Conformal vector fields for some vacuum classes of pp waves space times in ghost free infinite derivative gravity

2021 2150109 14 pagesc World Scientific Publishing Company DOI: 10.1142/S0219887821501097 Conformal vector fields for some vacuum classes of pp-waves space-times in ghost free infinite

(2021) 2150109 (14 pages)

c

 World Scientific Publishing Company

DOI: 10.1142/S0219887821501097

Conformal vector fields for some vacuum classes

of pp-waves space-times in ghost free infinite

derivative gravity

Fiaz Hussain∗, Ghulam Shabbir†,‡, Shabeela Malik†

and Muhammad Ramzan∗

∗ Department of Mathematics The Islamia University of Bahawalpur Bahawalpur, Punjab, Pakistan

† Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology Topi, Swabi, Khyber, Pakhtunkhwa, Pakistan

‡ shabbir@giki.edu.pk

Received 27 November 2020 Accepted 22 February 2021 Published 18 March 2021

The aim of this paper is to find conformal vector fields (CVFs) for some vacuum classes

of the pp-waves space-times in the ghost free infinite derivative gravity (IDG) In order

to find the CVFs of the above-mentioned space-times in the IDG, first, we deduce various classes of solutions by employing a classification procedure that in turn leads towards

10 cases By reviewing each case thoroughly by direct integration technique, we find that there exists only one case for which the space-time admits proper CVFs whereas

in rest of the cases, the space-time either becomes flat or it admits homothetic vector fields (HVFs) or Killing vector fields (KVFs) The overall dimension of CVFs for the pp-waves space-times in the IDG has turned out to be one, two, seven or fifteen.

Keywords: Conformal vector fields in infinite derivative gravity; pp-waves space-times;

infinite derivative gravity.

Mathematics Subject Classification 2020: 83C15, 83C40

1 Introduction

The accelerating behavior of astrophysical objects generate ripples in the fabric of

a space-time that points towards the idea of gravitational waves (GWs) A subclass

of the GWs representing vacuum solutions to the Einstein field equations (EFEs) is known as pp-waves The term pp-wave stands for the plane fronted GWs with the property that the generated waves move along the parallel direction The geometry

‡Corresponding author.

of pp-waves has been designed in such a way that for a suitable choice of the amplitude, space-time becomes flat or nearly flat after the passage of the wave [1] The class of solutions exhibiting such waves belongs to the space-times investigated

by Kundt and Ehlers [2] The slots of pp-waves are the owner of rich amount of physical aspects Nonlinear plane GWs are one of the key examples belonging to such waves [3] Another significant aspect of nonlinear GWs is their applicability

in describing the phenomenon of memory effect [4–7] For the prediction of GWs in space, memory effect has been found to be the key instrument with the possibility of enhancing a deep look into several internal physical aspects Especially, observing the motion of free particles, pp-waves put a great role to reduce the associated kinetic energy [8] and angular momentum [9] Due to this property, the space-times lying in the ring of pp-waves help in explaining the phenomenon of GWs that travel

in space for different intervals of time Recently, the GWs gained interest because

of direct observation of the twin LIGO [10] This observation has knocked the new door of astrophysics and cosmology conforming the exact nature of the GWs and Steller mass black holes [11] An astonishing fact about the GWs is that these are capable to test any gravitational theory [12] Especially, the GWs have been employed as a test tool to check the validity of recently explored theories called modified theories (MTs) of gravitation The MTs of gravitation got popularity

in the past few years due to having the potency of addressing several burning issues of recent cosmology Expansion of universe with accelerated phase, problem

of dark matter and dark energy, issue of space-time singularities, quantum nature

of gravitational interaction and the study of TeV-scale supersymmetry are some of the burning issues met by the general relativity (GR) These issues have activated the scientific community to design a structure where one can find the way in order

