Classification of the Lie and Noether point symmetries for the Wave and the Klein-Gordon equations in pp-wave spacetimes

TL;DR

This paper classifies Lie and Noether point symmetries of the Klein-Gordon and wave equations in pp-wave spacetimes, linking symmetries to conformal Killing vectors and identifying maximum symmetry cases.

Contribution

It provides a systematic classification of symmetries for these equations in pp-wave spacetimes, including the functional form of potentials and symmetry algebra dimensions.

Findings

Maximum Noether algebra dimension is seven in plane wave spacetimes.

Symmetries depend on the existence of conformal Killing vectors.

Explicit forms of potentials admitting symmetries are determined.

Abstract

We perform a classification of the Lie and Noether point symmetries for the Klein-Gordon and for the wave equation in pp-wave spacetimes. To perform this analysis we reduce the problem of the determination of the point symmetries to the problem of existence of conformal killing vectors on the pp-wave spacetimes. We use the existing results of the literature for the isometry classes of the pp-wave spacetimes and we determine in each class the functional form of the potential in which the Klein-Gordon equation admits point symmetries and Noetherian conservation law. Finally we derive the point symmetries of the wave equation and we find that the maximum Noether algebra has dimension seven, that is the case of plane wave spacetimes.

Tables

Table 1. Table 1: Commutators of the conformal algebras of isometry class 2 of the pp-wave spacetime ( 1 ) and of the subclasses (2i)-(2iii)

$[{𝐗}_I,{𝐗}_J]$	$k$	$X_2^{\left(2\right)}$	$H_3^{\left(2i\right)}$	$H_{3\left(-1\right)}^{\left(2i\right)}$	$H_3^{\left(2ii\right)}$	$H_{3\left(0\right)}^{\left(2ii\right)}$	$S_3^{\left(2iii\right)}$
$k$	$0$	$0$	$2\frac{q+1}{q+2}k$	$0$	$2k$	$k$	$0$
$X_2^{\left(2\right)}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$H_3^{\left(2i\right)}$	$-2\frac{q+1}{q+2}k$	$0$	$0$
$H_{3\left(-1\right)}^{\left(2i\right)}$	$0$	$0$		$0$
$H_3^{\left(2ii\right)}$	$-2k$	$0$			$0$
$H_{3\left(0\right)}^{\left(2ii\right)}$	$-k$	$0$				$0$
$S_3^{\left(2iii\right)}$	$0$	$0$					$0$

Table 2. Table 2: Commutators of the conformal algebras of isometry class 5 of pp-wave spacetime ( 1 ) and of the subclasses (5i), (5ii)

$[{𝐗}_I,{𝐗}_J]$	$k$	$X_2^{\left(5\right)}$	$X_3^{\left(5\right)}$	$S_4^{\left(5i\right)}$	$C_4^{\left(5ii\right)}$
$k$	$0$	$0$	$-k$	$0$	$0$
$X_2^{\left(5\right)}$	$0$	$0$	$0$	$0$	$0$
$X_3^{\left(5\right)}$	$k$	$0$	$0$	$S_4^{\left(5i\right)}-\zeta k$	$-\frac{4}{\left(\sigma -2\right)}{C}_4^{\left(5ii\right)}$
$S_4^{\left(5i\right)}$	$0$	$0$	$-S_4^{\left(5i\right)}+\zeta k$	$0$
$C_4^{\left(5ii\right)}$	$0$	$0$	$\frac{4}{\left(\sigma -2\right)}{C}_4^{\left(5ii\right)}$		$0$

Table 3. Table 3: Commutators of the conformal algebras of isometry class 6 of spacetime ( 1 ) and of the subclasses (6i)-(6iv)

$[{𝐗}_I,{𝐗}_J]$	$k$	$X_2^{\left(6\right)}$	$X_3^{\left(6\right)}$	$C_4^{\left(6i\right)}$	$C_5^{\left(6i\right)}$	$H_4^{\left(6ii\right)}$	$S_5^{\left(6ii\right)}$	$H_4^{\left(6iii\right)}$	$H_4^{\left(6iv\right)}$
$k$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$k$	$\left(1-\frac{\sigma }{2}\right)k$
$X_2^{\left(6\right)}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$X_3^{\left(6\right)}$	$0$	$0$	$0$	$C_5^{\left(6i\right)}$	$-C_4^{\left(6i\right)}$	$2X_3^{\left(6\right)}$	$H_4^{\left(6ii\right)}$	$X_3^{\left(6\right)}-\zeta k$	$\left(1+\frac{\sigma }{2}\right)X_3^{\left(6\right)}$
$C_4^{\left(6i\right)}$	$0$	$0$	$-C_5^{\left(6i\right)}$	$0$	$-X_3^{\left(6\right)}$
$C_5^{\left(6i\right)}$	$0$	$0$	$C_4^{\left(6i\right)}$	$X_3^{\left(6\right)}$	$0$
$H_4^{\left(6ii\right)}$	$0$	$0$	$-2X_3^{\left(6\right)}$			$0$	$2S_6^{\left(6ii\right)}$
$S_5^{\left(6ii\right)}$	$0$	$0$	$-H_4^{\left(6ii\right)}$			$2S_6^{\left(6ii\right)}$	$0$
$H_4^{\left(6iii\right)}$	$-k$	$0$	$-X_3^{\left(6\right)}+\zeta k$					$0$
$H_4^{\left(6iv\right)}$	$\left(\frac{\sigma }{2}-1\right)k$	$0$	$-\left(1+\frac{\sigma }{2}\right)X_3^{\left(6\right)}$						$0$

Table 4. Table 4: Commutators of the conformal algebras for the isometry class 8 for δ = 0 𝛿 0 \delta=0 and of the subclasses (8i)-(8iii) for the pp-wave spacetime ( 1 ).

$[{𝐗}_I,{𝐗}_J]$	$k$	$X_2^{\left(8\right)}$	$X_3^{\left(8\right)}$	$X_4^{\left(8\right)}$	$H_5^{\left(8i\right)}$	$X_5^{\left(8ii\right)}$	$X_6^{\left(8ii\right)}$	$S_6^{\left(8iii\right)}$
$k$	$0$	$0$	$0$	$0$	$\left(1+\frac{\gamma }{2}\right)k$	$0$	$0$	$0$
$X_2^{\left(8\right)}$	$0$	$0$	$0$	${\zeta }^2{X}_3^{\left(8\right)}$	$\left(1-\frac{\gamma }{2}\right)X_2^{\left(8\right)}$	$-\sqrt{2K}\zeta X_6^{\left(8ii\right)}$	$\sqrt{2K}\zeta X_5^{\left(8ii\right)}$	$H_5^{\left(8i\right)}$
$X_3^{\left(8\right)}$	$0$	$0$	$0$	$k$	$X_3^{\left(8\right)}$	$0$	$0$	$\frac{X_4^{\left(8\right)}}{{\zeta }^2}$
$X_4^{\left(8\right)}$	$0$	$-{\zeta }^2{X}_3^{\left(8\right)}$	$-k$	$0$	$\frac{\gamma X_4^{\left(8\right)}}{2}$	$0$	$0$	$0$
$H_5^{\left(8i\right)}$	$-\left(1+\frac{\gamma }{2}\right)k$	$\left(\frac{\gamma }{2}-1\right)X_2^{\left(8\right)}$	$-X_3^{\left(8\right)}$	$\frac{-\gamma X_4^8}{2}$	$0$	$-X_5^{\left(8ii\right)}$	$-X_6^{\left(8ii\right)}$	$2S_6^{\left(8iii\right)}$
$X_5^{\left(8ii\right)}$	$0$	$\sqrt{2K}\zeta X_6^{\left(8ii\right)}$	$0$	$0$	$X_5^{\left(8ii\right)}$	$0$	$\frac{\sqrt{2K}}{\zeta }k$
$X_6^{\left(8ii\right)}$	$0$	$-\sqrt{2K}\zeta X_5^{\left(8ii\right)}$	$0$	$0$	$X_6^{\left(8ii\right)}$	$-\frac{\sqrt{2K}}{\zeta }k$	$0$
$S_6^{\left(8iii\right)}$	$0$	$H_5^{\left(8i\right)}$	$-\frac{X_4^{\left(8\right)}}{{\zeta }^2}$	$0$	$-2S_6^{\left(8iii\right)}$			$0$

Table 5. Table 5: Commutators of the Killing algebra for the isometry class 9 of pp-wave spacetime ( 1 ).

