Vaidya–Tikekar Metric in Relativistic Astrophysics

• The Vaidya–Tikekar metric is a geometric ansatz that models spheroidal, non-uniform stellar interiors, adjusting curvature deviation via the parameter K.
• It provides analytic solutions to the Einstein field equations, yielding explicit density, pressure, and mass functions for compact star models.
• Applications span charged, anisotropic, and higher-dimensional configurations, enabling alignment of theoretical predictions with observed compact star data.

The Vaidya–Tikekar (VT) metric is a geometric ansatz developed for constructing analytic models of static, spherically symmetric relativistic stars, particularly those with interior matter distributions deviating from uniform density. The key structural feature of the VT metric is its prescription that the constant-time hypersurfaces of the interior metric possess spheroidal—not spherical—geometry, governed by a dimensionless parameter measuring the deviation from constant curvature. Since its introduction, the VT metric and its generalizations have played a central role in relativistic astrophysics, compact star modeling, gravitational theory extensions, and studies of charged, anisotropic, and higher-dimensional configurations.

1. Fundamental Formulation and Geometric Character

The canonical VT metric is constructed upon the static, spherically symmetric line element: $$ds^2 = -e^{\nu(r)}dt^2 + e^{\lambda(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2).$$ The VT prescription modifies the “radial” potential $e^{\lambda(r)}$—often denoted by $g_{rr}$ in Schwarzschild coordinates—so that on $t=$ const slices the spatial three-geometry is a spheroid embedded in (typically) four-dimensional Euclidean space. This is achieved with: $e^{\lambda(r)} = \frac{1-K\,r^2/R^2}{1-r^2/R^2}$ where $R$ is a geometric length scale and $K$ is the spheroidal (curvature-deviation) parameter. For $K=0$, the three-space is a perfect 3-sphere and the metric reduces to the constant-density Schwarzschild interior. For $K\neq0$, the constant-time space becomes prolate ($0$K<0$) spheroidal.

Alternative but equivalent forms appear throughout the literature using different parametrizations; e.g., with $e^{\psi(r)} = (b^2 - U r^2)/(b^2 - r^2)$ and $e^{\chi(r)} = [C+ D\sqrt{(1-U)(b^2 - r^2)}\,]^2$, with $U$ and $b$ related to $K$ and $R$ via embedding geometry (Awais et al., 20 Mar 2025).

The VT curvature parameter controls the departure from spherical symmetry:

• $K=0$: Spherical ($S^3$)
• $K>0$: Prolate spheroid
• $K<0$: Oblate spheroid

The choice of $e^{\lambda(r)}$ thereby geometrically encodes a one-parameter family of non-uniform density stellar interiors, ensuring analytic tractability of the resulting Einstein or modified gravity field equations (Sharma et al., 2020, Sharma et al., 2020, Khugaev et al., 2016).

2. Physical Interpretation and Mass Function

The VT ansatz generates a density profile which is generally non-uniform and monotonic outward: $$\rho(r) = \frac{(1-K)}{8\pi R^2}\frac{3 - K\,r^2/R^2}{(1-K\,r^2/R^2)^2}$$ in the uncharged, isotropic case. Monotonic decrease of density outward requires $K < 1$. The central density is increased by increasing the spheroidicity parameter for fixed $R$. Pressure (radial and, in anisotropic cases, transverse) follows analogously.

The interior mass function is: $m(r) = \frac{(1-K) r^3}{2(R^2 - K\,r^2)}$ yielding at the surface $r=R$ the total mass and a closed form expression for compactness: $u = \frac{M}{R} = \frac{(1-K)R^2}{2(R^2-K\,R^2)}$ This recovers the well-known Buchdahl limit $u < 4/9$ in the $K=0$ (Schwarzschild interior) case (Sharma et al., 2020, Khugaev et al., 2016).

VT stars satisfy mass-radius relations which can be tuned via $K$ to fit observational data for compact objects, including neutron stars and strange stars (Sharma et al., 2020, Sharma et al., 2020). Notably, negative $K$ (oblate spheroidal) is required in most realistic models given observational compactness and monotonic density.

3. Einstein Field Equations, Anisotropy, and Exact Solutions

Substitution of the VT ansatz into the Einstein field equations yields a closed system for the metric coefficients and matter variables. For isotropic matter, the solution for $e^{\nu(r)}$ reduces via an integrating factor to expressions involving algebraic or hypergeometric functions, depending on assumptions on the equation of state.

The pressure isotropy condition (equality of radial and tangential pressures) reduces to a Gauss-type equation, often presented in terms of a new variable $u^2 = \frac{K}{K+1}(1 - r^2/R^2)$. This ODE admits solutions in terms of hypergeometric, Gegenbauer, or associated Legendre functions. When anisotropy (difference between radial and tangential pressure) is introduced, the master equation retains hypergeometric solvability for suitable choices of the anisotropy profile: $\Delta(r) = a\,\frac{K\,r^2}{(R^2-K\,r^2)^2}$ with $a$ an anisotropy-strength parameter. Elementary function solutions exist for discrete series in $K, a$ (Thirukkanesh et al., 2020).

For charged configurations, the Einstein-Maxwell system generalizes the VT expressions. The solution still retains analytic form, with the electric field profile tied algebraically to the curvature parameter so as to maintain spherical symmetry in the charge distribution and preserve the reduction to the Schwarzschild interior in the neutral limit (Sharma et al., 2020, Kumar et al., 2017).

