New Conformally Invariant Born–Infeld Models and Geometrical Currents

Abstract

A new conformally invariant gravitational generalization of the Born–Infeld (BI) model is proposed and analyzed from the point of view of symmetries. Taking a geometric identity involving the determinant functions det$f\left(B_{\mu \nu },F_{\mu \nu }\right)$ with the Bach $B_{\mu \nu }$ and the electromagnetic field $F_{\mu \nu }$ tensors (with the 4-dimensional Greek letter indexes), two characteristic geometrical Lagrangian densities (Lagrangians) are derived: the first Lagrangian being the square root of the determinant function $\sqrt{\left|det\left(B_{\mu \nu }+F_{\mu \nu }\right)\right|}$ (reminiscent of the standard BI model) and the second Lagrangian being the fourth root $\sqrt[4]{gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }{F}_{\beta }^{\gamma }\right)}$. It is shown, after explicit computation of the gravitational equations, that the square-root model is incompatible with the inclusion of the electromagnetic tensor, consequently forcing the nullity of $F_{\mu \nu }$. In sharp contrast, the traceless fourth-root model is fully compatible and a natural ansatz of the type $B_{\mu \rho }{B}_{\nu }^{\rho }\propto \mathrm{\Omega }\left(x\right)g_{\mu \nu }$ (conformal-Killing), with $\mathrm{\Omega }$ the conformal factor and x the 4-coordinate, can be considered. Among other essential properties, the geometrical conformal Lagrangian of the fourth-root type is self-similar with respect to the determinant g of the metric tensor $g_{\mu \nu }$ and can be extended to non-Abelian fields in a way similar to the model developed by the author earlier. This self-similarity is related to the conformal properties of the model, such as the Bach currents or flows presumably of a topological origin. Possible applications and comparisons with other models are briefly discussed.

1. Introduction

Born–Infeld (BI) theory [1,2,3,4] has a particular appeal due to its simplicity and its serving as the basis for any unified theory. From the first investigations concerning obtaining solutions within the context of general relativity (GR) [5] and hydrodynamics [6] to the most complex aspects in string theory [7], D-branes [8], and supergravity [9], BI models are mainly relevant modern physics. From the point of view of exact solutions, the possibility of obtaining singularity-free solutions [10] (regularity concept) also allowed for establishing the equivalence between electromagnetic mass and gravitational mass. Later on, the possibility of a non-Abelian extension of the model was considered: one of the problems that occasionally reappears is the lack of a trace prescription, which removes the uniqueness of the Lagrangian of the model. Interestingly, in Ref. [11] it was shown that at least for SU(2) as a gauge group, the simple enough trace prescription was plausible. However, in Ref. [12] a non-Abelian BI model was considered starting from a geometric equivalence at the determinant level that established a fourth-root type Lagrangian density (Lagrangian): this fulfilled all the required topological conditions; the scalar invariant of the electromagnetic field is included as the basic block (proportional, for example, to the Yang-Mills action) and was remarkedly supersymmetrizable, and all this was possible because the fundamental building block is proportional to Maxwell’s energy–momentum tensor. In this paper, a new conformally invariant BI model is proposed and is fundamentally based on determinant functions depending on the Bach tensor $B_{\mu \nu }$ and the electromagnetic field $F_{\mu \nu }$ as det$f\left(B_{\mu \nu },F_{\mu \nu }\right)$, up to the fourth root, namely,

$$\sqrt[4]{gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }{F}_{\beta }^{\gamma }\right)},$$

(1)

with the 4-dimensional Greek letter indexes taken values 0 for the temporal and 1, 2, and 3 for the space components, g being the determinant of the metric tensor $g_{\mu \nu }$.

The reasons for the choice (1), as explained throughout this paper in detail, have to do with consistency with the dynamical (gravitational) equations that rule out a “square root” Lagrangian (reminiscent of that of standard BI theory). Another less fundamental but quite remarkable point at present is that in recent studies that consider $T\overline{T}$ [13], the stress tensor deformations, highlight the fourth-root Lagrangian [12] as the most plausible non-Abelian extension to be obtained in some approximation by the “$T\overline{T}$” flow equations. Additionally, the conformal invariance of the model, considered in this paper to be of primary importance, can establish a connection with more-consistent “Bach-flow” equations.

The paper is organized as follows. A new conformally invariant BI Lagrangian is proposed and it is based on a fourth-root type Lagrangian whose details are described in Section 3. Previously, in Section 2, the reasons why a square-root type Lagrangian is inconsistent are provided. Section 4 shows a crucial simplification that indicates that this fourth-root Lagrangian has several significant properties, such as a certain self-similarity and being proportional to the trace of the energy–momentum tensor. In Section 5 the topological bounds and the dynamics of the fields are given. Finally, Section 6 provides the concluding remarks and outlook.