$[{𝐗}_I,{𝐗}_J]$	$k$	$X_2^{\left(9\right)}$	$X_3^{\left(9\right)}$	$X_4^{\left(9\right)}$	$X_5^{\left(9\right)}$
$k$	$0$	$0$	$-k$	$0$	$0$
$X_2^{\left(9\right)}$	$0$	$0$	$X_2^{\left(9\right)}$	$X_5^{\left(9\right)}$	$0$
$X_3^{\left(9\right)}$	$k$	$-X_2^{\left(9\right)}$	$0$	$\frac{\eta }{{\sigma }^2}k+X_4$	$0$
$X_4^{\left(9\right)}$	$0$	$-X_5^{\left(9\right)}$	$-\frac{\eta }{{\sigma }^2}k-X_4$	$0$	$-\frac{{\eta }^2+{\sigma }^2}{{\sigma }^2}k$
$X_5^{\left(9\right)}$	$0$	$0$	$0$	$\frac{{\eta }^2+{\sigma }^2}{{\sigma }^2}k$	$0$

Table 6. Table 6: Lie symmetries and potentials for the Klein-Gordon ( 4 ) in isometry class 9.

Lie Sym.	$𝐕(u,v,y,z)$
$k$	$V(u,y,z)$
$X_2^{\left(9\right)}$	$V(v,y,z)$
$X_3^{\left(9\right)}$	$V(vu,y,u{e}^{-\sigma z})$
$X_4^{\left(9\right)}$	$V(u,z-\frac{\eta }{\sigma }y,2v{\sigma }^2-\frac{{\zeta }^2}{u}y-\frac{2\eta \left(\sigma z-\eta y\right)y}{2{\sigma }^{2}u})$
$X_5^{\left(9\right)}$	$V(u,v,z-\frac{\eta }{\sigma }y)$

Table 7. Table 7: Point symmetries and potentials for the Klein-Gordon equation ( 4 ) in the isometry classes 11-14

Class	$𝐀\left(u\right)$	$𝐁\left(u\right)$	$𝐂\left(u\right)$	Extra KV	Potential
11	$\alpha u^{-2}$	$\beta u^{-2}$	$\gamma u^{-2}$	$X_7^{\left(11\right)}=u{\partial }_u-v{\partial }_v$	$V(vu,y,z)$
12	$-c{u}^{-2}\left(\sin \varphi +\lambda \right)$	$c{u}^{-2}\left(\cos \varphi \right)$	$c{u}^{-2}\left(\sin \varphi -\lambda \right)$	$X_7^{\left(12\right)}=X_7^{\left(11\right)}+\delta {\partial }_{\theta }$	$V(vu,r,e^{\theta }{u}^{-\delta })$
13	$\alpha$	$\beta$	$c$	$X_7^{\left(13\right)}={\partial }_u$	$V(v,y,z)$
14	$-c\sin \left(2\delta u\right)+\lambda$	$-c\cos \left(2\delta u\right)$	$c\sin \left(2\delta u\right)+\lambda$	$X_7^{\left(14\right)}={\partial }_u$	$V(u,r,e^{\theta }{u}^{-\delta })$

Table 8. Table 8: Lie and Noether point symmetries for the wave equation in the pp-wave spacetime ( 1 ), for the isometry classes of [ 17 ]

Class	$\#$	Lie/Noether Sym.	Class	$\#$	Lie/Noether Sym.
1	$1$	$k$	6iv	$4$	$k,X_2^{\left(6\right)},X_3^{\left(6\right)},Y_4^{\left(6iv\right)}$
1i	$2$	$k,X_2^{\left(1i\right)}$	7	$3$	$k,X_2^{\left(7\right)},X_3^{\left(7\right)}$
2	$2$	$k,X_2^{\left(2\right)}$	8	$3$	$k,X_2^{\left(8\right)},X_3^{\left(8\right)}$
2i	$3$	$k,X_2^{\left(2\right)},Y_3^{\left(2i\right)}/Y_{3\left(-1\right)}^{\left(2i\right)}$	8(0)	$4$	$k,X_2^{\left(8\right)},X_3^{\left(8\right)},X_4^{\left(8\right)}$
2ii	$3$	$k,X_2^{\left(2\right)},Y_3^{\left(2ii\right)}/Y_{3\left(0\right)}^{\left(2ii\right)}$	8i	$5$	$k,X_2^{\left(8\right)},X_3^{\left(8\right)},X_4^{\left(8\right)},Y_5^{\left(8i\right)}$
2iii	$3$	$k,X_2^{\left(2\right)},Y_3^{\left(2iii\right)}$	8ii	$6$	$k,X_2^{\left(8\right)},X_3^{\left(8\right)},X_4^{\left(8\right)},X_5^{\left(8ii\right)},X_6^{\left(8ii\right)},Y_5^{\left(8i\right)}$
3	$2$	$k,X_2^{\left(3\right)}$	8iii	$6$	$k,X_2^{\left(8\right)},X_3^{\left(8\right)},X_4^{\left(8\right)},Y_5^{\left(8i\right)},Y_6^{\left(8iii\right)}$
4	$2$	$k,X_2^{\left(4\right)}$	9	$5$	$k,X_2^{\left(9\right)},X_3^{\left(9\right)},X_4^{\left(9\right)},X_5^{\left(9\right)}$
5	$3$	$k,X_2^{\left(5\right)},X_3^{\left(5\right)}$	10	$6$	$k,X_a^{\left(10\right)},Y_6^{\left(10\right)}$
5i	$4$	$k,X_2^{\left(5\right)},X_3^{\left(5\right)},Y_4^{\left(5i\right)}$	10i	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},Y_7^{\left(10i\right)}$
5ii	$4$	$k,X_2^{\left(5\right)},X_3^{\left(5\right)},Y_4^{\left(5ii\right)}$	10ii	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},Y_7^{\left(10ii\right)}$
6	$3$	$k,X_2^{\left(6\right)},X_3^{\left(6\right)}$	11	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},X_7^{\left(11\right)}$
6i	$5$	$k,X_2^{\left(6\right)},X_3^{\left(6\right)},Y_4^{\left(6i\right)},Y_5^{\left(6i\right)}$	12	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},X_7^{\left(12\right)}$
6ii	$5$	$k,X_2^{\left(6\right)},X_3^{\left(6\right)},Y_4^{\left(6ii\right)},Y_5^{\left(6ii\right)}$	13	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},X_7^{\left(13\right)}$
6iii	$4$	$k,X_2^{\left(6\right)},X_3^{\left(6\right)},Y_4^{\left(6iii\right)}$	14	$7$	$k,X_a^{\left(10\right)},H_6^{\left(10\right)},X_7^{\left(14\right)}$

