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

2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles International Journal of Geometric Methods in Modern Physics (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 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 2150109-1 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles F Hussain et al 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 2150109-2 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles Conformal vector fields for some vacuum classes of pp-waves 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 pp-waves solution of the theory As discussed earlier, the ppwaves 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 classical 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 φab as [50] c c + φac Z,b = 2ψφab , LZ φab ≡ φab,c Z c + φbc Z,a (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]: ⎧ HVF, if ψ = constant, ⎪ ⎪ ⎪ ⎪ ⎨Proper HVF, if ψ = constant = 0, Z= ⎪ KVF, if ψ = 0, ⎪ ⎪ ⎪ ⎩ Proper CVF, Otherwise It is important to mention here that with the aid of Eq (1), plenty of works regarding classification of various solutions via CVFs in MTs of gravitation have been 2150109-3 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 F Hussain et al Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles 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 Main Results The EFEs of IDG in the absence of gravitational contribution are given by [30] αk Gab + ⎡ ab ⎤ 4G F1 ( )R + φab RF1 ( )R − 4(∇a ∇b − φab )F1 ( )R ⎢ ⎥ ⎢ + 4RcaF2 ( )Rbc − φab Rce F2 ( )Rec − 4∇c ∇b (F2 ( )Rca ) ⎥ ⎢ ⎥ ⎢ + (F ( )Rab ) + 2φab ∇ ∇ (F ( )Rec ) − φab C ecρσ F ( )Cecρσ ⎥ e c ⎢ ⎥ ×⎢ ⎥ a becσ beca ab ⎢ + 4Cceσ ⎥ F3 ( )C − 4(Rec + 2∇e ∇c )(F3 ( )C ) − 2T1 ⎢ ⎥ ⎢ ⎥ ρ ⎣ + φab (T1ρ + T1 ) ⎦ ρ ρ + T2 ) − 4ω2ab − 2T3ab + φab (T3ρ + T3 ) − 8ω3ab − 2T2ab + φab (T2ρ = 0, (2) where Gab is the Einstein tensor, αk is the dimensionful parameter, Fi ( ) = n ∞ n=0 fin ( Ps2n ), with i = 1, 2, denoting form factor containing infinite derivative functions, fin are the dimensionless coefficients that help to overcome from the ghost like instabilities, n signify the nth derivative of De Alembert operator, Ps2n is the scale of nonlocality, R is the Ricci scalar, ∇ is the covariant derivative operator, Rab represents Ricci tensor, C ecρσ denotes the Wyle tensor, T1ab , T1 , ω2ab and ω3ab are symmetric tensors defined in [30] In Eq (2), when αk → 0, then one can recover the EFEs of GR without cosmological 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 successful 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 including 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 function 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 pp-waves space-time A ppwave space-time in the harmonic coordinates (u, x, y, v) is given by [2] ds2 = 2Hdu2 + dx2 + dy + du dv, 2150109-4 (3) 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Conformal vector fields for some vacuum classes of pp-waves Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles where the profile function H is an arbitrary function of u, x and y The above ∂ As discussed earlier, the profile space-times (3) admit only one KVF which is ∂v 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 H = H(x, y) Screw symmetric wave H = ln x2 + y 2 A(u)(x Cylindrically symmetric wave − y ) + B(u)xy H= H = 12 (x2 − y ) Plane wave 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]: