1 Introduction

In nonlinear field theory, Bogomol’nyi equations are closely related to attempts on finding exact solutions of the theory and their stabilities [1]. One of the earliest attempts in finding exact solutions were done by Prasad and Sommerfield in the case of SU(2) Yang–Mills–Higgs model [2]. In particular they found exact solutions of ’t Hooft–Polyakov monopoles and Julia-Zee dyons in the limit where both mass and quartic-coupling interaction of the Higgs fields go to zero. Bogomol’nyi then found by rewriting the energy functional into completed square form, so called Bogomol’nyi trick or decomposition, those exact solutions minimize the energy functional and are also solutions to first-order differential equations known as Bogomol’nyi equations. Furthermore he found that the total energy of these solutions are proportional to topological charge, and thus are stable, see [3, 4] for more detail calculations. These Bogomolny equations turn out to be usefull for studying stability of solutions in the classical field theory and one may be able to find some exact solutions since the problem of solving second-order differential equations, or Euler–Lagrange equations, now reduce to problem of solving these first-order differential equations. In supersymmetric theories, with central charge, the Bogomol’nyi equations can be obtained from variation of fermionic fields, that breaks some of supersymmetric charges and in this context the Bogomol’nyi equations are usually called BPS (Bogomol’nyi–Prasad–Sommerfield) equations [5].

The Bogomol’nyi trick does not have a systematic procedure, and hence is hard to be employed to more general models. There have been several methods developed in order to obtain the Bogomol’nyi equations: Strong Necessary Condition [6, 7], first-order formalism [8,9,10], On-Shell method [11, 12], FOEL (First-Order Euler–Lagrange) formalism [13], and BPS (Bogomol’nyi–Prasad–Sommerfield) Lagrangian method [14]. So far all these methods have been used mostly for (non-supersymmetric) field theories where the space-time metric is flat. Some of these methods have been used to find Bogomol’nyi equations for gravity, but only for a particular case of spatially flat universe metric in the scalar field inflation [9, 13]. There are some earliest attempts to find Bogomol’nyi-type equations in the system that include gravity either as a background metric or as dinamical field, such as incorporating the Einstein–Hilbert action, [15,16,17,18]. Unfortunately there were still no Bogomol’nyi-type equations found satisfying the Einstein equations. However there is an attempt to find Bogomol’nyi equations for black holes by using Bogomol’nyi trick which, as we mentioned before, may not be applicable to more general gravity models [19]Footnote 1. In this article we will try to find first-order differential equations that satisfy the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields) in some of gravity theories and, for an obvious reason, we will be using the BPS Lagrangian method. We shall call these first-order differential equations as Bogomol’nyi-like equations since they will be derived from the action, or to be more precise (effective) Lagrangian density, instead of total energy as in the original article by Bogomol’nyi [1].

This article is organized as follows. At first, in Sect. 1.1 we review in detail how to employ the BPS Lagrangian method to the SU(2) Yang–Mills–Higgs model without any ansatz and with Julia–Zee ansatz. Later in Sect. 2 we employ the BPS Lagrangian method to obtain and solve these Bogomol’nyi-like equations for (static spherically symmetric) vacuum Einstein gravity in Sect. 2.1, Einstein gravity coupled with U(1) gauge field in Sect. 2.2, and Einstein gravity coupled with a real scalar field in Sect. 2.3. Then in Sect. 3 we employ the BPS Langrangian method to more general n-dimensional, with \(n\ge 3\), gravity theory coupled with U(1) gauge and a real scalar fields. In particular, we obtain and solve explicitly the Bogomol’nyi-like equations for n-dimensional gravity theory coupled with U(1) gauge and for three dimensional gravity coupled with a real scalar field in Sects. 3.1 and 3.2 respectively. In the last Sect. 4 we give remarks about the results in this article, which are mainly about their relations with the black hole uniqueness theorems, and also other possible applications of the BPS Lagrangian method.

1.1 BPS Lagrangian method

The BPS Lagrangian method was first developed in [20] to reproduce the known Bogomol’nyi equations for BPS vortices in various three dimensional Maxwell–Higgs models. It was based on observation that the energy density of BPS vortices in these models has a similar form given by, what is called, BPS Lagrangian density. The BPS Lagrangian density found there contains only boundary terms, by means its Euler–Lagrange equations are trivially satisfied, so there are no additional constraint equations need to be considered. This type of BPS Lagrangian density, which shall be called boundary BPS Lagrangian density, was later used to find Bogomol’nyi equations for BPS monopoles and dyons in various model of four dimensional SU(2) Yang–Mills–Higgs models under Julia–Zee ansatz [21]. It is also possible to use another type of BPS Lagrangian density that may contain non-boundary terms as such its Euler–Lagrange equations are not trivially satisfied and we shall call it non-boundary BPS Lagrangian density. These Euler–Lagrange equations become constraint equations that must be considered in finding solutions. This non-boundary BPS Lagrangian density was first used in three dimensional generalized Born–Infeld–Higgs model and it was found that stress-tensor components of the BPS vortices are non-zero, or physically interpreted as having internal pressures, and hence unlike the usual BPS vortices [22]. However, non-boundary BPS Lagrangian density does not always result in BPS vortices with internal pressures. As an example in three dimensional Maxwell–Chern–Simons–Higgs model, using non-boundary BPS Lagrangian gave us the usual BPS vortices, without internal pressures, but with a less degree of freedom, by means one of the effective fields depend on the other effective field [23]. Another examples of using non-boundary BPS Lagrangian density have been discussed in the generalized Skyrme model and its higher dimensional extensions [24,25,26]. All those examples depend on some particular ansatzes which are mainly choosen by a requirement that the Bogomol’nyi equations must be spherically symmetric. An example of employing the BPS Lagrangian method without a priori imposing any particular ansatz, or independent of any ansatz, was shown in [27] for the case of generalized SU(2) Yang–Mills–Higgs models in order to obtain Bogomol’nyi equations for BPS dyons.

Let us consider an action of N scalar fields \(\phi _m\equiv (\phi _1,\ldots ,\phi _N)\) in \((d+1)\)-dimensions of spacetime,

$$\begin{aligned} S=\int d^{d+1}x\sqrt{-g}~\mathcal {L}(\phi _m,\partial _\mu \phi _m), \end{aligned}$$
(1)

with \(\partial _\mu \equiv {\partial \over \partial x^\mu }\equiv \left( {\partial \over \partial x^0},{\partial \over \partial x^1},\ldots ,{\partial \over \partial x^d}\right) \) and \(\mu =0,1,\ldots , d\) is the spacetime index. Similar to Bogomol’nyi’s trick, we shall rewrite the Lagrangian density into complete squared terms as such

$$\begin{aligned} \sqrt{-g}\mathcal {L}\equiv \mathcal {L}_{eff}=\left( \text{ squared } \text{ terms }\right) +\mathcal {L}_{BPS},\nonumber \\ \end{aligned}$$
(2)

where

$$\begin{aligned} \left( \text{ squared } \text{ terms }\right) = \sum _{n\!=\!1}^N h_n(\phi _m) \left( \partial _\nu \phi _n \!-\! f_n(\phi _m,\partial _\mu \phi _m^0) \right) ^2, \nonumber \\ \end{aligned}$$
(3)

with \(\partial _\mu \phi _m^0\equiv \{\partial _\mu \phi _m\}-\partial _\nu \phi _n\) for fixed \(\nu \) and n. Here \(\mathcal {L}_{BPS}\) is defined as BPS Lagrangian density that usually contain only boundary terms by means of its Euler-Lagrange equations are trivially satisfied. Once we determine the form of BPS Lagrangian density then we could find the “squared terms”. We then define a BPS limit in which \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\) and all terms in the “squared terms” are set to zero as such \(\partial _\nu \phi _n = f_n(\phi _m,\partial _\mu \phi _m^0)\) which shall be called Bogomol’nyi equations. Now one may ask what is the explicit form of BPS Lagrangian density?. One of the answer comes from the study of well-known Bogomol’nyi equations in various models using the On-Shell metode [11]. For spherically static cases, total energy is proportional to the action, \(E=-\int d^{d+1}x~\mathcal {L}_{eff}\), and in the BPS limit the energy of BPS solitons are determined by difference between values of a function \(Q\equiv Q(\phi _m)\), there is called BPS energy functional, on the boundary and on the origin,

$$\begin{aligned} E_{BPS}= & {} Q(r\rightarrow \infty )-Q(r=0)=\int _{r=0}^\infty dQ\nonumber \\= & {} \int dr \left( \sum _{n=1}^N{\partial Q\over \partial \phi _n}\phi _n'(r)\right) =\int dr \mathcal {L}_{BPS}. \end{aligned}$$
(4)

Here \(\mathcal {L}_{BPS}\) is a linear function of first-order derivative of fields \(\phi _m\) and one can simply show that its Euler–Lagrange equations are indeed trivial. The boundary terms are not restricted only to linear function of \(\phi _m'(r)\). In general there are also possible boundary terms contain \(\partial _\mu \phi _m\) with power higher than one. As an example for \(N=3\) and \(d=3\), we may have \(\mathcal {L}_{BPS}\) with boundary terms as follows [13]

$$\begin{aligned} \mathcal {L}_{BPS}=\sum _{n,o,p} Q^{[ijk]}_{[nop]} \partial _i\phi _n \partial _j\phi _o \partial _k\phi _p, \end{aligned}$$
(5)

where \( Q^{[ijk]}_{[nop]}\equiv Q^{[ijk]}_{[nop]}(\phi _m)\) is totally antisymmetric tensor in both up and down indices. Here we have used Einstein summation notation over the spatial coordinate indices i, j,  and k. However \(\mathcal {L}_{BPS}\) may also contain non-boundary terms such that its Euler–Lagrange equations,

$$\begin{aligned} {\partial \mathcal {L}_{bps}\over \partial \phi _m}-{\partial \over \partial x^\mu }\left( {\partial \mathcal {L}_{bps}\over \partial (\partial _\mu \phi _m)}\right) =0, \end{aligned}$$
(6)

are not trivially satisfied and must be considered as additional constraint equations in finding solutions to the Bogomol’nyi equations.