Equations

$$d{s}^2=-2dudv-2H\left(u,x^A\right)d{u}^2+{\delta }_{AB}d{x}^{A}d{x}^B$$

$${\Gamma }_{uu}^v=H_{,u},{\Gamma }_{uu}^A=H_{,A},{\Gamma }_{Au}^v=H_{,A}.$$

$$\Delta \Psi \equiv -2{\Psi }_{,uv}+2H\left(u,x^A\right){\Psi }_{,vv}+{\Delta }_{\delta }\Psi ,$$

$$-2{\Psi }_{,uv}+2H\left(u,x^A\right){\Psi }_{,vv}+{\Delta }_{\delta }\Psi +V\left(u,v,x^A\right)\Psi =0.$$

$$H_{,yy}+H_{,zz}=0.$$

$${\bar{x}}^i=x^i+\epsilon {\xi }^i\left(x^k\right).$$

$${\mathcal{L}}_{\xi }\Omega =\underset{\epsilon \to 0}{\lim }\frac{1}{\epsilon }\left[\Phi \left({\Omega }^k,x^k,x^{k^{\prime }}\right)-\Omega \right].$$

$$\bar{F}\left({\bar{x}}^i\right)=F\left(x^i+\epsilon {\xi }^i\right)=F\left(x^i\right)+\epsilon F_{,i}{\xi }^i+O\left({\epsilon }^2\right).$$

$${\mathcal{L}}_{\xi }F=F_{,i}{\xi }^i.$$

$${\mathcal{L}}_{\xi }\Omega =\Lambda$$

$${\mathcal{L}}_{\xi }{g}_{ij}=2\psi \left(x^k\right)g_{ij}.$$

$$G_{KV}\subseteq G_{HV}\subseteq G_{CV}$$

$$0\le \dim G_{H-K}\le 1$$

$$d{s}^2={\left(1+\frac{K}{4}{\eta }_{ij}{x}^i{x}^j\right)}^{-1}\left({\eta }_{ij}d{x}^{i}d{x}^j\right).$$

$$X={\xi }^i\left(x^k,\Psi \right){\partial }_i+\eta \left(x^k,\Psi \right){\partial }_{\Psi }.$$

$${\mathcal{L}}_{X^{\left[2\right]}}\Theta =\lambda \Theta ,$$

$$X^{\left[2\right]}=X+{\eta }_i{\partial }_{{\Psi }_{,i}}+{\eta }_{ij}{\partial }_{{\Psi }_{,i}},$$

$${\mathcal{L}}_{X^{\left[1\right]}}L+L{D}_i{\xi }^i=D_i{A}^i,$$

$$I_i={\xi }^j{\mathcal{H}}_{ij}+\eta p_i-A_i.$$

$${\Delta }_g\Psi +V\left(x^k\right)\Psi =0,$$

$$L\left(x^k,\Psi ,{\Psi }_{,k}\right)=\frac{1}{2}\sqrt{\left|g\right|}{g}^{ij}{\Psi }_{,i}{\Psi }_{,j}-\frac{1}{2}\sqrt{\left|g\right|}V\left(x^k\right){\Psi }^2$$

$$X={\xi }^i\left(x^k\right){\partial }_i+\left(\frac{2-n}{2}\psi \left(x^k\right)\Psi \right){\partial }_{\Psi },X_{\Psi }=\Psi {\partial }_{\Psi },X_b=b\left(x^k\right){\partial }_{\Psi }$$

$${\xi }^k{V}_{,k}+2\psi V-\frac{2-n}{2}{\Delta }_g\psi =0$$

$$A_i=\frac{2-n}{4}\sqrt{g}{\psi }_{,i}\left(x^k\right)u^2.$$

$${\mathcal{L}}_{k}V=0,$$

$$k={\partial }_v,X_2^{\left(1i\right)}={\partial }_y,X_3^{\left(1i\right)}=y{\partial }_v+u{\partial }_y.$$

$$X_2^{\left(1i\right)}$$

$$X_3^{\left(1i\right)}$$

$$V_G^{\left(1i\right)}=\left(u,x,\frac{2c_{1}y-2c_{2}v+c_3\left(y^2-2uv\right)}{2\left(c_2+c_{3}u\right)}\right).$$

$$\left[k,X_2^{\left(1i\right)}\right]=0,\left[k,X_3^{\left(1i\right)}\right]=0,\left[X_2^{\left(1i\right)},X_3^{\left(1i\right)}\right]=k$$

$$V_2^{\left(2\right)}=V\left(u,r,\theta -\frac{c_2}{c_1}v\right).$$

Full text

Keywords: Lie point symmetries; pp-waves spacetimes; Collineations; Klein-Gordon equation

1 Introduction

Lie point symmetries have been used in order to solve explicitly the Einstein field equations, and to find new exact solutions in modified theories of gravity (for instance see [1, 2, 3, 4, 5, 6, 7] and references therein). Furthermore, Lie point symmetries have also been used for the study of the geodesic equations and for the determination of exact solutions of the wave equation in various gravitation models [8, 9, 10, 11, 12].  As far as concerns the wave equation in Riemannian spacetimes, a symmetry analysis of wave equation in a power-law Bianchi III spacetime can be found in [13] and a symmetry analysis of the wave equation on static spherically symmetric spacetimes, with higher symmetries, was carried out in [14]. Recently, in [15], we started a research program where we performed the symmetry classification of the wave and the Klein-Gordon equation in Bianchi I spacetimes. In this work we would like to extend this analysis and perform a classification of the Lie and Noether point symmetries for the wave equation and the Klein-Gordon equation in pp-wave type N spacetimes.

The line element of a pp-wave spacetime is ($A,B=1,2)$ [16]

[TABLE]

where $x^{A}=\left(y,z\right)$ are the Cartesian coordinates and $\delta_{AB}=diag\left(1,1\right)$ is the two dimensional Euclidian metric. The non-zero connection coefficients of (1) are

[TABLE]

The Laplace operator for the spacetime (1) is

[TABLE]

where $\Delta_{\delta}~{}$is the Laplacian of the two dimensional Euclidian space. It follows that the Klein-Gordon equation in (1) has the following form

[TABLE]

The main property of a pp-wave spacetime (1) is that admits the null Killing vector field $k=\partial_{v}$. However for special forms of the function $H\left(u,x^{A}\right)$ (1) admits a greater conformal algebra. The function $H\left(u,x^{A}\right)$ is computed from the solution of Einstein field equations. For empty spacetime is given by the equation $\Delta_{\delta}H=0$, which in Cartesian coordinates is

[TABLE]

The classification of the Killing algebras of (1) has been done in [17], whereas in [18] are given the conformal algebras of (1). In the following we use the classification of [18], which means that the results we find hold also for non empty spacetimes.  The plan of the paper is as follows.

In Section 2 we give basic definitions and properties of Riemannian collineations and introduce the Lie and the Noether point symmetries of differential equations. Furthermore, we discuss the relation between the Lie and Noether symmetries of the Klein-Gordon equation with the conformal algebra of the underlying Riemannian manifold. In Section 3 we determine the functional form of the potential $V\left(u,v,x^{A}\right)$ and the function $H\left(u,x^{A}\right)$ of (1), in order the Klein-Gordon equation (4) to admit Lie and Noether point symmetries. The complete symmetry analysis for wave equation in the pp-wave spacetime (1) is given in Section 4. Finally, in Section 5 we discuss our results and draw our conclusions.