y2 −Ps 2 + x − e Ps2 H = E(u)Ps x2 −Ps 2 + y e + J(u)xy, Ps2 (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 uHu + 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 uHu + 2H = 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 uHu +xHx +yHy +2H = 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 Ps [( P22 + x2 )e s (ii) H = c1 Ps [( P22 + x2 )e y2 −Ps s (iii) H = u−4 Ps [( P22 + x2 )e s (iv) H = c1 Ps [( P22 + x2 )e s y2 −Ps s − ( P22 + y )e y2 −Ps y2 −Ps − ( P22 + y )e −Ps2 x2 s − ( P22 + y )e s − ( P22 + y )e s 2150109-5 x2 −Ps ] + u−2 xy, where c1 ∈ \{0}, x2 −Ps −Ps2 x2 ] + c1 xy, where c1 ∈ \{0}, ] + c1 xy, where c1 ∈ \{0}, ] + u−4 xy, where c1 ∈ \{0}, 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 F Hussain et al (v) H = u−4 Ps [( P22 + x2 )e s (vi) H = u−2 Ps [( P22 + x2 )e Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles s −Ps2 y2 −Ps2 y2 y2 −Ps − ( P22 + y )e s − ( P22 + y )e s x2 −Ps ], x2 −Ps ], x2 −Ps − ( P22 + y )e ], where c1 ∈ \{0}, (vii) H = c1 Ps [( P22 + x2 )e s s −2 −4 (viii) H = u xy (ix) H = u xy (x) H = c1 xy, 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) −Ps2 y2 Case (i) In this case, we have H = u−2 Ps [( P22 + x2 )e − ( P22 + y )e s s c1 xy, where c1 ∈ \{0} In this case the space-times (3) take the form ds2 = 2u−2 Ps y2 −Ps + x2 e − Ps −Ps2 x2 ]+ x2 −Ps + y2 e + 2c1 xy du2 Ps + dx2 + dy + du dv (5) 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 integration, one finds that ψ = 0, which indicate that no proper CVFs exist in this case ∂ The cases (ii)–(v) give the The CVFs reduce to a single KVF represented by ∂v same result Case (vi) Here, the value of H is u−2 Ps [( P22 + x2 )e s space-times (3) in this case become ds2 = 2u−2 Ps y2 −Ps + x2 e − Ps y2 −Ps − ( P22 + y )e x2 −Ps s ] The −Ps2 x2 + y2 e du2 Ps + dx2 + dy + du dv (6) 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 ∂ whereas the other is the boost rotation (6) admit two KVFs of which one is ∂v ∂ ∂ u ∂u − v ∂v [67] y2 −Ps Case (vii) The value of H i.e H = c1 Ps [( P22 + x2 )e − ( P22 + y )e s s where c1 ∈ \{0} For this case, the space-times (3) become ds2 = 2c1 Ps y2 −Ps + x2 e − Ps x2 −Ps ], x2 −Ps + y2 e du2 Ps + dx2 + dy + du dv (7) ∂ ∂u and The above space-times (7) also admit two KVFs (as ψ = 0) represented as ∂ It is important to note that in contrast to the previous case (vi), the boost ∂v 2150109-6 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Conformal vector fields for some vacuum classes of pp-waves ∂ ∂ ∂ 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 Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles Case (viii) In this case, the space-times (3) after utilizing H = u−2 xy take the form ds2 = 2u−2 xydu2 + dx2 + dy + du dv (8) Solving Eq (1) with the help of space-time (8) implies that ψ = c1 , \{0} This shows that the above space-time (8) does not where c1 ∈ admit proper CVFs, the CVFs become homothetic vector fields (HVFs) The dimension of HVFs √is seven of which six are isometries repre√ −β ∂ , u (η) + sented by u cos λ(γ) − 2√1 u [ 3(x sin λ − y sin λ) − x cos λ + y cos λ] ∂v √ √ β(x+y) −α ∂ ∂ ∂ u ∂v , u sin λ(γ) + 2√1 u [x sin λ − y sin λ + 3(x cos λ − y cos λ)] ∂v , ∂v , √ √ α α(x+y) β ∂ ∂ ∂ u ∂u − v ∂v and u (η) − u ∂v , where α = (1 + 5), β = (−1 + 5), √ ∂ ∂ ∂ ∂ γ = ( ∂y − ∂x ), η = ( ∂y + ∂x ) and λ = 32ln u The seventh is the proper HVF represented by x ∂ ∂ ∂ +y + 2v ∂x ∂y ∂v (9) 