Distinguishing a Slowly Accelerating Black Hole by Dierential Time Delays of Images Amjad Ashoorioonand Mohammad Bagher Jahani Poshtehy

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Distinguishing a Slowly Accelerating Black Hole

by Diﬀerential Time Delays of Images

Amjad Ashoorioon∗and Mohammad Bagher Jahani Poshteh†

School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

Robert B. Mann‡

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

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

Accelerating supermassive black holes, connected to cosmic strings, could contribute to structure

formation and get captured by galaxies if their velocities are small. This would mean that the

acceleration of these black holes is small too. Such a slow acceleration has no signiﬁcant eﬀect on

the shadow of such supermassive black holes. We also show that, for slowly accelerating black holes,

the angular position of images in the gravitational lensing eﬀects do not change signiﬁcantly. We

propose a method to observe the acceleration of these black holes through the gravitational lensing.

The method is based on the observation that diﬀerential time delays associated with the images are

substantially diﬀerent with respect to the case of non-accelerating black holes.

Over the past ﬁve years observations from

LIGO/VIRGO has provided us with qualitatively

new information about our universe via gravitational

waves. Black holes have so far been the primary source

of gravitational waves, but it is reasonable to expect

that future detectors will discover new sources, some of

which will yield information about the early universe.

Cosmic strings – line-like topological defects that can

emerge from some gauge theories during ﬁrst order phase

transitions [1,2] or at the end of brane inﬂation [3] –

are one such example. Such objects can break or fray to

produce pair of accelerating black holes [4,5], which can

also be produced in a background magnetic ﬁeld [6–8]

or in de Sitter space [9–13]. Alternatively, a network

of cosmic strings could capture primordial black holes

produced during their formation [14]. Such black holes

would be accelerating due to the tension of the cosmic

string.

It is therefore of interest to examine how we might de-

tect accelerating black holes. Although accelerating su-

permassive black holes connected to cosmic strings could

reside in the centers of galaxies [14,15], their velocities

must be small (<

∼100 km/s), so that they can contribute

to structure formation [14]. Measurement of their ac-

celeration is therefore a formidable challenge, since this

quantity must consequently be very small.

Here we propose a method for detection of black hole

acceleration via gravitational lensing, in which rays of

light coming from a source behind the black hole are de-

ﬂected near it and turn toward an observer. The ob-

server does not see the actual location of the source,

but rather sees images of it apparently located elsewhere.

Our method exploits the fact that diﬀerential time delays

associated with lensed images of accelerating black holes

∗amjad@ipm.ir

†jahani@ipm.ir

‡rbmann@uwaterloo.ca

substantially diﬀer with respect to their non-accelerating

counterparts.

Spacetime around uniformly accelerated black holes

can be described by

ds2=1

(1 + αr cos θ)2

×−Q(r)dt2+dr2

Q(r)+r2dθ2

P(θ)+P(θ)r2sin2θdφ2,

(1)

known as the C-metric [16,17], where

Q(r) = (1 −α2r2)1−2m

r,

P(θ) = 1 + 2αm cos θ. (2)

mis the mass parameter and αis interpreted as the ac-

celeration. We take the acceleration to be suﬃciently

small so that a ray of light passing the black hole to the

Earth, lies on the equatorial plane θ=π/2 of the black

hole during its evolution. On this plane, the line element

(1) reduces to 1

ds2=−Qdt2+dr2

Q+r2dφ2,(3)

where Q(r) is given by (2). In Fig. 1we schematically

illustrate the setup. Light coming from the source passes

the black hole and is deﬂected by an angle ˆα. The ob-

server sees the image of the object at angular position ϑ.

The equations governing the geodesics can be obtained

using the Lagrangian

L=1

2gµν ˙xµ˙xν=1

2−Q˙

t2+˙r2

Q+r2˙

φ2,(4)

1Within the small acceleration approximation, we can ﬁrst ﬁnd

the geodesic equations and then substitute θ=π/2 — the results

will be the same as presented here.

arXiv:2210.10762v2 [gr-qc] 19 Jan 2023

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摘要：

DistinguishingaSlowlyAcceleratingBlackHolebyDierentialTimeDelaysofImagesAmjadAshoorioonandMohammadBagherJahaniPoshtehySchoolofPhysics,InstituteforResearchinFundamentalSciences(IPM),P.O.Box19395-5531,Tehran,IranRobertB.MannzDepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,Ontario,N2L3G...

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Distinguishing a Slowly Accelerating Black Hole by Dierential Time Delays of Images Amjad Ashoorioonand Mohammad Bagher Jahani Poshtehy

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