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Exploring density dependent B as a suitable parameter in higher dimensional approach with a non-linear equation of state

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Pith's one-line read The paper constructs singularity-free higher-dimensional interior solutions for anisotropic compact stars from the spheroidal Vaidya–Tikekar ansatz, derives a causality bound on the spheroidal parameter, and fits PSR J1614-2230 with a…

desk verdict Useful causality bound and an apparently coherent exact solution, but the mass-radius validation is computed from a different (isotropic) model and the bag parameter is a fitted recasting, so the pulsar-mimicry claims don't stand. read the letter →

arxiv 2508.20634 v2 pith:MLJY5IC3 submitted 2025-08-28 gr-qc

Koushik Ballav Goswami , Debadri Bhattacharjee , Anirban Saha , Pradip Kumar Chattopadhyay This is my paper · ORCID
classification gr-qc MSC 83C1583C5583E1585A15 PACS 04.40.Dg04.50.-h97.60.Jd12.39.Ba
keywords higherdimensionspressureanisotropystrangequarkstarMITbagmodeldensitydependentparameterVaidya-Tikekaransatzcompactstarscausalitycondition
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

The paper tries to show that a specific geometric ansatz for stellar interiors, the Vaidya–Tikekar spheroid extended to $D\ge4$ spacetime dimensions with anisotropic pressures, produces regular, causal compact-star solutions. It derives a lower bound on the spheroidal parameter $\lambda$ from the causality of the radial sound speed, and shows the bound depends on dimension $D$ and anisotropy $\alpha$, reducing to the known $\lambda>3/17$ in four isotropic dimensions. Fitting the model to PSR J1614-2230, it finds that a fifth-degree polynomial radial equation of state $p_r(\rho)$ beats a linear one; this is reinterpreted as a density-dependent MIT bag parameter $B(\rho)$. The resulting mass-radius curves cover several observed pulsars and pass standard stability tests. A sympathetic reader would care because this connects a purely geometric ansatz to measurable pulsar properties and quark-matter phenomenology, while quantifying how extra dimensions and anisotropy are forced on each other by causality.

What carries the argument

The load-bearing object is the Vaidya–Tikekar ansatz for the $g_{rr}$ metric potential, $e^{2\mu}=(1+\lambda r^2/R^2)/(1-r^2/R^2)$, which makes each $t=\text{constant}$ slice a $(D-1)$-dimensional spheroid; $\lambda$ is the spheroidal parameter and $R$ sets the curvature scale. With the chosen anisotropy $\Delta=\alpha\lambda^2(1-x^2)(n-1)/[8\pi G_D R^2(1+\lambda(1-x^2))^2]$, the field equations reduce under $x^2=1-r^2/R^2$ and $z=\sqrt{\lambda/(\lambda+1)}\,x$ to a Legendre-type equation $(1-z^2)\psi_{zz}+z\psi_z+(n-1)(1+\lambda(1-\alpha))\psi=0$, whose closed-form solution $\psi$ supplies $\rho$, $p_r$, and $p_t$. This reduction is what turns the problem into a solvable linear equation, and the same expressions yield the sound-speed ratio $v_r^2=dp_r/d\rho$ that produces the parameter bound.

