--- title: Krori–Barua Interior Solution url: https://www.emergentmind.com/topics/krori-barua-solution type: topic --- # Krori–Barua Interior Solution 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 [1811.08112]. ## 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 \[ e^{\nu(r)}=e^{Br^2+C},\qquad e^{\lambda(r)}=e^{Ar^2}, \] or equivalently \(a(r)=Br^2+C\), \(b(r)=Ar^2\). The constants \(A\) and \(B\) have dimensions of \(\mathrm{length}^{-2}\), while \(C\) is dimensionless; in compact-star applications they are determined from the stellar mass \(M\) and radius \(R\) by junction conditions at the surface [1512.05202]. The same structural idea appears in other dimensions. In \((2+1)\) dimensions one uses \[ ds^2=-e^{2\nu(r)}dt^2+e^{2\mu(r)}dr^2+r^2d\theta^2, \qquad 2\mu(r)=Ar^2,\quad 2\nu(r)=Br^2+C, \] while in \(5\mathcal D\) Einstein–Gauss–Bonnet gravity the static spherically symmetric line element is written with \[ 2\nu(r)=Br^2+C,\qquad 2\lambda(r)=Ar^2 \] in the five-dimensional angular sector [1210.6346]. 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 \(g_{rr}\) through \(e^{\lambda(r)}=e^{Ar^2}\) and derives \[ e^{\nu(r)}=\Bigl(C+\frac{B}{A}e^{\tfrac12Ar^2}\Bigr)^2, \] while another \(f(T)\) construction uses \[ e^{\nu(r)}=(1+br^2)^2,\qquad e^{\lambda(r)}=\Bigl(A+\frac{B(2+br^2)^{3/2}}{3\sqrt b}\Bigr)^2. \] 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 [2105.15143]. ## 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 \(\kappa=8\pi G\), the Einstein equations yield \[ \rho(r)=\frac1{\kappa}\Bigl[2A\,e^{-Ar^2}+\frac{1-e^{-Ar^2}}{r^2}\Bigr], \qquad p(r)=\frac1{\kappa}\Bigl[2B\,e^{-Ar^2}-\frac{1-e^{-Ar^2}}{r^2}\Bigr]. \] The central values are finite, \[ \rho(0)=\frac{3A}{\kappa},\qquad p(0)=\frac{2B-A}{\kappa}, \] and \(e^{\nu(0)}=e^C\) is finite while \(e^{\lambda(0)}=1\), so the construction is nonsingular at the center [1811.08112]. Anisotropic versions replace the perfect-fluid stress tensor by \[ T_{\mu\nu}=(\rho+p_t)u_\mu u_\nu-p_tg_{\mu\nu}+(p_r-p_t)\chi_\mu\chi_\nu, \] with distinct radial and tangential pressures \(p_r\) and \(p_t\). The anisotropy is then measured by \[ \Delta(r)\equiv p_t(r)-p_r(r). \] In the \((2+1)\)-dimensional constant-\(\Lambda\) model one obtains \[ \rho(r)=\frac{1}{2\pi}\Bigl[Ae^{-Ar^2}-\Lambda\Bigr],\quad p_r(r)=\frac{1}{2\pi}\Bigl[Be^{-Ar^2}+\Lambda\Bigr],\quad p_t(r)=\frac{1}{2\pi}\Bigl[e^{-Ar^2}\bigl(B^2r^2+B-ABr^2\bigr)+\Lambda\Bigr], \] and \[ \Delta(r)=\frac{B(B-A)}{2\pi}\,r^2e^{-Ar^2}, \] which vanishes at the center and remains bounded everywhere [1210.6346]. A de Sitter generalization in four dimensions keeps the same quadratic potentials and adds a cosmological constant \(\Lambda\) to the Einstein equations. In that case the density and pressures are shifted according to \[ \rho(r)=\rho_{KB}(r)-\frac{\Lambda}{8\pi},\qquad p_r(r)=p_{r,KB}(r)+\frac{\Lambda}{8\pi},\qquad p_t(r)=p_{t,KB}(r)+\frac{\Lambda}{8\pi}, \] while the anisotropy \(p_t-p_r\) is unchanged by \(\Lambda\) [1201.5234]. ## 3. Junction conditions and determination of \(A\), \(B\), and \(C\) 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 \(r=R\). In the standard four-dimensional case the exterior is Schwarzschild, \[ ds^2=\Bigl(1-\frac{2M}{r}\Bigr)dt^2-\Bigl(1-\frac{2M}{r}\Bigr)^{-1}dr^2-r^2d\Omega^2. \] Continuity of \(g_{tt}\), \(g_{rr}\), and \(\partial_r g_{tt}\) at \(r=R\) gives \[ A=-\frac{1}{R^2}\ln\!