2. New Born–Infeld Model and Its Conformal Generalization

From the point of view of the action defined through the determinant of a fundamental tensor, in previous paper by the author [12], it was defined for a proposed non-Abelian BI Lagrangian that is purely determinantal, without trace prescriptions, independent of any gauge group, and supersymmetrizable, going to the standard BI Lagrangian in the Abelian limit. Mathematically, for tensors represented by matrices or hypermatrices, the algebraic invariant associated with a matrix may be obtained as traces of the powers of the given matrix. The possibility to construct the corresponding Lagrangian consequently considers, as a central point, the computation of the determinant. According to the Cayley–Hamilton theorem [14,15,16], only a finite number of these powers are linearly independent and, therefore, only a finite number of invariants are linearly independent. A more appropriate set of invariants is given by the discriminants that are suitable combinations of traces and are constructed in terms of alternating products with the unit matrix $\mathbb{I}$. As it is shown in this paper, the conceptual task to link a conformally invariant tensor with the electromagnetic field within the theoretical context of GR results in the non-trivial problem of defining the determinant of a function of a tensor object of type (1, 3) with another bivector of type (0, 2). The direct solution in this GR context is given by looking for a conformally invariant symmetric second-order tensor: the tensor that is here looking for in the first instance is the Bach tensor [17]. Consequently, the sum of the Bach tensor and the bivector corresponding to the electromagnetic field can be a solution. Specifically, a plausible proposal is (regarding conventions ($+++$) for the space components, the polynomial under the root must start with terms such as $1-F_{0i}^2/b^2+\dots$ when the magnetic components are zero):

$$S_{\mathrm{CBI}}=\int d^{4}x\sqrt{\left|det\left(B_{\mu \nu }+F_{\mu \nu }\right)\right|},$$

(2)

where, in order to find the conformal analog to the standard BI (CBI) action, the tensor under the square root, namely, ${\alpha }_{\mu \nu }=b{B}_{\mu \nu }+F_{\mu \nu }$, is formed by the Bach tensor $B_{\mu \nu }$ and the electromagnetic field $F_{\mu \nu }$. The absolute BI field b homogenizes the units and plays a role of the limit of the electromagnetic field strength. The parameter b is given explicitly in expressions where it is relevant to the consideration, and is set to $b=1$ in other cases. Apparently $S_{\mathrm{CBI}}$ (2) is the simplest combination consistent with the determinant of a rank-two tensor, the basis of a BI model in the context of GR. In this case, the determinant is obtained by the method performed in Refs. [12,15] (and the references therein), namely (see Appendix A),

$$\begin{array}{c}det{\alpha }_{\mu \nu }={\displaystyle \frac{g}{4!}}\left[{〈\alpha 〉}^4-6〈{\alpha }^2〉{〈\alpha 〉}^2+8〈\alpha 〉〈{\alpha }^3〉+3{〈{\alpha }^2〉}^2-6〈{\alpha }^4〉\right]={\displaystyle \frac{g}{8}}\left({〈{\alpha }^2〉}^2-2〈{\alpha }^4〉\right)\\ ={\displaystyle \frac{g}{8}}\left[{\left(B^{\alpha \beta }{B}_{\alpha \beta }+F^{\alpha \beta }{F}_{\alpha \beta }\right)}^2-2\left(B^{\alpha \beta }{B}_{\beta \gamma }{B}^{\gamma \delta }{B}_{\delta \alpha }+F^{\alpha \beta }{F}_{\beta \gamma }{F}^{\gamma \delta }{F}_{\delta \alpha }+4B^{\alpha \beta }{B}_{\beta \gamma }{F}^{\gamma \delta }{F}_{\delta \alpha }+2B^{\alpha \beta }{F}_{\beta \gamma }{B}^{\gamma \delta }{F}_{\delta \alpha }\right)\right],\end{array}$$

(3)

so that the BI Lagrangian density (Lagrangian) reads

$${\mathcal{L}}_{\mathrm{BI}}=\sqrt{det{\alpha }_{\mu \nu }}=\sqrt{g}\underset{L_{\mathrm{BI}}}{\underbrace{\sqrt{{\displaystyle \frac{1}{8}}\left({〈{\alpha }^2〉}^2-2〈{\alpha }^4〉\right)}}},$$

(4)

where $L_{\mathrm{BI}}$ denotes the BI Lagrangian.

Note that the form of the Lagrangian (4) is general for any traceless 4-dimensional tensor object.

Gravitational Equations and Symmetries

Now let us test the consistency of the model with the square-root Lagrangian given in (4). Varying with respect to the metric, the dynamic gravitational equations are obtained:

$${\displaystyle \frac{\delta \left(\sqrt{g}{L}_{\mathrm{BI}}\right)}{\delta g^{\alpha \beta }}}={\displaystyle \frac{\sqrt{g}}{2}}\left[-g_{\alpha \beta }{L}_{\mathrm{BI}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{〈{\alpha }^2〉{\alpha }^{\mu \nu }-2{\left({\alpha }^3\right)}^{\mu \nu }}{L_{\mathrm{BI}}}}{\displaystyle \frac{\delta B_{\mu \nu }}{\delta g^{\alpha \beta }}}\right]=0,$$

(5)

$${\displaystyle \frac{\delta B_{\alpha \beta }}{\delta g^{\rho \sigma }}}={\displaystyle \frac{1}{2}}\left[S_{\left(\rho \right.\nu }{C_{\alpha \left.\sigma \right)}}_{\beta }^{\nu }+S_{\nu \left(\rho \right.}{C}_{\alpha \beta \left.\sigma \right)}^{\nu }+2S^{ϵ\tau }{\left(C^{-1}\right)}_{\alpha ϵ\beta \tau }{B}_{\rho \sigma }\right].$$