2 Collineations and symmetries of differential equations

2.1 Collineations of Riemannian manifolds

Let a space with coordinates $\{x^{i}\}.$ Consider the one parameter point transformation

[TABLE]

in which $\xi^{i}\left(x^{k}\right)$ are the components of a vector field called the infinitesimal generator of (5).

Let $\Omega$ be a geometric object in $V^{n}$ with transformation law $\Omega^{a^{\prime}}=\Phi^{a}\left(\Omega^{k},x^{k},x^{k^{\prime}}\right)$.   Under the action of the point transformation $\Omega$ changes to $\Phi\left(\Omega^{k},x^{k},x^{k^{\prime}}\right).$ We define the Lie derivative $\mathcal{L}_{\xi}$ of $\Omega$ with respect to the vector field $\xi$ as follows [19]

[TABLE]

By definition the Lie derivative of a geometric object depends on its transformation law. For functions, the transformation law is $F^{\prime}\left(\bar{x}^{i}\right)=F\left(x^{i}\right)$, hence under the point transformation (5) we have

[TABLE]

Hence from (6) it follows

[TABLE]

We say that the function $F\left(x^{i}\right)$ is invariant under the action of (5) if $\mathcal{L}_{\xi}F=0$; In this case $\xi$ is called a symmetry of the function $F\left(x^{i}\right).$

In general we have

[TABLE]

where $\Lambda$ is a tensor which has the same number and symmetries of the indices with $\Omega$. We remark that $\Omega$ is not necessarily a tensor. In this case we say that the vector field $\xi$ is a collineation of $\Omega.$ The type of collineations depends on the tensor $\Lambda$.

In Riemannian Geometry (and in General Relativity) in general we are interested on geometrical objects $\Omega$ which are defined in terms of the metric111For the complete classification of the collineations of Riemannian manifold see [21, 20].. In particular in this work we shall consider the geometric object $\Omega=$ $g_{ij},$ and $\Lambda=$ $2\psi\left(x^{k}\right)g_{ij}$; that is, condition (8) becomes

[TABLE]

The vector field $\xi$ is called as conformal Killing vector (CKV). In general $\psi\left(x^{k}\right)=\frac{1}{n}\xi_{;i}^{i}$,  where $n$ is the dimension of the Riemann space $V^{n}$ and $";"$ denotes the covariant derivative with respect to the metric tensor $g_{ij}$. If $\psi_{;ij}=0$ the field $\xi$ is called special Conformal Killing vector (sp.CKV), when $\psi_{;i}=0$, i.e. $\psi\left(x^{k}\right)=\psi_{0}$, $\xi$ is called Homothetic vector (HV) and when $\psi\left(x^{k}\right)=0$, $\xi$ is a Killing vector (KV) of the metric tensor $g_{ij}$.

The CKVs of a metric form a Lie algebra and so do the KVs and the HV. If we denote by $G_{CV},G_{HV}$, $G_{KV}$ these algebras we have the inclusion relations

[TABLE]

and

[TABLE]

where G_{H-K}=G_{HV}-G_{HV}\cap G_{KV};\the last relation means that a Riemannian space admits at most one HV. Concerning the dimension of the conformal algebra $G_{CV}$ we have $\dim G_{CV}\leq\frac{1}{2}\left(n+1\right)\left(n+2\right).$

Collineations constitute a strong constraint on the geometric structure of a space. For example if a space admits $\frac{1}{2}n\left(n+1\right)$ then it must be a space of constant curvature and there are only three types of spaces with curvature $0,\pm K$ whose metric in Cartesian coordinates has the general form

[TABLE]

2.2 Lie point symmetries of differential equations

Consider the second order partial differential equation $\Theta\left(x^{k},\Psi,\Psi_{,i},\Psi_{,ij}\right)=0$, where $x^{k}$ are the independent variables, $\Psi=\Psi\left(x^{k}\right)$ is the dependent variable and, $\Psi_{,i}=\frac{\partial\Psi}{\partial x^{i}}$. Latin indices take the values $1,2,...,n$.  Let $\left(x^{i},\Psi\right)\rightarrow\left(\bar{x}^{i}\left(x^{k},\Psi,\varepsilon\right),\bar{\Psi}\left(x^{k},\Psi,\varepsilon\right)\right),$ be a one parameter point transformation of the independent and dependent variables with infinitesimal generator

[TABLE]

The differential equation $\Theta$ can be seen as a geometric object on the jet space $J=J\left(x^{k},\Psi,\Psi_{,i},\Psi_{,ij}\right).$ We say that $X$ defines a Lie point symmetry of $\Theta\left(x^{k},\Psi,\Psi_{,i},\Psi_{,ij}\right)=0,$ if the following condition is satisfied

[TABLE]

in which $X^{\left[2\right]},$ is the second extension/prolongation of $X$ in the space $J$, given by the formula

[TABLE]

where,$~{}\eta_{i}=D_{i}\left(\eta\right)-\Psi_{,k}D_{i}\left(\xi^{k}\right)$, $\eta_{ij}=D_{j}\left(\eta_{i}\right)-\Psi_{ki}D_{j}\xi^{k}$, and $D_{i}$ is the operator of the total derivative, i.e. $D_{i}=\frac{\partial}{\partial x^{i}}+\Psi_{,i}\frac{\partial}{\partial\Psi}+\Psi_{,ij}\frac{\partial}{\partial\Psi_{,j}}+...~{}$[22, 23].

The existence of a Lie point symmetry for a partial differential equation (PDE) means that there exist a ”coordinate” system in which the differential equation $\Theta$ is independent on one of the independent variables. In addition, Lie point symmetries can be used in order to transform solutions into solutions between different points of the space $\left(x^{i},\Psi\right)~{}$[24].

If the differential equation follows form a Lagrangian $L=L\left(x^{k},\Psi,\Psi_{,i}\right)$, that is $\Theta\equiv\mathbf{E}\left(L\right)=0$, where $\mathbf{E}$ is the Euler-operator, then one defines a special type of Lie symmetry by the condition

[TABLE]

where $A^{i}$ is a vector field and $X^{\left[1\right]}$  is the first prolongation of $X$. These Lie point symmetries are called Noether point symmetries222In the literature the vector fields which satisfy condition (16) have been called Noether Gauge symmetries. However, condition (16) is that of the standard Noether’s theorem [25] and the use of the term Gauge is unnecessar, for instance see [26, 27, 28]. and have the characteristic property that to the Lie symmetry $X$ there corresponds a conserved current $I^{i}\left(x^{k},\Psi,\Psi_{,i}\right)$, that is $D_{i}I^{i}=0$, where

[TABLE]

where $p^{i}=\frac{\partial L}{\partial\Psi_{,i}}$ and $\mathcal{H}_{~{}j}^{i}=\frac{\partial L}{\partial\Psi_{,i}}\Psi_{,j}-L$.

If (16) holds, the Lie symmetry $X$ is called Noether symmetry and the vector field $I^{i}$ Noether current. The Lie point symmetries of a PDE form a Lie algebra and the Noether point symmetries a subalgebra of this algebra.