to address such mysteries of physics [13–16] A list of prominent MTs of gravitation along with their field equations have been well defined in the plenty of renowned papers [17–26] The structure of these MTs is mainly based on the slight deviation from GR This deviation is either in its curvature invariants or by the involvement of torsion tensor leading to modified teleparallel theories A theory with comprehensive physical structure obtained from the small-scale deviation of GR is known to be infinite derivative gravity (IDG) [27–29] The IDG has been found as a potential theory especially dealing with the small UV region of nonlocality [30] The IDG has been derived from the Lagrangian density constituting the analytic form factors that help to express nonlocal physical interactions These analytic form factors offer extra degrees of freedom and thus provide resistance to the ghost like instabilities

as well as help to avoid cosmological singularities The form factors with infinite number of derivatives also have widely been used in quantum field theories For

a point like source even at small distances, the IDG helps to define nonsingular Newtonian potential [28, 31] Moreover, it has been expected that the IDG may have the potential to overcome cosmological big bang singularities both at linear and nonlinear scales [32, 33] The gravitational interaction in the IDG plays the role of toy model in order to control such singularities Instead of being a strong candidate

to resolve black hole singularities problem, IDG also has major influence to avoid

several topological defects like p-brans and cosmic strings [34] No doubt, the IDG

carries a rich amount of physical background but its equations of motion are very complicated, therefore exploring exact solutions in this theory is quite a challenging task Due to this discrepancy, a very few solutions of the theory have been found so far [30, 35–38] In [30], the author considered ghost-free mode of the IDG in order

to construct exact waves solution of the theory As discussed earlier, the pp-waves solutions are the owners of huge amount of physical background with many universal properties [39, 40] This physical background has motivated us to study

it from the symmetry point of view in the IDG Symmetries are fundamentally the transformations having capability of preserving the geometrical objects [41] These are defined through the Lie derivative of the associated geometric quantity

In order to find the solution of a given physical problem, symmetries help to reduce the dynamical variables thus leading towards the easy route for their solution The most fundamental Lie symmetry is the Killing symmetry that is associated with the Killing vector fields (KVFs) or isometries Physically, the KVFs help to define certain conservation laws of physics [42] Other sorts of the Lie symmetries include homothetic, conformal and curvature collineations, etc [43] Some useful reviews carrying work related to various classes of pp-waves on isometries [44], curvature collineations [3], Noether symmetries [45–47] and conformal symmetries [48, 49] have been performed in the literature The deficiency observed in their work was the consideration of problem without consuming any field equations

Particularly, the conformal symmetry obtained deliberation in the past few years owing to latent applications in several fields of mathematical physics [50] For instance, conformal symmetry forces Maxwell’s law of electromagnetic theory

to obey the invariance of dynamical system In quantum electrodynamics, the clas-sical field equations remain invariant under the action of a group of conformal motion [51] With the aid of conformal symmetry, one obtain conformal vector

fields (CVFs) A CVF say Z is defined by taking the Lie derivative L of the metric tensor φ ab as [50]

L Z φ ab ≡ φ ab,c Z c + φ bc Z ,a c + φ ac Z ,b c = 2ψφ ab , (1)

where comma denotes the partial derivative and ψ is a smooth conformal function The connection between the conformal function ψ and the CVF Z further helps to

outline its fundamental symmetries that could be best understood by the following relation [52]:

Z =

⎧

⎪

⎪

⎪

⎪

HVF, if ψ = constant, Proper HVF, if ψ = constant = 0,

Proper CVF, Otherwise.

It is important to mention here that with the aid of Eq (1), plenty of works regard-ing classification of various solutions via CVFs in MTs of gravitation have been

performed in [41, 52–66] In this paper, we want to categorize pp-waves solution via CVFs in the IDG The mathematical representation of pp-waves space-time along with equations of motion in the IDG will be discussed in the upcoming section of this paper

2 Main Results

The EFEs of IDG in the absence of gravitational contribution are given by [30]

G ab+α k

2

×

⎡

⎢

⎢

⎢

⎢

⎢

4G ab F1()R + φ ab RF1()R − 4(∇ a ∇ b − φ ab )F1()R + 4R a c F2()R bc − φ ab R e c F2()R c

e − 4∇ c ∇ b (F2()R ca)