1.1.1 Case without ansatz

Let us see how to employ explicitly the BPS Lagrangian method to the SU(2) Yang–Mills–Higgs model with the following action in Minskowski spacetime

$$\begin{aligned} S= & {} \int d^4x\sqrt{-g}~ \mathcal {L}=\int d^4x\left( -{1\over 2}\text {Tr}\left( F_{\mu \nu }F^{\mu \nu }\right) \nonumber \right. \\{} & {} \left. + \text {Tr}\left( D_\mu \Phi D^\mu \Phi \right) -V(|\Phi |)\right) , \end{aligned}$$
(7)

with \(|\Phi |=2\text {Tr}(\Phi ^2),~F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu -ie \left[ A_\mu ,A_\nu \right] , D_\mu \equiv \partial _\mu -ie\left[ A_\mu ,\right] \), \(A_\mu ={1\over 2}\tau ^a A^a_\mu ,~ \Phi ={1\over 2}\tau ^a \Phi ^a\), with internal index \(a=1,2,3\) and \(\tau ^a\) are the Pauli matrices. The Euler–Lagrange equations of action (7), with fundamental (real) fields \(\Phi ^a\) and \(A^a_\mu \), are written, respectively,

$$\begin{aligned} D_\mu D^\mu \Phi= & {} -2{\partial V\over \partial |\Phi |}\Phi , \end{aligned}$$
(8a)
$$\begin{aligned} D_\nu F^{\mu \nu }= & {} -ie[\Phi , D^\mu \Phi ]. \end{aligned}$$
(8b)

Rewriting the Lagrangian density, \(\mathcal {L}\), in terms of electric field strength, \(E_i=F_{0i}\), and magnetic field strength, \(B_i={1\over 2}\epsilon _{ijk}F_{jk}\) with \(i=1,2,3\), the effective Lagrangian density is given by

$$\begin{aligned} \mathcal {L}_{eff}\equiv \sqrt{-g}\mathcal {L}= & {} \text {Tr}\left( E_i\right) ^2-\text {Tr}\left( B_i\right) ^2\nonumber \\{} & {} +\text {Tr}\left( D_0\Phi \right) ^2-\text {Tr}\left( D_i\Phi \right) ^2-V, \end{aligned}$$
(9)

where the scalar potential \(V\ge 0\) is still arbitrary. Now consider BPS Lagrangian density, as defined in [27], as follows

$$\begin{aligned} \mathcal {L}_{BPS}=2\alpha ~\text {Tr}\left( E_iD_i\Phi \right) -2\beta ~\text {Tr}\left( B_iD_i\Phi \right) -\gamma ~\text {Tr}\left( D_i\Phi \right) ^2,\nonumber \\ \end{aligned}$$
(10)

where \(\alpha ,\beta ,\) and \(\gamma \) are arbitrary constants. From both Lagrangian densities, we may obtain

$$\begin{aligned} \mathcal {L}_{eff}-\mathcal {L}_{BPS}= & {} \text {Tr}\left( E_i-\alpha ~ D_i\Phi \right) ^2 - \text {Tr}\left( B_i-\beta ~ D_i\Phi \right) ^2\nonumber \\{} & {} +\text {Tr}\left( D_0\Phi \right) ^2 -\left( 1-\gamma +\alpha ^2-\beta ^2\right) \text {Tr}\nonumber \\ {}{} & {} \times \left( D_i\Phi \right) ^2-V. \end{aligned}$$
(11)

This form can be obtained by considering \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}\) as quadratic equation and completing the square in \(E_i,B_i,D_0\Phi ,\) and \(D_i\Phi \) subsequently. In the BPS limit, \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\), first three terms on the right hand side give us the Bogomol’nyi equations [1, 3, 4],

$$\begin{aligned} E_i=\alpha ~ D_i\Phi ,\qquad B_i=\beta ~ D_i\Phi ,\quad D_0\Phi =0. \end{aligned}$$
(12)

For the fourth term, we can not take \(D_i\Phi =0\) otherwise it will lead us to trivial solution and thus we must set \(\gamma =1 +\alpha ^2-\beta ^2\) and then \(V=0\). As we mentioned previously there are additional constraint equations since the BPS Lagrangian density contains non-boundary terms. The Euler–Lagrange equations of \(\mathcal {L}_{BPS}\), respectively, for fields \(\Phi \), \(A_i\), and \(A_0\) are simplified to

$$\begin{aligned} \left( 1-\beta ^2\right) D_iD_i\Phi= & {} 0, \end{aligned}$$
(13)
$$\begin{aligned} \left( 1-\alpha ^2-\beta ^2\right) \left[ D_i\Phi ,\Phi \right]= & {} 0, \end{aligned}$$
(14)
$$\begin{aligned} \alpha ~D_iD_i\Phi= & {} 0. \end{aligned}$$
(15)

There are two possible solutions without additional constraint equations:

  • BPS monopoles: \(\alpha =0, \beta =\pm 1\).

  • BPS dyons: \(\alpha \ne 0\Longrightarrow D_iD_i\Phi =\beta D_iB_i=0\) (by Bianchi Identity) and \(\alpha ^2+\beta ^2=1\).

By allowing \(\alpha ,\beta ,\) and \(\gamma \) in (10) to be functions of \(|\Phi |\), we may also obtain Bogomol’nyi equations for the generalized SU(2) Yang–Mills–Higgs model [27].

1.1.2 Case with ansatz

Working with PDEs (partial differential equations) are more difficult compared to ODEs (ordinary differential equations). Therefore sometimes it is more practical to turn the PDEs into PDEs with less number of variables, or even if possible to turn them into ODEs, by taking a particular ansatz. Here we will show how to employ the BPS Lagrangian method under (’t Hooft–Polyakov) Julia–Zee ansatz [28,29,30],

$$\begin{aligned} \Phi ^a= f(r) {x^a\over r},\quad A^a_0={j(r)\over e} {x^a\over r},\quad A^a_i={1-a(r)\over e} \epsilon ^{aij} {x^j\over r^2},\nonumber \\ \end{aligned}$$
(16)

where \(x^a\equiv (x,y,z)\), as well as \(x^{i,j}\equiv (x,y,z)\), denotes the Cartesian coordinates and \(\epsilon ^{aij}\) is the Levi–Civita symbol. Under this ansatz the Euler–Lagrange equations (8) are effectively written as

$$\begin{aligned} -{1\over r^2}(r^2 f')'+{2 a^2f\over r^2}&=-{dV(f)\over df}, \end{aligned}$$
(17a)
$$\begin{aligned} -{(r^2 j')'\over er^2}+{2a^2j\over er^2}&=\, 0, \end{aligned}$$
(17b)
$$\begin{aligned} {a(a^2-1)\over r^2}+a(e^2f^2-j^2)-a''&=\,0, \end{aligned}$$
(17c)

where we have defined a differential operator \('\equiv {d\over dr}\). Now we need to find the effective Lagrangian density and identify its effective fields. It turns out the effective Lagrangian density is proportional to \(\sqrt{-g}\mathcal {L}\) under the ansatz (16), in spherical coordinates,

$$\begin{aligned} \mathcal {L}_{eff}= & {} -r^2\left( \frac{f'^2}{2}+\frac{a^2 f^2}{r^2}\right) +{r^2\over e^2} \left( \frac{j'^2}{2}+\frac{a^2 j^2}{r^2}\right) \nonumber \\{} & {} -{r^2\over e^2} \left( \frac{a'^2}{r^2}+\frac{\left( a^2-1\right) ^2}{2 r^4}\right) -r^2V(f), \end{aligned}$$
(18)

and its effective fields are f, a,  and j such that Euler–Lagrange equations of \(\mathcal {L}_{eff}\left( r,f(r),a(r),j(r),{df(r)\over dr},{da(r)\over dr},\right. \left. {dj(r)\over dr}\right) \) for the effective fields f, a,  and j are given in (17) respectively. For this case we consider the effective BPS Lagrangian density, as defined in [27],

$$\begin{aligned} \mathcal {L}_{BPS}=- {\partial Q\over \partial f} f'- {\partial Q\over \partial a} a'- {\partial Q\over \partial j} j', \end{aligned}$$
(19)

where \(Q\equiv Q(f,a,j)\) is an auxiliary function of only the effective fields. We then rewrite \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}\) by completing the square in \(f',a',\) and \(j'\),

$$\begin{aligned} \mathcal {L}_{eff}-\mathcal {L}_{BPS}= & {} -{r^2\over 2}\left( f'-{Q_f \over r^2}\right) ^2\nonumber \\{} & {} -{1\over e^2}\left( a'-{e^2\over 2}Q_a \right) ^2+{r^2\over e^2}\left( j'+e^2{Q_j \over r^2}\right) ^2\nonumber \\{} & {} \times \frac{e^2 Q_f^2 -e^4 Q_j^2 -\left( a^2-1\right) ^2}{2r^2e^2 }\nonumber \\ {}{} & {} +\frac{4 a^2 \left( j^2 -e^2 f^2 \right) +e^4 Q_a^2}{4e^2 }-r^2V, \end{aligned}$$
(20)

with \(Q_f\equiv {\partial Q\over \partial f}, Q_a\equiv {\partial Q\over \partial a},\) and \(Q_j\equiv {\partial Q\over \partial j}\). In the BSP limit, \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\) implies Bogomol’nyi equations

$$\begin{aligned} f'={Q_f\over r^2},\quad a'={e^2\over 2} Q_a,\quad j'=-e^2 {Q_j \over r^2}. \end{aligned}$$
(21)

and

$$\begin{aligned}{} & {} \frac{e^2 Q_f^2 -e^4 Q_j^2 -\left( a^2-1\right) ^2}{2r^4e^2 }\nonumber \\ {}{} & {} \quad +\frac{4 a^2 \left( j^2 -e^2 f^2 \right) +e^4 Q_a^2}{4r^2 e^2 }-V=0 \end{aligned}$$
(22)

which can be solved by expanding the left hand side in explicit radial coordinate r and then setting all the “coefficients” to be zero, \(V=0\),

$$\begin{aligned}{} & {} V=0,\quad Q_a=\pm {2\over e^2} a \sqrt{e^2f^2-j^2},\nonumber \\ {}{} & {} e^2 Q_f^2-e^4 Q_j^2=\left( a^2-1\right) ^2. \end{aligned}$$
(23)

