Bogomol’nyi-like Equations in Gravity Theories

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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 propotional 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 equations [5].
The Bogomol'nyi trick does not have a systematic procedure, and hence is hard to be employed to more general models.There are several methods have been developed in order to obtain the Bogomol'nyi equations: Strong Necessary Condition [6,7], firstorder formalism [8][9][10], On-Shell method [11,12], FOEL (First-Order Euler-Lagrange) formalism [13], and BPS 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].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 [15] [16].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 which been applied to many field theories in various dimensions [17][18][19][20][21][22][23][24].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].
Let us consider an action of N scalar fields φ m ≡ (φ 1 , . . ., φ N ) in (d + 1)-dimensions of spacetime, with and µ = 0, 1, . . ., d is the spacetime index.Similar to Bogomol'nyi's trick, we shall rewrite the Lagrangian density into complete squared terms as such where with ∂ µ φ 0 m ≡ {∂ µ φ m } − ∂ ν φ n for fixed ν and n.Here 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 L e f f − L BPS = 0 and all terms in the "squared terms" are set to zero as such ∂ ν φ n = f n (φ m , ∂ µ φ 0 m ) 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 = − d d+1 x L e f f , and in the BPS limit the energy of BPS solitons are determined by difference between values of a function Q ≡ Q(φ m ), there is called BPS energy functional, on the boundary and on the origin, Here L BPS is a linear function of first-order derivative of fields φ 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 φ ′ m (r).In general there are also possible boundary terms contain ∂ µ φ m with power higher than one.As an example for N = 3 and d = 3, we may have L BPS with boundary terms as follows [13] where [nop] (φ 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 L BPS may also contain non-boundary terms such that its Euler-Lagrange equations, are not trivially satisfied and must be considered as additional constraint equations in finding solutions to the Bogomol'nyi equations.
Let us see how to employ explicitly the BPS Lagrangian method to the SU(2) Yang-Mills-Higgs model with the following Lagrangian density with , with a = 1, 2, 3 and τ a are the Pauli matrices.Rewrite it in terms of electric field strength E i = F 0i and magnetic field strength B i = 1 2 ε i jk F jk , with i = 1, 2, 3, the effective Lagrangian density is given by where the scalar potential V ≥ 0 is still arbitrary.Now consider a BPS Lagrangian density as follows where α, β , and γ are arbitrary constants.From both Lagrangian densities, we may obtain This form can be obtained by considering L e f f − L BPS as quadratic equation and completing the square in E i , B i , D 0 Φ, and D i Φ subsequently.In the BPS limit, L e f f − L BPS = 0, first three terms on the right hand side give us the Bogomol'nyi equations, For the fourth term, we can not take D i Φ = 0 otherwise it will lead us to trivial solution and thus we must set γ = 1 + α 2 − β 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 L BPS , respectively, for fields Φ, A i , and A 0 are simplied to There are two possible solutions without additional constraint equations: • BPS monopoles: α = 0, β = ±1.

II. FOUR-DIMENSIONAL GRAVITY
The four-dimensional Einstein-Hilbert action where Λ is the cosmological constant and κ = 8πG 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 [25,26].The total action is given by where h ab is the induced metric; K is the trace of the extrinsic curvature; and ε = +1 if the normal of boundary manifold ∂ M is spacelike and ε = −1 if the normal of boundary manifold ∂ M is timelike, with y a are the coordinates on the boundary manifold ∂ M .Here the last term is added to remove singularity part of the Gibbons-Hawking-York terms.

A. Static spherically symmetric
In this case the ansatz for the four-dimensional metric is where A, B,C ≥ 0. Non-trivial parts of the Einstein equations, R µν − 1 2 g µν R + Λg µν = 0, in this ansatz are simplified to 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 (18).To do so, we use the Gaussian normal coordinates and pick a boundary manifold as timelike hypersurface, with spacelike normal vector or ε = 1, at spatial infinity assumed to be at r → ∞.
The reduced Einsten-Hilbert action is given by One can check that the Euler-Lagrange equations of this reduced action are indeed the Einstein equations (18).Derivation of Euler-Lagrange equations ( 18) from the reduced action ( 19) 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 follows [27] where X 0 , X a , X b , and X c are auxilliary functions of A, B, and C. Our task now is to find explicit form of these functions.We then rewrite L e f f − L BPS as follows In the BPS limit, where L e f f − L BPS = 0, we can extract Bogomol'nyi-like equations for A and C, respectively, and the remaining terms is 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 Next, we must also consider additional constraint equations that are Euler-Lagrange equations of S = drL BPS .The constraint equation for B yields X 0 = 0.The remaining constraint equations can be simplified to 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 ( 22) can be simplified to which, together with the constraint equation ( 25), satisfy the Einstein equations (18).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.
In this case the solution for X a , from the Bogomol'nyi-like equation (26b), is given by X a = 4r √ AB .With this explicit function of X a , the Bogomol'nyi-like equation for A (26a) and the contraint equation ( 25) are now From the ratio A ′ (r)/B ′ (r), we may conclude the functions A and B are related by A = c a B , with c a is a real constant.Solutions to A and B are which are the Schwarzschild solutions.In particular the Schwarzschild black hole is obtained by setting Λ = 0, c a = 1, and c b = −2M, where M is the mass of gravitational source.

