Krori–Barua Interior Solution

• The Krori–Barua solution is a static interior spacetime metric characterized by quadratic functions of the radial coordinate to model compact stars.
• It provides closed-form expressions for density, pressure, and anisotropy through matching conditions with exterior solutions like Schwarzschild.
• The solution serves as a versatile template across GR and modified gravity theories, extending to different dimensions and addressing stellar stability.

The Krori–Barua solution is a static interior spacetime used for relativistic stellar modelling, especially for compact stars, in which the metric potentials are chosen as quadratic functions of the radial coordinate. In its standard four-dimensional form,

$$ds^2=e^{a(r)}dt^2-e^{b(r)}dr^2-r^2\bigl(d\theta^2+\sin^2\theta\,d\phi^2\bigr), \qquad a(r)=Br^2+C,\quad b(r)=Ar^2,$$

with constants $A,B,C$ fixed by boundary conditions. In later literature it is employed both as an exact interior solution in general relativity and as a seed metric or ansatz in $f(R)$, $f(R,T)$, $f(T)$, $f(\mathcal G)$, $f(\mathcal G,T)$, Rastall gravity, and lower- and higher-dimensional compact-star models (Sharif et al., 2018).

1. Metric ansatz and nomenclature

The standard Krori–Barua parametrization is written with either $(a,b)$, $(\nu,\lambda)$, or $(\alpha,\beta)$ as the two gravitational potentials. The most common notation is

$A,B,C$0

or equivalently $A,B,C$1, $A,B,C$2. The constants $A,B,C$3 and $A,B,C$4 have dimensions of $A,B,C$5, while $A,B,C$6 is dimensionless; in compact-star applications they are determined from the stellar mass $A,B,C$7 and radius $A,B,C$8 by junction conditions at the surface (Zubair et al., 2015).

The same structural idea appears in other dimensions. In $A,B,C$9 dimensions one uses

$f(R)$0

while in $f(R)$1 Einstein–Gauss–Bonnet gravity the static spherically symmetric line element is written with

$f(R)$2

in the five-dimensional angular sector (Rahaman et al., 2012).

The later literature also uses the labels “Krori–Barua potential” and “KB ansatz” for generalized interior parametrizations that retain only part of the original structure. One example fixes only $f(R)$3 through $f(R)$4 and derives

$f(R)$5

while another $f(R)$6 construction uses

$f(R)$7

This suggests that “Krori–Barua solution” functions in the literature both as a specific exact solution and as a broader template for regular interior metrics (Bhar, 2021).

2. Classical general-relativistic form

For a static, spherically symmetric perfect fluid in general relativity, the Krori–Barua interior gives closed-form density and pressure profiles. With $f(R)$8, the Einstein equations yield

$f(R)$9

The central values are finite,

$f(R,T)$0

and $f(R,T)$1 is finite while $f(R,T)$2, so the construction is nonsingular at the center (Sharif et al., 2018).

Anisotropic versions replace the perfect-fluid stress tensor by

$f(R,T)$3

with distinct radial and tangential pressures $f(R,T)$4 and $f(R,T)$5. The anisotropy is then measured by

$f(R,T)$6

In the $f(R,T)$7-dimensional constant-$f(R,T)$8 model one obtains

$f(R,T)$9

and

$f(T)$0

which vanishes at the center and remains bounded everywhere (Rahaman et al., 2012).

A de Sitter generalization in four dimensions keeps the same quadratic potentials and adds a cosmological constant $f(T)$1 to the Einstein equations. In that case the density and pressures are shifted according to

$f(T)$2

while the anisotropy $f(T)$3 is unchanged by $f(T)$4 (Kalam et al., 2012).

3. Junction conditions and determination of $f(T)$5, $f(T)$6, and $f(T)$7

The central practical step in the Krori–Barua construction is the matching of the interior metric to an exterior vacuum solution at the stellar surface $f(T)$8. In the standard four-dimensional case the exterior is Schwarzschild,

$f(T)$9

Continuity of $f(\mathcal G)$0, $f(\mathcal G)$1, and $f(\mathcal G)$2 at $f(\mathcal G)$3 gives

$f(\mathcal G)$4

Many papers also impose $f(\mathcal G)$5; in the classical isotropic derivation this is part of the determination of the constants (Sharif et al., 2018).