(6)

From Equations (5) and (6) one gets

$$\begin{array}{c}0=-g_{\rho \sigma }+\underset{{\left({\alpha }^{-1}\right)}^{\alpha \beta }}{\underbrace{4{\displaystyle \frac{〈{\alpha }^2〉{\alpha }^{\alpha \beta }-2{\left({\alpha }^3\right)}^{\alpha \beta }}{\left({〈{\alpha }^2〉}^2-2〈{\alpha }^4〉\right)}}}}{\displaystyle \frac{\delta B_{\alpha \beta }}{\delta g^{\rho \sigma }}},\\ 0=-g_{\rho \sigma }+{\displaystyle \frac{1}{2}}{\left({\alpha }^{-1}\right)}^{\alpha \beta }\left[S_{\left(\rho \right.\nu }{C}_{\alpha \left.\sigma \right)\beta }^{\nu }+S_{\nu \left(\rho \right.}{C}_{\alpha \beta \left.\sigma \right)}^{\nu }+2S^{ϵ\tau }{\left(C^{-1}\right)}_{\alpha ϵ\beta \tau }{B}_{\rho \sigma }\right].\end{array}$$

(7)

Tracing Equation (7) one obtains the constraint

$$0=-4+{\left({\alpha }^{-1}\right)}^{\alpha \beta }{B}_{\alpha \beta }.$$

(8)

The condition (8) is particularly strong: it enforces the condition $F_{\mu \nu }=0$ so it invalidates (turns inconsistent) the square-root conformal BI Lagrangian (4).

3. Abelian and Non-Abelian Lagragians from Geometry

Here, it is shown how to avoid the inconsistency presented in the Lagrangian (4). As in the non-Abelian case in [12], from the antisymmetric property of the indices of the tensors, one finds the following equality at the determinant level:

$$det\left(B_{\alpha \beta }+F_{\alpha \beta }\right)=det\left(B_{\alpha \beta }-F_{\alpha \beta }\right),$$

(9)

and then one immediately finds

$${\left(det\left(B_{\alpha \beta }+F_{\alpha \beta }\right)\right)}^2=gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }{F}_{\beta }^{\gamma }\right),$$

(10)

and consequently the exact determinant can be rewritten as

$$\sqrt{det\left(B_{\alpha \beta }+F_{\alpha \beta }\right)}=\sqrt[4]{gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }{F}_{\beta }^{\gamma }\right)},$$

(11)

Note that at this step of the study, the absolute field b is set to $b=1$ to facilitate the geometric investigation of the invariant Lagrangian. Notice that the Lagrangians from the equality (9) are classically equivalent but not from in the sense of the equation of motion. From Equation (11) one immediately finds that non-Abelian proposal is

$${\mathcal{L}}_{\mathrm{NA}}=\sqrt[4]{gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }^a{F}_{\beta a}^{\gamma }\right)},$$

(12)

which gives a conformal non-Abelian BI model; however, it can be simplified, as shown in Section 4 just below.

To represent the determinant in a compact form, let us define the following trace free tensors in their explicit form:

$$M_{\beta }^{\alpha }=B^{\alpha \gamma }{B}_{\gamma \beta }+{{\overline{N}}^{\alpha }}_{\beta },{N^{\alpha }}_{\beta }=F_a^{\alpha \gamma }{F}_{\beta \gamma }^a,{{\overline{N}}^{\alpha }}_{\beta }={N^{\alpha }}_{\beta }-{\displaystyle \frac{{{\delta }^{\alpha }}_{\beta }}{4}}{N}_{\gamma }^{\gamma },M_{\gamma }^{\gamma }\equiv M.$$

(13)

Consequently, the conformal and rotationally duality-invariant Lagrangian (12) reads

$${\mathcal{L}}_{\mathrm{NA}}=\sqrt[4]{gdet{M}_{\alpha \beta }}.$$

(14)

Then, the exact expression of the conformal non-Abelian Lagrangian as a function of barred (traceless) quantities takes the following final form as found for Equation (A6):

$${\mathcal{L}}_{\mathrm{NA}}=\sqrt[2]{g}\sqrt[4]{{\left({\displaystyle \frac{M}{4}}\right)}^4-{\displaystyle \frac{M^2}{32}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\alpha }^{\beta }+{\displaystyle \frac{M}{12}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\gamma }^{\beta }{\overline{M}}_{\alpha }^{\gamma }+{\displaystyle \frac{1}{8}}{\left({\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\alpha }^{\beta }\right)}^2-{\displaystyle \frac{1}{4}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\gamma }^{\beta }{\overline{M}}_{\delta }^{\gamma }{\overline{M}}_{\alpha }^{\delta }}.$$

(15)

