非线性电动力学映射：揭示广义相对论与非线性引力的对应关系

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本文探讨了如何通过非线性电动力学将非线性重力理论映射到广义相对论（GR）的背景下，进行理论物理研究的新途径。作者维克多·阿方索等人，来自巴西、西班牙、葡萄牙等不同机构的研究者，提出了一种创新的方法，即利用度量仿射法将一系列非线性重力理论与电动力学非线性理论结合起来。这种方法的关键在于通过代数变换，可以直接从广义相对论的解决方案中导出非线性重力理论的相应解，这种对应关系在爱丁顿启发的Born-Infeld引力理论中得到了明确展示。 Born-Infeld引力理论是研究对象之一，它是一种非线性引力模型，其耦合到麦克斯韦电动力学时，能够揭示出特殊的非线性电动力学性质。通过处理球对称的电真空情况，研究者展示了这种映射是如何直接应用的，从而生成正确的非线性重力理论解决方案。这种方法的运用，使得在广义相对论的分析和数值方法的强大支持下，非线性引力理论在天文学和宇宙学中的应用领域得以扩展，为探索这些复杂系统的性质开辟了新的可能性。值得注意的是，这项研究发表在《欧洲物理杂志C》上，2018年卷78，第866期，文章的DOI为10.1140/epjc/s10052-018-6356-1，属于开放获取资源。论文的接收日期为2018年7月22日，接受日期为10月18日，最终在线发布日期为10月27日。这标志着理论物理学界对于非线性重力与电动力学之间相互作用的深入理解，以及在实际问题解决中的应用价值正在不断得到提升。

Eur. Phys. J . C (2018) 78 :866 Page 3 of 11 866

i.e., ∇



αβ

= 0 (thus 

μν

is given by the Christoffel sym-

bols of q

μν

), while non-metricity is present for the space-

time metric, Q

μαβ

≡∇



αβ

= 0. In the above expression

the indices of the stress-energy tensor T

μν

−2

√

−g

δ(

√

−gL

)

δg

μν

are raised with the space-time metric, T

≡ g

μα

αν

, so that

T ≡ g

μν

denotes its trace. It is worth noting that the aux-

iliary metric q

μν

admits a nice interpretation in terms of an

analogy between some condensed matter systems and grav-

itational physics [39–42] in which the nonmetricity tensor

arises due to the existence of point-like defects in the under-

lying microscopic structure. In that analogy, g

μν

describes

the physical (defected) geometry, while q

μν

represents an

idealized structure without defects (vanishing nonmetricity).

Written in the form Eq. (2), the resulting ﬁeld equations

represent a set of second-order, Einstein-like equations for

the metric q

μν

, where the right-hand side depends on the

matter ﬁelds and, possibly, on g

μν

. This is particularly useful

from both a conceptual and an operational point of view,

since in RBGs the relation between the auxiliary, q

μν

, and

spacetime, g

μν

, metrics can always be written as

μν

= g

μα



. (3)

The deformation matrix 

(hereafter vertical bars will

denote its determinant) depends on the particular RBG cho-

sen but, likewise L

, it can always be written on-shell as

a function of the stress-energy tensor, T

(and possibly

on g

μν

as well). It should be pointed out that this q

μν

representation of the ﬁeld equations highlights the fact that

gravitational waves will propagate upon the background

deﬁned by q

μν

, rather than the one deﬁned by g

μν

(see [43]

for an explanation in the case of Born–Infeld-type theories

of gravity, and [44,45] for some related phenomenology).

An appealing feature of metric-afﬁne theories as follows

from the discussion above is that in vacuum, T

= 0, the

deformation matrix becomes the identity. Accordingly, one

has q

μν

= g

μν

(modulo a trivial re-scaling) and the ﬁeld

equations (2) of RBGs yield GR, possibly with a cosmologi-

cal constant term depending on the form of L

. This implies

that metric-afﬁne RBGs, as deﬁned by Eq. (1) do not propa-

gate extra degrees of freedom beyond the standard two polar-

izations of the gravitational ﬁeld travelling at the speed of

light (same as in GR). In turn, this allows these theories to suc-

cessfully pass both solar system experiments [46–48],

and

the recent gravitational wave observations from the LIGO-

VIRGO collaboration related to equality of speed of propa-

gation of gravitational and electromagnetic waves [8], as well

as the absence of additional polarization modes [9], results

which have either ruled out or placed strong constraints upon

For constraints of these theories from particle physics scattering

experiments, improving the solar system constraints by a few orders

of magnitude (depending on the particular RBG), see [49].

other proposals of gravitational schemes to extend GR [50–

55].

We stress that while the left-hand side of Eq. (2)isa

well deﬁned function of the metric q

μν

, the right-hand side

depends nonlinearly on the matter ﬁelds and g

μν

via the func-

tions L

and |

|. Though the dependence on g

μν

can be

explicitly solved in favour of q

μν

in speciﬁc cases, such as

in homogeneous and isotropic cosmological models and in

spherically symmetric scenarios, it is unclear that this can

be done in more general and less symmetric conﬁgurations.

In fact, from a numerical perspective, the inversion of that

relation is likely to be computationally very expensive. Addi-

tional efforts are thus necessary to bring the above equations

into a form that can be systematically worked out. Such is

the purpose of this paper.

In order to show that the Einstein frame representation of

the ﬁeld Eq. (2) can be written without making any reference

to the metric g

μν

, one must consider that there exists a new

matter source coupled to q

μν

such that the right-hand side of

that equation can be written in the standard Einstein form

(q) = κ

, (4)

where

≡ q

μα

αν

represents the stress-energy tensor

of a new set of matter ﬁelds of the same kind as the origi-

nal ones, i.e., ﬂuids turn into ﬂuids, scalar ﬁelds into scalar

ﬁelds, and so on. There is a simple reason to establish this

correspondence. Since the Einstein tensor on the left-hand

side is conserved by virtue of the contracted Bianchi identity,

∇

(q) = 0, the right-hand side must also be conserved

on-shell. Since the right-hand side represents a certain kind

of matter with deformations induced by the nonlinear grav-

ity theory, in order to be conserved under the action of ∇

should be possible to express it in the form of the same kind

of matter source (ﬂuid, scalar, vector, …), coupled to q

μν

in a way that guarantees its conservation. The simplest such

a choice is, evidently, the form of a standard stress-energy

tensor, which is the one we assume here. The equation above

thus supports the convenience of the q

μν

-representation of

the ﬁeld equations, which allows to transfer the problem of

generating solutions in RBGs from solving differential ﬁeld

equations to algebraic ones (albeit non-linear, in general).

Following the above discussion, in order to establish a

direct map between the spaces of solutions of RBGs and

GR, by comparison between Eqs. (2) and (4), one must have

|

1/2



− δ





, (5)

which relates the effective stress-energy tensor generated by

the matter coupled to the RBG, T

, with the one cou-

pled to GR,

. Note that this map works irrespective

of assumptions on symmetries of the problem or particu-

123

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