2.3 Collineations of Riemannian spaces as point symmetries of the

Klein-Gordon equation

One parameter point transformations in a Riemannian space define the collineations in that space which  characterize to a large extent the geometry of the space. But one parameter point transformations define also the Lie point symmetries of PDEs in that space. Therefore one should expect that there exists a relation between the collineations of a space and the Lie / Noether point symmetries of a PDE in that space. The reason for this is twofold. First one may ”see ” the defining equation of a collineation as a PDE in the space which remains invariant under the Lie point symmetry. Second the dynamical field equations which describe the evolution of a dynamical system the space should be affected by the geometry of the space. This is most vividly seen in the case of the geodesic equations which on one hand characterize the geometry of the space and on the other, in accordance to the Principle of Equivalence, describe the equations of motion of a ”free” particle in the space.

Indeed it has been shown that Lie point symmetries of the geodesic equations in a Riemannian space are elements of the special Projective algebra of the space [9], and that the Lie point symmetries form the projective algebra of an extended manifold, for details see [29, 30].

Similar results have been found for partial differential equations which involve the metric tensor $g_{ij}$.  Specifically, it has been shown that the Lie point symmetries of the Schrödinger equation are generated by the Homothetic algebra of the space which defines the Laplace operator [31]. Moreover, the Lie point symmetries of the wave and of the Poisson equation are elements of the conformal algebra of the Riemannian manifold [23, 32].

The Klein-Gordon equation in a general Riemannian space is defined as follows

[TABLE]

where $\Delta_{g}=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial x^{i}}\left(\sqrt{\left|g\right|}g^{ij}\frac{\partial}{\partial x^{j}}\right)$, is the Laplace operator defined by the metric tensor $g_{ij}$. Equation (18) arise from a variational principle given by the following Lagrangian function

[TABLE]

In [31], it has been shown that the Lie point symmetries for the Klein-Gordon equation (18) are

[TABLE]

where $\xi^{i}\left(x^{k}\right)$ is a CKV of $g_{ij}$ with conformal factor $\psi\left(x^{k}\right),$ $b\left(x^{k}\right)$ is a solution of the original equation (18) and the following condition holds

[TABLE]

where $n=\dim g_{ij}$. The two fields $X_{\Psi},X_{b}$ are called linear symmetries, because they exist for a general linear partial equation. For obvious reasons $X_{b}$ is called a solution symmetry.

Concerning the Noether point symmetries of (18) we have that for the Lagrange function (19), the Lie point symmetries of (18) (except the two trivial ones) are also Noether point symmetries of (19) and that the field $A^{i}$ of condition (16) has the following form

[TABLE]

In the following we apply these results in order to classify the Lie and the Noether point symmetries of the Klein-Gordon equation (18) and the wave equation, $V\left(x^{k}\right)=0$, in pp-wave spacetimes.

3 Lie and Noether point symmetries of the Klein-Gordon equation in

pp-wave spacetimes

The pp-wave spacetimes have been classified according to the admitted isometry algebra in [17]. The complete classification of the CKVs for the pp-wave spacetimes has been done in [18, 33]. However, as we discussed above the Lie/Noether point symmetries of the Klein-Gordon equation follow from the conformal algebra of the space which defines the Laplace operator which means that in order to perform the classification problem we follow the results of [18, 33] in order to determine all potentials $V\left(u,v,x^{k}\right)$ for which the resulting Klein-Gordon equation (4) admits Lie and Noether point symmetries.

From [17] and [18], we have 14 isometry classes with some subclasses, in which the spacetime (1) admits a greater conformal algebra. We remark that equation (4) is a linear equation therefore admits always the linear (trivial) symmetries $X_{\Psi},~{}X_{b}$.

3.1 Isometry class 1

This is the most general isometry class and $H=H\left(u,x^{A}\right)$ is an arbitrary function. The spacetime (1) admits a one dimensional conformal algebra given by the KV $k$. Hence we have that the Klein-Gordon equation (4) admits the vector field $k$ as a Lie or Noether point symmetry if and only if,

[TABLE]

from which follows that $V_{G}^{\left(1\right)}=V\left(u,x,y\right)$.

3.1.1 Subclass 1i

If, $H=H\left(u,z\right),$ space (1) admits a three dimensional conformal algebra spanned by the three KVs

[TABLE]

Therefore from conditions (21) we have that $X_{2}^{\left(1i\right)},$ $X_{3}^{\left(1i\right)}$ are Lie point symmetries when

[TABLE]

Furthermore from the linear combinations of the vector fields we have that the vector $X_{\left(1i\right)}=c_{1}k+c_{2}X_{2}^{\left(1i\right)}+c_{3}X_{3}^{\left(1i\right)}$ is a Lie point symmetry of (4) when

[TABLE]

The commutators of the Killing algebra are as follows

[TABLE]

3.2 Isometry class 2

When $H=H\left(u,r\right),$ the space admits a two dimensional conformal algebra spanned by the KVs $k$ and $X_{2}^{\left(2\right)}=\partial_{\theta}$. Therefore we have that the field $X_{\left(2\right)}=c_{1}k+c_{2}X_{2}^{\left(2\right)}$ is a Lie point symmetry of (4), if and only if,

[TABLE]

In this isometry class correspond four subclasses where the space (1) admits a greater conformal algebra. However in the fourth class the exact form of the function $H\left(u,r\right)$ is not exact, hence we study only the three subclasses.

3.2.1 Subclass 2i

Assume that $H=K\left(\alpha u+\beta\right)^{q}\ln r,$ with $\beta\in\mathbb{R}$ and $K,\alpha\in\mathbb{R}^{\ast}$. In this case the space (1) admits an extra HV.

For $q\neq-1$ the HV vector is

[TABLE]

whereas for $q=-1$ the HV is

[TABLE]

The corresponding factors $\psi_{H}^{\left(2i\right)}~{}$and $\psi_{H}^{\left(2i-1\right)}$ are equal to one.

We have that the vector fields $Y_{3}^{\left(2i\right)}=H_{3}^{\left(2i\right)}-\frac{1}{2}X_{\Psi}$, $Y_{3\left(-1\right)}^{\left(2i\right)}=H_{3\left(-1\right)}^{\left(2i\right)}-\frac{1}{2}X_{\Psi}$ are Lie point symmetries of (4) provided the potential has the form

[TABLE]

and

[TABLE]

The generic fields $X_{\left(2i\right)}=c_{1}k+c_{2}X_{2}^{\left(2\right)}+c_{3}H^{\left(2i\right)}-\frac{1}{2}c_{3}X_{\Psi}$ and $X_{\left(2i-1\right)}$ $=c_{1}k+c_{2}X_{2}^{\left(2\right)}+c_{3}H_{\left(-1\right)}^{\left(2i\right)}-\frac{1}{2}c_{3}X_{\Psi}$ are Lie point symmetries when the potential is

[TABLE]

or

[TABLE]

respectively; the function $f\left(u,v\right)$ is

[TABLE]

3.2.2 Subclass 2ii

When $H=Ke^{-\frac{\Theta}{\beta}u}\ln r$, where$~{}\Theta\in\mathbb{R}$ and $\beta,K\in\mathbb{R}^{\ast},$ the space (1) admits an extra HV. For $\Theta\neq 0$ the HV is

[TABLE]

and for $\Theta=0$ the HV is

[TABLE]

Hence, we have that $Y_{3}^{\left(2ii\right)}=H^{\left(2ii\right)}-\frac{1}{2}X_{\Psi},~{}Y_{3\left(0\right)}^{\left(2ii\right)}=H_{\left(0\right)}^{\left(2ii\right)}-\frac{1}{2}X_{\Psi}$ are Lie point symmetries of (4) if and only if the potential is

[TABLE]