4. Simplified Lagrangian

It should be noted that the Lagrangian (14) can be simplified considering the entire block $M_{\alpha \beta }$ from Equation (13), which is fully traceless, namely,

$${\mathcal{L}}_{\mathrm{NA}}=\sqrt[4]{gdet{\overline{M}}_{\alpha \beta }},$$

(16)

with

$${\overline{M}}_{\alpha \beta }=M_{\alpha \beta }-g_{\alpha \beta }{\displaystyle \frac{M}{4}}\equiv {\overline{K}}_{\alpha \beta }+{\overline{N}}_{\alpha \beta },$$

and

$${\overline{K}}_{\alpha \beta }=K_{\alpha \beta }-g_{\alpha \beta }.$$

Then, Equation (16) takes the following explicit form:

$$\begin{array}{c}{\mathcal{L}}_{\mathrm{NA}}={\displaystyle \frac{\sqrt{g}}{2}}\sqrt[4]{2\left({〈{\overline{M}}^2〉}^2-2〈{\overline{M}}^4〉\right)}\\ ={\displaystyle \frac{\sqrt{g}}{2}}\sqrt[4]{2\left[{\left(M_{\beta }^{\alpha }{M}_{\alpha }^{\beta }\right)}^2+{\displaystyle \frac{5M^4}{32}}-2M_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\delta }^{\gamma }{M}_{\alpha }^{\delta }+2M{M}_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\alpha }^{\gamma }-{\displaystyle \frac{1}{4}}{M}^2{M}_{\beta }^{\alpha }{M}_{\alpha }^{\beta }\right]},\end{array}$$

(17)

where the definitions

$$\begin{array}{c}{〈{\overline{M}}^2〉}^2={\left(M_k^i{M}_i^k+{\displaystyle \frac{M^2}{4}}\right)}^2\\ \mathrm{and}〈{\overline{M}}^4〉=\left(M_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\delta }^{\gamma }{M}_{\alpha }^{\delta }-{\displaystyle \frac{3}{4}}{{\displaystyle \frac{M}{16}}}^4-M{M}_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\alpha }^{\gamma }+{\displaystyle \frac{3}{8}}{M}^2{M}_{\beta }^{\alpha }{M}_{\alpha }^{\beta }\right)\end{array}$$

(18)

are used.

The simplification becomes crucial for the some conformal and dynamical properties of the model, as one can see in what follows. Restoring the absolute field b in the Lagrangian density (17), one finds

$${\mathcal{L}}_{\mathrm{NA}}={\displaystyle \frac{b^2\sqrt{g}}{8\pi }}\sqrt[4]{2\left({〈{\overline{M}}^2〉}^2-2〈{\overline{M}}^4〉\right)},$$

(19)

with the corresponding redefined $F_{ij}$: $F_{ij}\to F_{ij}/b.$

4.1. Flat Limits

As one can find, the flat limit for the Lagrangian (19) (or (14)) is ($B_{jk}\to 0$) as follows:

$${\left.{\mathcal{L}}_{\mathrm{NA}}\right|}_{\mathrm{flat}}\to \sqrt[2]{g}\sqrt[4]{{\displaystyle \frac{1}{8}}{\left({\overline{N}}_{\beta }^{\alpha }{\overline{N}}_{\alpha }^{\beta }\right)}^2-{\displaystyle \frac{1}{4}}{\overline{N}}_{\beta }^{\alpha }{\overline{N}}_{\gamma }^{\beta }{\overline{N}}_{\delta }^{\gamma }{\overline{N}}_{\alpha }^{\delta }},$$

(20)

and Abelian limit is tested passing to U(1) gauge fields:

$${\left.{\mathcal{L}}_{\mathrm{NA}}\right|}_{\mathrm{flat}}\to {\left.L_A\right|}_{\mathrm{flat}}={\displaystyle \frac{\sqrt{g}}{4}}\sqrt{{\left(F^{\alpha \beta }{F}_{\alpha \beta }\right)}^2+{\left(*F^{\alpha \beta }{F}_{\alpha \beta }\right)}^2}.$$

(21)

(where the asterisk denotes the dual: $*F^{\alpha \beta }={\displaystyle \frac{1}{2}}{\epsilon }^{\alpha \beta \gamma \delta }{F}_{\gamma \delta }$). That is, one sees that the fourth-root action factorizes to a square root.

4.2. Discussion: Abelian Limit, Relation with Other Proposals

In order to analyze possible relation with the ModMax Lagrangian (see, e.g., [18,19] for details), which is invariant under the dualities in the definition of Ref. [20] (which is not the standard duality—see, e.g., [21,22,23,24]), a linear term is added to the Lagrangian (14) along with the hyperbolic functions, taking the form

$$\begin{array}{c}{\mathcal{L}}_{\mathrm{NA}}=\sqrt[2]{g}\left[cosh\gamma \left(B^{\alpha \beta }{B}_{\alpha \beta }-{\displaystyle \frac{F_a^{\alpha \beta }{F}_{\alpha \beta }^a}{4}}\right)\right.\\ +sinh\gamma \left.\sqrt[4]{{\left({\displaystyle \frac{M}{4}}\right)}^4-{\displaystyle \frac{M^2}{32}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\alpha }^{\beta }+{\displaystyle \frac{M}{12}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\gamma }^{\beta }{\overline{M}}_{\alpha }^{\gamma }+{\displaystyle \frac{1}{8}}{\left({\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\alpha }^{\beta }\right)}^2-{\displaystyle \frac{1}{4}}{\overline{M}}_{\beta }^{\alpha }{\overline{M}}_{\gamma }^{\beta }{\overline{M}}_{\delta }^{\gamma }{\overline{M}}_{\alpha }^{\delta }}\right],\end{array}$$