\Bigl(1-\frac{2M}{R}\Bigr),\qquad B=\frac{M}{R^2(R-2M)},\qquad C=\ln\!\Bigl(1-\frac{2M}{R}\Bigr)-BR^2. \] Many papers also impose \(p_r(R)=0\); in the classical isotropic derivation this is part of the determination of the constants [1811.08112]. 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, \(\partial_r g_{tt}\), together with surface conditions such as \(p_r(R)=0\) or \(p_r^{\rm eff}(R)=0\) [2105.15144]. | Setting | Exterior metric | Surface conditions | |---|---|---| | Four-dimensional GR and many modified-gravity models | Schwarzschild | continuity of \(g_{tt}\), \(g_{rr}\), \(\partial_r g_{tt}\); often \(p_r(R)=0\) | | de Sitter interior | Schwarzschild–de Sitter | continuity of \(g_{tt}\), \(g_{rr}\), \(\partial_r g_{tt}\) | | Charged \(f(R,T)\) model | Reissner–Nordström | continuity of \(g_{tt}\), \(g_{rr}\), \(\partial_r g_{tt}\), \(p_r(R)=0\) | | \((2+1)\)-dimensional model | BTZ | continuity of \(g_{tt}\), \(g_{rr}\) | | \(5\mathcal D\) Einstein–Gauss–Bonnet model | Boulware–Deser | continuity of \(g_{tt}\), \(g_{rr}\), \(\partial_r g_{tt}\), \(p_r(R)=0\) | In \((2+1)\) dimensions, for example, matching to the BTZ exterior gives \[ e^{BR^2+C}=-M_0-\Lambda R^2,\qquad e^{AR^2}=\frac{1}{-M_0-\Lambda R^2}, \] or, with \(\Lambda=-K^2\), \[ A=-\frac{1}{R^2}\ln\bigl(K^2R^2-M_0\bigr),\qquad B=\frac{K^2}{K^2R^2-M_0},\qquad C=B-\ln\bigl(K^2R^2-M_0\bigr) \] [1210.6346]. ## 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 [1512.05202]. For anisotropic compact stars the regularity requirements are typically stated as \[ \rho(0),\;p_r(0),\;p_t(0)\ \text{finite},\qquad \frac{d\rho}{dr}\Big|_{r=0}=0,\qquad \frac{dp_r}{dr}\Big|_{r=0}=0, \] together with \[ \frac{d^2\rho}{dr^2}\Big|_{r=0}<0,\qquad \frac{d^2p_r}{dr^2}\Big|_{r=0}<0. \] These conditions ensure that the central density and pressure are finite and maximal at the center. The energy conditions are usually written as \[ \text{NEC: }\rho+p_r\ge0,\;\rho+p_t\ge0, \] \[ \text{WEC: }\rho\ge0,\;\rho+p_r\ge0,\;\rho+p_t\ge0, \] \[ \text{SEC: }\rho+p_r+2p_t\ge0, \] \[ \text{DEC: }\rho\ge|p_r|,\;\rho\ge|p_t|. \] Causality requires \[ 0\le v_{sr}^2=\frac{dp_r}{d\rho}\le1,\qquad 0\le v_{st}^2=\frac{dp_t}{d\rho}\le1, \] and Herrera’s stability criterion is \[ |v_{st}^2-v_{sr}^2|<1 \] everywhere inside the star [1512.05202]. The surface redshift is generally expressed in terms of the compactness \(u=M/R\) as \[ Z_s=(1-2u)^{-1/2}-1. \] Many papers also decompose the equilibrium equation into force terms. In the anisotropic general-relativistic and modified-gravity literature one encounters \[ F_g+F_h+F_a=0 \] or, when charge and matter–curvature coupling are present, \[ F_g+F_h+F_a+F_e+F_m=0. \] This decomposition makes explicit the balance of gravitational, hydrostatic, anisotropic, electric, and coupling contributions [1803.00442]. The sign of \(\Delta=p_t-p_r\) is not universal. Several models interpret \(\Delta>0\) as a repulsive or outward anisotropy that helps support higher masses, but in the \(f(\mathcal G,T)\) model for