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 + dy + 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 x−y x−y 1 ∂ 1 ∂ u cos( u )[α] + ( u )[u cos( u ) + sin( u )] ∂v , u sin( u )[α] + ( u )[u sin( u ) − cos( u )] ∂v , ∂ and u sinh( u1 )[β] − γ[u sinh( u1 ) − cosh( u1 )] u cosh( u1 )[β] − γ[u cosh( u1 ) − sinh( u1 )] ∂v x+y ∂ ∂ ∂ ∂ ∂ ∂v , where α = (− ∂x + ∂y ), β = ( ∂x + ∂y ) and γ = ( u ) One is proper HVF which is defined in Eq (9) whereas one is proper CVF given by u2 ∂ ∂ ∂ + ux + uy − ∂u ∂x ∂y x2 + y 2 ∂ ∂v (11) The conformal factor in this case is ψ = (c1 u + c2 ), where c1 , c2 ∈ (c1 = 0) Case (x) Here, we have H = c1 xy, where c1 ∈ 2 \{0} The space-times (3) become ds = 2c1 xy du + dx + dy + 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 √ √ √ ∂ ∂ ∂ − ∂x ] + c1 cos( c1 u)(x − y) ∂v , with six isometries represented by sin( c1 u)[ ∂y √ √ √ √ √ ∂ ∂ ∂ ∂ ∂ ∂ ∂ e c1 u [ ∂x + ∂y − c1 (x + y) ∂v ], ∂u , ∂v , cos( c1 u)[ ∂y − ∂x ]− c1 sin( c1 u)(x − y) √ √ ∂ ∂ ∂ − c1 u ∂ [ ∂x + ∂y + c1 (x + y) ∂v ] The seventh is proper HVF already ∂v and e defined in Eq (9) 2150109-7 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles F Hussain et al 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 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , ∂v , ∂x , ∂y , u ∂y − y ∂v , x ∂y − y ∂x , u ∂x − x ∂v , u ∂u − v ∂v , represented by ∂u ∂ ∂ ∂ ∂ 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 ∂ x2 − y ∂ + xy − uv + ∂u ∂x x2 + y 2 ∂ ∂ + vy , ∂y ∂v ∂ ∂ ∂ ∂ − vx − vy − v2 , ∂u ∂x ∂y ∂v ∂ y − x2 ux − uv + ∂u ∂ ∂ ∂ + xy + vx ∂x ∂y ∂v u2 ∂ ux ∂ uy ∂ + + − ∂u ∂x ∂y x2 + y (13) and ∂ ∂v For the case when the space-time become flat, the conformal factor ψ is given by ψ = ( c12u + c3 x + c4 y − c5 v + c2 ), where c1 , c2 , c3 , c4 , c5 ∈ (c1 , c3 , c4 , c5 = 0) Summary and Discussion The discovery of GWs has put a great role in the development of modern cosmology Particularly, the GWs have been employed in the MTs of gravitation that have been designed in order to address several confusing phenomenon of current theoretical 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 ) gravity [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 specific 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 provide 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 gravitation [88–92] as they provide additional constraints for investigating the solutions 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, 2150109-8 2nd Reading March 17, 2021 22:36 WSPC/S0219-8878 IJGMMP-J043 2150109 Conformal vector fields for some vacuum classes of pp-waves Int J Geom Methods Mod Phys Downloaded from www.worldscientific.com by NANJING UNIVERSITY on 06/05/21 Re-use and distribution is strictly not permitted, except for Open Access articles 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 ∂ ∂ , ∂u vanishing conformal factor ψ The KVFs admitted by the space-times are ∂v ∂ ∂ 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 symmetry 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 cosmology [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 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gravity, J Cosmol... 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. .. 2150109 Conformal vector fields for some vacuum classes of pp-waves ∂ ∂ ∂ isometry u ∂u − v ∂v is replaced with the translational isometry ∂u This is due to the fact that the function H is independent