(22)

and, similarly for the simplified case, the full traceless Equation (19) reads

$$\begin{array}{c}{\mathcal{L}}_{\mathrm{NA}}=\sqrt[2]{g}\left[cosh\gamma \left(B^{\alpha \beta }{B}_{\alpha \beta }-{\displaystyle \frac{F_a^{\alpha \beta }{F}_{\alpha \beta }^a}{4}}\right)\right.\\ +sinh\gamma \left.{\displaystyle \frac{1}{2}}\sqrt[4]{2\left({\left(M_{\beta }^{\alpha }{M}_{\alpha }^{\beta }\right)}^2+{\displaystyle \frac{5M^4}{32}}-2M_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\delta }^{\gamma }{M}_{\alpha }^{\delta }+2M{M}_{\beta }^{\alpha }{M}_{\gamma }^{\beta }{M}_{\alpha }^{\gamma }-{\displaystyle \frac{1}{4}}{M}^2{M}_{\beta }^{\alpha }{M}_{\alpha }^{\beta }\right)}\right].\end{array}$$

(23)

Then, the non-Abelian, conformal, and duality-invariant (at the Abelian level) Lagrangians (22) and (23) containing gravity in the flat Abelian limit belong to the ModMax model of Refs. [18,19].

In summary:

(1)

The equivalence Lagrangians (22) and (23) containing gravity with ModMax are only in the flat case (for example, $B_{ji}=0$).

(2)

ModMax is prior to Refs. [18,19]; in the study by the author in Ref. [11], the origin of that type of Lagrangian was thoroughly elucidated—it is nothing but a particular eigenvalue of the corresponding characteristic (secular) equation of the tensor $F_{\mu \nu }$, given as

$$\det (F_{\mu \nu }-\lambda I_4)={\lambda }^4+2\mathcal{F}{\lambda }^2-{\mathcal{G}}^2=0,$$

where $I_4$ is the unit 4-matrix, $\mathcal{F}={\displaystyle \frac{1}{2}}{F}_{\mu \nu }{F}^{\mu \nu }$ and $\mathcal{G}={\displaystyle \frac{1}{2}}{F}_{\mu \nu }*F^{\mu \nu }$, resulting in four eigenvectors corresponding to four main directions according to the Newman and Penrose [25] or Segre classification (similarly to the gravitational field Petrov classification [26]).

5. Dynamical Equations, Symmetries, and Topological Bounds

Let us now calculate the variation of the geometric traceless fourth-root Lagrangian with respect to the metric. Let us start with the variation $\delta det{\overline{M}}_{\alpha ,\beta }=$$det{\overline{M}}_{\alpha ,\beta }{\left({\overline{M}}^{-1}\right)}^{\alpha ,\beta }\delta {\overline{M}}_{\alpha ,\beta }$, for simplicity $det{\overline{M}}_{\alpha ,\beta }\equiv \overline{M}$ and the scalar product $a^{-1}·\delta a={\left(a^{-1}\right)}^{\alpha \beta }·\delta a_{\alpha \beta }$; then,

$$\delta \sqrt[4]{g\overline{M}}=\sqrt[4]{g\overline{M}}\left[g^{-1}·\delta g+{\overline{M}}^{-1}·\delta \overline{M}\right].$$

(24)

Now, the result (24) (detailed in Appendix B) is used to define the Bach operator in four dimensions as

$$S_{\mu \nu }\equiv \left({\nabla }_{\mu }{\nabla }_{\omega }+{\displaystyle \frac{1}{2}}{R}_{\mu \omega }\right),$$

where $R_{\mu \omega }$ is the Ricci tensor.

Then, one computes

$$\begin{array}{cc}\hfill \delta {\overline{M}}_{\mu \nu }& ={\displaystyle \frac{1}{2}}\{2S^{\delta \lambda }{\left(C^{-1}\right)}_{\omega \delta \tau \lambda }{B}_{\nu \rho }{B}_{\mu }^{\rho }+B_{\nu }^{\sigma }\left[S_{\omega \lambda }\left(C_{\mu \tau \sigma }^{\lambda }+C_{\sigma \tau \mu }^{\lambda }\right)\right]+B_{\mu }^{\sigma }\left[S_{\omega \lambda }\left(C_{\nu \tau \sigma }^{\lambda }+C_{\sigma \tau \nu }^{\lambda }\right)\right]\hfill \\ \hfill & +B_{\mu \omega }{B}_{\nu \tau }+B_{\mu \tau }{B}_{\nu \omega }+F_{\mu \omega }{F}_{\nu \tau }+F_{\mu \tau }{F}_{\nu \omega }-{\displaystyle \frac{1}{4}}\left(g_{\mu \omega }{g}_{\nu \tau }+g_{\mu \tau }{g}_{\nu \omega }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }+F_{\alpha \beta }{F}^{\alpha \beta }\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}{g}_{\mu \nu }\left[2B^{\alpha \beta }\left(S_{\omega \lambda }\left(C_{\alpha \tau \beta }^{\lambda }+C_{\beta \tau \alpha }^{\omega }\right)+S^{\delta \lambda }{\left(C^{-1}\right)}_{\alpha \delta \beta \lambda }{B}_{\omega \tau }\right)+4\left(B_{\omega \beta }{B}_{\tau }^{\beta }+F_{\omega \beta }{F}_{\tau }^{\beta }\right)\right]\}\delta g^{\mu \tau }.\hfill \end{array}$$