+ 2(F2()R ab ) + 2φ ab ∇ e ∇ c (F2()R ec)− φ ab C ecρσ F3()C ecρσ

+ 4C ceσ a F3()C becσ − 4(R ec+ 2∇ e ∇ c )(F3()C beca)− 2T ab

1

+ φ ab (T 1ρ ρ + T1)

− 2T ab

2 + φ ab (T 2ρ ρ + T2)− 4ω ab

2 − 2T ab

3 + φ ab (T 3ρ ρ + T3)− 8ω ab

3

⎤

⎥

⎥

⎥

⎥

⎥

where G ab is the Einstein tensor, α k is the dimensionful parameter, F i() = ∞

n=0 f in(n

P 2n

s ), with i = 1, 2, 3 denoting form factor containing infinite deriva-tive functions, f in are the dimensionless coefficients that help to overcome from

the ghost like instabilities,n signify the nth derivative of De Alembert operator,

P s 2n is the scale of nonlocality, R is the Ricci scalar, ∇ is the covariant derivative operator, R ab represents Ricci tensor, C ecρσ denotes the Wyle tensor, T1ab , T1, ω ab2 and ω3abare symmetric tensors defined in [30]

In Eq (2), when α k → 0, then one can recover the EFEs of GR without

cos-mological constant and in the absence of matter distribution Moreover, Eq (2)

is highly nonlinear showing that finding its general solution is significantly hard This issue has created a necessity to develop a technique which may clear the way towards seeking their solution One way is to select a suitable space-time geometry and then peruse for the solution of a given problem In the literature, some success-ful attempts have been made in this regard considering flat background geometry that help them to solve these equations at the linearized level [35–38] PP wave metrics belong to the space-times that represent variety of physical aspects includ-ing the GWs A very familiar example in this regard is the class of generalized plane GWs [3] The plane GWs propagate by varying the associated profile func-tion which further leads towards screw symmetric, cylindrically symmetric, plane wave and plane wave linearly polarized [3] With this physical background, it seems interesting to explore the solution of Eq (2) choosing waves space-time A

pp-wave space-time in the harmonic coordinates (u, x, y, v) is given by [2]

ds2= 2Hdu2+ dx2+ dy2+ 2 du dv, (3)

where the profile function H is an arbitrary function of u, x and y The above

space-times (3) admit only one KVF which is ∂

∂v As discussed earlier, the profile function H may exhibit a key role with the property of generating different types

of waves Some of the important forms of waves corresponding to H are tabulated

as follows [3]:

Sr No Form of the profile function H Type of wave

3 H = 12A(u)(x2− y2) + B(u)xy. Plane wave

4 H = 12(x2− y2). Plane wave linearly polarized

In the above table, A(u) and B(u) are functions of u only It is necessary to indicate that in [54], different vacuum classes of pp-waves in the f (R) gravity along

with their CVFs have been explored The aim of this paper is to find CVFs of the pp-waves space-times (3) in IDG, therefore firstly we need the value of profile

function H in the said theory By utilizing Eq (3) in the EFEs of IDG defined in

Eq (2), one reaches the following solution [30]:

H = E(u)P s



2

P2+ x

2

e −P 2 s y

2

4 −

 2

P2 + y

2

e −P 2 s x

2 4



+ J(u)xy, (4)

where E(u) and J(u) are the functions of u only and are characterized as amplitudes

of the wave A particular value of E(u) and J(u) will help to identify the shape of

exact gravitational wave In order to understand the internal structure and physical

feature of the source generating the waves, one must seek for the functions E(u) and J(u) In this context, some varieties of functions producing amplitude of the

pp-waves along with the isometries of resulting metrics have been found in [44] It

has been observed that if the function H satisfies the condition uH u + 2H = 0, [67] with the assumption that J(u) vanishes, then one obtain E(u) = u −2 Similarly, vanishing of E(u) along with the condition uH u + 2H = 0 yields J(u) = u −2 Such behavior of wave function H helps to obtain additional isometries and hence additional conservation laws [45] Similarly, the condition uH u +xH x +yH y +2H = 0 implies that J(u) = E(u) = u −4 [65] In view of this observation and hoping to better judge the geometry of pp waves, we are making a classification of the above

solution (4) by varying the functions E(u) and J(u) This classification involves the

following cases:

