^2 = -A^2 dt^2 + B^2 dr^2 + C^2 dOmega^2 with C(r)=r for most applications
    Imposed in Section 2 and used for photon sphere analysis.
  • domain assumption For zero-complexity spacetimes the general solution of Y_TF=0 with C=r is B^2 = alpha^2 (A'/r)^2 (Eq. 37)
    Derived in Section 7 from Eq. (23); the no-photon-sphere conclusion depends on this family.
  • ad hoc to paper Null energy condition rho + p_r >= 0 required for physically viable zero-complexity configurations
    Invoked in Section 7 to rule out photon spheres for zero-complexity metrics; the paper does not defend this as a universal requirement and elsewhere uses NEC-violating wormholes.

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Pith. "Pith review of Herrera Complexity and Shadows of Spherically Symmetric Compact Objects." pith.science (2026). https://pith.science/paper/27MXLQEH

@misc{pith2026250114282,
  author       = {Pith},
  title        = {Pith review of: Herrera Complexity and Shadows of Spherically Symmetric Compact Objects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27MXLQEH}},
  note         = {Machine review of arXiv:2501.14282}
}
read the original abstract

In this work we investigate the effect of complexity factor on the formation of photon spheres for spherically symmetric compact objects. The complexity factor obtained from the orthogonal splitting of the Riemann curvature tensor connects the geometric attributes of a compact spherically symmetric gravitating object with its matter inhomogeneity and pressure anisotropy via a scalar term. The novelty of the complexity factor is the inherent simple definition that identifies the evolution of matter tensors inside a given region of space-time. Such identification helps to obtain an equivalence class of gravitating compact objects based on their degree of complexity with zero complexity identified as the simplest system. On the other hand shadows and photon rings have become essential for identifying compact regions of space time characterised by massive gravity. Advanced observational data analysis tools augments the hope for identification of exotic gravitational objects, like the so called ``black hole mimickers" and may serve as testing ground for other gravity theories. In this context we explore how complexity of compact objects (a fundamentally theoretical classification) is connected to the photon ring (an astrophysical observable in the universe) and its stability. We consider zero complexity systems and discuss its significance with respect to (wrt) formation of photon rings and hence shadows.

Figures

Figures reproduced from arXiv: 2501.14282 by the authors.

Figure 1
Figure 1. FIG. 1: (a) This shows the light rings of the Schwarzschild black hole obtained for [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) This shows the light rings of the naked singularity obtained for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) This shows the light rings of the wormhole obtained for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Reference graph

Works this paper leans on

80 extracted references · 58 canonical work pages

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    Herrera Complexity and Shadows of Spherically Symmetric Compact Objects

    INTRODUCTION Symmetric compact structures in the universe are important source of information, that serve as objects for testing general relativity (GR). It is natural that research on these objects are of paramount importance. The key factor in GR research is the link between theoretical formulations and observational evidence. Several mechanism has been...

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    COMPLEXITY F ACTOR: A BRIEF DESCRIPTION In this section, we will discuss about the complexity of static spherically symmetric geometric structure expressed by the line element: ds2 =−A2dt2 +B2dr2 +C 2(dθ2 +sin2θdΦ2), (1) where the functionsA(r), B(r), andC(r) are functions of the radial coordinater. Further, asr→∞, A(r), B(r)→ 1. The matter tensors are de...

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    THE SCHW ARZSCHILD BLACK HOLE METRIC We consider a Schwarzschild black hole like metric by replacing A(r) = 1 B(r) as: ds2 =−B−2(r)dt2 +B2(r)dr2 +r2(dθ2 +sin2θdΦ2), (29) where B(reh)→∞ corresponding to the black hole event horizon at reh < rph. Here the Veff (r) = 1 r2B2(r) and V′ eff = 0 gives B′ B =− 1 r. Using the above metric (27equation.4.27) reduces...

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    Here the function φ(r) is the redshift function and the function b(r) is called the wormhole shape function

    THE ST A TIC MORRIS-THORNE TRA VERSABLE WORMHOLE We now consider the static Morris-Thorne Traversable Wormhole (MTTW) space time, where A2(r) = e2φ(r) and B2(r) = ( 1− b(r) r )−1 . Here the function φ(r) is the redshift function and the function b(r) is called the wormhole shape function. This is because the shape of a wormhole is synonymous to its throat...

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    THE CLASS OF ZERO COMPLEXITY SP ACE-TIME We consider all those generic class of space-times for which complexity vanishes. Zero complexity space times are considered as simple space times, which are equivalent to the homogeneous and isotropic FRW space. We have already seen that the zero-tidal force class of the MTTW is a zero-complexity space-time. In fa...

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    Forn = 1 2, the anisotropic analogue of the Finch-Skea star can be obtained [47] while for n =− 1 2 a dark energy star is obtained. The matter tensors for n =− 1 2 become homogeneous and isotropic with pr = pt =−ρ = 3R. Thus the resulting object is supported by dark energy ene...

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    (b) This shows the light rings of the black hole obtained for M = 0.5, q = 1.5. The corresponding photon sphere radius is rph = 2.25, while event horizon occurs at reh = 1. The shadow radius is 4.691. (c) This shows the evolution of YTF corresponding to the Schwarzschild black...

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