Consequently, due to ${\left({\overline{M}}^{-1}\right)}_{\mu }^{\mu }=0$, one finds

$$\begin{array}{c}{\left({\overline{M}}^{-1}\right)}^{\mu \nu }\delta {\overline{M}}_{\mu \nu }\\ ={\left({\overline{M}}^{-1}\right)}^{\mu \nu }{\displaystyle \frac{1}{2}}\{2S^{\delta \lambda }{\left(C^{-1}\right)}_{\omega \delta \tau \lambda }{B}_{\nu \rho }{B}_{\mu }^{\rho }+B_{\nu }^{\sigma }\left[S_{\omega \lambda }\left(C_{\mu \tau \sigma }^{\lambda }+C_{\sigma \tau \mu }^{\lambda }\right)\right]+B_{\mu }^{\sigma }\left[S_{\omega \lambda }\left(C_{\nu \tau \sigma }^{\lambda }+C_{\sigma \tau \nu }^{\lambda }\right)\right]\\ +B_{\mu \omega }{B}_{\nu \tau }+B_{\mu \tau }{B}_{\nu \omega }+F_{\mu \omega }{F}_{\nu \tau }+F_{\mu \tau }{F}_{\nu \omega }-{\displaystyle \frac{1}{4}}\left(g_{\mu \omega }{g}_{\nu \tau }+g_{\mu \tau }{g}_{\nu \omega }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }+F_{\alpha \beta }{F}^{\alpha \beta }\right)\}\delta g^{\mu \tau },\end{array}$$

and using $g^{\mu \nu }\delta g_{\mu \nu }=-g_{\mu \nu }\delta g^{\mu \nu }$, finally:

$$\begin{array}{c}{\displaystyle \frac{\delta \sqrt[4]{g\overline{M}}}{\delta g^{\omega \tau }}}=0={\displaystyle \frac{1}{4}}\sqrt[4]{gM}\{-g_{\omega \tau }\\ +{\displaystyle \frac{1}{2}}{\left({\overline{M}}^{-1}\right)}^{\mu \nu }\left[2S^{\delta \lambda }{\left(C^{-1}\right)}_{\omega \delta \tau \lambda }{B}_{\nu \rho }{B}_{\mu }^{\rho }+B_{\nu }^{\sigma }\left(S_{\omega \lambda }\left(C_{\mu \tau \sigma }^{\lambda }+C_{\sigma \tau \mu }^{\lambda }\right)\right)+B_{\mu }^{\sigma }\left(S_{\omega \lambda }\left(C_{\nu \tau \sigma }^{\lambda }+C_{\sigma \tau \nu }^{\lambda }\right)\right)\right.\\ \begin{array}{cc}& \left.+B_{\mu \omega }{B}_{\nu \tau }+B_{\mu \tau }{B}_{\nu \omega }+F_{\mu \omega }{F}_{\nu \tau }+F_{\mu \tau }{F}_{\nu \omega }-{\displaystyle \frac{1}{4}}\left(g_{\mu \omega }{g}_{\nu \tau }+g_{\mu \tau }{g}_{\nu \omega }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }+F_{\alpha \beta }{F}^{\alpha \beta }\right)\right]\}.\end{array}\end{array}$$

(25)

Tracing in $\left(\omega \tau \right)$, in four dimensions the following condition is obtained:

$${\left({\overline{M}}^{-1}\right)}^{\mu \nu }{B}_{\nu \rho }{B}_{\mu }^{\rho }=0.$$

(26)

Inserting Equation (26) into Equation (25), one finally obtains the gravitational equation for the fourth-root Lagrangian:

$$\begin{array}{c}{\displaystyle \frac{\delta \sqrt[4]{g\overline{M}}}{\delta g^{\omega \tau }}}=0={\displaystyle \frac{1}{4}}\sqrt[4]{gM}\{-g_{\omega \tau }\\ +{\displaystyle \frac{1}{2}}{\left({\overline{M}}^{-1}\right)}^{\mu \nu }\left[B_{\nu }^{\sigma }\left(S_{\omega \lambda }\left(C_{\mu \tau \sigma }^{\lambda }+C_{\sigma \tau \mu }^{\lambda }\right)\right)+B_{\mu }^{\sigma }\left(S_{\omega \lambda }\left(C_{\nu \tau \sigma }^{\lambda }+C_{\sigma \tau \nu }^{\lambda }\right)\right)\right.\\ \begin{array}{cc}& \left.+B_{\mu \omega }{B}_{\nu \tau }+B_{\mu \tau }{B}_{\nu \omega }+F_{\mu \omega }{F}_{\nu \tau }+F_{\mu \tau }{F}_{\nu \omega }-{\displaystyle \frac{1}{4}}\left(g_{\mu \omega }{g}_{\nu \tau }+g_{\mu \tau }{g}_{\nu \omega }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }+F_{\alpha \beta }{F}^{\alpha \beta }\right)\right]\}\end{array}\end{array}$$