(i) H = u −2 P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] + c1xy, where c1∈ \{0}, (ii) H = c1P s[(P22 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] + u −2 xy, where c1∈ \{0}, (iii) H = u −4 P s[(P22 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] + c1xy, where c1∈ \{0}, (iv) H = c1P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] + u −4 xy, where c1∈ \{0},

(v) H = u −4 P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ], (vi) H = u −2 P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ], (vii) H = c1P s[(P22 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ], where c1∈ \{0}, (viii) H = u −2 xy (ix) H = u −4 xy (x) H = c1xy, c1∈ \{0}.

Now, our goal is to find the CVFs for each of the above class of solutions (i)–(x) The process of finding the CVFs will be completed in two steps Initially, we will

generate exact pp-waves space-times by substituting the values of H in Eq (3) As

the space-times will be generated, we will solve Eq (1) for each of the space-time via direct integration approach Following is the brief procedure of finding the CVFs for the cases (i)–(x)

Case (i) In this case, we have H = u −2 P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] +

c1xy, where c1∈ \{0} In this case the space-times (3) take the form

ds2=



2u −2 P s



2

P2+ x

2

e −P 2 s y

2

4 −

 2

P2 + y

2

e −P 2 s x

2 4



+ 2c1xy



du2

Utilizing the above space-times (5) in Eq (1) will generate 10 coupled nonlinear partial differential equations Solving those equations by the aid of direct

integra-tion, one finds that ψ = 0, which indicate that no proper CVFs exist in this case.

The CVFs reduce to a single KVF represented by ∂

∂v The cases (ii)–(v) give the

same result

Case (vi) Here, the value of H is u −2 P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ] The

space-times (3) in this case become

ds2= 2u −2 P s



2

P2 + x

2

e −P 2 s y

2

4 −

 2

P2 + y

2

e −P 2 s x

2 4



du2

Adoptinging the similar procedure as we did in the previous case, we come to

know that ψ = 0, ensuring the nonexistence of proper CVFs Here, the space-times

(6) admit two KVFs of which one is ∂

∂v whereas the other is the boost rotation

u ∂u ∂ − v ∂

∂v [67].

Case (vii) The value of H i.e H = c1P s[( 2

P2 + x2)e −P 2 s y

2

4 − ( 2

P2 + y2)e −P 2 s x

2

4 ], where c1∈ \{0} For this case, the space-times (3) become

ds2= 2c1P s



2

P2+ x

2



e −P 2 s y

2

4 −

 2

P2 + y

2



e −P 2 s x

2 4



du2

The above space-times (7) also admit two KVFs (as ψ = 0) represented as ∂u ∂ and

∂

∂v It is important to note that in contrast to the previous case (vi), the boost

isometry u ∂u ∂ − v ∂

∂v is replaced with the translational isometry ∂u ∂ This is due to the fact that the function H is independent of the coordinate u.

Case (viii) In this case, the space-times (3) after utilizing H = u −2 xy take the

form

ds2= 2u −2 xydu2+ dx2+ dy2+ 2 du dv. (8)

Solving Eq (1) with the help of space-time (8) implies that ψ = c1, where c1 ∈ \{0} This shows that the above space-time (8) does not

admit proper CVFs, the CVFs become homothetic vector fields (HVFs) The dimension of HVFs is seven of which six are isometries repre-sented by √

u cos λ(γ) −2√1

u[

√ 3(x sin λ − y sin λ) − x cos λ + y cos λ] ∂v ∂ , u −β2 (η) +

β(x+y)