Therefore we can fix the auxiliary function \(Q=\pm {1\over e^2}(a^2-1)\sqrt{e^2f^2-j^2}\) and the Bogomol’nyi equations now become [1, 3, 4]

$$\begin{aligned}{} & {} f'=\pm {(a^2-1)f\over r^2\sqrt{e^2f^2-j^2}},\quad a'=\pm a \sqrt{e^2f^2-j^2},\nonumber \\ {}{} & {} j'=\pm {(a^2-1)j \over r^2\sqrt{e^2f^2-j^2}}. \end{aligned}$$
(24)

There are no additional constraint equations since the Euler–Lagrange equations of \(\mathcal {L}_{BPS}\) (19) are trivially satisfied. Furthermore from the Bogomol’nyi equations we may have an equation \({j'/ f'}={j/f}\) whose solution is \(j(r)=\sigma f(r)\), with \(\sigma \) is an integration constant.

2 Four-dimensional gravity

The four-dimensional Einstein–Hilbert action

$$\begin{aligned} S_{EH}={1\over 2\kappa }\int d^4x\sqrt{-g} \left( R-2\Lambda \right) , \end{aligned}$$
(25)

where \(\Lambda \) is the cosmological constant and \(\kappa =8\pi G\) is the Einstein gravitational constant. In general, it requires additional Gibbons–Hawking–York boundary terms as such the resulting Euler–Lagrange equations are the Einstein equations [31, 32]. The total action is given by

$$\begin{aligned} S=&S_{EH}+S_{GHY}\nonumber \\ =&{1\over 2\kappa }\int _\mathcal {M} d^4x\sqrt{-g}\left( R-2\Lambda \right) \nonumber \\&\quad +{1\over \kappa }\int _{\partial \mathcal {M}}d^3y\epsilon \sqrt{h}K-{1\over \kappa }\int _{\partial \mathcal {M}}d^3y\epsilon \sqrt{h}K_0, \end{aligned}$$
(26)

where \(h_{ab}\) is the induced metric; K is the trace of the extrinsic curvature; and \(\epsilon =+1\) if the normal of boundary manifold \(\partial \mathcal {M}\) is spacelike and \(\epsilon =-1\) if the normal of boundary manifold \(\partial \mathcal {M}\) is timelike, with \(y^a\) are the coordinates on the boundary manifold \(\partial \mathcal {M}\). Here the last term is added to remove singularity part of the Gibbons–Hawking–York terms.

2.1 Static spherically symmetric

In this case the ansatz for the four-dimensional metric is

$$\begin{aligned} ds^2=-A(r)dt^2+B(r)dr^2+C(r)d\Omega _2^2, \end{aligned}$$
(27)

where \(A,B,C\ge 0\). Non-trivial parts of the Einstein equations, \(R_{\mu \nu }-{1\over 2}g_{\mu \nu }R+\Lambda g_{\mu \nu }=0\), in this ansatz are simplified to

$$\begin{aligned}{} & {} 2 C(r) A'(r) C'(r)+A(r) \left( C'(r)^2-4 B(r) C(r)\right. \nonumber \\ {}{} & {} \quad \left. \left( 1-\Lambda ~ C(r)\right) \right) =0, \end{aligned}$$
(28a)
$$\begin{aligned}{} & {} 2 C(r) B'(r) C'(r)+B(r) \left( C'(r)^2-4 C(r) C''(r)\right) \nonumber \\{} & {} \quad +4 B(r)^2 C(r)\left( 1-\Lambda ~ C(r)\right) =0, \end{aligned}$$
(28b)
$$\begin{aligned}{} & {} B(r) \left( -A(r) C(r) \left( A'(r) C'(r)+2 A(r) C''(r)\right) \nonumber \right. \\{} & {} \quad \left. +C(r)^2 \left( A'(r)^2-2 A(r) A''(r)\right) +A(r)^2 C'(r)^2\right) \nonumber \\{} & {} \quad +A(r) C(r) B'(r) \left( C(r) A'(r)+A(r) C'(r)\right) \nonumber \\ {}{} & {} \quad -4\Lambda ~ A(r)^2 B(r)^2 C(r)^2=0. \end{aligned}$$
(28c)

Now we are going to find the Bogomol’nyi-like equations, or first-order derivative equations, for functions A, B,  and C using the BPS Lagrangian method [14]. The BPS Lagrangian method requires a Lagrangian density, which shall be called effective Lagrangian density, that give us the Einstein equations (28). To do so, we use the Gaussian normal coordinates and pick a boundary manifold as timelike hypersurface, with spacelike normal vector or \(\epsilon =1\), at spatial infinity assumed to be at \(r\rightarrow \infty \). The reduced Einsten–Hilbert action is given by

$$\begin{aligned}{} & {} S_{RedEH}\propto \int dr\nonumber \\ {}{} & {} \quad \frac{2 C(r) A'(r) C'(r) +A(r) \left( 4 B(r) C(r)\left( 1-\Lambda ~ C(r)\right) +C'(r)^2\right) }{\sqrt{A(r) B(r) C(r)^2}}\nonumber \\ {}{} & {} \quad =\int dr \mathcal {L}_{eff}. \end{aligned}$$
(29)

One can check that the Euler–Lagrange equations of this reduced action are indeed the Einstein equations (28). Derivation of Euler–Lagrange equations (28) from the reduced action (29) is rather naive by assuming the radial coordinate r to act as time coordinate and the variation is taken over the effective fields (A, B, C). We will also use these assumptions when applying the BPS Lagrangian method further. Following prescriptions of the BPS Lagrangian method, we pick a standard BPS Lagrangian density which contains linear terms in \(A'(r),B'(r),\) and \(C'(r)\) as followsFootnote 2,

$$\begin{aligned} \mathcal {L}_{BPS}= & {} X_0(A,B,C)+X_a(A,B,C) A'(r)\nonumber \\{} & {} + X_b(A,B,C) B'(r)+ X_c(A,B,C) C'(r), \end{aligned}$$
(30)

where \(X_0,X_a,X_b,\) and \(X_c\) are auxiliary functions of A, B,  and C. Our task now is to find explicit form of these functions. We then rewrite \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}\) as follows

$$\begin{aligned}{} & {} -{C\over A\sqrt{AB}} \left( A'(r)-\frac{\sqrt{A B}}{2 C}\left( X_cC-X_aA\right) \right) ^2\nonumber \\{} & {} \quad +\sqrt{A\over B}\left( {C'(r)\over C}+{1\over A}\left( A'(r)-\frac{X_c}{2}\sqrt{A B}\right) \right) ^2\nonumber \\{} & {} \quad -B'(r) X_b-X_0+{\sqrt{AB}\over 4C}\left( X_a^2A-2(X_a X_c-8\left( 1-\Lambda ~ C\right) )C\right) . \nonumber \\ \end{aligned}$$
(31)

In the BPS limit, where \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\), we can extract Bogomol’nyi-like equations for A and C, respectively,

$$\begin{aligned} A'(r)= & {} \frac{\sqrt{A B}}{2 C}\left( X_cC-X_aA\right) , \end{aligned}$$
(32a)
$$\begin{aligned} C'(r)= & {} -{C\over A}\left( A'(r)-\frac{X_c}{2}\sqrt{A B}\right) \end{aligned}$$
(32b)

and the remaining terms is

$$\begin{aligned} {\sqrt{AB}\over 4C}\left( X_a^2A-2(X_a X_c-8\left( 1-\Lambda ~ C\right) )C\right) -B'(r) X_b=0. \nonumber \\ \end{aligned}$$
(33)

Ones can consider this equation as first-order differential equation of B which could be later identified as Bogomol’nyi-like equation for B. This however contradicts with our previous results that there are only Bogomol’nyi-like equations for A and C. In order to avoid this contradiction we must set \(X_b=0\) which later implies

$$\begin{aligned} X_c=\frac{A B \left( X_a^2 A +16 C\left( 1-\Lambda ~ C\right) \right) -4X_0C \sqrt{A B} }{2X_a A B C }. \end{aligned}$$
(34)

Next, we must also consider additional constraint equations that are Euler–Lagrange equations of \(S=\int dr \mathcal {L}_{BPS}\). The constraint equation for B yields \(X_0=0\). The remaining constraint equations can be simplified to

$$\begin{aligned}{} & {} {4 C\over \sqrt{A B}} {\partial X_a\over \partial B} X_a~ B'(r)\nonumber \\{} & {} \quad =X_a^3-2C {\partial X_a\over \partial C}X_a^2-\left( 16 C\left( 1-\Lambda ~ C\right) -X_a^2 A \right) {\partial X_a\over \partial A}.\nonumber \\ \end{aligned}$$
(35)

This simplified constraint equation is a first-order differential equation of B. Using explicit functions of \(X_0,X_b,\) and \(X_c\) the Bogomol’nyi-like equations (32) can be simplified to

$$\begin{aligned} A'(r)= & {} \sqrt{A B} \left( \frac{4\left( 1-\Lambda ~ C\right) }{X_a(A,B,C)}-\frac{A}{4 C}X_a(A,B,C)\right) , \end{aligned}$$
(36a)
$$\begin{aligned} C'(r)= & {} \frac{\sqrt{A B}}{2} X_a(A,B,C), \end{aligned}$$
(36b)

which, together with the constraint equation (35), satisfy the Einstein equations (28). However those equations are impractical when we want to find their explicit solutions because there is still one auxilliary function \(X_a\) needs to be determined.