What would settle it

Integrate the full $D$-dimensional anisotropic TOV equations using the paper's radial equation of state $p_r(\rho)$ together with the explicit anisotropy $\Delta(r)$ and check whether the fitted point $M=1.908\,M_\odot$, $R=11.93$ km for PSR J1614-2230 lies on the resulting mass-radius curve; the paper's own Table 3 peaks at $1.77\,M_\odot$ in $D=4$, so any mismatch shows the fitted object is not a solution of the model.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is a closed-form family of exact interior solutions of the Einstein field equations for anisotropic matter in $D\ge4$, built from the Vaidya–Tikekar ansatz $e^{2\mu}=(1+\lambda r^2/R^2)/(1-r^2/R^2)$ and a chosen anisotropy profile $\Delta=p_t-p_r$. The metric potential $\psi=e^\nu$ is obtained as a closed trigonometric expression after transforming to $z=\sqrt{\lambda/(\lambda+1)}\,\sqrt{1-r^2/R^2}$, and the solution matches a higher-dimensional Schwarzschild exterior at the boundary where $p_r=0$. The causality condition $0<v_r^2<1$ forces $\lambda$ above a dimension- and anisotropy-dependent bound. The paper then fits the model to PSR J1614-2230, finds that a fifth-degree polynomial $p_r(\rho)$ is the best radial equation of state, converts this into a density-dependent bag parameter through the MIT-bag-style relation $p_r=(\rho-4B)/3$, and reports that the resulting mass-radius curves and energy-per-baryon stability windows reproduce a range of known pulsars and satisfy the generalized TOV, Herrera cracking, and adiabatic-index conditions.

Load-bearing premise

The load-bearing assumption is that the spheroidal Vaidya–Tikekar geometry with the chosen anisotropy function is an exact description of the stellar interior, and that the radial equation of state taken from that geometry is sufficient to compute the mass-radius relation.

Editorial extensions

If this is right

  • If the central claim holds, the four-dimensional isotropic limit $\lambda>3/17$ is only the first member of a family: in $D=5,\dots,8$ causality requires increasingly large $\lambda$, and for $D>8$ a nonnegative bound demands substantial anisotropy $\alpha$.
  • The observed mass and radius of PSR J1614-2230 can be reproduced by this model in $D=4,5,6,7$ only with a nonlinear fifth-degree radial equation of state; a constant-bag MIT linear equation of state is excluded, so density-dependent $B(\rho)$ becomes the natural quark-matter description.
  • The model's mass-radius relation spans the measured bands of several compact objects and respects the $D$-dimensional Buchdahl bound at least up to $D=11$.
  • Strange-quark matter in this model is absolutely stable in $D=4$, while in $D=5,6,7$ stability requires exceeding threshold anisotropy values $\alpha_{\rm crit}\approx0.16,0.61,0.97$, respectively.
  • The model passes the generalized TOV force-balance equation, the Herrera cracking condition, and the adiabatic-index criterion, so within the assumed geometry the configurations are dynamically stable.

Reading between the lines

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

  • A direct way to stress-test the model would be to integrate the full anisotropic TOV equations with the explicit $\Delta(r)$, rather than using only the radial equation of state $p_r(\rho)$ for the mass-radius curves; the paper's Table 3 gives $M_{\max}=1.77\,M_\odot$ in $D=4$, while the fitted pulsar has $1.908\,M_\odot$, so the two could disagree.
  • The construction effectively inverts observed mass-radius data into a density-dependent bag function $B(\rho)$; applied to other pulsars, the same pipeline would produce a family of bag functions whose mutual consistency could be checked against nuclear-physics constraints.
  • The causality bound acts as a selection rule for higher-dimensional stars: if a compact object were ever observed whose inferred mass-radius point requires $D>4$, the model predicts it must be strongly anisotropic, a testable prediction for future gravitational-wave or X-ray measurements.
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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

3 major / 4 minor

Summary. The manuscript constructs an exact static, spherically symmetric interior solution of the Einstein field equations in D>=4 spacetime dimensions with anisotropic pressures, using the Vaidya-Tikekar ansatz for the g_rr metric function and a specific form of pressure anisotropy. It derives a causality bound on the spheroidal parameter lambda, shows that the bound reduces to the known lambda>3/17 for D=4, alpha=0, fits the model to the observed mass and radius of PSR J1614-2230 with a fifth-order polynomial equation of state, converts that equation of state into a density-dependent MIT bag parameter B(rho), computes mass-radius curves by integrating TOV equations, and checks energy conditions and stability criteria such as the generalized TOV equation, Herrera cracking, and the adiabatic index. The paper claims that the model is singularity-free and that its mass-radius relation mimics a wide range of observed pulsars in four and higher dimensions.