General solutions
Using the Bogomol'nyi-like equations (26), we can recast the constraint equation ( 25) to be Comparing it with the Bogomol'nyi-like equation for C (26b), we obtain X a = cx a √ C, with cx a is a real constant.Furthermore using explicit solution of X a and comparing both Bogomol'nyi-like equations (26), we may obtain a differential equation whose solution is with c a is a real constant.In order to find solution for B, we can not use the constraint equation ( 25) since it becomes trivial when we substitute the explicit solution of X a .We can use the Bogomol'nyi's equation (26b) to obtain solution for B as follows So the solution for B depends on explicit function of C(r).Here we can see B ∝ A −1 .

B. Static Electrovac
Now consider incorporating electromagnetic field into the Hilbert-Einstein action such that the total action is with F µν = D µ A ν − D ν A µ and D µ is the covariant derivative.For our convinient we set κ = 1 from now on.A general static spherical symmetric ansatz for electromagnetic field is given by with A θ and A φ are real constants.Within this ansatz the function A r (r) and constant A θ do not appear anywhere in the action so we may safetly set them to zero, A r (r) = A θ = 0.Under the ansatzes ( 17) and ( 35), the reduced action are given by Again one can check that Euler-Lagrange equations of this effetive Lagrangian L e f f satisfy the Einstein equations and the Maxwell equations, D µ F µν = 0.The BPS Lagrangian density is taken to be 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 L e f f −L BPS = 0 can be rewritten to be In the BPS limit, the first line of equation (38) 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 Substituting X b and X c , the Bogomol'nyi-like equations for A t ,C, and A now become The remaining unknown functions X a and X t can be determined from the Euler-Lagrange equations of BPS Lagrangian density (37).The constraint equation for B yields X 0 = 0 which further simplifies the constraint equation for A t to be dX t dr = 0, or X t = cx t is a real constant.The remaining constraint equations, for A and C, can be simplified to Using the Bogomol'nyi-like equations (40), we can recast this constraint equation to be whose solution is X a = cx a √ C, with cx a is the integration constant.Using the explicit solutions X a and X t , and comparing the Bogomol'nyi-like equations (40a) and (40b) we then obtain with c t is the integration constant.Furthermore comparing the Bogomol'nyi-like equations (40b) and (40c), we may obtain a differential equation whose solution is given by with c a is a real constant.As in the previous case, the solution for B can be obtained from the Bogomol'nyi equation (40b), If we take C(r) = r 2 and set cx a = 4, cx t = 4Q, and c a = −2M then we get the Reissner-Nordström black hole solutions, where M is the black hole mass, Q is the electric charge, and A φ is the magnetic charge.

C. Einstein-Scalar Gravity
An action for four-dimensional Einstein-Scalar gravity is given by with V is a generic scalar potential.We take an ansatz for the real scalar field φ ≡ φ (r) such that, together with the ansatz (17), the reduced action is given by In this case we consider BPS Lagrangian density as follows where X 0 , X a , X b , X c , and X φ are auxilliary functions of A(r), B(r),C(r), and φ (r).In the BPS limit, L e f f − L BPS = 0 can be rewritten as The first line of equation (51) 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 Substituting explicit functions X b and X c , the Bogomol'nyi-like equations for φ ,C, and A become The remaining unknown functions X 0 , X a , and X φ can be determined from Euler-Lagrange equations of the BPS Lagrangian density L e f f .The constraint equation for B yields X 0 = 0. Furthermore using this explicit X 0 , the constraint equations for A and φ can be simplified to, respectively, These equations further implies the constraint equation for C to be trivial.
This type of solutions are obtained by setting X a = 4 C A B which further simplifies the Bogomol'nyi-like equation for A and , from the constraint equation (54a), implies a constraint equation for B, respectively Using these equations we can write d log(AB) a. Coulomb-like form: φ = Q s r , with a real constant Q s .This implies X φ = 4φ A C B and whose solution is with c a is an integration constant.Since all the auxiliary functions have been fixed, the remaining equations to be solved, in terms of φ , are whose solution is with c a is an integration constant.The remaining equations to be solved are

