大重力与Born-Infeld理论：紫外线引力的红外启示

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本文主要探讨了在宇宙学和引力理论领域的一个重要课题——紫外线重力与大重力中的Born-Infeld作用。紫外线重力是指在量子引力理论中，高能（或短距离）物理现象下的表现，而Born-Infeld理论则是一种修正的电磁理论，旨在解决经典电动力学中场强度无穷大的问题。作者将这两个概念结合起来，试图从大重力的经验中提取关于UV引力的洞察。文章的核心内容是基于Palatini形式主义，这是一种将重力与几何联系起来的理论框架。在这个框架下，作者展示了如何将非线性的Born-Infeld作用与黎曼几何中的度量联系起来，这使得理论包含了所有基本对称多项式。通过分析代数方程，理论产生了多个分支，其中的一个在低曲率情况下可以归约到广义相对论，表现出良好的低能行为。研究的重点放在了理论的最小形式上，特别是在存在完美流体的情境下探讨了宇宙学演化。在真空状态下，理论能够恢复Ricci-flat解以及与爱因斯坦空间相关的物理特性。对于非真空解决方案，有两个物理分支，其中与广义相对论相接的分支表现出对能量密度的特定不敏感性。在非真空情况下，哈勃函数的行为根据流体状态方程有所不同。有三种可能性：一是能量密度最大且与真空连续，二是能量密度可变但高密度时哈勃函数饱和，三是能量密度无限制且哈勃函数增长超过广义相对论。第二种情况特别引人注意，因为它可能为早期宇宙的加速膨胀提供一种解释，即使在包含尘埃成分的系统中也是如此。文章还讨论了在最小模型中如何避免某些类型的奇点，这是理论物理中的关键问题，因为奇点往往暗示着现有理论的有效性局限。通过分析这些结构，研究者希望能够更深入地理解重力的本质及其在极端条件下的行为。这篇论文深入剖析了紫外线重力与Born-Infeld作用在大重力背景下的结合，提供了新的视角来研究引力理论的UV完成，并在宇宙学和奇异点的处理上展示了理论的可能性。这是一项对当前引力理论前沿的重要贡献。

JCAP11(2014)004

generalizes the linear Fierz-Pauli action without propagating ghost degrees of freedom and

contains only second order equations of motion. Its successful realization relies on a very

speciﬁc structure of a 2-parameter family interactions. Very soon after that Hassan and

Rosen have realized that this very speciﬁc structure of the interactions was automatically

fulﬁlled by writing the interactions in terms of a deformed determinant det (

+ β

K) [22],

where the matrix

K =

ˆg

−1

f −

encodes the potential interactions of the graviton with the

square root structure. This deformed determinant on the other hand can be written in terms

of the symmetric polynomials of the matrix

K. The determinant plays a very crucial role. An

equivalent description of the interactions in terms of the Stueckelberg ﬁelds requires that the

scalar Stueckelberg ﬁeld π in zeroth order of metric ﬂuctuations h

µν

= 0 must appear in the

action only in total derivative terms. Exactly the antisymmetry property of the Levi-Civita

tensors in the determinant det (δ

µν

+ ∂

∂

π) guarantees that. Furthermore, the theory of

massive gravity can be promoted to a theory of bigravity by invoking explicit dynamics for the

reference metric f [23]. The ghost-free interactions between the two metrics are still given by

the determinant det (

K) with the diﬀerence that there is also an additional kinetic term

for the f metric. There is nothing special about one metric versus the other since the theory

is symmetric under the exchange of the two metrics. Therefore naively one could attempt

to couple both metrics to external matter ﬁelds even though there might be restrictions due

to the existence of ghost degrees of freedom [19–21, 24, 25, 40–42]. Another crucial point

is that massive gravity in a given limit reproduces a given important class of scalar-tensor

theories: the Galileon theory [26]. The Galileon interactions represent an interesting subclass

of Horndeski interactions [27] with shift and Galileon symmetry and they have only second

order equations of motion. The decoupling limit of massive gravity contains these interactions

automatically [18, 28]. Exactly as in massive gravity the Galileon interactions can also be

written in terms of a deformed determinant det (δ

µν

+ a∂

∂

π + b∂

π∂

π) [29]. As can be

seen, in all these theories, the interactions appear in a very speciﬁc form of a deformed

determinant, to which we will pay a speciﬁc attention when we construct our generalization

of the Born-Infeld gravity theory. In this work we will combine the Born-Infeld gravity with

the ideas of massive gravity and extend the Born-Infeld action to include all of the symmetric

polynomials.