Significance. If the exact solution and the stability checks were correct, the paper would provide a higher-dimensional anisotropic generalization of the Vaidya-Tikekar construction with a closed-form interior solution and a simple causality bound; the recovery of the known isotropic four-dimensional limit is a useful check. The solution algebra in Sec. 2 appears transparent, and the paper offers several standard physical checks (energy conditions, sound speeds, matching). However, the main phenomenological claims, namely the fit to PSR J1614-2230, the density-dependent bag model, and the mass-radius mimicry of pulsars, are not supported by the calculations as presented, and the paper's own numbers are internally inconsistent. The paper also does not ship reproducible code or machine-checked derivations, so the numerical claims rest entirely on the written text and figures.

major comments (3)
  1. [Sec. 7, Eq. (31), Table 3] The mass-radius curves and Table 3 are obtained by "solving the TOV equations" with the radial equation of state p_r(rho) from Eq. (31), but the TOV system used is not written down and no anisotropic term appears. The exact solution constructed in Sec. 2 is anisotropic, with Delta = p_t - p_r nonzero according to Eqs. (10), (15), and (16), and its hydrostatic equilibrium is governed by the generalized TOV equation (36), which contains the anisotropic force (n/r)(p_t - p_r). Integrating an isotropic TOV equation with p_r(rho) alone discards the anisotropy that was essential to the fitted exact solution. Consequently, Figs. 15-17 and Table 3 do not follow from the anisotropic solution presented in Sec. 2, and the abstract's claim that the mass-radius relation shows the model mimics a wide range of observed pulsars is unsupported.
  2. [Sec. 6 vs. Table 3] There is a direct numerical contradiction between the fitting section and the mass-radius table. Table 3 lists a D=4 maximum mass of 1.77 M_sun at b_max=10.09 km, while Sec. 6 fits the same model to PSR J1614-2230 with M=1.908^{+0.016}_{-0.016} M_sun and R=11.93^{+0.50}_{-0.50} km. Since 1.908 M_sun exceeds the declared maximum mass of the D=4 sequence, the fitted configuration cannot lie on the mass-radius curve shown in Fig. 15. The paper's validation against PSR J1614-2230 and its own mass-radius curves are therefore mutually inconsistent.
  3. [Sec. 6.2, Eq. (32), Figs. 12-14] The density-dependent bag parameter B(rho) is not independently determined. It is constructed by equating the linear MIT bag relation p_r=(rho-4B)/3 with the polynomial equation of state (31), whose coefficients in Table 2 are fits to the (rho,p_r) profile generated from Eqs. (13)-(14) for PSR J1614-2230. The stability window of strange quark matter in Fig. 12 and the critical anisotropy alpha_crit in Fig. 14 are therefore consequences of that fitting procedure rather than independent predictions of the model. The claim that the density-dependent MIT bag model is "useful" for the correct description of compact objects in this model needs to be reframed as a consistency check, not a validation.
minor comments (4)
  1. [Table 2 caption and Sec. 6] The table caption says the coefficients a_i are "obtained from Eq. (30)"; the coefficients actually come from fitting Eq. (31). The same typographical slip appears in the text near Table 2.
  2. [Fig. 20 caption] The caption of Fig. 20 states that the figure shows |v_t^2-v_r^2|, but the figure plots the adiabatic index Gamma and should also identify the anisotropic limit gamma of Eq. (40).
  3. [Introduction, after Eq. (1)] The sentence about the energy per baryon is garbled: "E_B of such system is 934 B_{145}^{1/4} MeV where,B_{145}^{1/4}=B_{145}^{1/4}" needs rewriting.
  4. [Sec. 5, Eq. (23)] The parentheses in the denominator of Eq. (23) are visually ambiguous; the expression should be typeset with explicit brackets so that the intended trigonometric ratio is unambiguous.

Circularity Check

2 steps flagged · score 6.0 of 10

Density-dependent bag parameter and 'wide pulsar mimicry' reduce to the same polynomial EoS fitted to PSR J1614-2230; exact solution and causality bound are self-contained.