2.1.1 Schwarzschild black hole: \(C=r^2\)

In this case the solution for \(X_a\), from the Bogomol’nyi-like equation (36b), is given by \(X_a={4r\over \sqrt{AB}}\). With this explicit function of \(X_a\), the Bogomol’nyi-like equation for A (36a) and the constraint equation (35) are now

$$\begin{aligned} A'(r)= & {} {A}{B\left( 1-\Lambda ~ r^2\right) -1\over 2r}, \end{aligned}$$
(37a)
$$\begin{aligned} B'(r)= & {} -{B}{B\left( 1-\Lambda ~ r^2\right) -1\over 2r}. \end{aligned}$$
(37b)

From the ratio \(A'(r)/B'(r)\), we may conclude the functions A and B are related by \(A={c_a\over B}\), with \(c_a\) is a real constant. Solutions to A and B are

$$\begin{aligned} A= & {} c_a{(r-{r^3\over 3}\Lambda +c_b)\over r },\end{aligned}$$
(38)
$$\begin{aligned} B= & {} {r\over r-{r^3\over 3}\Lambda + c_b}, \end{aligned}$$
(39)

which are the Schwarzschild solutions. In particular the Schwarzschild black hole is obtained by setting \(\Lambda =0, c_a=1,\) and \(c_b=-2 M\), where M is the mass of gravitational source.

2.1.2 General solutions

Using the Bogomol’nyi-like equations (36), we can recast the constraint equation (35) to be

$$\begin{aligned} {dX_a\over dr}={\sqrt{AB}\over 4C}X_a^2. \end{aligned}$$
(40)

Comparing it with the Bogomol’nyi-like equation for C (36b50b), we obtain \(X_a=cx_a\sqrt{C}\), with \(cx_a\) is a real constant. Furthermore using explicit solution of \(X_a\) and comparing both Bogomol’nyi-like equations (36), we may obtain a differential equation

$$\begin{aligned} 2C{dA\over dC}={16\over cx_a^2}\left( 1-\Lambda C\right) -A \end{aligned}$$
(41)

whose solution is

$$\begin{aligned} A=\frac{16}{3 cx_a^2}\left( 3-\Lambda ~ C\right) +\frac{c_a}{\sqrt{C}}, \end{aligned}$$
(42)

with \(c_a\) is a real constant. In order to find solution for B, we can not use the constraint equation (35) since it becomes trivial when we substitute the explicit solution of \(X_a\). We can use the Bogomol’nyi’s equation (50b) to obtain solution for B as follows

$$\begin{aligned} B={4\over cx_a^2}{1 \over A~ C}C'(r)^2. \end{aligned}$$
(43)

So the solution for B depends on explicit function of C(r). Here we can see \(B\propto A^{-1}\).

2.2 Static electrovac

Now consider incorporating electromagnetic field into the Hilbert–Einstein action such that the total action is

$$\begin{aligned} S=\int d^4x \sqrt{-g}\left( {1\over 2\kappa }\left( R-2\Lambda \right) -{1\over 4}F_{\mu \nu }F^{\mu \nu }\right) , \end{aligned}$$
(44)

with \(F_{\mu \nu }=D_\mu A_\nu -D_\nu A_\mu \) and \(D_\mu \) is the covariant derivative. For our convenient we set \(\kappa =1\) from now on. A general static spherical symmetric ansatz for electromagnetic field is given by

$$\begin{aligned} A_\mu =\left( A_t(r),A_r(r), A_\theta \sin \theta , A_\phi \cos \theta \right) , \end{aligned}$$
(45)

with \(A_\theta \) and \(A_\phi \) are real constants. Within this ansatz the function \(A_r(r)\) and constant \(A_\theta \) do not appear anywhere in the action so we may safety set them to zero, \(A_r(r)=A_\theta =0\). Under the ansatzes (27) and (45), the reduced action are given by

$$\begin{aligned} S_{Red}= & {} \int dr\mathcal {L}_{eff}\nonumber \\= & {} \int dr \frac{2 C(r) \left( A'(r) C'(r)+C(r) A_t'(r)^2\right) +A(r) \left( C'(r)^2-2 B(r) \left( A_\phi ^2+2 C(r) (\Lambda C(r)-1)\right) \right) }{\sqrt{A(r) B(r)}C(r)}. \end{aligned}$$
(46)

Again one can check that Euler–Lagrange equations of this effective Lagrangian \(\mathcal {L}_{eff}\) satisfy the Einstein equations and the Maxwell equations, \(D^\mu F_{\mu \nu }=0\). The BPS Lagrangian density is taken to be

$$\begin{aligned} \mathcal {L}_{BPS}= & {} X_0(A,B,C,A_t)+X_a(A,B,C,A_t) A'(r)\nonumber \\ {}{} & {} + X_b(A,B,C,A_t) B'(r)+ X_c(A,B,C,A_t) C'(r)\nonumber \\{} & {} + X_t(A,B,C,A_t) A_t'(r), \end{aligned}$$
(47)

where \(X_0,X_a,X_b,X_c,\) and \(X_t\) are auxilliary functions of A(r), B(r), C(r),  and \(A_t(r)\). The equation \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\) can be rewritten to be

$$\begin{aligned}{} & {} \frac{2 C}{\sqrt{AB}}\left( A_t'(r)-\frac{\sqrt{A B} X_t}{4 C}\right) ^2\nonumber \\{} & {} \quad +\frac{1}{C}\sqrt{A\over B}\left( C'(r)+\frac{1}{2} C \sqrt{\frac{B}{A}} \left( \frac{2 A'(r)}{\sqrt{A} \sqrt{B}}-X_c\right) \right) ^2\nonumber \\{} & {} \quad -\frac{C}{A^{3/2} \sqrt{B}}\left( A'(r)-\frac{\left( A^{3/2} \sqrt{B}\right) }{2 C}\left( \frac{C X_c}{A}-X_a\right) \right) ^2\nonumber \\{} & {} \quad -{\sqrt{AB}\over 8 C} \left( 4 C (X_a X_c+8 \Lambda C-8)+X_t^2+16 A_\phi ^2\right) \nonumber \\ {}{} & {} \quad -X_0+{A^{3/2} \sqrt{B}\over 4 C} X_a^2-B'(r) X_b=0. \end{aligned}$$
(48)

In the BPS limit, the first line of Eq. (48) will give us Bogomol’nyi-like equations for \(A_t, C,\) and A while the second line must zero such that, for the same reason as in the previous case, \(X_b=0\) and thus

$$\begin{aligned} X_c= & {} -\frac{\sqrt{AB} \left( -2 A~ X_a^2+X_t^2+16 \left( A_\phi ^2+2 C (C~ \Lambda -1)\right) \right) +8 C~ X_0}{4 \sqrt{A} \sqrt{B} C~X_a}.\nonumber \\ \end{aligned}$$
(49)

Substituting \(X_b\) and \(X_c\), the Bogomol’nyi-like equations for \(A_t, C,\) and A now become

$$\begin{aligned} A_t'(r)= & {} \frac{\sqrt{A~ B}}{4 C} X_t, \end{aligned}$$
(50a)
$$\begin{aligned} C'(r)= & {} \frac{1}{2} \sqrt{A~B}~ X_a, \end{aligned}$$
(50b)
$$\begin{aligned} A'(r)= & {} -\frac{\sqrt{A~B} \left( 2 A~ X_a^2+X_t^2+16 \left( A_\phi ^2+2 C (C \Lambda -1)\right) \right) }{8 C X_a}\nonumber \\{} & {} + {X_0\over X_a}. \end{aligned}$$
(50c)

The remaining unknown functions \(X_a\) and \(X_t\) can be determined from the Euler–Lagrange equations of BPS Lagrangian density (47). The constraint equation for B yields \(X_0=0\) which further simplifies the constraint equation for \(A_t\) to be \({dX_t\over dr}=0\), or \(X_t=cx_t\) is a real constant. The remaining constraint equations, for A and C, can be simplified to

$$\begin{aligned}{} & {} 8 C~ X_a {\partial X_a \over \partial B}B'(r)\nonumber \\ {}{} & {} \quad =\sqrt{A~B} \left( {\partial X_a \over \partial A} \left( 2 A~ X_a^2+16 A_\phi ^2+32 C (C \Lambda -1)+cx_t^2\right) \nonumber \right. \\{} & {} \qquad \left. +2 X_a \left( -cx_t {\partial X_a \over \partial A_t}-2 C {\partial X_a \over \partial C} X_a+X_a^2\right) \right) . \end{aligned}$$
(51)

