广义引力理论下的重力波研究：LIGO事件GW150914和GW151226

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"LIGO GW150914和GW151226重力波检测和广义重力理论（MOG）" 这篇研究文章深入探讨了广义重力理论（Modified Gravity Theory，简称MOG）在重力波现象中的应用。作者J.W. Moffat针对2015年LIGO探测到的GW150914和GW151226两个重力波事件进行了分析，并尝试用广义引力理论来解释这些观测结果。LIGO和Virgo合作组的数据表明，这两个事件可能涉及到了质量相对较小的黑洞二进制系统。文章首先介绍了广义引力理论的基本概念，这是一个超越了爱因斯坦的广义相对论的理论框架，旨在解决某些天体物理问题，例如星系旋转曲线的平坦化问题以及宇宙的大尺度结构。MOG理论试图通过引入新的引力修正项，来改变引力的强度和范围，从而可能解释无需暗物质即可理解的天文观测现象。研究中，作者推导了线性场方程，这是在弱引力场条件下广义引力理论的一个关键组成部分。这些方程用于描述重力波在空间时间中的传播。同时，还计算了度量张量四极矩功率，这是描述双黑洞系统合并时能量损失的重要量，这个过程伴随着重力波的发射。此外，文中还讨论了二进制系统中两个紧凑物体（如黑洞）轨道半径的减小，这是由于它们之间的引力相互作用和能量释放。特别地，文章引入了描述旋转黑洞的广义Kerr度量。Kerr度量是广义相对论中一个非常重要的解，它刻画了一个具有角动量（自旋）的黑洞的时空几何。在MOG框架下，黑洞的质量M和自旋参数a（由角动量S和引力常数G、黑洞质量M的关系决定）共同决定了旋转黑洞的特性。 LIGO-Virgo合作组的观测数据支持了这种解释，因为他们发现的黑洞质量与电磁观测中X射线双星系统的黑洞质量上限相吻合，即大约小于10个太阳质量（M≲10M⊙）。这表明，即使在低质量的黑洞系统中，广义引力理论也能提供合理的预测。总结来说，这篇文章通过结合LIGO的重力波数据，对广义引力理论进行了实证检验，尤其是针对黑洞二进制系统的行为。通过对MOG的运用，作者提供了关于重力波性质的新见解，这对于理解宇宙中的极端天体物理学现象以及可能替代暗物质的理论模型具有重要意义。

Physics Letters B 763 (2016) 427–433

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

LIGO GW150914 and GW151226 gravitational wave detection

and generalized gravitation theory (MOG)

J.W. Moffat

a,b,∗

Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada

a r t i c l e i n f o a b s t r a c t

Article history:

Received

27 May 2016

Received

in revised form 16 August 2016

Accepted

3 October 2016

Available

online 4 November 2016

Editor:

M. Trodden

The nature of gravitational waves in a generalized gravitation theory is investigated. The linearized

ﬁeld equations and the metric tensor quadrupole moment power and the decrease in radius of an

inspiralling binary system of two compact objects are derived. The generalized Kerr metric describing

a spinning black hole is determined by its mass M and the spin parameter a = cS/GM

. The LIGO-

Virgo

collaboration data is ﬁtted with smaller binary black hole masses in agreement with the current

electromagnetic, observed X -ray binary upper bound for a black hole mass, M  10M



(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

A century after the fundamental prediction by Einstein, based

on his gravitational ﬁeld equations [1] of gravitational waves [2,3],

and Schwarzschild’s derivation of his solution to the general rel-

ativity

(GR) ﬁeld equations [4], which led to the prediction of

black holes, the ﬁrst direct detections of gravitational waves has

been reported [5,6]. The gravitational waves are produced by the

merging of a binary black hole system to form a single black

hole. The measurements lead to a new access to the physi-

cal

properties of spacetime and strong ﬁeld gravity. The obser-

vations

of the energy loss by Taylor and Weisberg [7], follow-

ing

the discovery of the binary pulsar system PSR B1913+16 by

Taylor and Hulse [8] demonstrated the existence of gravitational

waves.

the following, we will investigate the nature of gravitational

waves in a generalized gravitation theory calle d Scalar–Tensor–

Vector–Gravity

(STVG), also known in the literature as MOG (mod-

iﬁed

gravity) [9]. The theory has been studied as an alternative

to GR without (detectable) dark matter in the present universe,

and ﬁts to galaxy rotation curves and galaxy clusters have been

obtained [10–12]. Moreover, the theory has been applied to cos-

mology

with an explanation for the growth of structure in the

early universe and ﬁts have been obtained to the CMB data [14,

15].

In the early universe cosmology, the mass of the vector ﬁeld

Correspondence to: Perimeter Institute for Theoretical Physics, Waterloo, ON

N2L 2Y5, Canada.

E-mail

address: jmoffat@perimeterinstitute.ca.

is m

 10

−22

eV and acts as a cold dark matter particle

with density ρ

> ρ

, where ρ

and ρ

denote the density of

the boson particle and baryons, respectively. As the universe ex-

pands

and enters the era of the formation of stars and galaxies

the effective boson mass decreases to the value m

∼ 10

−28

eV,

and due to the weak gravitational coupling of the particle to or-

dinary

matter the particle will be unobservable in the late-time

universe.

alternative early universe cosmology without dark matter is

formulated in [16].

our generalized gravitational theory electromagnetic waves

(photons) and gravitational waves (gravitons) move with the speed

of light. The null geodesic equation for photon paths is determined

in a Jordan frame conformal metric, and in the Einstein frame met-

ric

the gravitational constant for photon paths is screened, yielding

the deﬂection of light by the Sun, and the Shapiro time delay in

agreement with GR. The enhanced gravitational interaction expe-

rienced

by photons in the lensing of galaxies and galactic clusters

leads to an explanation of gravitational lensing data without dark

matter [13].

The

paper is organized, as follows. In Section 2, we present the

STVG ﬁeld equations, while in Section 3, we investigate the lin-

earized

weak ﬁeld approximation of the ﬁeld equations, the mod-

iﬁed

two-body acceleration law and the tensor gravitational wave

equations for a binary system. Section 4, presents the generalized

Kerr solution of the gravielectric vacuum STVG ﬁeld equations,

while in Section 5, we investigate the inspiralling merger of two

black holes and the LIGOGW150914 and GW151226 detections of

gravitational waves. In Section 5, we discuss the measurements of

black hole masses, and we end with conclusions in Section 6.

http://dx.doi.org/10.1016/j.physletb.2016.10.082

0370-2693/

SCOAP

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