  1. fitted input called prediction [Sec. 6.2, Eqs. (31)-(33), Fig. 12]
    "B(ρ) = lX i=0 kiρi, where the coefficients ki s are related to ai s through the relations k0 =− 34a0, k1 = (1−3a1)4 , kj =− 34aj, where j runs from 2 to s. ... The energy per baryon of strange matter reads as [9] EB = 2√3(3π2B(ρ)/4)^{1/4}."

    Eq. (32) for B(ρ) is obtained by eliminating p_r between the MIT bag relation p_r = (ρ−4B)/3 and the polynomial EoS Eq. (31), whose coefficients a_i were fitted in this same section to ρ and p_r values computed from the exact solution at the observed mass and radius of PSR J1614-2230. B(ρ) is therefore a re-expression of the fitted EoS, not an independent density-dependent bag parameter. The E_B(ρ) curves and the claimed stability window (absolute stability for D=4, metastability/instability in higher D) are consequences of that fit, not independent predictions.

  2. fitted input called prediction [Sec. 6.2 and Sec. 7, Eqs. (31), Fig. 15, Table 3]
    "we use the observed mass, radius data of PSR J1614-2230 to find the values of ρ and pr at various interior points of the compact object using Eqs. (13) and (14). Then, we fit the obtained data with the predicted EoS given by Eq. (31). ... To solve the TOV equation, we have made use of the EoS given in Eq. (31)."

    The mass-radius curves and maximum masses are outputs of a TOV integration whose input is the same polynomial EoS Eq. (31) that was fitted, earlier in the paper, to the interior ρ(r), p_r(r) profile of PSR J1614-2230 built from that object's observed mass and radius. Thus the J1614-based validation in Sec. 6 is in-sample by construction, and the abstract's claim that 'the mass radius relation shows that our model mimics a wide range of recently observed pulsars' is a statement about curves generated from that fitted EoS, not an independent prediction of the anisotropic exact solution.

full rationale

The core exact-solution derivation is self-contained: starting from the Vaidya-Tikekar metric ansatz (8) and the explicit anisotropy choice (10), the paper solves Eqs. (9)-(12), obtains density and pressures (13)-(16), imposes boundary conditions (20)-(24), and derives the causality bound (30), which reduces to the known Mukherjee et al. limit for D=4, α=0. No circularity attaches to that part. The self-citation to Goswami et al. [72] for the form of Δ is a modeling choice stated in the text, not a load-bearing external theorem. The circularity is confined to the phenomenological layer: Eq. (32) defines B(ρ) by substituting the fitted polynomial EoS (31) into the MIT bag relation, so the energy-per-baryon stability window is a relabeling of the fit; and the Sec. 7 mass-radius curves are generated by integrating that same fitted EoS. Consequently, the abstract's validation claims for PSR J1614-2230 and the 'wide range' of pulsars are not independent tests of the exact solution. Table 3's D=4 maximum mass (1.77 M_sun) also falls short of the fitted 1.908 M_sun point, an internal inconsistency that underscores that the M-R sequence does not directly reproduce the fitted configuration. Overall score 6: partial circularity in the phenomenological predictions, with the exact-solution core intact.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central solution depends on a set of chosen parameters (lambda, alpha, R) and on several modeling assumptions: the Vaidya-Tikekar metric ansatz, the specific anisotropy profile, and the assumed polynomial EoS. The density-dependent bag parameter is not an independent physical entity; it is a mathematical rearrangement of the fitted polynomial EoS. No new particles, forces, or conserved quantities are introduced.