The generic vector fields $X_{\left(2ii\right)}=c_{1}k+c_{2}X_{2}^{\left(2\right)}+c_{3}Y_{3}^{\left(2ii\right)}$ and $X_{\left(2ii0\right)}=c_{1}k+c_{2}X_{2}^{\left(2\right)}+c_{3}Y_{3\left(0\right)}^{\left(2ii\right)},$ are Lie point symmetries of the Klein Gordon equation if

[TABLE]

3.2.3 Subclass 2iii

When $H=e^{g\left(u\right)}\ln r$ with $g\left(u\right)=-\ln\left(\rho u^{2}+\alpha u+\beta\right)-\Theta\int\left(\rho u^{2}+\alpha u+\beta\right)^{-1}du$ and $\rho,\alpha,\beta\in\mathbb{R}^{\ast},~{}\Theta\in\mathbb{R}$ the space (1) admits a sp.CKV. For simplicity we study the case $\Theta=0$. The sp.CKV is

[TABLE]

where the conformal factor is $\psi_{S}^{\left(2iii\right)}=2\rho u+\alpha,$ with $\left(\psi_{S}^{\left(2iii\right)}\right)_{;\mu\nu}=0$.

Therefore from the sp.CKV $S_{3}^{\left(2iii\right)},$ the generated Lie/Noether point symmetry vector is $Y_{3}^{\left(2iii\right)}=S_{3}^{\left(2iii\right)}-\frac{1}{2}\psi_{S}^{\left(2iii\right)}X_{\Psi}$ with corresponding potential

[TABLE]

Furthermore the vector field $Y_{\left(2iii\right)}=c_{1}k+c_{2}X_{\left(2\right)}^{2}+c_{3}Y_{3}^{\left(2iii\right)}$ is a Lie point symmetry vector of (4) if

[TABLE]

The commutators of the elements of the conformal algebras of the isometry class 2 with the subclasses (2i)-(2iii) are given in table 1.

3.3 Isometry class 3

In the isometry class 3, the function $H\left(u,x^{A}\right)$ is of the form $H=u^{-2}W\left(s,t\right)$ where the coordinates $\left\{s,t\right\}$ are

[TABLE]

The pp-wave spacetime (1) admits two KVs, the fields $k$ and $X_{2}^{\left(3\right)}=u\partial_{u}-v\partial_{v}$ with commutator $\left[k,X_{2}^{\left(3\right)}\right]=-k$.  Hence the vector field $X_{3}=c_{1}k+c_{2}X_{2}^{\left(3\right)}$ is a Lie point symmetry of (4) provided that

[TABLE]

3.4 Isometry class 4

When $H=W\left(\bar{s},\bar{t}\right)$ with

[TABLE]

the pp-wave spacetime (1) admits a two dimensional conformal algebra with elements the two KVs $k,~{}X_{2}^{\left(4\right)}=\partial_{u}$ with commutator $\left[k,X_{2}^{\left(4\right)}\right]=0$. Therefore we have that the field $X_{4}=c_{1}k+c_{2}X_{2}^{\left(4\right)}$ is a Lie point symmetry of (4), if and only if,

[TABLE]

3.5 Isometry class 5

In this case the spacetime (1) admits a three dimensional Killing algebra with commutators

[TABLE]

where $X_{2}^{\left(5\right)}=\partial_{\theta}$, $X_{3}^{\left(5\right)}=u\partial_{u}-v\partial_{v}$ and $H=u^{-2}W\left(r\right)$.

Therefore, the generic vector field $X_{5}=c_{1}k+c_{2}X_{2}^{\left(5\right)}+c_{3}X_{3}^{\left(5\right)}$ is a Lie point symmetry of (4), if and only if

[TABLE]

Moreover, for the subclasses (5i) with $W\left(r\right)=\zeta\ln r,$ and (5ii) with $W\left(r\right)=\left(\delta r^{-\sigma}-\sigma\left(2-\sigma\right)^{-2}r^{2}\right),~{}\left|\sigma\right|\neq 0,2$, the spacetime (1) admits a four dimensional conformal algebra.

3.5.1 Subclass 5i

When $H=\zeta u^{-2}\ln r$ the spacetime (1) admits the extra sp.CKV

[TABLE]

Hence the field $Y_{4}^{\left(5i\right)}=S_{4}^{\left(5i\right)}-\frac{1}{2}uX_{\Psi}$ is a Lie point symmetry of (4) provided

[TABLE]

Similarly the field $X_{\left(5i\right)}=c_{1}k+c_{2}X_{2}^{\left(5\right)}+c_{3}X_{3}^{\left(5\right)}+c_{4}Y_{4}^{\left(5i\right)}$ is a Lie point symmetry of (4) when

[TABLE]

where

[TABLE]

3.5.2 Subclass 5ii

Contrary to the subclass (5i), this subclass admits the proper CKV

[TABLE]

with conformal factor $\psi_{4}^{\left(5ii\right)}=\frac{2}{\sigma-2}u^{\frac{\sigma+2}{2-\sigma}}$. Since $C_{4}^{\left(5ii\right)}$ is a not a sp.CKV holds that $\left(\psi_{4}^{\left(5ii\right)}\right)_{;ij}\neq 0$; however we note that $\Delta\left(\psi_{4}^{\left(5ii\right)}\right)=0$ which means that $\psi_{4}^{\left(5ii\right)}$ is a solution of the wave equation. Therefore, we have that the vector field $Y_{4}^{\left(5ii\right)}=C_{4}^{\left(5ii\right)}-\frac{1}{2}\psi_{4}^{\left(5ii\right)}$ is a point symmetry of (4) when

[TABLE]

Furthermore, the field $Y_{\left(5ii\right)}=c_{1}k+c_{2}X_{2}^{\left(5\right)}+c_{3}X_{3}^{\left(5\right)}+Y_{4}^{\left(5ii\right)}$ is a point symmetry of (4) when

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

The commutators of the elements of the conformal algebras of the isometry class 5 with the subclasses (5i) and (5ii) are given in table 2.

3.6 Isometry class 6

In the isometry class 6, the spacetime (1) admits three KVs, the field $k$ and the two vector fields $X_{2}^{\left(6\right)}=\partial_{\theta}~{},~{}X_{3}^{\left(6\right)}=\partial_{u},$ where $H\left(u,x^{A}\right)=W\left(r\right)$.

Hence, we have that the generic field $X_{6}=c_{1}k+c_{2}X_{2}^{\left(6\right)}+c_{3}X_{3}^{\left(6\right)},$ is a Lie point symmetry of (4), if

[TABLE]

For special functions $H\left(r\right)$ the spacetime (1) admits a greater conformal algebra. There exist four possible subclasses in which the pp-wave spacetime admits a greater conformal algebra.

3.6.1 Subclass 6i

When $W\left(r\right)=\frac{N}{4}r^{2}+\delta r^{-2}$, the spacetime (1) admits two proper non sp.CKVs, the fields

[TABLE]

with conformal factors $\psi_{4}^{\left(6i\right)}=\frac{\sqrt{2N}}{2}\cos\left(\sqrt{2N}u\right)$ and $\psi_{5}^{\left(6i\right)}=-\frac{\sqrt{2N}}{2}\sin\left(\sqrt{2N}u\right)$ which satisfy the wave equation, i.e. $\Delta\psi_{4-5}^{\left(6i\right)}=0$.