III. EINSTEIN-MAXWELL-SCALAR GRAVITY IN n−DIMENSIONS
The action for Einstein-Maxwell-Scalar gravity is taken to be the n-dimensional Einstein's equations, Maxwell's equations, and Scalar's equation are respectively given by with D µ is the covariant derivative.We take ansatz for the n-dimensional metric to be where A, B,C ≥ 0 and n > 4. For the metric of unit hyper-sphere, we follow convention in [28] as below: Here we take ansatz for the n-dimensional gauge field as follows [28,29] and for the scalar field to be static and spherically symmetric, φ ≡ φ (r).Using those ansatzes, the equations (64) can be obtained from the following reduced action Here again without loos of generality we may set κ = 1.We consider the BPS Lagrangian density to be where X 0 , X a , X b , X c , X t , and X φ are auxilliary functions of A(r), B(r),C(r), At(r), and φ (r).In the BPS limit, L e f f − L BPS = 0 can be rewritten as The first three terms in left hand side of equation (71) will give us Bogomol'nyi-like equations while the remaining terms must be set to zero such that X b = 0 and ) Substituting X b and X c , the Bogomol'nyi-like equations for A t , φ ,C, and A respectively become The remaining auxilliary functions X 0 , X a , X t , and X φ can be determined from the Euler-Lagrange equations of BPS Lagrangian density L e f f .The constraint equation for B yields X 0 = 0 which further implies, from the constraint equation for A t , dX t 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 φ implies, respectively, These equations further implies the constraint equation for C to be trivial.

A. 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 φ and its potential V to be zero which also imply X φ = 0 such that the constraint equation ( 75) is trivially satisfied and the constraint equation ( 74) is simplified to whose solution is X a = cx a C n−3 2 , with cx a is the integration constant.Using additional explicit function of X a and comparing the Bogomol'nyi-like equations (73a) and (73c), we obtain with c t is the integration constant.Further comparing the Bogomol'nyi-like equations (73d) and (73c), we may obtain a differential equation whose solution is given by 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 (73c), If we take C(r) = r 2 and set cx a = (n−2) 2 , cx t = −Q, and c a = −2M then we get the black hole type solutions, where M is the black hole mass and Q is the electric charge.

B. Three dimensional V -scalar vacuum
Here n = 3 without gauge field, A t = X t = 0, such that the Bogomol'nyi-like equations for φ ,C, and A are, respectively, and the constraint equations for A and φ , respectively, As an example we fix the functions C = r 2 and φ = c φ r , with c φ is a real constant, and Λ = 0 as in [30] such that X a = 1 2 √ AB and X φ = φ A B .Using these additional explicit functions of X a and X φ , the constraint equation (83) can be rewritten as Furthermore we find that whose solution is A = c a B e −φ 2 , with c a is an integration constant.On the other hand the constraint equation (84) can also be rewritten, by using V ′ (φ ) = V ′ (r) φ ′ (r) , as What remain to be solved are the constraint equations ( 85) and (87) whose one of the solutions are given in [30], with c b and c v are real constants.

IV. 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 multipliying 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 direclty.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 (17), after identifying the effective fields, into the action (16).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 section (II A 2), and of electrovac, in section (II B), both solutions for B, (33) and (46) 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.Here by using the BPS Lagrangian method we have shown a simple alternative proof of the black hole uniqueness theorems [31,32].A rather complete proof of those black hole uniqueness theorems would be done by considering all possible terms in the BPS Lagrangian density, L BPS .In section (III A) 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 .Here again we may have shown a simple (incomplete) proof of the black hole uniqueness theorem in higher dimensions as proposed in [33].On the other hand, the solutions for V −scalar vacuum in section (II C) can not be completely determined by functions C(r) and C ′ (r).Even after we fixed C = r 2 and the scalar field φ ≡ φ (r), there were stil remaining first-order differential equations, (59) or (62), that need to be solved.One can easily check that the Schwarzschild black hole, with , is not solutions to those first-order differential equations.This may show a violation of the "no-scalar-hair" theorem formulated in [34].There might be black hole solutions that are not Schwarzschild such as the one found in [30] for n = 3, which is also solution to the equations (85) and (87) in section (III B).
In this article, we only considered the BPS Lagrangian densities 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 (49), is X i jklm (A, B,C) A ′ (r) j B ′ (r) k C ′ (r) l φ ′ (r) m (91) that would allow us to completing the square L e f f − L BPS in A ′ (r), B ′ (r),C ′ (r), and φ ′ (r).However the maximum auxilliary functions X i jkl 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 L e f f − 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 many of literatures about black holes with scalar hair [35,36].Unfortunately this relation can not be realized using the linear BPS Lagrangian density (50) since it leads to relations (58) and (61).