free parameters (4)
  • spheroidal parameter lambda = 40 in the application; varied in Table 1 and Figs. 13-14
    A free geometric parameter in the Vaidya-Tikekar ansatz Eq. (8). It is not derived from microphysics and is chosen by hand to generate the plotted configurations.
  • anisotropy parameter alpha = 0.3 for D=4,5 and 0.98 for D=6,7 in the application; varied elsewhere
    A free parameter in the anisotropy ansatz Eq. (10). The values are selected to satisfy physical conditions such as positive energy per baryon and causality, not determined independently.
  • curvature parameter R = not tabulated explicitly; fixed by matching to the observed mass and radius of PSR J1614-2230
    Sets the density scale through Eq. (21) and is determined by boundary conditions, so it is effectively fitted to the observed pulsar data.
  • polynomial EoS coefficients a0...a5 = listed in Table 2 for D=4,5,6,7
    Obtained by fitting Eq. (31) to the model-generated (rho, p_r) data. No uncertainties are reported, and no out-of-sample validation is given.
assumptions (7)
  • domain assumption The interior spacetime is static, spherically symmetric in D dimensions, and filled with an anisotropic perfect fluid described by Eqs. (2) and (3).
    This is the starting point for the field equations (5)-(7) and is not derived in the paper.
  • ad hoc to paper The g_rr metric function takes the Vaidya-Tikekar spheroidal form in Eq. (8).
    This ansatz is chosen to make the equations solvable and represents a spheroidal geometry for the t=constant hypersurface. It is an assumption, not a consequence of the Einstein equations.
  • ad hoc to paper The pressure anisotropy Delta has the specific form in Eq. (10).
    This form is selected so that the differential equation (9) becomes integrable and the anisotropic force is regular at the center. It is an ad hoc modeling choice with no independent physical justification.
  • domain assumption The exterior spacetime is the D-dimensional Schwarzschild/Tangherlini vacuum metric in Eq. (18), with mass related by Eq. (19).
    Standard matching condition for compact stars; the paper assumes the exterior is vacuum and spherically symmetric.
  • ad hoc to paper The equation of state is a finite polynomial in rho, Eq. (31), truncated at fifth order.
    The polynomial form is assumed for fitting; it is not derived from quark matter or nuclear physics. The density-dependent bag parameter B(rho) is then defined by requiring the MIT bag relation p = (rho - 4B)/3, which is also assumed.
  • domain assumption Causality requires 0 <= v_r^2 < 1, and this condition is used to derive the lambda bound in Eqs. (27)-(30).
    The paper explicitly notes that superluminal sound speeds are debated at high density, but it assumes the causality condition as a physical requirement.
  • domain assumption The energy per baryon of strange quark matter is given by the MIT-bag formula Eq. (33).
    This is an external phenomenological formula from the cited literature used to assess quark matter stability.

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Pith. "Pith review of Exploring density dependent B as a suitable parameter in higher dimensional approach with a non-linear equation of state." pith.science (2026). https://pith.science/paper/MLJY5IC3

@misc{pith2026250820634,
  author       = {Pith},
  title        = {Pith review of: Exploring density dependent B as a suitable parameter in higher dimensional approach with a non-linear equation of state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLJY5IC3}},
  note         = {Machine review of arXiv:2508.20634}
}
abstract

In this investigation, we present a singularity free interior solution of the Einstein field equation for a class of anisotropic compact objects in dimensions $D\geq4$. In accordance with the concept of Vaidya and Tikekar, the geometry of the physical $(D-1)$-space of a star corresponding to $t=constant$ hypersurface is assumed to be of a $(D-1)$ spheroid. For the fulfilment of causality condition, a limit of the spheroidal parameter ($\lambda$) is noted depending on the values of amount of anisotropy ($\alpha$) and space-time dimensions ($D$). We note that by switching off the extra parameters ($\alpha$ and $D$), previously obtained limit of $\lambda$ can be generated. To validate our findings, we compare the results obtained from our model with observational data of PSR J1614-2230 (mass=$1.908^{+0.016}_{-0.016}M_{\odot}$, radius=$11.93^{+0.50}_{-0.50}km$). It is noted that the best fit equation of state corresponds to polynomial equation of state of the order of five. We use this finding to develop a density dependent MIT bag model which seems to be useful for the correct description of compact object in our model. The mass radius relation shows that our model mimics a wide range of recently observed pulsars in four and higher dimensions. Furthermore, we also found that our model exhibits stability according to Generalised TOV equation, Herrera cracking condition, and the adiabatic index.