(27)

Note that the expression (26) suggests a possible ansatz of the type $B_{\nu \rho }{B}_{\mu }^{\rho }\propto \mathrm{\Omega }\left(x\right)g_{\nu \mu }$, with $\mathrm{\Omega }\left(x\right)$ the conformal factor. There also exists a singular solution given by the condition (also suggested by Equation (26))

$${\left({\overline{M}}^{-1}\right)}^{\mu \nu }=0.$$

5.1. Electromagnetic Field Equations for the Abelian Case

For completeness, let us consider the structure of the dynamic equations of the electromagnetic field from the fourth-root Lagrangian (1) in the Abelian case. Then, the geometric equations read

$${\nabla }_{\sigma }*F^{\rho \sigma }=0\left(dF=0\right)$$

and the field equations are

$${\nabla }_{\sigma }{\mathbb{F}}^{\rho \sigma }=0\left(d*\mathbb{F}=0\right)$$

with

$${\mathbb{F}}^{\rho \sigma }=\sqrt{g}{\displaystyle \frac{2\left(〈{\overline{M}}^2〉{\overline{M}}^{\rho \beta }{F}_{\beta }^{\sigma }-4〈{\overline{M}}^3〉F^{\rho \sigma }\right)}{{\left[2\left({〈{\overline{M}}^2〉}^2-2〈{\overline{M}}^4〉\right)\right]}^{3/4}}},$$

where the non-linear character of the model is well seen.

5.2. The Metric Energy–Momentum Tensor

As mentioned in Section 2, for the simplest action that is full traceless ${\overline{M}}_{\alpha ,\beta }$, the energy–momentum tensor can be determined from the gravitational Equation (25) geometrically and by inspection. The Lagrangian is proportional given the practical effects by Hilbert’s definition that are consistently given for the GR case of Einstein’s (linear) theory (for example, $T^{\mu \nu }\equiv -{\displaystyle \frac{2}{\sqrt[2]{g}}}{\displaystyle \frac{\partial \left(\sqrt{g}L\right)}{\partial g_{\mu \nu }}}).$ Note that the action functional contains $B_{\nu \omega }$ and $F_{\mu \omega }$ in a non-linear way; then, a pure-matter Lagrangian does not exist in the case considered.

5.2.1. Fundamental Function and Self-Similarity

From Equation (25), the energy–momentum tensor can be determined by inspection:

$$\begin{array}{c}{\mathcal{T}}_{\mu \nu }={\displaystyle \frac{\partial \stackrel{{\mathcal{L}}_{\mathrm{NA}}}{\overbrace{\left(\sqrt{g}L\right)}}}{\partial g^{\mu \nu }}}={\displaystyle \frac{1}{4}}\sqrt[4]{gM}[-g_{\omega \tau }\\ +{\left({\overline{M}}^{-1}\right)}^{\mu \nu }\left(B_{\mu \omega }{B}_{\nu \tau }+B_{\mu \tau }{B}_{\nu \omega }+F_{\mu \omega }{F}_{\nu \tau }+F_{\mu \tau }{F}_{\nu \omega }-{\displaystyle \frac{1}{4}}\left(g_{\mu \omega }{g}_{\nu \tau }+g_{\mu \tau }{g}_{\nu \omega }\right)\left(B_{\alpha \beta }{B}^{\alpha \beta }+F_{\alpha \beta }{F}^{\alpha \beta }\right)\right)],\end{array}$$

(28)

which, taking its trace, leads to the proposed non-linear Lagrangian density

$${\mathcal{T}}_{\mu }^{\mu }={\mathcal{L}}_{\mathrm{NA}}.$$

(29)

From Equation (29) and using the known expression of the differentiation of the metric determinant $dg=g{g}^{\mu \nu }d{g}_{\mu \nu }$, one obtains the relation valid in four dimensions:

$${\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{NA}}}{\partial lng}}={\displaystyle \frac{1}{4}}{\mathcal{L}}_{\mathrm{NA}}\to det{\overline{M}}_{\alpha \beta }=g_0,$$

(30)

where $g_0$ is a constant. The relationship (30) is crucial, since it implies a self-similarity property of the proposed Lagrangian density that can be translated into a constraint between the Lagrangian density and the determinant of the metric.

Note that the self-similarity condition (which operates at $T_{\mu }^{\mu }$(29) level) is preserved at both the Abelian and non-Abelian levels, disregard the gauge group.

