Orthogonal splitting of the Riemann curvature tensor and its implications in modeling compact stellar structures

This paper asks how to quantify “complexity” inside compact stellar objects when gravity is modeled not by standard general relativity (GR) but by the extended theory of gravity known as $f(R,L_m,T)$, where the Lagrangian depends on the Ricci scalar $R$, the matter Lagrangian $L_m$, and the trace of the energy-momentum tensor $T$. The motivation matters because compact stars (neutron stars, quark stars, charged anisotropic configurations) are extreme laboratories for both strong-field gravity and dense-matter physics. In GR, Herrera’s complexity framework links complexity to the coexistence of energy-density inhomogeneity and pressure anisotropy. The authors’ central contribution is to transplant this idea into $f(R,L_m,T)$ gravity by using the orthogonal splitting of the Riemann curvature tensor, which produces “structure scalars.” In this construction, the trace-free scalar $Y_{TF}$ (their notation $Y_{TF}$ and $Y_{TF}^F$) becomes the complexity factor (CF) because it encodes anisotropy and inhomogeneity; in $f(R,L_m,T)$ gravity, additional curvature–matter coupling terms act like “dark source terms” that can modify whether complexity vanishes.

Methodologically, the paper is a theoretical derivation rather than an empirical study. The authors assume a static, spherically symmetric interior spacetime with anisotropic matter and an electromagnetic field. The metric is written as $$d{s}^2=e^{\xi (r)}d{t}^2-e^{\chi (r)}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2).$$ They start from the action $$S=\int \sqrt{-g}\left[\frac{f(R,L_m,T)}{16\pi }+L_E+L_m\right]d^{4}x,$$ where $L_E$ is the electromagnetic Lagrangian. Varying the action yields modified field equations of the form $$G_{\mu \nu }=8\pi \left(T_{\mu \nu }^{(\mathrm{eff})}+E_{\mu \nu }\right),$$ with effective matter terms split into the “standard” matter contribution and correction terms $T_{\mu \nu }^{(cr)}$ that depend on derivatives of $f$ with respect to $R$, $L_m$, and $T$. The electromagnetic sector is treated via the Maxwell tensor and charge function $q(r)$ obtained from the charge density.

The authors derive the modified Einstein equations’ non-vanishing components (their Eqs. (14)–(16)) for energy density $\rho$, radial pressure $P_r$, tangential pressure $P_{\perp }$, charge $q(r)$, and correction terms $T_{\mu \nu }^{(cr)}$. They then derive a generalized hydrostatic equilibrium condition (a modified TOV equation) that includes the non-conservation of the matter stress tensor typical of $f(R,L_m,T)$ theories. Junction conditions are imposed using Darmois matching at the stellar surface $r=r_{\Sigma }$, matching the interior to an exterior Reissner–Nordström spacetime. This yields relations among interior metric functions and the exterior mass $M$ and charge $Q$, including continuity constraints and a surface condition on $P_r$.

The complexity framework is built by orthogonally splitting the Riemann tensor relative to the fluid four-velocity. Using tensors constructed from the Riemann tensor and its dual (their $Y_{\omega }$, $Z_{\omega }$, $X_{\omega }$), the authors express the trace-free scalar $Y_{TF}$ in terms of anisotropy $\Pi$, charge contributions, and an additional term involving the conformal/Weyl scalar $\mathcal{E}$ (their $\mathcal{E}$ and $\mathcal{E}$-dependent scalar $\in$). In their charged $f(R,L_m,T)$ setting, they obtain an explicit expression for the complexity factor (their $Y_{TF}^F$, Eq. (64)) that includes: 1) a term proportional to $\Pi$ (anisotropy), 2) a term proportional to $q^2/r^4$ (electromagnetic contribution), 3) integrals of the radial derivative of $\rho$ plus correction terms (inhomogeneity plus $f$-gravity effects), and 4) additional integral terms involving $q(r)$ and $q^{\prime }(r)$.