Figures

Figures reproduced from arXiv: 2508.20634 by the authors.

Figure 1
Figure 1. Radial variation of energy density (˜ρ) inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. 0 2 4 6 8 10 12 0 100 200 300 400 r HKmL pr Ž HMeVfm 3 L D=7 D=6 D=5 D=4 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Radial variation of radial pressure ( ˜pr [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Radial variation of tangential pressure ( ˜pt [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Radial variation of pressure anisotropy ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Radial variation of e 2µ (panel a) and e 2ν (panel b) inside PSR J1614-2230 in different space-time dimensions (D). The dashed lines correspond to exterior Schwarzschild solution. The vertical line represents boundary of the star. Here, the pressure anisotropy paramete…
Figure 6
Figure 6. Figure 6: Radial variation of (˜ρ + ˜pr) inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. 0 2 4 6 8 10 12 500 1000 1500…
Figure 7
Figure 7. Figure 7: Radial variation of (˜ρ + ˜pt) inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. 0 2 4 6 8 10 12 200 400 600 8…
Figure 8
Figure 8. Figure 8: Radial variation of (˜ρ − p˜r) inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. polynomial relation between p…
Figure 9
Figure 9. Figure 9: Radial variation of (˜ρ − p˜t) inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. 0 2 4 6 8 10 12 2000 4000 600…
Figure 10
Figure 10. Figure 10: Radial variation of (n−1)˜ρ+ ˜pr +np˜t inside PSR J1614-2230 in different space-time dimensions (D). Here, the pressure anisotropy parameter α = 0.3 for D = 4, 5 and 0.98 for D = 6, 7 respectively. Value of the spheroidal parameter λ is set to 40. Now, we use the obse…
Figure 11
Figure 11. Figure 11: Equation of state for PSR J1614-2230 in different space-time dimensions ( [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Variation of energy per baryon (EB) of 3-flavour quark matter with density (ρ) of star for PSR J1614-2230 in different space-time dimensions (D). The red and green lines represent binding energy and mass correspond to 56F e (930.4 MeV ) and nucleon (939 MeV ) respecti…
Figure 13
Figure 13. Figure 13: Variation of central energy per baryon ( [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: Variation of central energy per baryon ( [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: Variation of mass (M) of compact objects with radius (b) for different values of D. Here the shaded regions correspond to the observed masses and radii of some compact objects such as, Yellow region: Her X-1, Magenta region: 4U 1820-30, Cyan region: Cen X-3, Orange re…
Figure 16
Figure 16. Figure 16: Variation of mass (M) of compact objects with central density (ρc) for different values of D. on compactness is greater than the radius until D = 6 then the effect reverses i.e. the effect of radius dominates over mass for D > 6. 15 [PITH_FULL_IMAGE:figures/full_fig_…
Figure 17
Figure 17. Figure 17: Variation of maximum mass (Mmax) of compact objects with space-time dimensions (D). 8 Stability Analysis We have checked the stability of our model from the following points of view: (i) Generalised TOV equation, (ii) Herrera cracking condition, and (iii) Variation of…
Figure 18
Figure 18. Figure 18: Variation of different forces inside PSR J1614-2230 for [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]
Figure 19
Figure 19. Figure 19: Radial variation of |v 2 t − v 2 r | inside PSR J1614-2230 for D = 4 and 6 respectively. Abreu’s inequality given by Eq. (38) is maintained in our model. 8.3 Adiabatic index The adiabatic index (Γ) acts as a basic ingredient of the condition of dynamical stability pos…
Figure 20
Figure 20. Figure 20: Radial variation of |v 2 t − v 2 r | inside PSR J1614-2230 for D = 4 and 6 respectively. 9 Conclusions In this article, we have presented a method of generating exact solution to the Einstein field equation for a spherically symmetric perfect fluid distribution taking…

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