{'fraction_non_alphanumeric': 0.029381641275124822, 'fraction_numerical': 0.01107412623223659, 'mean_word_length': 4.0922425032594525, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Born-Infeld-Einstein theory with matter

The field equations associated with the Born-Infeld-Einstein action including matter are derived using a Palatini variational principle. Scalar, electromagnetic, and Dirac fields are considered. It is shown that an action can be chosen for the scalar field that produces field equations identical to the usual Einstein field equations minimally coupled to a scalar field. In the electromagnetic and Dirac cases the field equations reproduce the standard equations only to lowest order. The spherically symmetric electrovac equations are studied in detail. It is shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory. It is also shown that point charges are not allowed. All particles must have a finite size. Mass terms for the fields are also considered.


Introduction
In the 1930's Born and Infeld [1] attempted to eliminate the divergent self energy of the electron by modifying Maxwell's theory. Born-Infeld electrodynamics follows from the Lagrangian where g µν is the metric tensor and F µν is the electromagnetic field tensor. In the weak field limit this Lagrangian reduces to the Maxwell Lagrangian plus small corrections. For strong fields the field equations deviate significantly from Maxwell's theory and the self energy of the electron can be shown to be finite. The Born-Infeld action also appears in string theory. The action for a D-brane is of the Born-Infeld form with two fields, a gauge field on the brane and the projection of the Neveu-Schwarz B-field onto the brane [2].
In a recent paper [3] I considered using a Palatini variational approach to derive the field equations associated with the action where R µν is the Ricci tensor, and κ = 8πG. This action has also been examined using a purely metric variation by Deser and Gibbons [12], Feigenbaum, Freund and Pigli [5] and Feigenbaum [6]. In this paper I will consider adding matter to the theory by including it in the determinant in equation (2). The forms of matter that I investigate in this paper include scalar, electromagnetic, and Dirac fields. In all the cases except for the scalar field there are corrections to the standard equations. In the scalar field case it is shown that it is possible to chose a Lagrangian that gives the standard equations exactly. The fields produced by a spherically symmetric charge distribution are also studied. It is shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory and that point charges are not allowed in the theory.

The Field Equations
The field equations for the theory follow from the Born-Infeld-Einstein action where R µν is the Ricci tensor, κ = 8πG, b is a constant, and M µν is the matter contribution. For a simple example of the matter contribution consider a scalar field φ. In this case M µν = ∇ µ φ∇ ν φ. The Ricci tensor is given by and the connection is taken to be symmetric. Note that R µν is not symmetric in general. If the curvature and matter terms are small an expansion of (3) gives Thus, if M does not vanish, the lowest order contribution to the matter Lagrangian is In the case of the electromagnetic field M vanishes, so higher order terms will be needed.
Varying the action with respect to g µν gives where P µν = g µν +bR µν +κbM µν , P −1 is the inverse of P , (P −1 ) (µν) is the symmetric part of P −1 , P = −det(P µν ) and g = −det(g µν ). If M µν is independent of the metric, as will be the case for massless scalar and electromagnetic fields, the second term on the left hand side in equation (7) will vanish and the field equations will simplify significantly.
Varying with respect to Γ α µν gives Here I have assumed that M µν is independent of the connection. This will be true for scalar and electromagnetic fields, but not for the Dirac field. The Dirac field will be studied in detail later. Contracting over α and ν gives where (P −1 ) [µν] is the antisymmetric part of P −1 . From equation (9) we see that Taking M µν to be independent of the metric and substituting equations (7) and (10) into (8) gives Since the trace of the above system of equations vanishes there are four too few equations and the system is under determined. Thus, we expect four arbitrary functions in the solution. Such a solution is given by where V µ is an arbitrary vector. The connection that follows from this set of equations is given by [3] Γ α µν = α µν where the first term on the right hand side is the Christoffel symbol. This new "geometrical" vector field was discussed in detail in [3] and will not be considered here. I will therefore set V µ = 0 and the connection is the Christoffel symbol. Thus, the gravitational field equations are given by (7) with the connection given by the Christoffel symbol if M µν is independent of the metric. The matter field equations are given by varying the action with respect to the matter fields. I will study these equations in the subsequent sections, once a specific type of matter field has been chosen.

