N维f(T)重力下的旋转黑洞解析解：非对角vielbein与热力学一致性

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本文主要探讨了在N维f(T)重力理论背景下，通过采用非对角vielbein ansatz方法来处理场方程。该研究关注于一个特殊的二次多项式函数f(T) = T + ϵT^2，这是一个常见的f(T) gravity模型，其中T代表张量张量，而ϵ是与引力修正相关的耦合常数。作者首先推导出了一个解析的真空解，这个解的独特性在于它包含了非对角线分量，这表明了空间的非均匀性，即存在旋转性质。由于非对角线分量的存在，这些解决方案不能仅仅通过坐标变换来消除旋转，从而引入了额外的旋转参数。这一特性使得这些旋转黑洞解与传统的广义相对论中的静态或旋转解有所不同，它们可能反映了f(T)引力理论中的新型几何特性。研究者深入分析了由此得到的度量的曲率和扭转标量不变量，这些量是衡量时空曲率和扭转变换的物理量。通过对这些不变量的计算，作者试图揭示解的奇异性和视界的性质。值得注意的是，与广义相对论中通常的视界概念不同，f(T)理论中的视界可能会显示出高阶的扭转效应，这可能预示着新的物理现象。此外，文章还探讨了f(T)重力能量动量矢量的一般表达式，这是用来计算系统总能量的重要工具。通过这种方法，作者能够评估黑洞的物理属性，如质量和旋转，以及它们如何受到f(T)引力修正的影响。最后，一项关键的理论检验是验证这种解是否符合热力学第一定律。在f(T)引力理论的框架下，作者成功地证明了这一解确实满足热力学的基本原理，这进一步巩固了该理论的物理合理性。总结来说，这项研究不仅提供了N维f(T)重力下的一种旋转黑洞解析解，而且深入分析了其几何特性和物理意义，同时验证了理论的自洽性。这不仅丰富了对f(T)引力的理解，也为未来的黑洞物理和宇宙学研究提供了新的视角和可能。

Eur. Phys. J. C (2017) 77:90

DOI 10.1140/epjc/s10052-017-4663-6

Regular Article - Theoretical Physics

Analytic rotating black-hole solutions in N-dimensional f (T)

gravity

G. G. L. Nashed

1,2,3,a

, W. El Hanafy

1,3,b

Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43, 11837 Cairo, Egypt

Faculty of Science, Mathematics Department, Ain Shams University, Cairo, Egypt

Egyptian Relativity Group (ERG), Cairo, Egypt

Received: 31 October 2016 / Accepted: 31 January 2017 / Published online: 11 February 2017

Abstract A non-diagonal vielbein ansatz is applied to the

N -dimension ﬁeld equations of f (T ) gravity. An analyti-

cal vacuum solution is derived for the quadratic polynomial

f (T ) = T + T

and an inverse relation between the cou-

pling constant  and the cosmological constant . Since the

induced metric has off-diagonal components, it cannot be

removed by a mere coordinate transformation, the solution

has a rotating parameter. The curvature and torsion scalars

invariants are calculated to study the singularities and hori-

zons of the solution. In contrast to general relativity, the

Cauchy horizon differs from the horizon which shows the

effect of the higher order torsion. The general expression of

the energy-momentum vector of f (T ) gravity is used to cal-

culate the energy of the system. Finally, we have shown that

this kind of solution satisﬁes the ﬁrst law of thermodynamics

in the framework of f (T ) gravitational theories.

1 Introduction

General relativity is constructed in Riemann geometry, which

reduces to Minkowski spacetime in the absence of a gravita-

tional ﬁeld. In the Riemannian geometry, the differentiable

manifold is described by a metric tensor which measures the

distances between different space points. However, gravita-

tion in this theory is a manifestation of the curvature of the

spacetime, where the Riemann Christoffel tensor is respon-

sible for this curvature. All quantities in this manifold can

be deﬁned in terms of the metric tensor [1]. At present time,

to extend GR to include torsion is considered as an urgent

issue because many questions depend on whether the space-

time connection is symmetric or not. General relativity is an

orthodox theory that does not allow quantum effects. How-

e-mail: nashed@bue.edu.eg

e-mail: waleed.elhanafy@bue.edu.eg

ever, these effects should be taken into account in any theory

that accompanies gravity. Changing geometry from V

(4-

dimensional Riemannian manifolds) to U

(4-dimensional

Weitzenböck manifolds) is considered as a ﬁrst direct exten-

sion that attempts to incorporate the spin ﬁelds of matter

into the same geometrical scheme of GR. As an example

of this, we have the mass-energy in which curvature is the

source of gravity, while the spin of torsion. Einstein–Cartan–

Sciama–Kibble theory is considered as one of the most sig-

niﬁcant trails in this direction. Nevertheless, the task of spin-

matter ﬁelds does not explain the function of the torsion ten-

sor that seems to play important tasks in any fundamental

theory.

The accelerated expansion of our Universe is conﬁrmed

by many separated cosmological experiments [3–21]. This

expansion can be explained in GR by taking into account the

dark energy component in the total energy of our Universe.

It is widely accepted, without real justiﬁcations, to interpret

these components by inserting a cosmological constant 

into Einstein’s ﬁeld equation which is called CDM model

[2]. Also, this model can be extended by adding a speculated

dynamical ﬂuid with a negative pressure, e.g. quintessence

ﬂuid. Therefore, we can consider the accelerated expansion

of our Universe as an indicator of the failure of our informa-

tion to the principles of gravitational ﬁeld, and thus a modiﬁ-

cation of GR is necessary. This modiﬁcation can be done by

making the action a function of the curvature scalar R, i.e.,

f (R) [22–24].

Using a different processor we can set up another method

and use the Weitzenböck connection, which includes torsion

as an alternative to curvature. This process was suggested by

Einstein who called it a “teleparallel equivalent of general

relativity” (TEGR) [1,25–30]. This theory, TEGR, is closely

related to GR and differs only by a total derivative term in

the action. The dynamical objects in such geometry are the

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