2 u −α2 ∂

∂v,

√

u sin λ(γ) + 2√1u [x sin λ − y sin λ + √

3(x cos λ − y cos λ)] ∂v ∂ , ∂v ∂ ,

u ∂u ∂ − v ∂

∂v and u α2(η) − α(x+y)2 u β2 ∂

∂v , where α = (1 + √

5), β = (−1 + √

5),

γ = ( ∂y ∂ − ∂

∂x ), η = ( ∂y ∂ + ∂

∂x ) and λ = √ 3 ln u2 The seventh is the proper HVF

represented by

x ∂

∂x + y

∂

∂y + 2v

∂

Case (ix) Here, we have H = u −4 xy, which forces the space-times (3) to take the

following form:

ds2= 2u −4 xy du2+ dx2+ dy2+ 2 du dv. (10) This is the case, where the space-time (10) admits proper CVFs The dimension of CVFs has turned out to be seven of which five are the KVFs represented as ∂

∂v ,

u cos( u1)[α] + ( x−y u )[u cos( u1) + sin(1

u)]∂v ∂ , u sin( u1)[α] + ( x−y u )[u sin(1u)− cos(1

u)]∂v ∂ ,

u cosh( u1)[β] − γ[u cosh( u1)− sinh(1

u)]∂v ∂ and u sinh( u1)[β] − γ[u sinh(1u)− cosh(1

u)]

∂

∂v , where α = (− ∂x ∂ + ∂

∂y ), β = ( ∂x ∂ + ∂

∂y ) and γ = ( x+y u ) One is proper HVF

which is defined in Eq (9) whereas one is proper CVF given by

u2 ∂

∂u + ux

∂

∂x + uy

∂

∂y −



x2+ y2

2



∂

The conformal factor in this case is ψ = (c1u + c2), where c1, c2∈ (c1= 0).

Case (x) Here, we have H = c1xy, where c1∈ \{0} The space-times (3) become

ds2= 2c1xy du2+ dx2+ dy2+ 2 du dv. (12)

Solving Eq (1) with the aid of space-time (12) implies ψ = c2, where c2∈ \{0}.

This shows that the space-time (12) does not admit proper CVFs In fact, the CVFs become HVFs The dimension of HVFs in this case has turned out to be seven with six isometries represented by sin(√

c1u)[ ∂y ∂ − ∂

∂x] +√

c1cos(√

c1u)(x − y) ∂v ∂ ,

e √c1u[ ∂

∂x+∂y ∂ − √c1(x + y) ∂v ∂ ], ∂u ∂ , ∂v ∂ , cos( √

c1u)[ ∂y ∂ − ∂

∂x]−√c1sin(√

c1u)(x − y)

∂

∂v and e − √ c1u[∂

∂x+∂y ∂ +√

c1(x + y) ∂v ∂ ] The seventh is proper HVF already

defined in Eq (9)

It is important to mention here that if c1 becomes zero in the above space-time

(12), then the space-time will become flat and admit 15 CVFs with 10 isometries represented by ∂

∂u , ∂v ∂ , ∂x ∂ , ∂y ∂ , u ∂y ∂ − y ∂

∂v , x ∂y ∂ − y ∂

∂x , u ∂x ∂ − x ∂

∂v , u ∂u ∂ − v ∂

∂v ,

y ∂u ∂ − v ∂

∂y and x ∂u ∂ − v ∂

∂x The remaining five CVFs are classified as one proper

HVF which is given in Eq (9) while the other four are proper CVFs given by

uy ∂

∂u + xy

∂

∂x −



uv + x

2− y2

2



∂

∂y + vy

∂

∂v ,



x2+ y2

2



∂

∂u − vx ∂

∂x − vy ∂

∂y − v2 ∂

∂v ,

ux ∂

∂u −



uv + y

2− x2

2



∂

∂x + xy

∂

∂y + vx

∂

∂v and

u2

2

∂

∂u+

ux

2

∂

∂x+

uy

2

∂

∂y −



x2+ y2

4



∂

∂v .

(13)

For the case when the space-time become flat, the conformal factor ψ is given

by ψ = ( c1u

2 + c3x + c4y − c5v + c2), where c1, c2, c3, c4, c5∈ (c1, c3, c4, c5= 0).