5.2.2. Topological Bounds

From Minkowski inequality,

$${\left[det\left(B_{\alpha \beta }{B}_{\gamma }^{\beta }+F_{\alpha \beta }{F}_{\gamma }^{\beta }\right)\right]}^{1/4}\ge {\left[det\left(B_{\alpha \beta }{B}_{\gamma }^{\beta }\right)\right]}^{1/4}+{\left[det\left(F_{\alpha \beta }{F}_{\gamma }^{\beta }\right)\right]}^{1/4},$$

(31)

and taking into account Equations (9) and (10), one finds the following (using the determinant properties mentioned in Section 3 above):

$$\sqrt{det\left(B_{\alpha \beta }+F_{\alpha \beta }\right)}\ge \sqrt{det{B}_{\alpha \beta }}+\sqrt{det{F}_{\alpha \beta }}.$$

Consequently, as in the case considered in Refs. [12,27] (and the references therein), there is a topological bound for the action given because, as is known, $\sqrt{det{F}_{\alpha \beta }}$ = ${\displaystyle \frac{1}{4}}\left|\sqrt{g}{F}_{\alpha \beta }*F^{\alpha \beta }\right|$ is a topological invariant.

The following conclusions can be made.

(1)

The saturation of the bound in the action (31) is given by

$$B_{\alpha \beta }{B}_{\gamma }^{\beta }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }.$$

(32)

(2)

What concerns Equation (26), a conformal-Killing ansatz can be considered as $B_{\nu \rho }{B}_{\mu }^{\rho }\propto \mathrm{\Omega }\left(x\right)g_{\nu \mu }$, that is, the bound saturation (32) now takes the form

$$\mathrm{\Omega }\left(x\right)g_{\nu \mu }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }$$

(33)

in contrast to the saturation of the bound $g_{\nu \mu }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }$ of the reference cases [12,27] where the determinant function was of the (non-conformal) type with $g_{\nu \mu }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }$.

(3)

One can find then that, from Refs. [12,27], the non-conformal models (with metric and electromagnetic field tensors) satisfy the condition

$$\sqrt[4]{det\left(g_{\alpha \gamma }+F_{\alpha \beta }{F}_{\gamma }^{\beta }\right)}\to g_{\nu \mu }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }⟹F^{\alpha \beta }={\left(F_{\alpha \beta }\right)}^{-1}.$$

(34)

In the case under study here, in the new conformal BI-type model here considered (with the Bach tensor and electromagnetic field tensor), the condition (34) includes the conformal factor $\mathrm{\Omega }\left(x\right)$ and reads

$$\sqrt[4]{det\left(B_{\alpha \beta }{B}_{\gamma }^{\beta }+F_{\alpha \beta }{F}_{\gamma }^{\beta }\right)}\to \mathrm{\Omega }\left(x\right)g_{\nu \mu }\propto F_{\alpha \beta }{F}_{\gamma }^{\beta }⟹F^{\alpha \beta }={\mathrm{\Omega }}^{-1}\left(x\right){\left(F_{\alpha \beta }\right)}^{-1}.$$

(35)

From the argument (35), one observes that in a general curved spacetime manifold, the new conformal BI Lagrangian (31) is bounded by a topological quantity, and the bound realized depends on the gauge fields and the conformal factor $\mathrm{\Omega }\left(x\right)\propto F_{\alpha \beta }{F}^{\alpha \beta }/4.$

6. Concluding Remarks

In summary, the current study concludes with the following observations concerning the new proposal here for the new conformally invariant gravitational Born–Infeld type Lagrangian density (Lagrangian)

(1)

The square-root form of the proposed Lagrangian, being traceless and containing the antisymmetric bivector and the symmetric tensor (Bach tensor), which is conformally invariant, is inconsistent since the trace of gravitational equations imposes the condition $F=0$.

(2)

The fourth-root form of the proposed Lagrangian, being traceless and containing the antisymmetric bivector and the symmetric tensor (Bach tensor), which is conformally invariant, is fully consistent since the trace of the gravitational equations leads to the condition ${\left({\overline{M}}^{-1}\right)}^{\mu \nu }{B}_{\nu \rho }{B}_{\mu }^{\rho }=0$ suggesting an ansatz $B_{\nu \rho }{B}_{\mu }^{\rho }\propto \mathrm{\Omega }\left(x\right)g_{\nu \mu }$, where $\mathrm{\Omega }\left(x\right)$ is a conformal factor: consequently, it is to be associated with a conformal-Killing vector equation.

(3)

The consistent Lagrangian is self-similar with respect to the determinant of the metric tensor, suggesting a possible conformal Bach flow of topological character.

In this paper, a fourth-root, conformally invariant nonlinear Lagrangian of the BI type has been presented where the determinant function (which contains a symmetric tensor and the antisymmetric one) replaces the metric tensor with the Bach tensor. This presentation differs from the twistor Lagrangianof the previous paper by the author [28], where the Lagrangian was considered being based on the gauge-invariant twistor connection, where the fundamental role (besides the electromagnetic field tensor) is directly played by the Weyl tensor (of rank four).

The model presents a self-similarity with respect to variations with respect to the determinant of the metric, which can be crucial in astrophysical scenarios, for example, where the processes of accretion, ejection (jets), and gravitational waves with conformal invariance and self-similarity play a fundamental role.

It is quite of interest to continue similar study in a non-Riemannian model that allows for a generalization of the Lagrangian presented here, where, in contrast to the model proposed by the author in Ref. [29], the generalized Ricci tensor is replaced by a generalized Weyl tensor (as was proposed previously in Ref. [30]).