Her X-1 the reported result is \(\Delta<0\) for most of the star, indicating \(p_t0. \] The exterior is then Reissner–Nordström, and the matching conditions fix \(A\), \(B\), and \(C\) from \(M\), \(R\), and \(Q\) [2105.15144]. In \(f(T)\) gravity, the off-diagonal equation forces \[ f_{TT}=0, \qquad f(T)=\beta T+\beta_1. \] This makes the teleparallel Krori–Barua interior analytically tractable, with explicit \(\rho(r)\), \(p_r(r)\), \(p_t(r)\), and \(\Delta(r)\) after substitution of \(a(r)=Br^2+C\) and \(b(r)=Ar^2\) [1501.05829]. Other extensions retain the same KB structure in distinct modified theories. The Starobinsky model uses \[ f(R)=R+\lambda R^2 \] and produces explicit anisotropic density and pressure profiles for Her X-1, SAX J 1808.4–3658, and 4U 1820–30 [1412.2120]. Gauss–Bonnet-based models use either \[ f(\mathcal G)=\alpha \mathcal G^n \] or \[ f(\mathcal G,T)=\alpha\mathcal G^n+\lambda T, \] with the extra source terms organized through \(f_{\mathcal G}\), \(f_{\mathcal G}'\), and \(f_{\mathcal G}''\) or \(f_G\), \(f_{GG}\), and \(f_{GGG}\) [1501.00427]. The Krori–Barua construction has also been generalized to \(5\mathcal D\) Einstein–Gauss–Bonnet gravity, where baryonic matter obeys \[ p_r=\beta\rho+\gamma,\qquad 0<\beta<1,\;\beta\neq\tfrac13,\;\gamma>0, \] and strange quark matter obeys the MIT bag model \[ p_q=\tfrac13(\rho_q-4B_g). \] In that setting one solves for effective baryonic-plus-quark density and pressures and matches to the Boulware–Deser exterior [2312.08225]. ## 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 [1705.06910]. In most studies the observed \((M,R)\) pair fixes \(A\), \(B\), and \(C\), 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 \[ C\equiv \frac{2GM}{Rc^2}. \] After imposing surface matching, all interior functions become functions of \(x=r/R\) and \(C\) alone. The same study reports that the strongest constraint comes from the strong energy condition, giving \[ C\le C_{\max}=0.715. \] Using NICER and LIGO/Virgo mass-radius information, it finds \[ \rho(R)\approx 2.7\times10^{14}\,\mathrm{g/cm^3}, \] described there as exactly the nuclear-saturation density, and with the boundary condition \(\rho(R)=\rho_{\rm sat}\) the maximum mass is \[ M_{\max}\approx4.1\,M_\odot \] at \[ R_{\max}\approx16.8\,\mathrm{km} \] [2007.09797]. 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 \(\xi\); for \(\xi\in[0.01,0.09]\) the reported values are \(M_{\max}=2.24\text{–}2.36\,M_\odot\) and \(R=9.48\text{–}10.15\) km, together with satisfaction of causality, energy conditions, and several stability criteria [2505.21583]. The literature also shows that the Krori–Barua ansatz is not, by itself, a guarantee of physical acceptability. One \(f(T)\) model reports that the configuration is marginally unstable according to Herrera’s criterion, and another \(f(T)\) construction reports mixed behavior in the no-cracking test; in a gravitational-decoupling analysis in \(f(\mathcal G)\) gravity, one anisotropic solution is physically viable while the second is unstable at the core of the compact star [1501.05829]. 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. Source: https://www.emergentmind.com/topics/krori-barua-solution