Therefore, from the fields $C_{4}^{\left(6i\right)},$ $C_{5}^{\left(6i\right)}$ we have the possible point symmetries $Y_{4}^{\left(6i\right)}=C_{4}^{\left(6i\right)}-\frac{1}{2}\psi_{4}^{\left(6i\right)}X_{\Psi}$ and $Y_{5}^{\left(6i\right)}=C_{5}^{\left(6i\right)}-\frac{1}{2}\psi_{5}^{\left(6i\right)}X_{\Psi}$ for the Klein-Gordon equation (4) provided the potential has the following forms

[TABLE]

[TABLE]

Moreover, if the general vector field $X_{\left(6i\right)}=X_{6}+c_{4}Y_{4}^{\left(6i\right)}+c_{5}Y_{5}^{\left(6i\right)}$ is a point symmetry of (4), then

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

3.6.2 Subclass 6ii

When $N=0$, i.e. $W\left(r\right)=\delta r^{-2}$, the spacetime (1) admits a HV and a proper sp.CKV. These fields are respectively

[TABLE]

[TABLE]

Note that $\left(\psi_{5}^{\left(6ii\right)}\right)_{;\mu\nu}=0$. Therefore from the fields $H_{4}^{\left(6ii\right)}$ and $S_{5}^{\left(6ii\right)}$ we have the possible Lie point symmetries of (4) $Y_{4}^{\left(6ii\right)}=H_{4}^{\left(6ii\right)}-\frac{1}{2}X_{\Psi}$ and $Y_{5}^{\left(6ii\right)}=S_{5}^{\left(6ii\right)}-\frac{1}{2}\psi_{5}^{\left(6ii\right)}X_{\Psi}$ respectively.

This means that the generic point symmetry vector of (4) is $X_{\left(6ii\right)}=X_{\left(6\right)}+c_{4}Y_{4}^{\left(6ii\right)}+c_{5}Y_{5}^{\left(6ii\right)}$ provided the potential has the form

[TABLE]

where

[TABLE]

3.6.3 Subclass 6iii

When $W\left(r\right)=\zeta\ln r$, the spacetime (1) admits the extra HV

[TABLE]

It follows that $Y_{4}^{\left(6iii\right)}=H_{4}^{\left(6iii\right)}-\frac{1}{2}X_{\Psi}$ is a Lie point symmetry of (4) when

[TABLE]

Moreover, the vector field $X_{\left(6iii\right)}=X_{6}+c_{4}Y_{4}^{\left(6iii\right)}$ is a Lie and Noether point symmetry of (4) if and only if the potential has the following form

[TABLE]

3.6.4 Subclass 6iv

When $W\left(r\right)=\delta r^{-s}$ with $s\neq 2,0$,   the spacetime (1) admits the extra HV

[TABLE]

Hence, the vector field $X_{\left(6iv\right)}=X_{6}+c_{4}\left(H_{5}^{\left(6iv\right)}-\frac{1}{2}X_{\Psi}\right)$ is a Lie and Noether point symmetry of (4) when

[TABLE]

where $f_{1}\left(u\right)=\left(2c_{3}+c_{4}u\left(2+\sigma\right)u\right)^{-\frac{4}{2+\sigma}}$.

In table 3, we give the commutators of the vector fields which form the conformal algebras of the isometry class 6 and the subclasses (6i)-(6iv).

3.7 Isometry class 7

In the isometry class 7 $H\left(u,x^{A}\right)=e^{2\omega\theta}W\left(r\right).$ For this $H\left(u,x^{A}\right)$ spacetime (1) admits as KVs the fields $k$ and

[TABLE]

The commutators of the Killing algebra are

[TABLE]

The field $X_{2}^{\left(7\right)}$ is a point symmetry of (4) when

[TABLE]

Moreover, the generic KV $\ X_{7}=c_{1}k+c_{2}X_{2}^{\left(7\right)}+c_{3}X_{3}^{\left(7\right)}$, is a  Lie symmetry for equation (4) if and only if,

[TABLE]

We note, that there exist subclasses of the isometry class 7 in which the spacetime (1) admits a greater conformal algebra; however, these subclasses are not vacuum pp-wave spacetimes. We continue with the isometry class 8.

3.8 Isometry class 8

Consider $H=\exp\left(2\delta t^{\prime}\right)W\left(s^{\prime}\right)$ where

[TABLE]

$\delta=-\frac{c}{\eta^{2}+\sigma^{2}}$ and $c,\eta,\sigma\in\mathbb{R}-\left\{\zeta^{2}\equiv\eta^{2}+\sigma^{2}=0\right\}$.

For $\delta\neq 0$, the spacetime (1) admits the three KVs

[TABLE]

and for $\delta=0$ admits four KVs, the extra KV is

[TABLE]

Therefore, for $\delta\neq 0$, the general form of the potential $V\left(u,v,s^{\prime},t^{\prime}\right)$ for which the Klein-Gordon equation admits as a Lie point symmetry the vector field $X_{8}=c_{1}k+c_{2}X_{2}^{\left(8\right)}+c_{3}X_{3}^{\left(8\right)},$ is as follows:

[TABLE]

Finally, the commutators of the Killing algebra are as follows

[TABLE]

Similarly, for $\delta=0$ we have that the vector field $X_{8}^{\left(0\right)}=X_{8}+c_{3}X_{4}^{\left(8\right)}$ is a Lie point symmetry of (4), if and only if

[TABLE]

3.8.1 Subclass 8i

When $\delta=0$ and $W\left(s\right)=Ks^{\gamma}$, $K,\gamma\in\mathbb{R}^{\ast}$ and $\gamma\neq\pm 2,1$, the spacetime (1) admits a five dimensional homothetic algebra. The extra HV is

[TABLE]

where $\psi_{5}^{\left(8i\right)}=1$. Hence, the field $Y_{5}^{\left(8i\right)}=H_{5}^{\left(8i\right)}-\frac{1}{2}X_{\Psi}$ is a Lie point symmetry of (4) when

[TABLE]

Finally the generic field $X_{\left(8i\right)}=X_{8}^{\left(0\right)}+c_{5}Y_{5}^{\left(8i\right)}$, is a Lie point symmetry for equation (4)), when

[TABLE]

where functions $g_{1},g_{2}$ are

[TABLE]

[TABLE]

We continue with the subclasses (8ii) and (8iii), where $\gamma=2$ and $\gamma=-2$ respectively. As we have discussed, when $\gamma=1$ the space (1) is flat, and we do not consider that case.

3.8.2 Subclass 8ii

When $\gamma=2$, i.e. $W\left(s\right)=Ks^{2}$, the pp-wave spacetime (1) admits a seven dimensional conformal algebra. In particular admits six KVs and a HV. The KVs are the fields $k,~{}X_{2}^{\left(8\right)},~{}X_{3}^{\left(8\right)},X_{4}^{\left(8\right)}~{}$and

[TABLE]

where the HV is the $H_{5}^{\left(8i\right)}$.  Therefore, the fields $X_{5}^{\left(8ii\right)}$ and $X_{6}^{\left(8ii\right)}$ are Lie point symmetries of (4) when

[TABLE]

We continue with the next subclass, where $\gamma=-2.$

3.8.3 Subclass 8iii

When $W\left(s\right)=Ks^{-2}$, the pp-wave spacetime (1) admits four KVs, one HV and the sp.CKV

[TABLE]

Therefore, from the sp.CKV we have that vector field $Y_{6}^{\left(8iii\right)}=S_{6}^{\left(8iii\right)}-\frac{1}{2}\psi_{6}^{\left(8iii\right)}X_{\Psi}$ is a Lie point symmetry of the Klein-Gordon equation when

[TABLE]

The commutators of the elements of the conformal algebras of the isometry class 8 for $\delta=0,~{}$and of the subclasses (8i)-(8iii) are given in table 4. We would like to remark that for the subclasses (8ii) and (8iii), we did not give the form of the potential for which the Klein-Gordon equation (4) admits as Lie point symmetry the generic symmetry vector, which follows from the linear combination of the CKVs, because in this case the potential has a complex functional form.