The Scalar Field
For the scalar field I will take M µν = ∇ µ φ∇ ν φ. In this case M µν is symmetric and is independent of the metric. Note that R µν is symmetric since the connection is just the Christoffel symbol. This implies that P µν is symmetric.
Consider the general case in which M µν is symmetric and independent of the metric. Taking the determinant of equation (7) gives P = g and after some simple algebra we find that Thus, for the scalar field we have Note that these are the standard Einstein field equations minimally coupled to a scalar field. The scalar field equation can be found by varying the action with respect to φ. The resulting field equation is Using (7) gives the usual field equations Note that the lowest order matter Lagrangian given in equation (6) is the standard Lagrangian for a scalar field. Therefore, the higher order terms in both the gravitational and matter Lagrangians do not modify the field equations. This extends the universality of the Einstein vacuum equations discovered by Ferraris et al. [7,8] Such choices lead to more complicated theories in which the connection is no longer the Christoffel symbol.

The Electromagnetic Field
For the electromagnetic field take M µν = αF µν , where α is a constant. In this case M µν is independent of the metric, but P µν is no longer symmetric. This implies that the connection is the Christoffel symbol. The field equations are difficult to solve exactly, but if the fields are sufficiently weak they can be expanded to second order in the fields. The following expressions, computed to quadratic order in the field variables, will be useful in finding the field equations and The field equations up to and including quadratic terms in the fields are given by (20) Note that electromagnetic terms on the right hand side correspond to the energymomentum tensor of the electromagnetic field. Thus, to obtain the Einstein field equations to lowest order we must take α 2 = 1/(κb). This implies that b must be positive. Also note that there are correction terms to the Einstein field equations that only vanish in the limit b → 0. In fact, certain cubic and quartic terms need to be included for the approximation to be consistent. To see this note that R ∼ κF 2 which implies that the quadratic corrections in (20) are ∼ b(κF 2 ) 2 . However, terms like bκRF 2 and bκ 2 F 4 are of the same order and therefore need to be included in the approximation. Since the exact form of the corrections is not important I will not consider these additional terms further.
The field equations for the electromagnetic field are derived by varying the action with respect to A µ and are given by To lowest order the above reduces to the usual field equations The lowest order contribution to the electromagnetic field Lagrangian cannot be computed from (6) since M = 0. The Lagrangian to first order in the curvature and second order in the electromagnetic field is given by Thus, to this order, we obtain the standard Einstein-Maxwell Lagrangian.
It is interesting to note that the inclusion of the electromagnetic field F µν excludes the possibility of the "geometric" vector field discussed earlier. To see this note that (9) and (21) imply that This together with (7) and (11) implies that which tells us that the connection is the Christoffel symbol and that there is no room for an extra "geometric" vector field.
In general it seems that it is difficult to write the field equations in the form G µν = −κT µν . However, it is possible to write them in this form for spherically symmetric fields. In this case we can take where φ depends only on r and the metric is diagonal. A short calculation shows that and where q µν = g µν + bR µν , ∆ = q 00 q 11 + (Φ ′ ) 2 and Φ = √ κbφ. Taking the determinant of both sides of (7) gives g = −q 00 q 11 q 22 q 33 .
The field equations can then be written as The three independent equations are q 00 = − q 00 q 11 ∆ g 11 , and From (31) and (32) we find that ∆ = g 00 g 11 and g 00 R 11 = g 11 R 00 .
Using these two equations gives The solution to this equation is Similarly we have and The Einstein field equations can be written in covariant form as where I have taken b = a 2 /κ with a constant and F 2 = F αβ F αβ . Note that this equation is not general. I have only shown that it is valid if P µν is of the form given in (27). The energy-momentum tensor is identical to the Born-Infeld energy-momentum tensor, except for the sign in the square roots and the sign in front of the metric. The electromagnetic field equations given in (21) can be written as These field equations are the same as the Born-Infeld field equations with the sign in the square root changed. In Born-Infeld theory the square root 1 − 1 2 a 2 F 2 , that appears in the above equations, is replaced by 1 + 1 2 a 2 F 2 . This difference is important in terms of the allowed magnitude of the electric field. In flat space-time the Born-Infeld square root is √ 1 − a 2 E 2 , so that the maximum magnitude of the electric field is 1/a. In the theory presented here the square root is √ 1 + a 2 E 2 , so that the square root does not, by itself, constrain the magnitude of the electric field. For example consider the electromagnetic field equations in a vacuum outside a spherically symmetric charge distribution in flat space-time. The electromagnetic field equation is and the solution is where Q is a constant. Note that although E can be arbitrarily large the left hand side in the above equation has a maximum magnitude of 1/a as E → ∞. Thus, we cannot let r → 0 on the right hand side and the particle has to have a finite size. In fact, any charged particle will have to have a radius that is greater than or equal to a|Q|. Now consider the solution in curved space-time. The general spherically symmetric solution for any theory with a Lagrangian of the form is given by [9,10,11,12,13,14] and where s = − 1 4 D µν D µν = Q 2 /(2r 4 ) and H is the "Hamiltonian" given by The electromagnetic field equations are and the energy-momentum tensor is The field equations (39) and (40) can be obtained if we take For this Lagrangian and equation (47) can be written as Thus, just as in the flat space-time case, the particle has to have a coordinate radius that is greater than or equal to a|Q|.
To see if the space-time becomes singular as we approach a|Q| consider the Ricci tensor which is given by From (54) we have Thus, there is a curvature singularity at r = a|Q| and the particle has to have a coordinate radius greater than a|Q| to avoid the singularity.