The same logic extends to other exteriors and theories. The exterior may be Schwarzschild–de Sitter, Reissner–Nordström, BTZ, or the five-dimensional Boulware–Deser vacuum, depending on the field equations and matter content. The matching data remain the induced metric and, when required, $f(\mathcal G)$6, together with surface conditions such as $f(\mathcal G)$7 or $f(\mathcal G)$8 (Bhar, 2021).

Setting	Exterior metric	Surface conditions
Four-dimensional GR and many modified-gravity models	Schwarzschild	continuity of $f(\mathcal G)$9, $f(\mathcal G,T)$0, $f(\mathcal G,T)$1; often $f(\mathcal G,T)$2
de Sitter interior	Schwarzschild–de Sitter	continuity of $f(\mathcal G,T)$3, $f(\mathcal G,T)$4, $f(\mathcal G,T)$5
Charged $f(\mathcal G,T)$6 model	Reissner–Nordström	continuity of $f(\mathcal G,T)$7, $f(\mathcal G,T)$8, $f(\mathcal G,T)$9, $(a,b)$0
$(a,b)$1-dimensional model	BTZ	continuity of $(a,b)$2, $(a,b)$3
$(a,b)$4 Einstein–Gauss–Bonnet model	Boulware–Deser	continuity of $(a,b)$5, $(a,b)$6, $(a,b)$7, $(a,b)$8

In $(a,b)$9 dimensions, for example, matching to the BTZ exterior gives

$(\nu,\lambda)$0

or, with $(\nu,\lambda)$1,

$(\nu,\lambda)$2

(Rahaman et al., 2012).

4. Matter variables, anisotropy, and physical acceptability

The Krori–Barua framework is normally evaluated through a standard set of viability requirements. The most frequently imposed conditions are regularity at the center, positivity and monotonic decrease of density and pressure, pointwise energy conditions, causality of sound speeds, equilibrium under the Tolman–Oppenheimer–Volkoff equation, Herrera’s cracking condition, adiabatic-index bounds, and a physically acceptable surface redshift (Zubair et al., 2015).

For anisotropic compact stars the regularity requirements are typically stated as

$(\nu,\lambda)$3

together with

$(\nu,\lambda)$4

These conditions ensure that the central density and pressure are finite and maximal at the center. The energy conditions are usually written as

$(\nu,\lambda)$5

$(\nu,\lambda)$6

$(\nu,\lambda)$7

$(\nu,\lambda)$8

Causality requires

$(\nu,\lambda)$9

and Herrera’s stability criterion is

$(\alpha,\beta)$0

everywhere inside the star (Zubair et al., 2015).

The surface redshift is generally expressed in terms of the compactness $(\alpha,\beta)$1 as

$(\alpha,\beta)$2

Many papers also decompose the equilibrium equation into force terms. In the anisotropic general-relativistic and modified-gravity literature one encounters

$(\alpha,\beta)$3

or, when charge and matter–curvature coupling are present,

$(\alpha,\beta)$4

This decomposition makes explicit the balance of gravitational, hydrostatic, anisotropic, electric, and coupling contributions (Biswas et al., 2018).

The sign of $(\alpha,\beta)$5 is not universal. Several models interpret $(\alpha,\beta)$6 as a repulsive or outward anisotropy that helps support higher masses, but in the $(\alpha,\beta)$7 model for Her X-1 the reported result is $(\alpha,\beta)$8 for most of the star, indicating $(\alpha,\beta)$9. The KB ansatz therefore does not fix the sign of anisotropy by itself; the result depends on the field equations and the matter sector (Shamir et al., 2017).

5. Extensions in modified gravity and alternative dimensions

A major part of the modern literature uses the Krori–Barua ansatz inside modified field equations. In $A,B,C$00 gravity one common choice is

$A,B,C$01

and, for $A,B,C$02, $A,B,C$03, substitution of the KB metric yields closed-form but lengthy expressions for $A,B,C$04, $A,B,C$05, $A,B,C$06, and $A,B,C$07. The physical analysis then proceeds through regularity, energy conditions, causality, Herrera’s stability criterion, and the surface redshift (Zubair et al., 2015).

