引力波探测与帕拉蒂尼引力理论探索
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"这篇论文探讨了使用引力波探测来测试重力理论，特别是Palatini型引力理论。文章关注的是在这些理论中带电黑洞的轴向扰动的准正态模（QNMs）。研究涉及两个理论：一是Palatini f(R)引力与Born-Infeld非线性耦合电动力学的结合，二是Eddington启发的Born-Infeld引力（EiBI）与Maxwell理论的选择。作者包括来自不同大学和研究机构的物理学家。该资源可在Eur.Phys.J.C(2019)79:63上找到，DOI为10.1140/epjc/s10052-019-6585-y，并且是开放访问的。"
在物理学中，引力波探测已经成为验证和测试各种引力理论的重要工具。Palatini型引力理论是一种相对论性的引力理论，它与广义相对论有所区别，主要在于如何处理度量和联络的几何关系。在Palatini形式主义中，这两个量被视为独立的，而广义相对论则将它们视为相互关联的。这种差异可能导致不同的物理预测，尤其是在极端条件下，如黑洞或大爆炸等。
本研究中提到的QNMs是黑洞在受到扰动后振荡并逐渐衰减的频率模式，这些模式反映了黑洞的固有属性。QNMs对于理解黑洞动力学至关重要，因为它们提供了关于黑洞物理特性的无反射振荡，可以作为测试引力理论的有效指标。在这种情况下，作者研究了带电黑洞的轴向QNMs，这涉及到黑洞的电荷和旋转对振荡模式的影响。
论文中涉及的两种Palatini型理论扩展了标准的重力模型。Palatini f(R)引力理论是广义相对论的修改版本，其中引力作用由R的某个函数f(R)代替了R本身来描述。Born-Infeld电动力学是一种非线性电磁理论，旨在解决经典电动力学中的无穷大问题，当电子接近光速时，其能量和动量会变得无限大。将这两者结合考虑，可以探究非线性效应如何影响黑洞的QNMs。
另一方面，Eddington启发的Born-Infeld引力（EiBI）是另一种引力理论，它试图在保持局域洛伦兹不变性的同时，消除某些奇异性。EiBI理论与Maxwell理论的结合则涉及了电磁场在黑洞背景下的行为，这可能会改变QNMs的性质。
通过对这两种理论的QNMs分析，作者旨在寻找可能与广义相对论预测不同的特征，从而提供对这些替代引力理论的实验验证。引力波探测器如LIGO和Virgo已经能够检测到引力波事件，未来可能的观测将有助于确认或排除这些理论，并进一步深化我们对宇宙基本力的理解。这项工作对于理论物理和天体物理学的研究具有重要意义，因为它推动了对引力理论边界的探索。
63 Page 4 of 17 Eur. Phys. J. C (2019) 79 :63
where φ
X
= dφ/dXand Q
∗
is an integration constant which
can be regarded as the charge of the black hole. Note that the
last equality in Eq. (3.3) can be obtained from the conser-
vation equation of NED at the background level [45]. From
Eq. (3.3), we get
F
(0)(2)
=
Q
∗
r
2
φ
X
. (background level) (3.4)
In the general case where the perturbations are taken into
account, the metric functions and the field strength could
depend on t, r, and θ . In this case, the Bianchi identity of the
field strength F
[(a)(b)|(c)]
= 0 leads to
e
ψ+μ
2
F
(1)(2)
,3
+
e
ψ+μ
3
F
(3)(1)
,2
= 0, (3.5)
e
ψ+ν
F
(0)(1)
,2
+
e
ψ+μ
2
F
(1)(2)
,0
= 0, (3.6)
e
ψ+ν
F
(0)(1)
,3
+
e
ψ+μ
3
F
(1)(3)
,0
= 0, (3.7)
e
μ
2
+ν
F
(0)(2)
,3
+
e
μ
3
+ν
F
(3)(0)
,2
+
e
μ
2
+μ
3
F
(2)(3)
,0
=−e
ψ+ν
q
3,2
− q
2,3
F
(0)(1)
+e
ψ+μ
2
σ
,3
−q
3,0
F
(1)(2)
+ e
ψ+μ
3
σ
,2
− q
2,0
F
(3)(1)
. (3.8)
Note that the comma here denotes the partial derivative with
respect to the tensor indices. This derivative is related to the
directional derivative shown in Eq. (2.7)byH
(a)(b),(c)
=
e
μ
(c)
H
(a)(b),μ
[59]. In addition, Eq. (3.5) is a redundant equa-
tion because it is just an integrability condition for Eqs. (3.6)
and (3.7).
