Kerr–Newman-NUT黑洞：视界与奇异性的新特性
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"这篇论文探讨了Kerr–Newman-NUT时空的一种特殊类型，其中黑洞的视界位于$r = 2M$，条件是$Q^2 = l^2 - a^2$，这里的M、l、a和Q分别表示黑洞的质量、NUT参数、旋转参数和电荷。这种配置导致了因果结构的根本改变，因为没有柯西视界，暗示了当存在奇点时其具有类似于空间的特性。对于$l^2 > a^2$的情况，虽然没有曲率奇点，但可能存在圆锥奇点。此外，该黑洞的旋转参数$a/M$不受限制，甚至可以超过1，而不会破坏视界。"
在论文中，作者们详细研究了这类特殊黑洞的多种特性，包括赤道内外的时空结构、零地线学、超辐射和彭罗斯过程中的能量提取，以及热力学性质和准周期振荡。研究表明，与Kerr–Newman-NUT家族中的其他黑洞相比，这个特殊成员通过超辐射模式和彭罗斯过程损失的能量较少。
Kerr–Newman-NUT黑洞是广义相对论中一个重要的理论模型，它结合了旋转（Kerr）、电荷（Newman）和纽特恩（NUT）参数，形成一个复杂的解。在这个特殊情况下，由于奇点的特殊性质，黑洞的物理行为表现出独特性。例如，没有柯西视界意味着常规的黑洞事件视界不适用，这可能影响到时空的局部和全局结构。
超辐射是黑洞物理学中的一个重要现象，指的是粒子从黑洞中获得能量并逃离的过程，而彭罗斯过程则涉及利用黑洞的旋转能量来提取能量。论文中提到，这个特殊Kerr–Newman-NUT黑洞在这些过程中表现出的效率较低，可能对理解黑洞动力学和能量提取机制有深远的影响。
此外，热力学性质的研究揭示了黑洞的熵和温度等关键参数如何变化。准周期振荡是黑洞附近物质运动的一种表现，它们可能与黑洞吸积盘的物理过程有关。对这些振荡的研究有助于深入理解黑洞周围的动力学环境。
这项工作深化了我们对Kerr–Newman-NUT时空的理解，特别是对于那些视界位置和参数选择导致奇点性质显著改变的黑洞。这些发现对于黑洞物理、引力理论和宇宙学的未来研究具有重要意义，并可能推动我们对黑洞基本性质的新认识。
Eur. Phys. J. C (2019) 79 :161 Page 5 of 24 161
A =−Et + Lφ + A
r
(r) + A
θ
(θ) (6)
Here A
r
(r) and A
θ
(θ) correspond to the parts of the action
dependent on radial coordinate r and angular coordinate θ
respectively. Given the above structure of the action asso-
ciated with a particle moving in the Kerr–Newman-NUT
spacetime, the Hamilton–Jacobi equation becomes separa-
ble. Thus one can obtain separate equations for A
r
(r) and
A
θ
(θ) respectively, having the following structures,
dA
θ
dθ
2
+
(EP − L)
2
sin
2
θ
+ m
2
(l + a cos θ)
2
= K (7)
dA
r
dr
2
+ m
2
r
2
−
1
{E(r
2
+ a
2
+l
2
) − aL}
2
=−K (8)
In the above expressions the quantity K acts as the sepa-
ration constant and m is the mass of the orbiting particle.
Among others l is the NUT charge, a is the rotation parame-
ter and P = a sin
2
θ −2l cos θ . Since the components of the
momentum four vector of the orbiting particle correspond
to, ∂A/∂ x
μ
, it is possible to rewrite Eqs. (7) and (8)in
a more explicit form. This involves writing the separation
constant K appearing in both Eqs. (7) and (8), such that
K = λ +(L −aE)
2
, with λ as the Carter constant. Thus we
obtain the following geodesic equations,
m
2
ρ
4
dr
dτ
2
={E(r
2
+ a
2
+l
2
) − aL}
2
− (λ+m
2
r
2
) −
(
L − aE
)
2
(9)
m
2
ρ
4
sin
2
θ
dθ
dτ
2
= λ sin
2
θ +
(
L − aE
)
2
sin
2
θ
−
[
EP−L
]
2
−m
2
sin
2
θ(l+a cos θ)
2
(10)
where τ represents the affine parameter for a timelike
geodesic. Similar to the above case with massive particles, for
photons with zero rest mass, the geodesic equations become,
ρ
4
dr
dν
2
={E(r
2
+ a
2
+l
2
) − aL}
2
− λ −
(
L − aE
)
2
(11)
ρ
4
sin
2
θ
dθ
dν
2
=λ sin
2
θ +
(
L − aE
)
2
sin
2
θ −
[
EP − L
]
2
(12)
where ν stands for affine parameter along the null geodesic.
Given the above two geodesic equations one can proceed
towards solving these equations, which will ultimately lead to
the trajectory of a particle moving in the Kerr–Newman-NUT
spacetime. With the complicated structure of the geodesic
equations as presented above, it is very difficult to solve
them in general circumstances. However in some specific
situations, e.g., in the equatorial plane (located at θ = π/2)
it is indeed possible to solve the above equations analyti-
cally, which we will discuss next. Having grasped the analyt-
ical structure of the trajectory associated with the equatorial
plane, we will consider the general scenario later on.
3.1 Orbits confined on a given plane
The addition of the NUT charge will introduce nontrivial
difficulties while obtaining the orbital dynamics for a mas-
sive or even a massless particle. This can be understood by
employing the angular equations given in Eqs. (10) and (12)
respectively. For any trajectory to be confined on a particu-
lar plane θ = θ
0
, we bound to have
˙
θ =
¨
θ = 0. While the
condition
˙
θ = 0 indicates a trajectory moving on a constant
plane, the additional second derivative ensures that the first
derivative remains null throughout its motion. Therefore both
of these conditions are essentials to determine any planner
orbit in the presence of a NUT charge and unlike the Kerr
black hole, this constraint would introduce stringent bound
on the particle trajectories in Kerr–Newman-NUT spacetime.
In the upcoming discussions, we shall explore any possible
scenarios in which the conditions for a planner orbit can be
satisfied in a certain ranges of parameters. Regarding that, we
first introduce the massless particles and following that, the
phenomenon involving massive particles will be addressed.
3.1.1 The massless particles
For a massless particle, the effective radial potential can be
written as
V
eff
(r) ={E(r
2
+ a
2
+l
2
) − aL}
2
− {λ +
(
L − aE
)
2
}.
(13)
Remember, the value of the Carter constant λ would only be
fixedfromtheangularequations,i.e,
˙
θ =
¨
θ = 0. Eventhough
for a general E and L, the angular equations are compli-
cated to solve and often subjected to numerical calculations,
we start with a simple exercise which corresponds to the
following relation between angular momentum and energy:
L = aE. In this case the impact parameter, defined as the
ratio of angular momentum and energy, turns out to be D
sp
≡
L/E = a. On other hand, the angular equations become
V
eff
(θ) = λ sin
2
θ −{aE sin
2
θ − 2El cos θ − aE}
2
= 0,
V
eff
(θ)
dθ
= 2sinθ cos θλ − 2{aE sin
2
θ − 2El cos θ − aE}
123
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