Using the Bogomol’nyi-like equations (50), we can recast this constraint equation to be

$$\begin{aligned} {dX_a \over dr}={\sqrt{A~B}\over 4 C}X_a^2={X_a\over 2C} {dC \over dr} \end{aligned}$$
(52)

whose solution is \(X_a=cx_a\sqrt{C}\), with \(cx_a\) is the integration constant. Using the explicit solutions \(X_a\) and \(X_t\), and comparing the Bogomol’nyi-like equations (50a) and (36b) we then obtain

$$\begin{aligned} A_t=c_t-{cx_t \over cx_a}C^{-1/2}, \end{aligned}$$
(53)

with \(c_t\) is the integration constant. Furthermore comparing the Bogomol’nyi-like equations (50b) and (50c), we may obtain a differential equation

$$\begin{aligned} -4 cx_a^2 C^2{dA\over dC}=16 \left( A_\phi ^2+2 C (\Lambda ~C-1)\right) +2 cx_a^2 A~C+cx_t^2\nonumber \\ \end{aligned}$$
(54)

whose solution is given by

$$\begin{aligned} A=\frac{1}{6 cx_a^2 C}\left( 48 A_\phi ^2-32 C (\Lambda ~C -3)+3 cx_t^2\right) +\frac{c_a}{\sqrt{C}}, \nonumber \\ \end{aligned}$$
(55)

with \(c_a\) is a real constant. As in the previous case, the solution for B can be obtained from the Bogomol’nyi equation (36b),

$$\begin{aligned} B={4\over cx_a^2}{1\over A~C}C'(r)^2. \end{aligned}$$
(56)

If we take \(C(r)=r^2\) and set \(cx_a=4,cx_t=4 Q,\) and \(c_a=-2M\) then we get the Reissner–Nordström black hole solutions,

$$\begin{aligned} ds^2= & {} -\left( 1-{r^2\over 3}\Lambda -{2M\over r}+{Q^2+A_\phi ^2\over 2 r^2}\right) dt^2\nonumber \\{} & {} +\left( 1-{r^2\over 3}\Lambda -{2M\over r}+{Q^2+A_\phi ^2\over 2 r^2}\right) ^{-1}dr^2+r^2 d\Omega _2^2, \nonumber \\ \end{aligned}$$
(57)

where M is the black hole mass, Q is the electric charge, and \(A_\phi \) is the magnetic charge.

2.3 Einstein-scalar gravity (V-scalar-vacuum)

An action for four-dimensional Einstein-scalar gravity is given by

$$\begin{aligned} S=\int ~dx^4 \sqrt{-g}\left( {1\over 2\kappa }\left( R-2\Lambda \right) -{1\over 2}D_\mu \phi D^\mu \phi -V(\phi )\right) ,\nonumber \\ \end{aligned}$$
(58)

with V is a generic scalar potential. We take an ansatz for the real scalar field \(\phi \equiv \phi (r)\) such that, together with the ansatz (27), the reduced action is given by

$$\begin{aligned} S_{red}= & {} \int dr\mathcal {L}_{eff}\nonumber \\ {}= & {} \int dr \left( \frac{2 C~ A'(r) C'(r)+ 4 A B C \left( 1- \Lambda C\right) \nonumber +A~ C'(r)^2}{\sqrt{A~B}~C}\nonumber \right. \\{} & {} \left. -\sqrt{A B}C \left( \frac{2}{B}\phi '(r)^2+4V(\phi )\right) \right) . \end{aligned}$$
(59)

In this case we consider BPS Lagrangian density as follows

$$\begin{aligned} \mathcal {L}_{BPS}= & {} X_0(A,B,C,\phi )+X_a(A,B,C,\phi ) A'(r)\nonumber \\ {}{} & {} + X_b(A,B,C,\phi ) B'(r)+ X_c(A,B,C,\phi ) C'(r)\nonumber \\ {}{} & {} +X_\phi (A,B,C,\phi ) \phi '(r), \end{aligned}$$
(60)

where \(X_0,X_a,X_b,X_c,\) and \(X_\phi \) are auxilliary functions of A(r), B(r), C(r),  and \(\phi (r)\). In the BPS limit, \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\) can be rewritten as

$$\begin{aligned}{} & {} -\frac{2\sqrt{A}C}{\sqrt{B}} \left( \phi '(r)+\frac{\sqrt{B}}{4\sqrt{A}C} X_\phi \right) ^2\nonumber \\{} & {} \quad -\frac{C}{A^{3/2} B^{1/2}}\left( A'(r)-\frac{A^{3/2} B^{1/2}}{2 C}\left( \frac{C}{A}X_c -X_a\right) \right) ^2\nonumber \\{} & {} \quad +\frac{A^{1/2}}{B^{1/2}~C}\left( C'(r)+\frac{B^{1/2}C}{2 A^{1/2}}\left( \frac{2}{\sqrt{A~B}}A'(r)-X_c\right) \right) ^2\nonumber \\{} & {} \quad -B'(r) X_b-X_0+\frac{\sqrt{B}}{8 \sqrt{A} C} \left( -4 A C (X_a X_c\nonumber \right. \\{} & {} \quad \left. +8 C (\Lambda +V)-8)+2 A^2 X_a^2+X_\phi ^2\right) =0. \end{aligned}$$
(61)

The first line of Eq. (61) will give us Bogomol’nyi-like equations while the second line must be set to zero such that, similar to previous cases, \(X_b=0\) and

$$\begin{aligned}{} & {} X_c=\frac{1}{4 A C X_a}\nonumber \\ {}{} & {} \times \left( 2 A^2 X_a^2-\frac{8 \sqrt{A} C}{\sqrt{B}} X_0\!+\!X_\phi ^2\!-\!32 A C^2 V\!-\!32 A C ( \Lambda C -1)\right) . \nonumber \\ \end{aligned}$$
(62)

Substituting explicit functions \(X_b\) and \(X_c\), the Bogomol’nyi-like equations for \(\phi , C,\) and A become

$$\begin{aligned} \phi '(r)= & {} -{1\over 4 C}\sqrt{B\over A} X_\phi , \end{aligned}$$
(63a)
$$\begin{aligned} C'(r)= & {} \frac{\sqrt{A~ B}}{2} X_a, \end{aligned}$$
(63b)
$$\begin{aligned} A'(r)= & {} \frac{-2 A^2 \sqrt{B}X_a^2-8 \sqrt{A} C X_0+\sqrt{B}X_\phi ^2-32 A \sqrt{B} C (C (\Lambda +V)-1)}{8 \sqrt{A} C X_a}.\nonumber \\ \end{aligned}$$
(63c)

The remaining unknown functions \(X_0, X_a,\) and \(X_\phi \) can be determined from Euler–Lagrange equations of the BPS Lagrangian density \(\mathcal {L}_{eff}\). The constraint equation for B yields \(X_0=0\). Furthermore using this explicit \(X_0\), the constraint equations for A and \(\phi \) can be simplified to, respectively,

$$\begin{aligned} {X_a}'(r)= & {} -\frac{\sqrt{B}}{8A^{3/2}C}\left( X_\phi ^2-2 A^2 X_a^2\right) , \end{aligned}$$
(64a)
$$\begin{aligned} {X_\phi }'(r)= & {} -4\sqrt{AB} C~ V'(\phi ), \end{aligned}$$
(64b)

These equations further implies the constraint equation for C to be trivial.

Let us consider solution for which \(C=r^2\). In this case solutions are obtained by setting \(X_a=4\sqrt{C\over A~B}\) which further simplifies the Bogomol’nyi-like equation for A and, from the constraint equation (64a), implies a constraint equation for B, respectively

$$\begin{aligned} A'(r)= & {} \frac{1}{32 C^{3/2}}\left( B X_\phi ^2-32 A C (B (C \Lambda -1)+B C V+1)\right) , \nonumber \\ \end{aligned}$$
(65)
$$\begin{aligned} B'(r)= & {} \frac{B}{32 A C^{3/2}}\left( B X_\phi ^2+32 A C (B (C \Lambda -1)+B C V+1)\right) .\nonumber \\ \end{aligned}$$
(66)

Using these equations we can write

$$\begin{aligned} {d\log (AB) \over dr}=\sqrt{C}\phi '(r)^2. \end{aligned}$$
(67)

Now take an example where solution for the scalar field is a Coulomb-like solution where \(\phi ={Q_s\over r}\) with \(Q_s\) is a real constant. This implies \(X_\phi =4\phi \sqrt{A~C\over B}\) and

$$\begin{aligned} {d\log (AB) \over dr}={\phi ^2\over \sqrt{C}}=Q_s^2 {1\over r^3} \end{aligned}$$
(68)

whose solution is

$$\begin{aligned} A=c_a{e^{-{\phi ^2\over 2}}\over B}, \end{aligned}$$
(69)

with \(c_a\) is an integration constant which we will set to unity, \(c_a=1\). Since all the auxiliary functions have been fixed, the remaining equations to be solved, in terms of \(\phi \), are

$$\begin{aligned} B'(\phi )= & {} -\frac{B}{2 \phi ^3} \left( 2Q_s^2 B(\Lambda +V)-2 (B-1) \phi ^2+\phi ^4\right) ,\nonumber \\ V'(\phi )= & {} \phi \left( -\frac{(B-1) \phi ^2}{Q_s^2 B}+\Lambda +V\right) . \end{aligned}$$
(70)