A second $A,B,C$08 line of work adopts

$A,B,C$09

together with the MIT bag-model radial equation of state

$A,B,C$10

In that case the KB ansatz gives especially compact expressions,

$A,B,C$11

The same paper reports that for observed compact stars with $A,B,C$12 and $A,B,C$13 km, a positive coupling $A,B,C$14 yields stable models with bag constants $A,B,C$15\,MeV/fm$A,B,C$16, while in the GR limit $A,B,C$17 one recovers $A,B,C$18\,MeV/fm$A,B,C$19 (Biswas et al., 2018).

Charged strange-star models add the Maxwell sector and use

$A,B,C$20

together with the generalized Chaplygin equation of state

$A,B,C$21

The exterior is then Reissner–Nordström, and the matching conditions fix $A,B,C$22, $A,B,C$23, and $A,B,C$24 from $A,B,C$25, $A,B,C$26, and $A,B,C$27 (Bhar, 2021).

In $A,B,C$28 gravity, the off-diagonal equation forces

$A,B,C$29

This makes the teleparallel Krori–Barua interior analytically tractable, with explicit $A,B,C$30, $A,B,C$31, $A,B,C$32, and $A,B,C$33 after substitution of $A,B,C$34 and $A,B,C$35 (Abbas et al., 2015).

Other extensions retain the same KB structure in distinct modified theories. The Starobinsky model uses

$A,B,C$36

and produces explicit anisotropic density and pressure profiles for Her X-1, SAX J 1808.4–3658, and 4U 1820–30 (Zubair et al., 2014). Gauss–Bonnet-based models use either

$A,B,C$37

or

$A,B,C$38

with the extra source terms organized through $A,B,C$39, $A,B,C$40, and $A,B,C$41 or $A,B,C$42, $A,B,C$43, and $A,B,C$44 (Abbas et al., 2014).

The Krori–Barua construction has also been generalized to $A,B,C$45 Einstein–Gauss–Bonnet gravity, where baryonic matter obeys

$A,B,C$46

and strange quark matter obeys the MIT bag model

$A,B,C$47

In that setting one solves for effective baryonic-plus-quark density and pressures and matches to the Boulware–Deser exterior (Karmakar et al., 2023).

6. Astrophysical applications, constraints, and limitations

The constants of the Krori–Barua interior are routinely calibrated from observed masses and radii of compact stars. Recurrent stellar candidates in this literature include Her X-1, SAX J 1808.4–3658, 4U 1820–30, PSR J 1614–2230, Vela X-1, Cen X-3, 4U 1538–52, RX J 1856–37, and Vela X-12 (Shamir et al., 2017). In most studies the observed $A,B,C$48 pair fixes $A,B,C$49, $A,B,C$50, and $A,B,C$51, after which one checks regularity, energy conditions, equilibrium, and stability.

A particularly systematic analysis treats the KB spacetime as a one-parameter family controlled by the compactness

$A,B,C$52

After imposing surface matching, all interior functions become functions of $A,B,C$53 and $A,B,C$54 alone. The same study reports that the strongest constraint comes from the strong energy condition, giving

$A,B,C$55

Using NICER and LIGO/Virgo mass-radius information, it finds

$A,B,C$56

described there as exactly the nuclear-saturation density, and with the boundary condition $A,B,C$57 the maximum mass is

$A,B,C$58

at

$A,B,C$59

(Roupas et al., 2020).

Modified-gravity applications use the same KB machinery to obtain alternative mass-radius limits. In Rastall gravity, for example, the maximum mass increases with the Rastall parameter $A,B,C$60; for $A,B,C$61 the reported values are $A,B,C$62 and $A,B,C$63 km, together with satisfaction of causality, energy conditions, and several stability criteria (Biswas et al., 27 May 2025).

The literature also shows that the Krori–Barua ansatz is not, by itself, a guarantee of physical acceptability. One $A,B,C$64 model reports that the configuration is marginally unstable according to Herrera’s criterion, and another $A,B,C$65 construction reports mixed behavior in the no-cracking test; in a gravitational-decoupling analysis in $A,B,C$66 gravity, one anisotropic solution is physically viable while the second is unstable at the core of the compact star (Abbas et al., 2015). This suggests that the role of the Krori–Barua solution is best understood as providing a tractable singularity-free interior geometry whose astrophysical viability remains contingent on the matter sector, the field equations, and the junction conditions.