dRGT重力中黑弦的准正规模式：标量摄动影响分析
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更新于2024-07-16
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"这篇论文详细探讨了de Rham–Gabadadze–Tolley (dRGT) 大型引力理论中的中性黑弦的准正规模式（QNMs）。研究集中于标量摄动如何影响这些黑弦，并利用准正规模式来理解这种影响。dRGT重力理论是一种引入额外度自由度以解决引力理论中的超距作用问题的理论。在该理论中，引力不再是由纯度度规描述，而是通过包含额外的大质量场来修正。
通过对准正规模态的分析，研究人员使用了渐近迭代方法（AIM）和WKB方法来数值计算具有正宇宙常数的黑弦背景下的标量摄动的QNM。高动量QNM的行为可以通过一阶WKB方法得到精确的近似，这对于理解和预测这些模式的动态至关重要。此外，对于事件视界非常接近宇宙视界的近乎极端的黑弦，研究者应用了Pöschl-Teller技术，得到了QNM的精确解析公式。
研究发现，随着质量重力参数 $$ \gamma $$ 的增加，标量模式的振动频率提高，衰减速度也更快。这表明更重的引力质量导致了更快的能量耗散。同时，当 $$ \gamma $$ 从正值变为负值时，全息声极的运动发生碰撞并形成扩散极，揭示了负宇宙常数下时空结构的变化。
文章还使用频谱方法全面探讨了 $$ \gamma $$ 在所有可能范围内的情况，特别是在负宇宙学常数背景下的中性黑弦的QNM。然而，无论宇宙学常数是正还是负，研究者都没有发现中性黑弦不稳定的迹象，这为dRGT引力理论提供了一定的稳定性证据。
这篇开放获取的文章由Supakchai Ponglertsakul、Piyabut Burikham和Lunchakorn Tannukij合作完成，分别来自泰国朱拉隆功大学的高能理论物理组和韩国汉阳大学的物理系。它发表于《欧洲物理期刊C》（Eur.Phys.J.C, 2018, 78:584），并可通过提供的DOI链接（https://doi.org/10.1140/epjc/s10052-018-6057-9）进行访问。这项工作对于理解大规模引力理论中的复杂物理现象以及黑洞和弦理论的相互作用具有重要意义。"
Eur. Phys. J. C (2018) 78:584 Page 3 of 15 584
α
4
=
β
4
+
1 − α
12
. (4)
U
2
=[K]
2
−[K
2
], (5)
U
3
=[K]
3
− 3[K][K
2
]+2[K
3
], (6)
U
4
=[K]
4
− 6[K]
2
[K
2
]+8[K][K
3
]+3[K
2
]
2
− 6[K
4
].
(7)
α and β are free parameters. K
μ
ν
= δ
μ
ν
−
g
μσ
f
ab
∂
σ
φ
a
∂
ν
φ
b
.
[K]=K
μ
μ
and [K
n
]=(K
n
)
μ
μ
. We will work in unitary
gauge for which the four Stückelberg fields take the form
φ
a
= x
μ
δ
a
μ
. The fiducial metric f
ab
is chosen to be,
f
ab
= diag(0, 0,α
2
g
h
2
0
, h
2
0
), (8)
for the coordinates (t, r, z,ϕ)where α
g
and h
0
are arbitrary
constants.
2.1 Field equations
1. Einstein’s equations
R
μν
−
1
2
Rg
μν
=−m
2
g
X
μν
+ T
μν
(9)
where the full expression for X
μν
can be found in [11,17].
The energy-momentum tensor is given by,
T
μν
=∇
μ
∇
ν
−
1
2
g
μν
g
σρ
∇
σ
∇
ρ
+ m
2
s
2
(10)
2. Scalar-field equation
∇
a
∇
a
− m
2
s
= 0. (11)
2.2 Static solutions
The Einstein field equation (9) admits cylindrically symmet-
ric solution or “dRGT black string” in the absence of scalar
field, i.e., T
μν
= 0. The dRGT black string solution is defined
as [17],
ds
2
=−f (r)dt
2
+ f
−1
dr
2
+r
2
α
2
g
dz
2
+r
2
dϕ
2
, (12)
where,
f (r) = α
2
m
r
2
−
4M
α
g
r
+ γ r + , (13)
α
2
m
= m
2
g
(1 + α + β) ≡−
3
, (14)
γ =−
α
2
m
h
0
(1 + 2α + 3β)
1 + α + β
, (15)
=
α
2
m
h
2
0
(α + 3β)
1 + α + β
. (16)
M is mass per unit length in z direction of black string. One
can clearly see that the graviton mass m
g
naturally generates
the effect of cosmological constant. With these definitions,
<0 case associates with α
2
m
> 0 while >0 is obtained
by letting α
2
m
< 0. This is the unique character of black
string in dRGT massive gravity. Since there is no de-Sitter
(dS) analogue of black string in standard four dimensional
general relativity [15]. The presence of linear term γ and
constant term makes black string with positive possible.
As will be shown later, these two branches of black string
solutions have different horizon and asymptotic structures.
Therefore, to determine their QNMs, we need to consider
each branch of solution separately. Note that, in the limit,
γ = 0, = 0,α
g
= α
m
, this metric (12) reduces to four
dimensional black string solution in general relativity found
by Lemos [15].
We shall now investigate the root structure of dRGT black
string with negative α
2
m
. Generally speaking, the metric func-
tion (13) has three zeros. With a proper parameter choice, it is
possible that all three roots will be real number. More specif-
ically, for α
2
m
< 0 there are two positive real roots and one
negative real root. These two positive roots will be treated as
black string’s event horizon r
h
and cosmological horizon r
c
,
where r
h
< r
c
. As an example, the roots structure of met-
ric (13) is displayed in Fig. 1. In this plot, the black string
mass M, cosmological constant α
2
m
, α
g
and are fixed to be
M = 1,α
2
m
=−0.01,α
g
= 1, = 0 respectively. Each
curve represents different values of γ . These black strings
shown in this figure have two positive roots. The inner root
is black string’s event horizon and the outer root is cosmic
horizon. As γ increases, the cosmic horizon increases, but the
event horizon slightly decreases. In our previous work [19],
we showed that the charged dRGT black hole has different
properties ranging from naked singularity to a black hole and
extremal black hole. However, this is not the case for neutral
black string with positive . Without the charge term, the
metric (13) cannot develop a naked singularity or extremal
scenario (for charged black string [17], naked singularity can
exist just like in the black hole case). Note that throughout
this work, will be fixed to zero [19]. However, this setting
prevents us from having black string solution with negative
and vanishing γ .
An example of black string solution with α
2
m
> 0 ( < 0)
can be found in Fig. 2. In this plot, we set the black string
mass, cosmological constant and other constant to be M =
1,α
2
m
= 4,α
g
= 1, = 0 and consider the effect of γ on the
spacetime. As can be seen from the plot, the horizon structure
dramatically changes from the positive case. There is only
one positive real root, i.e., black string’s event horizon, for
vanishing [26]. Unlike the positive case, the black string
exists with all possible value of γ (negative, zero and posi-
tive). We can see that the black string becomes large when γ
is more negative. It is clear from the metric (13) that, the black
123
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