Solutions to these equations are

$$\begin{aligned} B^{-1}= & {} {1\over \phi ^2}\left( e^{\frac{\phi ^2}{4}} \phi \left( -2 \sqrt{\pi } \text {erf}\left( \frac{\phi }{2}\right) \nonumber +c_2 Q_s^2\right) \nonumber \right. \\{} & {} \left. +e^{\frac{\phi ^2}{2}} \left( -\sqrt{\pi } c_2 Q_s^2 \text {erf}\left( \frac{\phi }{2}\right) +\pi \text {erf}\left( \frac{\phi }{2}\right) ^2+2 c_1 Q_s^2\right) -4\right) ,\nonumber \\ \end{aligned}$$
(71)
$$\begin{aligned} V= & {} -\Lambda +{\phi ^2+12\over Q_s^2}-{3\over Q_s^2} e^{\frac{\phi ^2}{4}} \phi \left( -2 \sqrt{\pi } \text {erf} \left( \frac{\phi }{2}\right) +c_2 Q_s^2\right) \nonumber \\{} & {} +{\left( \phi ^2-6\right) \over 2 Q_s^2}e^{\frac{\phi ^2}{2}} \left( -\sqrt{\pi } c_2 Q_s^2 \text {erf}\left( \frac{\phi }{2}\right) +\pi \text {erf}\left( \frac{\phi }{2}\right) ^2+2 c_1 Q_s^2\right) ,\nonumber \\ \end{aligned}$$
(72)

where \(\text {erf}(z)={2\over \sqrt{\pi }}\int _0^z e^{-t^2}dt\) is the Error function, with \(c_1\) and \(c_2\) are constant. At \(r\rightarrow \infty \), we have

$$\begin{aligned} A(r\rightarrow \infty )\sim & {} -\lambda r^2 +1-{Q_s^3 c_2\over 6 r}+O\left( \left( 1\over r\right) ^3\right) , \end{aligned}$$
(73)
$$\begin{aligned} V(r\rightarrow \infty )\sim & {} 3 \lambda -\Lambda +\frac{\lambda Q_s^2}{r^2}+\frac{\lambda Q_s^4}{8 r^4}-\frac{c_2 Q_s^5}{30 r^5}+ O\left( \left( 1\over r\right) ^6\right) ,\nonumber \\ \end{aligned}$$
(74)

with \(\lambda =\frac{4}{Q_s^2}-2 c_1\) is an effective cosmological constant. The existence of asymptotically flat black holes, with \(\lambda =0\), depend on the value of \(c_2\) (Figs. 1, 2).

Fig. 1
figure 1

All curves in the region \(r/Q_s<0\) correspond to \(Q_s<0\). The horizon is larger as we increase (decrease) the value of \(c_2\) for fixed \(Q_s>0\) (\(Q_s<0\)). The horizon radius for \(c_2=|4|/Q_s^2\) is \(r_h\approx 1.03 x 10^{-8} |Q_s|\) and it increases by an order of \(10^7\) for \(c_2=|5|/Q_s^2\), which is \(r_h\approx 0.6~|Q_s|\)

Full size image
Fig. 2
figure 2

In the left figure, with \(\Lambda =0\) and \(Q_s>0\), the potential is always negative in the whole space even though the space is cut-off from below at \(r_h\approx 0.6~Q_s\) in the presence of horizon. However it can be lift-up such that it can have positive value for all \(r>r_h\) when the cosmological constant is negative, \(\Lambda \lesssim -2.2/Q_s^2\), as shown in the right figure

Full size image

3 Einstein–Maxwell-scalar gravity in \(n-\)dimensions

The action for Einstein–Maxwell-scalar gravity is taken to be

$$\begin{aligned} S= & {} \int ~dx^n \sqrt{-g}\nonumber \\ {}{} & {} \times \left( {1\over 2\kappa }\left( R-2\Lambda \right) -{1\over 4}F_{\mu \nu }F^{\mu \nu }-{1\over 2}D_\mu \phi D^\mu \phi -V(\phi )\right) .\nonumber \\ \end{aligned}$$
(75)

the n-dimensional Einstein’s equations, Maxwell’s equations, and Scalar’s equation are respectively given by

$$\begin{aligned} R_{\mu \nu }-{1\over 2}g_{\mu \nu }R+\Lambda g_{\mu \nu }= & {} \kappa \left( {F_\mu }^\gamma F_{\nu \gamma }-{g_{\mu \nu }\over 4}F_{\gamma \sigma }F^{\gamma \sigma }\right. \nonumber \\ {}{} & {} \left. +D_\mu \phi D_\nu \phi -{g_{\mu \nu }\over 2}D_\gamma \phi D^\gamma \phi \right. \nonumber \\ {}{} & {} \left. -g_{\mu \nu } V(\phi )\right) , \end{aligned}$$
(76a)
$$\begin{aligned} D^\mu F_{\mu \nu }= & {} 0, \end{aligned}$$
(76b)
$$\begin{aligned} D^\mu D_\mu \phi= & {} V'(\phi ), \end{aligned}$$
(76c)

with \(D_\mu \) is the covariant derivative. We take ansatz for the n-dimensional metric to be

$$\begin{aligned} ds^2=-A(r)dt^2+B(r)dr^2+C(r)d\Omega _{n-2}^2, \end{aligned}$$
(77)

where \(A,B,C\ge 0\) and \(n>4\). For the metric of unit hyper-sphere, we follow convention in [33] as below:

$$\begin{aligned} d\Omega _{n-2}^2=\sum _{i=1}^{n-2}\left( \prod _{l=1}^{i-1}\sin ^2\theta _l~ d\theta _i^2\right) . \end{aligned}$$
(78)

Here we take ansatz for the n-dimensional gauge field as follows [33, 34]

$$\begin{aligned} A_\mu =\left( A_t(r),0,\dots ,0\right) \end{aligned}$$
(79)

and for the scalar field to be static and spherically symmetric, \(\phi \equiv \phi (r)\). Using those ansatzes, the Eq. (76) can be obtained from the following reduced action

$$\begin{aligned} S_{red}= & {} \int dr\mathcal {L}_{eff} \end{aligned}$$
(80)
$$\begin{aligned}= & {} \int dr \left( \frac{2 (n-2) C~ A'(r) C'(r)+ 4 A B C \left( (n-3)(n-2)-2 \Lambda C\right) +(n-3) (n-2) A~ C'(r)^2}{8\kappa \sqrt{A~B}~C^{3-\frac{n}{2}} }\right. \nonumber \\{} & {} \left. +\sqrt{A B C^{n-2}} \left( \frac{A_t'(r)^2}{2 A B}-\frac{\phi '(r)^2}{2 B}-V(\phi )\right) \right) . \end{aligned}$$
(81)

Here again without loos of generality we may set \(\kappa =1\). We consider the BPS Lagrangian density to be

$$\begin{aligned} \mathcal {L}_{BPS}= & {} X_0(A,B,C,A_t,\phi )+X_a(A,B,C,A_t,\phi ) A'(r)\nonumber \\{} & {} + X_b(A,B,C,A_t,\phi ) B'(r)+ X_c(A,B,C,A_t,\phi ) C'(r)\nonumber \\{} & {} +X_t(A,B,C,A_t,\phi ) A_t'(r)+X_\phi (A,B,C,A_t,\phi ) \phi '(r),\nonumber \\ \end{aligned}$$
(82)

where \(X_0,X_a,X_b,X_c,X_t,\) and \(X_\phi \) are auxilliary functions of A(r), B(r), C(r), At(r),  and \(\phi (r)\). In the BPS limit, \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}=0\) can be rewritten as

$$\begin{aligned}{} & {} \frac{C^{\frac{n}{2}-1}}{2\sqrt{A~B}} \left( A_t'(r)-\frac{\sqrt{A~B}}{C^{\frac{n}{2}-1}} X_t\right) ^2\nonumber \\ {}{} & {} \quad -\frac{\sqrt{A}C^{\frac{n}{2}-1}}{2\sqrt{B}} \left( \phi '(r)+\frac{\sqrt{B}}{\sqrt{A}C^{\frac{n}{2}-1}} X_\phi \right) ^2 \nonumber \\{} & {} \quad -\frac{(n-2) C^{\frac{n}{2}-1}}{8 (n-3) A^{3/2} B^{1/2}}\left( A'(r)\nonumber \right. \\{} & {} \quad \left. -\frac{4(n-3) A^{3/2} B^{1/2}}{(n-2) C^{\frac{n}{2}-1}}\left( \frac{C}{(n-3) A}X_c -X_a\right) \right) ^2\nonumber \\{} & {} \quad +\frac{(n-3) (n-2) A^{1/2}}{8 B^{1/2}~C^{3-\frac{n}{2}}}\left( C'(r)\nonumber \right. \\{} & {} \quad \left. +\frac{4B^{1/2}C^{3-\frac{n}{2}}}{(n-3) (n-2) A^{1/2}}\left( \frac{(n-2) C^{\frac{n}{2}-2}}{4\sqrt{A~B}}A'(r)-X_c\right) \right) ^2\nonumber \\ \quad{} & {} -B'(r) X_b-X_0+\frac{\sqrt{B}~C^{-\frac{n}{2}-2}}{2 (n-2)\sqrt{A}}\nonumber \\ {}{} & {} \left( C^3 \left( A \left( 4 (n-3) A X_a^2-(n-2) X_t^2-8 C X_a X_c \right) +(n-2) X_\phi ^2\right) \nonumber \right. \\{} & {} \quad \left. +(n-2) A C^n \left( -2 C (\Lambda +V)+(n-3)(n-2)\right) \right) =0. \end{aligned}$$
(83)

