Stationary charged Zipoy-Voorhees metric from colliding wave spacetime Mustafa HalilsoyMert Mangutand Chia-Li Hsieh

2025-05-03 0 0 314.16KB 13 页 10玖币

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Stationary, charged Zipoy-Voorhees metric from colliding wave

spacetime

Mustafa Halilsoy,∗Mert Mangut †,‡and Chia-Li Hsieh§

Department of Physics, Faculty of Arts and Sciences,

Eastern Mediterranean University, Famagusta,

North Cyprus via Mersin 10, Turkey

Abstract

Through the Ernst formalism we provide expression for a class of colliding Einstein-Maxwell (EM)

metrics with cross polarization. Local isometry is imposed as a means to transform interaction

region of the spacetime into stationary, charged Zipoy-Voorhees (ZV) metric in Schwarzschild

coordinates. This is known as the Chandrasekhar-Xanthopoulos (CX) duality which maps the

plane of double-null coordinates (with two spacelike cyclic coordinates) to the static/stationary

spacetime. The ZV-metric is known to describe planetary/stellar objects with arbitrary distortion

parameter. Asymptotic behaviour of the metric for practical use is provided.

”The world may be seen in a grain of sand” - William Blake.

†Corresponding author

∗Electronic address: mustafa.halilsoy@emu.edu.tr

‡Electronic address: mert.mangut@emu.edu.tr

§Electronic address: galise@gmail.com

arXiv:2210.15007v3 [gr-qc] 15 Sep 2023

I. INTRODUCTION

Surely a grain of sand may not encompass the world , but Chandrasekhar-Xanthopoulos

(CX)-duality between colliding plane waves (CPWs) spacetime and black holes (BHs), in

analogy with the wave-particle duality [1, 12] may do the trick. This was the case of our

recently found [2] static, charged, ZV-metric [3, 4]. Such a process implicates a diﬀerent kind

of holography from the one between 4D inner region of a BH and 3D boundary at inﬁnity.

The AdS/CFT correspondence [5] emerges as the resulting physics from the fact that the

inner gravity is taken into account by the ﬂat boundary of AdS, apt for quantum ﬁeld theory.

We note that our CX- duality, equivalently and under the title of Chandrasekhar-Ferrari-

Xanthopoulos has been employed in [18] to obtain the graviton scattering amplitudes at

Planckian length. Our analysis in this study will be entirely classical.

In the CX-duality/holography, distinct from the familiar one of diﬀerent dimensionalities,

both spacetimes are in 4D and is valid also between CPW and non-BH spacetimes as we

shall undertake in the present study. However, CPW geometry depends only on the two null

coordinates (u, v) or their useful combinations (η(u, v), µ(u, v)) whereas it does not depend

on the other coordinates (x, y). The (u, v), or (η, µ)−cell in the CPW spacetime which is

curved lies at a ﬁnite distance and maps to a solid metric in Schwarzschild (S) coordinates to

represent arbitrarily shaped objects. In this duality electromagnetic (em) and gravitational

wave parameters transform into charge and mass, respectively. In general relativity (GR)

matter (energy) and charge derive from the bending and warping of spacetime which is

manifested in the present study through the CX duality. In brief, a ’cell’ of the (η, µ)

coordinates with proper signature represents the ’whole’, as expected from a hologram.

Once the collision process is physically well-deﬁned with proper boundary conditions their

situation is reminiscent of the classical, i.e. non-quantum, version of the Breit-Wheeler

formalism [6] of colliding photons proposed long ago to create mass. In general relativity

(GR) with colliding null sources such a process takes place naturally at the level of spacetimes

and geometrically.

Organization of the paper is as follows. In section II by using the Ernst formalism we

obtain a new metric representing colliding waves with cross-polarization. In section III we

transform the metric into Schwarzschild coordinates to obtain the generalized version of the

ZV-metric. Asymptotic expansions and closed timelike curves are discussed in section IV.

(The details of expansions are tabulated in the Appendix). Conclusion and discussion in

section V completes the paper.

II. NEW METRIC IN THE SPACE OF CPWs

The spacetime of colliding Einstein-Maxwell (EM) waves is described by the line element

[1, 7, 10]

ds2=eν+µ3√∆dη2

∆−dµ2

δ−√∆δχdx2+1

χ(dy −q2dx)2(1)

where χ,q2and ν+µ3are metric functions depending only on (η, µ) with ∆ = 1 −η2

and δ= 1 −µ2. Note that the coordinates (η, µ) are functions of the null coordinates (u, v)

which are introduced for convenience. The Ernst equations [8] are expressed in complex

potentials (Z, H) as

ReZ − |H|2∇2Z= (∇Z)2−2¯

H∇Z.∇H(2)

ReZ − |H|2∇2H=∇Z.∇H−2¯

H(∇H)2(3)

where the diﬀerential operators are deﬁned appropriately on a basis manifold and a bar

denotes complex conjugation. Here

Z= Ψ + |H|2−iΦ (4)

in which Ψ and Φ are auxiliary complex potentials in the construction of the metric. For

details one may consult [1, 9, 10]. The vacuum case, i.e. H= 0,admits an integral [9]

Z= Ψ −iΦ = 1

D(1 −iq0cosh2X) (5)

where |q0| ≤ 1 is a constant that makes the metric stationary, which was denoted by

q0=sinα in [9]. The function Dis given by

D=q1 + q2

0cosh2X−sinh2X(6)

where X(η, µ) is a harmonic function satisfying

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摘要：

Stationary,chargedZipoy-VoorheesmetricfromcollidingwavespacetimeMustafaHalilsoy,∗MertMangut†,‡andChia-LiHsieh§DepartmentofPhysics,FacultyofArtsandSciences,EasternMediterraneanUniversity,Famagusta,NorthCyprusviaMersin10,TurkeyAbstractThroughtheErnstformalismweprovideexpressionforaclassofcollidingEins...

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Stationary charged Zipoy-Voorhees metric from colliding wave spacetime Mustafa HalilsoyMert Mangutand Chia-Li Hsieh

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