By comparing the structures of the basic building blocks of Born-Infeld gravity and

the potential term in massive gravity, the Born-Infeld Lagrangian can be written either as

√

−g

det

Ω [16] or as

√

−g det(

Ω). The identiﬁcation of the matrix

Ω as the basic

element of the action and the fact that det

Ω is nothing but the highest order elementary

symmetric invariant polynomial of this matrix motivates us to propose an extension involving

other invariant polynomials of

Ω. After explaining in detail this construction 2, we derive

the ﬁeld equations of the resulting theory 3. Then we concentrate in more detail on the speciﬁc

case of the minimal extension of the theory, which consists only of the ﬁrst polynomial 4,

and consider diﬀerent applications aimed at exploring the behavior of these new theories in

scenarios with cosmological singularities 4.4.

Throughout the paper, we use the metric signature convention (−, +, +, +) and units in

which the speed of light and the Planck constant are unity c = ~ = 1, and the reduced Planck

mass is M

= 1/

√

8πG. We follow the convention R

αβ

= ∂

βα

−∂

µα

+ Γ

µν

βα

−Γ

βν

µα

for the Ricci tensor and T

µν

= Γ

[µν]

for the torsion. Some contractions of rank-2 tensors are

denoted by M

=[M]=Tr(M), M

=[M

]=Tr(M

), M

=[M

], and so on.

– 3 –

JCAP11(2014)004

2 The theory

2.1 Born-Infeld electromagnetism

Motivated by ﬁnding a regularization scheme for the singularities that appear in classical

electromagnetism at small scales (UV regime), Born and Infeld suggested to construct a

nonlinear extension of the Maxwell action in the form [30]

BIE

= −λ



−det (η

µν

+ λ

−2

µν

) − 1



, (2.1)

where η

µν

is the Minkowski metric, F

µν

= ∂

− ∂

the strength tensor of the elec-

tromagnetic potential A

and λ some energy scale. The form of this action follows a close

analogy to that of a relativistic point particle S

= −mc

1 − v

. Very much like

in special relativity the speed of light serves as an upper limit for speeds, in Born-Infeld

electromagnetism the scale λ gives the maximum allowed value for electromagnetic ﬁelds.

The above action can be alternatively written as

BIE

= −λ

1 +

2λ

µν

−

16λ

µν

)

− 1

(2.2)

= −λ





1 −

− B

−

(E · B)

− 1





with

µν

≡

µναβ

αβ

the Hodge dual of F

αβ

and in the second line we have used the usual

deﬁnitions for the electric and magnetic ﬁelds E

= −F

and B



ink

. For small

electromagnetic ﬁelds |F

µν

|  λ

, the leading order contribution to the action reads

BIE

(|F

µν

|  λ

) ' −

µν

(2.3)

and we see that the theory reduces to usual Maxwell electrodynamics. However, for ﬁelds

of order λ

the self-interaction terms become important and prevent the unlimited growth

of electric and magnetic ﬁelds so that UV singularities are regularized. In particular, the

self-energy of point-like charges becomes ﬁnite. One important feature of Born-Infeld elec-

tromagnetism is that it preserves the hyperbolicity of the ﬁeld equations, thus, guaranteeing

causal propagation.

2.2 Born-Infeld inspired gravity

Inspired by Born-Infeld electrodynamics, Deser and Gibbons [3] suggested a similar construc-

tion for gravity. They started by considering an action of the form

BIG

= λ

−det (g

µν

+ λ

−2

µν

+ c

µν

) (2.4)

where g

µν

is the spacetime metric tensor, R

µν

is the metric Ricci tensor and X

µν

contains

terms of quadratic and higher orders in R

µν

and must be chosen to remove ghosts from the

spectrum of the theory. The scale λ must be tuned to recover GR at low curvatures.

It was argued in [3] that, unlike in the original Born-Infeld construction for electro-

magnetism, there was no clear criterion to make an analogous construction for gravity. The

– 4 –

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