The first three terms in left hand side of Eq. (83) will give us Bogomol’nyi-like equations while the remaining terms must be set to zero such that \(X_b=0\) and

$$\begin{aligned} X_c= & {} \tfrac{\sqrt{B} \left( 4 A^2 C^3 (n-3) X_a^2+(n-2) \left( C^3 \left( X_\phi ^2-A X_t^2\right) +A C^n ((n-3)(n-2)-2 C \Lambda )\right) \right) -2 \sqrt{A} (n-2) C^{\frac{n}{2}+2}\left( X_0+\sqrt{AB} C^{\frac{n}{2}+2} V\right) }{8 A \sqrt{B} C^4 X_a} \end{aligned}$$
(84)

Substituting \(X_b\) and \(X_c\), the Bogomol’nyi-like equations for \(A_t,\phi ,C,\) and A respectively become

$$\begin{aligned} A_t'(r)= & {} \sqrt{A~ B}~C^{1-{n\over 2}} X_t, \end{aligned}$$
(85a)
$$\begin{aligned} \phi '(r)= & {} -\sqrt{B\over A}C^{1-\frac{n}{2}} X_\phi , \end{aligned}$$
(85b)
$$\begin{aligned} C'(r)= & {} \frac{4\sqrt{A~ B}}{(n-2)}C^{2-{n\over 2}} X_a, \end{aligned}$$
(85c)
$$\begin{aligned} A'(r)= & {} \frac{\sqrt{B}\left( (n-2)A C^{-2+\frac{n}{2}} ((n-3)(n-2)-2 C(V+\Lambda ))-C^{1-\frac{n}{2}} \left( 4 (n-3) A^2 X_a^2+(n-2) \left( A X_t^2-X_\phi ^2\right) \right) \right) }{2 \sqrt{A} (n-2) X_a}-{X_0\over X_a}.\nonumber \\ \end{aligned}$$
(85d)

The remaining auxilliary functions \(X_0, X_a, X_t,\) and \(X_\phi \) can be determined from the Euler–Lagrange equations of BPS Lagrangian density \(\mathcal {L}_{eff}\). The constraint equation for B yields \(X_0=0\) which further implies, from the constraint equation for \(A_t\), \({dX_t\over dr}=0\), or \(X_t=cx_t\) is a real constant. With these additional explicit functions of \(X_0\) and \(X_t\), the constraint equations for A and \(\phi \) implies, respectively,

$$\begin{aligned} {X_a}'(r)= & {} -\frac{\sqrt{B} C^{1-\frac{n}{2}}}{2(n-2)A^{3/2}}\nonumber \\{} & {} \times \left( (n-2) X_\phi ^2-4(n-3) A^2 X_a^2\right) , \end{aligned}$$
(86)
$$\begin{aligned} {X_\phi }'(r)= & {} -\sqrt{AB} C^{-1+\frac{n}{2}} V'(\phi ), \end{aligned}$$
(87)

These equations further implies the constraint equation for C to be trivial.

3.1 Tangherlini black holes

In this case the action only contains the gravity and gauge fields such that we need to remove all terms in the effective and BPS Lagrangian densities that contain the scalar field. This can be done by taking \(\phi \) and its potential V to be zero which also imply \(X_\phi =0\) such that the constraint equation (87) is trivially satisfied and the constraint equation (86) is simplified to

$$\begin{aligned} {dX_a \over dr}= & {} {2(n-3)\over (n-2)}\sqrt{A~B}~C^{1-{n\over 2}} X_a^2\nonumber \\= & {} {(n-3)X_a\over 2C} {dC \over dr} \end{aligned}$$
(88)

whose solution is \(X_a=cx_a C^{n-3\over 2}\), with \(cx_a\) is the integration constant. Using additional explicit function of \(X_a\) and comparing the Bogomol’nyi-like Eqs. (85a) and (85c), we obtain

$$\begin{aligned} A_t=c_t-{(n-2)\over 2(n-3)}{cx_t \over cx_a}C^{-{n-3\over 2}}, \end{aligned}$$
(89)

with \(c_t\) is the integration constant. Further comparing the Bogomol’nyi-like Eqs. (85d) and (85c), we may obtain a differential equation

$$\begin{aligned}{} & {} -8 cx_a^2 C{dA\over dC}=cx_t^2 (n-2)C^{3-n}+(n-3)\nonumber \\{} & {} \quad \times \left( 4Acx_a^2-(n-2)^2\right) +2(n-2) C\Lambda \end{aligned}$$
(90)

whose solution is given by

$$\begin{aligned} A= & {} \frac{(n-2)}{4(n-3)cx_a^2}\left( \frac{cx_t^2}{C^{n-3}}+{(n-3)\over (n-1)} \left( (n-2)(n-1)-2 C \Lambda \right) \right) \nonumber \\ {}{} & {} \quad +{c_a\over C^{n-3\over 2}}, \end{aligned}$$
(91)

with \(c_a\) is a real constant. As in the previous case, the solution for B can be obtained from the Bogomol’nyi-like equation (85c),

$$\begin{aligned} B={(n-2)^2\over 16cx_a^2}{1\over A~C}C'(r)^2. \end{aligned}$$
(92)

If we take \(C(r)=r^2\) and set \(cx_a={(n-2)\over 2},cx_t=-Q,\) and \(c_a=-2M\) then we get the black hole type solutions,

$$\begin{aligned} ds^2= & {} -\left( 1-{2M\over C^{n-3\over 2}}+\frac{Q^2}{(n-3)(n-2)C^{n-3}}-{2C\Lambda \over (n-2)(n-1)}\right) dt^2\nonumber \\ \nonumber \\{} & {} +\left( 1-{2M\over C^{n-3\over 2}}+\frac{Q^2}{(n-3)(n-2)C^{n-3}}-{2C\Lambda \over (n-2)(n-1)}\right) ^{-1}dr^2\nonumber \\{} & {} +r^2 d\Omega _{n-2}^2, \end{aligned}$$
(93)

where M is the black hole mass and Q is the electric charge.

3.2 Three dimensional V-scalar-vacuum

Here \(n=3\) without gauge field, \(A_t=X_t=0\), such that the Bogomol’nyi-like equations for \(\phi ,C,\) and A are, respectively,

$$\begin{aligned} \phi '(r)= & {} -\sqrt{B\over A~C} X_\phi , \end{aligned}$$
(94a)
$$\begin{aligned} C'(r)= & {} 4\sqrt{A~ B~C}~ X_a, \end{aligned}$$
(94b)
$$\begin{aligned} A'(r)= & {} \sqrt{B\over A~C}\frac{1}{2 X_a}\left( X_\phi ^2-2A C(V+\Lambda )\right) , \end{aligned}$$
(94c)

and the constraint equations for A and \(\phi \), respectively,

$$\begin{aligned} {X_a}'(r)= & {} -\frac{1}{2A^{3/2}}\sqrt{B\over C}X_\phi ^2, \end{aligned}$$
(95)
$$\begin{aligned} {X_\phi }'(r)= & {} -\sqrt{A~B~C}~ V'(\phi ). \end{aligned}$$
(96)

As an example we fix the functions \(C=r^2\) and \(\phi ={c_\phi \over r^p}\), with \(p>0\) and \(c_\phi \) is a real constant, such that \(X_a={1\over 2\sqrt{AB}}\) and \(X_\phi =p~\phi \sqrt{A\over B}\). Using these additional explicit functions of \(X_a\) and \(X_\phi \), the constraint equation (95) can be rewritten as

$$\begin{aligned} B'(r)=\frac{B}{\sqrt{C}}\left( 2 B C (V+\Lambda )+p^2\phi ^2\right) . \end{aligned}$$
(97)

Furthermore we find that

$$\begin{aligned} {d\log (A~B)\over dr}=-2p\phi {d\phi \over dr} \end{aligned}$$
(98)

whose solution is \(A={c_a\over B} e^{-p\phi ^2}\), with \(c_a\) is an integration constant. Without loos of generality we can set \(c_a=1\). On the other hand the constraint equation (96) can also be rewritten, by using \(V'(\phi )={V'(r)\over \phi '(r)}\), as

$$\begin{aligned} V'(r)=-{p^2\phi ^2\over \sqrt{C}}\left( {p\over B~C}+2(V+\Lambda )\right) . \end{aligned}$$
(99)