On the other hand, the conservation equation for NED
η
(b)(c)
(F
(a)(b)
φ
X
)
|(c)
= 0 can be written explicitly as follows
e
ψ+μ
3
F
(0)(2)
φ
X
,2
+
e
ψ+μ
2
F
(0)(3)
φ
X
,3
= 0, (3.9)
e
ψ+μ
2
F
(0)(3)
φ
X
,0
+
e
ψ+ν
F
(3)(2)
φ
X
,2
= 0, (3.10)
e
ψ+ν
F
(2)(3)
φ
X
,3
+
e
ψ+μ
3
F
(0)(2)
φ
X
,0
= 0, (3.11)
e
ν+μ
3
F
(1)(2)
φ
X
,2
+
e
ν+μ
2
F
(1)(3)
φ
X
,3
+
e
μ
2
+μ
3
F
(0)(1)
φ
X
,0
=
e
ψ+μ
3
σ
,2
−q
2,0
F
(0)(2)
+e
ψ+μ
2
σ
,3
−q
3,0
F
(0)(3)
+ e
ψ+ν
q
3,2
− q
2,3
F
(2)(3)
φ
X
. (3.12)
Again, it can be shown that Eq. (3.9) is a redundant equation
since it is an integrability condition for Eqs. (3.10) and (3.11).
To linearize the equations above, the scalar field φ
X
and
the gauge field invariant X should be decomposed into the
background and the 1st order parts:
φ
X
→ φ
X
+ δφ
X
, X → X + δ X, (3.13)
where
δφ
X
= φ
XX
δ X = 2φ
XX
F
(0)(2)
δF
(0)(2)
. (3.14)
In the above expressions, we have decomposed F
(0)(2)
as
F
(0)(2)
→ F
(0)(2)
+δ F
(0)(2)
in which the background F
(0)(2)
is given in
1
Eq. (3.4). Note that the quantity X at the back-
ground level is given by X = F
2
(0)(2)
.
After the decomposition, the linearized Eqs. (3.6), (3.7),
and (3.8)are
re
ν
sin θ F
(0)(1)
,r
+re
μ
2
sin θ F
(1)(2),0
= 0, (3.15)
re
ν
F
(0)(1)
sin θ
,θ
+r
2
sin θ F
(1)(3),0
= 0, (3.16)
e
μ
2
+ν
δF
(0)(2),θ
+ F
(0)(2)
(
δμ
2
+ δν
)
,θ
+
re
ν
F
(3)(0)
,r
+re
μ
2
F
(2)(3),0
= 0. (3.17)
On the other hand, the linearized Eqs. (3.10), (3.11) and
(3.12) can be written as
φ
X
re
μ
2
F
(0)(3),0
+
φ
X
re
ν
F
(3)(2)
,r
= 0, (3.18)
φ
X
δF
(0)(2),0
+ F
(0)(2)
(
δψ + δμ
3
)
,0
+
φ
X
e
ν
r sin θ
F
(2)(3)
sin θ
,θ
+ δφ
X,0
F
(0)(2)
= 0, (3.19)
re
ν
φ
X
F
(1)(2)
,r
+ e
ν+μ
2
φ
X
F
(1)(3),θ
+ re
μ
2
φ
X
F
(0)(1),0
= r
2
sin θ F
(0)(2)
φ
X
σ
,2
− q
2,0
,
(3.20)
where δν, δμ
2
, δμ
3
, and δψ are the 1st order parts of the
metric functions ν, μ
2
, μ
3
, and ψ, respectively.
For the sake of abbreviation, we define the field perturba-
tion
B ≡ F
(0)(1)
sin θ. (3.21)
After differentiating Eq. (3.20) with respect to x
0
= t and
using Eqs. (3.15) and (3.16), we have
φ
X
e
ν−μ
2
re
ν
B
,r
,r
+ φ
X
e
2ν+μ
2
r
B
,θ
sin θ
,θ
sin θ
− φ
X
re
μ
2
B
,00
=−r
2
F
(0)(2)
φ
X
σ
,20
− q
2,00
sin
2
θ.
(3.22)
Recall that in the derivation of Eq. (3.22), we have only used
the Maxwell equation of the NED source. Eq. (3.22) would
become one of the coupled master equations in the sense
that the linear order metric functions on the right hand side
of Eq. (3.22) are coupled with the field perturbation B (or
the 1st order field strength F
(0)(1)
) on the left hand side. In
1
We only use a delta into the linear order perturbations of quantities
whose values at the background level do not vanish. The quantities that
vanish at the background level, such as the metric functions σ , q
2
,and
q
3
, and the Maxwell tensor components F
(i)( j)
(ij = 02 or 20), shall
be regarded as linear order perturbation quantities directly.
123
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