A key theoretical identity is that the sum of two trace-free scalars satisfies $$X_{TF}+Y_{TF}=\frac{16\pi }{3}\Pi -\frac{2q^2}{r^4}+L_{\omega },$$ where $L_{\omega }$ is a contribution from the correction terms. The authors further relate $Y_{TF}$ to the Tolman mass $m_T$ (their inertial mass) and show that the vanishing of a particular integral combination implies that $Y_{TF}$ governs how anisotropy and inhomogeneity (including dark source terms) affect $m_T$.

For the “minimal model” of $f(R,L_m,T)$, they adopt $$f(R,L_m,T)=R+2c_1{L}_m+c_{2}T,$$ with constants $c_1$ and $c_2$. They then illustrate the radial behavior of $X_{TF}$ and $Y_{TF}$ using a specific compact star, 4U 1820–30, quoting observationally motivated values $M=1.58\pm 0.06M_{\odot }$ and $R=9.1\pm 0.4\mathrm{km}$. The paper’s qualitative numerical/graphical result (Figure 1) is that $Y_{TF}$ starts at zero at the center, reaches a maximum at intermediate radius, and returns to (approximately) zero at the stellar surface. They interpret this as “minimum complexity” at both the center and the surface, with maximum complexity in between. They emphasize that this behavior contrasts with GR and with other modified gravity models cited in their references, where CF does not vanish at the surface.

They then analyze conditions for vanishing complexity. In their framework, imposing CF disappearance yields a constraint (their Eq. (70)) that can be satisfied either by (i) uniform energy density and isotropic pressure, or (ii) a cancellation between non-uniform energy density and anisotropic pressure, but now modified by the $f(R,L_m,T)$ correction terms. They interpret this as a non-local equation of state (EoS) in $f(R,L_m,T)$ gravity.

To produce explicit classes of solutions with vanishing complexity, they consider two fluid-model strategies. First, they use the Gokhroo–Mehra ansatz for the energy density profile, $$\rho =\left(1-\frac{Y{r}^2}{r_{\Sigma }^2}\right){\rho }_0,$$ and derive corresponding metric and pressure relations consistent with the modified field equations and matching conditions. Second, they impose a polytropic EoS for the radial pressure, $P_r=K{\rho }^{\tau }$ with $\tau =1+\frac{1}{n}$, and reformulate the system in dimensionless variables to solve the modified TOV and mass equations under the vanishing-complexity constraint. They discuss physically motivated parameter regimes (e.g., values of $K$ corresponding to matter/radiation/stiff-fluid-like behavior and phantom/quintessence-like eras) and note that neutron-star-relevant polytropic indices include $n=0.5$ and $n=1$.

Limitations are not presented as a formal “limitations” section, but they are apparent from the methodology: the work is analytic and semi-numerical, relying on assumed symmetry (static, spherical), assumed matter content (anisotropic fluid plus electromagnetic field), and a specific minimal $f(R,L_m,T)$ model. The “complexity factor” behavior is demonstrated via plots rather than a systematic parameter scan with quantitative error bars, and the paper does not provide explicit p-values or statistical confidence intervals (appropriate for a theoretical derivation). Another limitation is that the observational comparison is largely qualitative: the star parameters are quoted, but the paper does not show how the internal anisotropy profile and correction-term functions are uniquely constrained by data.

Practically, the results matter for researchers modeling compact stars in alternative gravity. The authors’ main implication is that in $f(R,L_m,T)$ gravity, the complexity factor $Y_{TF}$ can vanish even when density inhomogeneity and anisotropy would produce nonzero complexity in GR, because dark source terms can enable cancellations. This means that using complexity as a diagnostic of internal structure may require accounting for modified-gravity couplings. Who should care: (1) relativists studying structure scalars and curvature decomposition, (2) astrophysicists building interior models for neutron stars and charged compact objects, and (3) gravitational-wave modelers, since internal structure affects tidal deformability and thus waveform signatures. The paper also suggests that future multi-messenger constraints could test whether the predicted complexity behavior (e.g., CF vanishing at the surface) is realized.

Overall, the core contribution is the derivation of a charged, anisotropic, $f(R,L_m,T)$-modified complexity factor from orthogonal splitting of the Riemann tensor, its relation to Tolman mass, and the identification of conditions and example models where the complexity factor vanishes—highlighting how modified gravity’s correction terms can qualitatively change the complexity diagnostics compared with GR.