Anisotropic Compact Star Model

• Anisotropic compact star models are characterized by distinct radial and tangential pressures, captured using general relativistic field equations and 5D embedding techniques.
• The framework uses a Schwarzschild-like metric and a Lake-type temporal potential to derive exact solutions that align with observed properties of neutron stars, strange stars, and white dwarfs.
• Pressure anisotropy, defined as the difference between tangential and radial pressures, enhances stellar stability and allows higher compactness compared to isotropic models.

An anisotropic compact star model characterizes stellar interiors in which radial and tangential pressures differ, reflecting intrinsic microphysical or macroscopic stress anisotropy often present at supranuclear densities. Such models extend beyond idealized isotropic treatments to capture physical effects essential for describing observed properties of neutron stars, strange stars, and white dwarfs at high compactness. The most comprehensive class of these models couples general relativistic field equations with sophisticated metric and embedding conditions, enabling both exact closure and physical viability across a broad parameter space (Maurya et al., 2017).

1. Metric Structure and Embedding Approach

A central aspect of the generalized anisotropic compact star model is its use of the Schwarzschild-like spherically symmetric metric: $$ds^2 = -e^{\nu(r)}\, dt^2 + e^{\lambda(r)}\, dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2)$$ The closure of the system employs a five-dimensional embedding, serving as an alternative to the Karmarkar class-1 constraint. Specifically, mapping four-dimensional spacetime into a 5D flat metric yields: $$e^{\lambda(r)} = 1 + \frac{K\, e^{\nu(r)}}{4} [\nu'(r)]^2$$ where $K$ is a constant related to the embedding. The monotonic temporal potential is chosen following Lake: $$e^{\nu(r)} = B\, (1 - A\, r^2)^n, \quad A > 0,\, n \le -3,\, B > 0$$ with the radial potential derived from the embedding condition: $$e^{\lambda(r)} = 1 + D\, A\, r^2\, (1 - A\, r^2)^{n-2}\,,\qquad D = n^2 A B K$$ This ansatz provides a flexible framework capable of generating both standard and ultra-dense star models (Maurya et al., 2017).

2. Einstein Field Equations and Matter Constituents

The matter distribution is modeled as an anisotropic fluid with energy-momentum tensor: $$T^{\mu}{}_{\nu} = \mathrm{diag}(-\rho, p_r, p_t, p_t)$$ The Einstein field equations, in geometrized units ($G=c=1$), read: $$8\pi\rho = \frac{1 - e^{-\lambda}}{r^2} + e^{-\lambda} \frac{\lambda'}{r}$$

$$8\pi p_r = \frac{e^{-\lambda} - 1}{r^2} + e^{-\lambda} \frac{\nu'}{r}$$

$$8\pi p_t = e^{-\lambda} \left[\frac{\nu''}{2} + \frac{{\nu'}^2}{4} - \frac{\nu' \lambda'}{4} + \frac{\nu' - \lambda'}{2r}\right]$$

Substituting metric functions yields explicit closed forms for $\rho(r),\ p_r(r),\ p_t(r)$, and confirms that solutions for central objects such as LMC X-4 attain densities and pressures fully commensurate with strange star expectations (central $\rho\sim8.4\times10^{14}$ g/cm³, $p_c\sim10^{35}$ dyn/cm²) (Maurya et al., 2017).

3. Pressure Anisotropy: Definition and Physical Role

Anisotropy is quantified by: $\Delta(r) = p_t(r) - p_r(r)$ The analytic form derived from the exact solution: $$8\pi\Delta(r) = \frac{A^2 r^2}{\left[(1 - A r^2)^2 + A D r^2 (1 - A r^2)^n\right]^2} \left[D(1 - A r^2)^n + (n-2)(1 - A r^2)\right] \left[D(1 - A r^2)^n + n(1 - A r^2)\right]$$ Key properties:

• $\Delta(0) = 0$: anisotropy vanishes centrally.
• $\Delta(r)$ increases outward, peaking at the surface ($r=R$).
• Outward-directed anisotropy ($p_t > p_r$) supports higher mass for a given radius and enhances stability, in line with theoretical predictions for dense nuclear matter.

4. Boundary Matching and Equation of State Parameters

Physical acceptability requires matching the interior solution at $r = R$ to the exterior Schwarzschild geometry: $e^{\nu(R)} = e^{-\lambda(R)} = 1 - 2M/R$ with the condition $p_r(R) = 0$ to ensure mechanical equilibrium at the surface. These conditions fix constants: $D = \frac{2n (A R^2 - 1)}{(1 - A R^2)^n}$

$$B = \frac{1}{(1 - A R^2)^n [1 + D A R^2 (1 - A R^2)^{n-2}]}$$

The compactness ratio emerges as: $$\frac{M}{R} = \frac{D A R^2 (1 - A R^2)^{n-2} }{2 [1 + D A R^2 (1 - A R^2)^{n-2}]}$$ Effective equation of state (EOS) parameters: $$\omega_r(r) = \frac{p_r(r)}{\rho(r)}, \quad \omega_t(r) = \frac{p_t(r)}{\rho(r)}$$ are shown to satisfy $0 < \omega_r,\,\omega_t < 1$ throughout the star, indicating physically reasonable causal sound speeds (Maurya et al., 2017).