3 Summary and Discussion

The discovery of GWs has put a great role in the development of modern cosmol-ogy Particularly, the GWs have been employed in the MTs of gravitation that have been designed in order to address several confusing phenomenon of current theoret-ical physics Recently, they made tremendous contribution towards searching the solutions of such mysteries with a plenty of physical prospective Several important

considerations of the GWs in the frame work of f (R) gravity [68–71], f (T ) grav-ity [72, 73], higher order gravities [74–79], scalar tensor gravities [80, 81] and f (T , B) gravity [82] reflect that the phenomenon has potentially been accepted by the

cosmologists Most specifically, the plane GWs have contributed a lot carrying a larger amount of physical background with plenty of useful properties [83] From the symmetry point of view, pp-waves admit an additional planner symmetry along the wave fronts The most inherent property of the pp-waves admitting covariantly constant null KVF has made it more significant as it point towards the vanishing

of all curvature invariants Due to this property, classes of solutions representing the plane GWs provide a classical background for string theory [84, 85] Some spe-cific sorts of the plane waves also help in order to study the kinetic energy of free particles, center of mass density and the memory effect [86, 87] The pp-waves pro-vide basement of all the above physical aspects that enhances our encouragement

to make a flash in it from the symmetry prospective The space-time symmetries have remained as the back bones of the GR as well as teleparallel theory of gravita-tion [88–92] as they provide addigravita-tional constraints for investigating the solugravita-tions of several problems in both the theories Isometries, being a candidate of space-time symmetries, help to develop various conservation laws of physics In this paper,

we deal with a generalized version of the isometries in order to classify different modes of the plane fronted GWs in the ghost free IDG via their CVFs During the classification of considered space-time, there arose 10 cases A careful analysis of each case by means of direct integration reveals the following results:

(a) In cases (i)–(vii), the space-times either admit one or two KVFs due to the

vanishing conformal factor ψ The KVFs admitted by the space-times are ∂v ∂ , ∂u ∂ and u ∂u ∂ − v ∂

∂v The former two KVFs are the translations generated by the null coordinates u and v thus leading towards the conservation of linear momentum

whereas the latter isometry represents boost rotation giving law of conservation of angular momentum [67]

(b) In cases (viii) and (x), the space-times admit proper HVFs due to the constant

value of conformal factor ψ The dimension of HVFs for each of the cases (viii) and

(x) has turned out to be seven with six isometries and one proper homothetic sym-metry For both the cases, the proper HVF has turned out to be same and is shown

in Eq (9) Due to the invariance of metric tensor up to the constant scale factor, the HVFs have been employed in order to study constant of motion which help to observe trajectory of particle in the vicinity of a space-time [52] The singularity problem in the GR is also found to be well tackled by utilizing the properties of HVFs The HVFs also help to produce various self-similar solutions of the EFEs [63] (c) In case (ix), the space-time admits proper CVFs The dimension of CVFs for the case (ix) is seven with five isometries, one proper HVF while the remaining one

is proper CVF which is expressed in Eq (11) The CVFs resulted from conformal symmetry help to produce several cosmological models in the loop quantum cos-mology [93] With the aid of conformal motion, one can inspect various classes of compact stars, gravstars as well as dense stars conforming their role in the field of astrophysics

It is significant to make an indication over here that in order to complete the

classification, we have also discussed the case where the profile function H(u, x, y)

vanishes, under this situation, the pp-waves space-times (3) become flat and admit

15 CVFs with 10 isometries, one proper HVF and remaining four are the proper CVFs which could be seen in Eq (13) Moreover, the prescribed classification proce-dure has enabled us to obtain an important subclass of screw symmetric pp-wave [3]

We would like to make an end with the aside that this study may help to study various conservation laws which reflect the role of symmetries in the IDG

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function H in the said theory By utilizing Eq (3) in the EFEs of. .. Plane wave linearly polarized

In the above table, A(u) and B(u) are functions of u only It is necessary to indicate that in [54], different vacuum classes of pp- waves in the f (R) gravity. .. the property of generating different types

of waves Some of the important forms of waves corresponding to H are tabulated

as follows [3]:

Sr No Form of the profile