What remain to be solved are the constraint equations (51) and (99) whose solutions for \(p=1\) are given by,

$$\begin{aligned} B(r)= & {} \frac{c_b}{r^2 e^{\frac{c_\phi ^2}{2 r^2}}\left( 1+c_\phi ^2 c_v e^{\frac{c_\phi ^2}{2r^2}}\right) }, \end{aligned}$$
(100)
$$\begin{aligned} A(r)= & {} \frac{r^2}{c_b} e^{-\frac{c_\phi ^2}{2 r^2}}\left( 1+c_\phi ^2 c_v e^{\frac{c_\phi ^2}{2r^2}}\right) , \end{aligned}$$
(101)
$$\begin{aligned} V(r)= & {} -\Lambda -{e^{\frac{c_\phi ^2}{2 r^2}}\over c_b}\left( 1+c_\phi ^2 c_v e^{\frac{c_\phi ^2}{2 r^2}}\right) +\frac{c_\phi ^4 c_v }{2 c_b}{e^{\frac{c_\phi ^2}{r^2}}\over r^2}, \end{aligned}$$
(102)

with \(c_b>0\) and \(c_v<0\) are real constants. Here black hole solutions exist if \(c_\phi ^2c_v>-1\), with the effective cosmological constant \(\lambda =-{1+c_\phi ^2c_v\over c_b}\), and hence there are only asymptotically Anti-de Sitter black holes. These solutions are equal to solutions found in [35], for \(\Lambda =0\), and has been discussed there in great detail. For \(p=2\) and \(\Lambda =0\), the exact solutions are given by

$$\begin{aligned} B(r)= & {} \frac{c_b e^{-\frac{2 c_\phi ^2}{r^4}}}{r^2 \left( -\sqrt{\pi }~ \text {erf}\left( \frac{c_\phi }{r^2}\right) +4 c_v c_\phi \right) }, \end{aligned}$$
(103)
$$\begin{aligned} A(r)= & {} \frac{r^2}{c_b}\left( -\sqrt{\pi }~ \text {erf}\left( \frac{c_\phi }{r^2}\right) +4 c_v c_\phi \right) , \end{aligned}$$
(104)
$$\begin{aligned} V(r)= & {} -{2 c_\phi \over c_b r^2} e^{\frac{c_\phi ^2}{r^4}}+{e^{\frac{2 c_\phi ^2}{r^4}}\over c_b r^4} \left( 2 c_\phi ^2-r^4\right) \left( -\sqrt{\pi }~ \text {erf}\left( \frac{c_\phi }{r^2}\right) +4 c_v c_\phi \right) ,\nonumber \\ \end{aligned}$$
(105)

where \(c_v>0\) and \(c_b\ne 0\) are real constants, with \(\text{ sign }(c_b)=\text{ sign }(c_\phi )\). The black holes exist if \(4c_v|c_\phi |<\sqrt{\pi }\), with the effective cosmological constant \(\lambda =-{4c_\phi c_v\over c_b}\), and thus again there are only asymptotically Anti-de Sitter black holes (Figs. 3, 4). Near the boundary, \(r\rightarrow \infty \), the solutions behave as

$$\begin{aligned} A(r\rightarrow \infty )\sim & {} -\lambda r^2 -{2 c_\phi \over c_b}+O\left( \left( 1\over r\right) ^3\right) , \end{aligned}$$
(106)
$$\begin{aligned} V(r\rightarrow \infty )\sim & {} c_\phi \lambda -{8 c_\phi ^4 \over 3 c_b r^6} + O\left( \left( 1\over r\right) ^7\right) . \end{aligned}$$
(107)
Fig. 3
figure 3

The horizon is determined by the value of \(c_v c_\phi \), with fixed \(c_b/c_\phi \). Its radius is larger as we increase \(c_v c_\phi \)

Full size image
Fig. 4
figure 4

Here we can see the horizon radius does not change if we vary the value of \(c_b/c_\phi \). For \(c_v=0.1/c_\phi \), the horizon radius is \(r_h\approx 2.22088\sqrt{c_\phi }\)

Full size image

4 Remarks

The first important step in the BPS Lagrangian method is to determine the effective Lagrangian density. For general solutions, without a priori imposing any ansatz, the effective Lagrangian density is simply its original Lagrangian density multiplying with the square root of determinant of the metric tensor. However in many cases, such as discussed in this article, it is simpler to work on solutions under particular ansatz directly. In these cases we need to identify the effective Lagrangian density and its effective fields such that its Euler–Lagrange equations, for the effective fields, are effectively equal to the original Euler–Lagrange equations for the fundamental fields written under the corresponding ansatz. For the cases considered here the effective Lagrangian density for gravity parts are simply obtained by imposing the ansatz (27), after identifying the effective fields, into the action (26). Knowing the correct effective fields is important for deriving the constraint equations from the BPS Lagrangian density.

In the case of static spherically symmetric vacuum, in Sect. 2.1.2, and of electrovac, in Sect. 2.2, both solutions for B, (43) and (56) respectively, are given by the same function of C(r) and \(C'(r)\) while the remaining functions can be written as functions of C. Therefore once the function C is fixed the other functions can be determined completely. Taking \(C=r^2\) gives us the Schwarzschild and Reissner–Nordström black holes, and hence other solutions of C are related to those black holes. To see this, using (55) and (56), let us rewrite the metric (27), by taking \(C=\bar{c}^2\) and \(C'(r)^2 dr^2=4 \bar{c}^2 d\bar{c}^2\), to be

$$\begin{aligned} ds^2=-A(\bar{c})dt^2+{16\over cx_a^2}{1\over A(\bar{c})}d\bar{c}^2+\bar{c}^2d\Omega _2^2. \end{aligned}$$
(108)

Here we have written the metric to be independent of radial coordinate. Suppose \(\bar{c}=r\) in a radial coordinate r such that the metric becomes Reissner–Nordström black hole (57) and in another radial coordinate R we have \(\bar{c}=f(R)\ne R\) such that the metric is given by another black hole

$$\begin{aligned} ds^2=-A(f(R))dt^2+{16\over cx_a^2}{1\over A(f(R))}\left( {\partial f\over \partial R}\right) ^2dR^2+f(R)^2d\Omega _2^2\nonumber \\ \end{aligned}$$
(109)

which is related to Reissner–Nordström black hole (57) by a radial coordinate transformation \(r=f(R)\). Here by using the BPS Lagrangian method we have shown a simple alternative proof of the black hole uniqueness theorems [36, 37]. A rather complete proof of those black hole uniqueness theorems would be done by considering all possible terms in the BPS Lagrangian density, \(\mathcal {L}_{BPS}\). In Sect. 3.1 we found that static spherically symmetric electrovac in higher dimensions \(n>4\) are completely solved in terms of functions C(r) and \(C'(r)\), and hence other solutions are related to the Tangherlini black holes in which we fixed \(C=r^2\). Again using similar arguments as in the four-dimensional case (Reissner–Nordström black hole), since B in (92) is equal to (56) up to a constant, we can show a simple (incomplete) proof of the black hole uniqueness theorem in higher dimensions as proposed in [38]. On the other hand, the solutions for \(V-\)scalar vacuum in section (2.3) can not be completely determined by functions C(r) and \(C'(r)\). Even after we fixed \(C=r^2\), we still need to fix the form of scalar field \(\phi (r)\) and then solve remaining first-order differential equations (70). One can easily check that the Schwarzschild black hole, with \(B=\left( 1-{r^2\over 3}\Lambda -{2M\over r}\right) ^{-1}\), is not solution to those first-order differential equations. This may show a violation of the black hole“no-scalar-hair” theorem formulated in [39]. The existance of these scalar hair black holes have been studied numerically for asymptotically flat metric, with \(\Lambda =0\), in [40,41,42]. In Sect. 2.3 we indeed found exact solutions of these scalar hair black holes given in (71). Although the scalar potential has similar profile, with negative value in whole space, as asymptotically AdS-black holes found in [43], we nevertheless able to show that there exist asymptotically flat black holes in the Einstein-scalar gravity (58). We also found exact solutions (103) of scalar hair black holes in three dimensional spacetime other then the one found in [35] for \(n=2\).

In this article, we only considered the BPS Lagrangian density that are linear in first-order derivative of the effective fields. The most general BPS Lagrangian density that we can consider, e.g. for the effective Lagrangian density (59), is

$$\begin{aligned} \mathcal {L}_{BPS}=\sum ^2_{\begin{array}{c} i=0\\ j+k+l+m=i \end{array}} X_{ijklm}(A,B,C) ~A'(r)^j B'(r)^k C'(r)^l\phi '(r)^m \nonumber \\ \end{aligned}$$
(110)

that would allow us to completing the square \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}\) in \(A'(r),B'(r),C'(r),\) and \(\phi '(r)\). However the maximum auxilliary functions \(X_{ijkl}\) that can be considered is five otherwise the set of equations could be underdetermined since there are only four constraint equations and one equation from the remaining terms in \(\mathcal {L}_{eff}-\mathcal {L}_{BPS}\) after competing the square. Using this general BPS Lagrangian density, we may able to obtain a relation for functions A and B, that is \(A=1/B\), which is usually assumed in some of literatures on scalar hair black holes [44].

The BPS Lagrangian opens up a new way to find gravity, or in particular black hole, solutions. We have shown in detail how this method works for Einstein gravity theory. With its systematic procedures, we believe the BPS Lagrangian method can be employed to find gravity solutions in modified gravity theories such as scalar–tensor, Bimetric, f(R), Hořava–Lifschitz, and others [45]. So far we have discussed only static spherically symmetric black hole solutions. One can also consider stationary black hole solutions by means of taking diagonal metric with an additional cross-term component \(dtd\varphi \) between the time t and the azimuthal angle \(\varphi \), or namely stationary spacetime in spherical-type coordinates, with all the metric components depend on the radial r and the polar \(\theta \) coordinates. In this way the effective Lagrangian density will depend on these two coordinates and so the BPS Lagrangian density. Since the uniqueness of stationary black hole has been long proven in [46,47,48], while the extended charge version in [49, 50], we expect to show similar conclusion using the BPS Lagrangian method. As we mentioned in the case of static spherically symmetric black holes, in the context of BPS Lagrangian method this uniqueness theorems indicated by all the metric components and the U(1) gauge field being functions of one of the metric components and its derivatives. Furthermore we need to check that different solutions are related by coordinate transformations. This will be done in the future work. Other application of the BPS Lagrangian method is to find BPS and non-BPS solutions in supersymmetric theories. An example of this has been shown in the case of \(\mathcal {N}=2\) Supersymmetric Baby–Skyrme model [51]. It will be interesting to extend its application to supergravity theories.