In $f(Q)$ gravity, the VT ansatz inserted into the relevant field equations with $f(Q)=\beta + jQ$ yields analytic closed-form expressions for $\rho$, $P_r$, and $P_t$, with the VT parameters matched to the external Schwarzschild metric at the boundary (Awais et al., 20 Mar 2025). All standard regularity, causality, and stability conditions can be enforced explicitly.

4. Boundary Conditions, Matching, and Physical Acceptability

A key strength of the VT approach is the direct solvability of the matching problem at the star's surface $r=R$. The internal VT solution is matched to the external Schwarzschild (or Reissner–Nordström if charged) metric by enforcing:

• Continuity of $g_{tt}$ and $g_{rr}$ (i.e., $e^{\nu(R)}$, $e^{\lambda(R)}$)
• Vanishing of the radial pressure $p_r(R)=0$
• Equality of total mass $m(R)=M$ and total charge $q(R)=Q$ (where applicable)

This sets algebraic relations between the interior constants ($K$, $R$, etc.) and the physically measurable parameters ($M$, $R$, $Q$), ensuring that physical acceptability criteria (regularity, positivity of density and pressure, causality, energy conditions, TOV equilibrium, Buchdahl limit, and surface redshift bounds) can be satisfied throughout the object (Awais et al., 20 Mar 2025, Kumar et al., 2021, Chattopadhyay et al., 2012).

Explicit criteria include:

• Central regularity and positivity: $e^{\lambda(0)}=1$, $e^{\nu(0)}>0$, $\rho(0)>0$, $p(0)>0$
• Monotonicity: $d\rho/dr<0$, $dp/dr<0$
• Causality: $0 < dp/d\rho < 1$
• Energy conditions: WEC, SEC, DEC hold throughout interior
• Stability: adiabatic index $\Gamma > 4/3$, $dM/d\rho_c>0$, fulfillment of cracking criteria

5. Extensions: Electromagnetic, Anisotropic, and Modified Gravity Models

The VT metric serves as the geometric backbone for a wide range of physically enriched models:

• Electromagnetic extension: Inclusion of charge is achieved by coupling the VT metric to the Einstein–Maxwell equations. The charge function $q(r)$ is constructed such that the metric reduces to the neutral case when $k\to0$. The resulting models permit analytic expressions for the charged Buchdahl limit, showing, for example, that the maximal compactness can exceed $4/9$ for charged spheres, reaching up to $8/9$ for certain charge-to-mass ratios (Sharma et al., 2020, Kumar et al., 2017, Chattopadhyay et al., 2012).
• Anisotropic matter: The VT ansatz accommodates anisotropy via an explicit anisotropy function in the pressure difference, yielding master ODEs solvable in terms of hypergeometric functions. As anisotropy increases, the mass–radius relation, equation of state stiffness, and surface redshift exhibit significant changes (Thirukkanesh et al., 2020).
• Modified gravity: In $f(Q)$ gravity—a symmetric teleparallel formulation—the VT potentials lead to closed-form expressions for energy density and pressures given $f(Q)=\beta + jQ$, and all relevant physical and stability criteria can be explicitly checked (Awais et al., 20 Mar 2025).
• Conformal symmetry and exotic fields: When combined with conformal motion or the presence of a quintessence field, the VT metric gives rise to models admitting conformal Killing vectors, with effective density and pressure including the exotic component. Such models can satisfy all standard energy, regularity, and stability criteria (Bhar, 2014).

6. Generalizations: Higher Dimensions and Pure Lovelock Gravity

The analytic structure induced by the VT ansatz in the pressure isotropy equation admits direct extension to higher dimensions. Specifically, for static, spherically symmetric fluid stars in $n$-dimensional spacetime, the pressure isotropy equation preserves the Gauss form, provided the curvature parameter is redefined by

$K_n = \frac{K_4 - n + 4}{n-3}$

where $K_4$ is the 4D VT parameter. Physical density profiles require $K_n \ge 0$, yielding a maximum allowed dimension $n=K_4 + 4$; equality marks the constant-density limit (Schwarzschild interior solution in $n$D) (Khugaev et al., 2016).

In pure Lovelock gravity of order $N$, the same universality emerges: VT-type solutions yield the same hypergeometric structure for the pressure isotropy equation in all $n\geq 2N+2$, with appropriate rescaling of the curvature parameter to maintain physical acceptability (Molina et al., 2016).

7. Astrophysical Implications and Observational Applications

The VT metric is widely used for phenomenological modeling of compact stars (neutron stars, quark stars, and superdense objects). Tuning $K$ allows matching observed mass–radius relations, total charge, surface redshifts, and other observables for individual sources, such as the LMC X-4 pulsar (Awais et al., 20 Mar 2025), RX J1856–37, SAX J1808.4, Her X-1 (Sharma et al., 2020), and PSR B0943+10 (Kumar et al., 2021).

In realistic stellar modeling, the spheroidicity parameter $K$ (or equivalents, such as $k$, $U$, $a$) is typically negative (oblate) and less than unity in value, reflecting both the deviation from uniform density and the observed compactness of astrophysical objects. The method provides a unified and tractable framework for analyzing regularity, matching, energy conditions, stability, causality, and the effect of charge and anisotropy on macroscopic observables—including explicit dependence on parameters relevant to electromagnetic and modified gravity theories.

A recurring implication is that the VT approach provides an analytic, geometry-based algorithm for generating entire families of stellar solutions, interpolating between the most compact (Schwarzschild—$K=0$) and least compact (Finch–Skea—$K\to\infty$) stars within a single parametrization (Molina et al., 2016). This universality extends to both general relativity and alternative metric theories, highlighting the centrality and adaptability of the Vaidya–Tikekar construction in gravitational physics.