The Dirac Field
In this section I will examine the Dirac field in Born-Infeld-Einstein theory. To deal with spinors it is convenient to introduce a tetrad e a µ satisfying e a µ e b ν η ab = g µν and e a µ e b ν g µν = η ab .
The Riemann tensor is defined to be where ω ab µ is the spin connection (see [15] for a discussion of spinors) The action will be taken to be where g µν is to be expressed in terms of the tetrad, and Σ ab = 1 4 [γ a , γ b ] are the generators of the Dirac representation. Note that the Dirac Lagrangian depends on the spin connection. Of course, other choices for M µν are possible. For example one could symmetrize on the indices µ and ν. However, since R µν is not symmetric this does not simplify the field equations. The Lagrangian, to lowest order, corresponding to this choice for M µν is given by (6) and is the usual Dirac Lagrangian.
Varying the action with respect to the fieldΨ gives To lowest order we obtain the standard Dirac equation γ α D α Ψ = 0. Varying the action with respect to the tetrad gives, to lowest order, the field equations where the field equation, to lowest order, for Ψ has been used. The field equation for the connection is found by varying the action with respect to the connection. Note that the connection appears in both the Ricci tensor and in M µν . The details are rather messy, but to lowest order we get the usual result (see [16] for details). This is not surprising since to lowest order the matter Lagrangian is the standard Dirac Lagrangian.

Mass Terms
In this section I will consider the inclusion of mass terms into the theory. LetM µν be the matter contribution without a mass terms. A simple approach to include a mass term would be to let where Ω depends on the field variable. For example, in the case of a scalar field Ω = 1 4 m 2 φ 2 and for the Dirac field Ω = 1 4 mΨΨ, where m is the mass of the field. For simplicity consider the case of the scalar field. Equation (7) gives the field equations The field equations given by varying the action with respect to the connection are still given by (8). Thus the connection is no longer the Christoffel symbol with respect to the metric g µν . In fact, the connection is the Christoffel symbol with respect to the metric h µν = (1 + 1 4 κbm 2 φ) −1 g µν . Thus, R µν and P µν will be symmetric. One can also show that the Einstein equations are given by These are not the standard Einstein equations with a mass term. However, if κbm 2 φ 2 << 1 we get the standard equations to lowest order. There is nothing wrong with these field equations, but a simpler theory can be found. This theory follows from the action (67) Varying this action with respect to the connection gives the same equations as before and varying it with respect to the metric gives √ P P −1 (µν) = √ gg µν .
Thus, the resulting field equations are exactly the Einstein equations coupled to a massive scalar field. A similar approach can be used for the Dirac and electromagnetic fields. For the Dirac field the mass terms will be of the form m 2Ψ Ψg µν and for the electromagnetic field the mass terms will be of the form m 2 A µ A ν . The electromagnetic mass term, m 2 A µ A ν , is actually simpler to deal with than the scalar field mass term since it is independent of the metric.

Conclusion
The field equations for the Born-Infeld-Einstein action including matter were derived using the first order or Palatini formalism. For the scalar field it was shown that it is possible to chose a Lagrangian that gives the standard Einstein equations minimally coupled to a scalar field even in the massive case. For the electromagnetic and Dirac fields the standard equations are only reproduced to lowest order.
The spherically symmetric electrovac equations were studied in detail. It was shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory. It was also shown that point charges are not allowed in the theory. All particles must have a finite size.