5. Energy Conditions and Stability Criteria

The model passes standard energy conditions for all physically relevant parameters:

• Null Energy Condition (NEC): $\rho \geq 0$
• Weak Energy Condition (WEC): $\rho + p_r \geq 0,\, \rho + p_t \geq 0$
• Strong Energy Condition (SEC): $\rho + p_r + 2p_t \geq 0$

Stability is analyzed via:

• The generalized Tolman-Oppenheimer-Volkoff (TOV) equation:

$F_g + F_h + F_a = 0$

where contributions $F_g$ (gravitational), $F_h$ (hydrostatic), $F_a$ (anisotropic) balance numerically.

• Causality: $0 \le v_r^2, v_t^2 \le 1$, no cracking ($|v_t^2 - v_r^2| \le 1$).
• Adiabatic indices:

$$\Gamma_r = \frac{\rho + p_r}{p_r} \cdot v_r^2\,;\quad \Gamma_t = \frac{\rho + p_t}{p_t} \cdot v_t^2$$

Both exceed $4/3$, ensuring stability against radial perturbations.

• Buchdahl compactness bound: $u = 2M/R < 8/9$ is satisfied for all parametric choices.

6. Parametric Families and Astrophysical Applications

Model flexibility is governed by the parameter $n$:

• $n \approx -3$ yields white dwarf models.
• $n \approx -20$ to $-10^4$ gives ultra-dense compact and strange star regimes.
• As $n \to -\infty$, the metric approaches a Gaussian-type potential ($e^\nu \sim e^{Cr^2}$), providing a natural link to stiff equations of state.

Specific fits to LMC X-4 ($M \approx 1.29 M_\odot,\, R = 9.48\,\mathrm{km}$) utilize values $n = -23, -40, -10^2, -10^3, -10^4$ to match observed central density ($\rho_c \sim 8.4 \times 10^{14}$ g/cm³) and confirm applicability to both white dwarfs and strange stars.

Key physical insights:

• The metric ansatz with tunable $A,\,n$ and embedding constant $K$ generates a wide class of anisotropic stellar models.
• Surface-peaked anisotropy enhances stability, enabling higher compactness than isotropic models.
• EOS is emergent, not imposed, with effective parameters derived from the solution structure.

Main Physical Limitations and Research Directions

• The model is phenomenological: microphysical sources of anisotropy (superfluidity, strong magnetic fields) are not incorporated.
• Parametric n must be adjusted to fit individual observed stars or classes.
• Confrontation with high-precision mass-radius data and nuclear/strange-matter equations of state is pending.
• Rotational and magnetic-field stability have yet to be addressed in the present analytic framework (Maurya et al., 2017).

Summary Table: Key Model Outputs for LMC X-4 (n = −23, −40, −10², −10³, −10⁴)

n-value	A	K	D	ρ_c (g/cm³)	p_c (dyne/cm²)
–23	0.00576	...	...	8.4e14	1.0e35
–40	0.00567	...	...	8.4e14	1.0e35
–10²	0.00565	...	...	8.4e14	1.0e35
–10³	0.00565	...	...	8.4e14	1.0e35
–10⁴	0.00565	...	...	8.4e14	1.0e35

Constants K and D are found explicitly for each n in Table 2 of (Maurya et al., 2017).

7. Comparative Context and Model Extensions

Subsequent studies employing Karmarkar embedding, metric ansätze (Lake, Durgapal-Fuloria, Tolman IV, Finch-Skea), or quadratic EOS reinforce the central features of this model—regular central behavior, controlled anisotropy, and robust satisfaction of all physical and stability criteria (Bhar et al., 2016, Bhar et al., 2015, Maurya et al., 2015, Ratanpal et al., 22 Aug 2025). In the context of alternative gravity theories, e.g., $f(T)$ and $\mathcal{F}(\mathcal{Q})$ gravity, similar techniques and physical checks confirm the generality and viability of embedding-based anisotropic compact star modeling (Momeni et al., 2016, Paul et al., 20 May 2025).

In sum, the anisotropic compact star model anchored in embedding-class metric closure and Lake-type monotone potentials offers a rigorous, highly parametric, and physically viable interior solution suitable for the entire class of compact stellar objects. Its primary strengths are analytic tractability, broad applicability, and explicit exhibition of the stabilizing role of tangential pressure anisotropy in ultra-dense regimes.