3.9 Isometry class 9

In isometry class 9, $H=Ke^{\eta y-\sigma z}$ with $K,\eta,\sigma~{}\in\mathbb{R}-\left\{\zeta^{2}\equiv\eta^{2}+\sigma^{2}=0\right\}.$ In this class, spacetime (1) admits a five dimensional Killing algebra. The KVs are the field $k$ and

[TABLE]

[TABLE]

The commutators of the Lie algebra are given in table 5. In table 6 we give the form of the potential $V\left(u,v,x,y\right)$ for which any of the elements of the Killing algebra of (1) is a point symmetry of (4).

Furthermore, the generic vector field $X_{9}=c_{1}k+c_{2}X_{2}^{\left(9\right)}+...+c_{5}X_{5}^{\left(9\right)}$ is a Lie point symmetry of (4) if and only if

[TABLE]

where $C_{3}=\frac{c_{2}c_{4}}{c_{3}\sigma^{2}}\left[\zeta^{2}c_{2}c_{4}-\eta c_{3}\left(c_{3}+\eta c_{5}\right)-\sigma^{2}c_{3}c_{5}\right]$,

[TABLE]

[TABLE]

and

[TABLE]

3.10 Plane wave spacetime: Isometry class 10

When the function $H\left(u,x^{A}\right)$ has the form

[TABLE]

the spacetime (1) is a plane wave spacetime.

The spacetime (1) with (101) is vacuum when $A\left(u\right)+C\left(u\right)=0$. Moreover, admits a six dimensional homothetic algebra. The four KVs are given by the vector field [17, 18, 33]

[TABLE]

where $\dot{d}_{a}=\frac{d}{du}\left(d_{a}\right),$ and the functions $d_{a}\left(u\right),e_{a}\left(u\right)$ satisfy the following system of equations

[TABLE]

The fifth KV is the field $k,$ and the proper HV is

[TABLE]

Hence, we have that the fields $X_{a}^{\left(10\right)}$ are Lie point symmetries of (4), if and only if

[TABLE]

Moreover, from the HV, we have that the field $Y_{6}^{\left(10\right)}=H_{6}^{\left(10\right)}-\frac{1}{2}X_{\Psi}$ is a Lie point symmetry of (4) provided

[TABLE]

Finally, the generic field $X_{10}=c_{1}k+c_{a}X_{a}^{\left(10\right)}+c_{6}Y_{6}^{\left(10\right)}$ is a Lie point symmetry of (4) when

[TABLE]

where

[TABLE]

The commutators of the homothetic algebra are [18]

[TABLE]

where Q_{ab}\are constants. Furthermore, there are three subclasses, for special form of the functions $A\left(u\right),B\left(u\right),C\left(u\right)$ of (101), for which the plane symmetric spacetime admits a greater conformal algebra. In the following we consider the two subclasses for which the spacetime (1) admits extra sp.CKV.

3.10.1 Subclass 10i

When the functions $A\left(u\right),B\left(u\right)$ and $C\left(u\right)$ of (101) are

[TABLE]

where $\phi\left(u\right)=2\gamma\int\frac{du}{u^{2}+\beta},~{}$the spacetime (1) admits the extra sp.CKV

[TABLE]

where $\psi_{7}^{\left(10i\right)}=u$. Since $S_{7}^{\left(10i\right)}$ is a sp.CKV the $\left(\psi_{7}^{\left(10i\right)}\right)_{;\mu\nu}=0.$ Therefore we from $S_{7}^{\left(10i\right)}$ we have that the vector field $Y_{7}^{\left(10i\right)}=S_{7}^{\left(10i\right)}-\frac{1}{2}\psi_{7}^{\left(10i\right)}X_{\Psi}$, is a point symmetry of (4) when

[TABLE]

3.10.2 Subclass 10ii

When $A\left(u\right)=-\alpha\left(u^{2}+\beta\right)^{-2}$, $B\left(u\right)=-b\left(u^{2}+\beta\right)^{-2}$ and $C\left(u\right)=-c\left(u^{2}+\beta\right)^{-2}$, the extra sp.CKV of (4) is

[TABLE]

which is the field $S_{7}^{\left(10i\right)}$ for $\gamma=0$. Hence, the field $Y_{7}^{\left(10ii\right)}=S_{7}^{\left(10ii\right)}-\frac{1}{2}\psi_{7}^{\left(10ii\right)}X_{\Psi}$ is a Lie point symmetry of integral when the potential has the form

[TABLE]

There are also four more isometry classes in which the plane wave spacetime (1) admits a seven dimensional homothetic algebra [18], where there exists a six dimensional subalgebra and it is the homothetic algebra of isometry class 10.

For these isometry classes, in table333In the isometry class 12, $\phi=2\delta\ln u.$ 7, we give the functional form of $A,B,C$, the extra KV and the form of the potential for which the corresponding KV is a Lie point symmetry of the Klein-Gordon equation (4).

4 Symmetry classification for the Wave equation

When the potential in (4) vanishes the Klein Gordon equation becomes

[TABLE]

which is the wave equation in spacetime (1).

Contrary to the Klein-Gordon equation, a CKV of the metric which defines the Laplace operator generates a Lie/Noether symmetry for the wave equation only when the conformal factor is a solution of the original equation (see condition (21)). Therefore, the KVs, the HV and the sp.CKVs generate always point symmetries for the wave equation. Furthermore, in section 3 we showed that when the pp-wave spacetime admits a proper CKV, then the conformal factor is a solution of the original equation, which means that the proper CKVs, when there exist, generate always Lie and Noether point symmetries for the wave equation (119).

The Lie and Noether point symmetries of the wave equation (except the trivial ones) for the isometry classes 1 to 14, of section 3 are given in table 8.

5 Conclusions

In this work we performed a complete classification of the Lie /Noether point symmetries for the Klein-Gordon and the wave equation in pp-wave spacetimes using three results: (a) The general results of [31] and [15] concerning the relation between the Lie / Noether point symmetries of the Klein-Gordon equation with the conformal algebra of the underlying space; (b) the classification of the Klein Gordon equation based on the isometries of (1) done in [17], and (c) The classification of the conformal algebra of the pp-wave spacetimes (1) done in [18] and [33].

In addition we used these results in order to calculate the Lie and the Noether point symmetries of the wave equation (119) in a pp-wave spacetime.  We found that the Lie point symmetries form a Lie algebra $G_{W}~{}$(except the trivial symmetries), of dimension $\dim G_{W}\leq 7$ where the equality holds for the case where the space (1) is a plane wave spacetime. In addition we noted that due to the fact that the conformal factors of the CKVs of (1) are solutions of wave equation (119) all CKVs of (1) give rise to a Lie point symmetry give a Lie point symmetry of (119). Because we have followed the classification of [18], the symmetry classification holds and for non-empty spacetimes.

A further use of the results obtained in this work is that they can be used in order one to reduce and possibly to solve analytically the Klein-Gordon equation (4) and wave equation (119) in a pp-wave spacetime.

Acknowledgements

The research of AP was supported by FONDECYT postdoctoral grant no. 3160121.