1. No category

PRAMANA
c Indian Academy of Sciences
— journal of
physics
Vol. 77, No. 3
September 2011
pp. 555–570
Invariance analysis and conservation laws of the wave
equation on Vaidya manifolds
R NARAIN and A H KARA∗
School of Mathematics and Centre for Differential Equations, Continuum Mechanics and
Applications, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050,
South Africa
∗ Corresponding author. E-mail: [email protected]
Abstract. In this paper we discuss symmetries of classes of wave equations that arise as a consequence of some Vaidya metrics. We show how the wave equation is altered by the underlying
geometry. In particular, a range of consequences on the form of the wave equation, the symmetries
and number of conservation laws, inter alia, are altered by the manifold on which the model wave
rests. We find Lie and Noether point symmetries of the corresponding wave equations and give some
reductions. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with
the standard wave equations on a flat geometry. Finally, we pursue the existence of higher-order
variational symmetries of equations on nonflat manifolds.
Keywords. Wave equations on nonflat manifolds; symmetry analysis; conservation laws.
PACS Nos 02.20.Sv; 02.30.Jr; 02.30.Xx; 02.40.Vh; 04.20.Jb
1. Introduction
The well-known Vaidya metric representing a model for the spherically symmetric solution
of the Einstein equations with geometrical optics stress energy tensor of radiation is widely
discussed in the literature [1–3]. A special case of the metric is the well-known Papapetrou
model [4] and a study involving the Carter constant and Petrov classification is conducted
in [5]. The references in the latter includes studies on the nature of the Killing tensors and
well-known notion of the isometries of the metric which are the diffeomorphisms of the
manifold onto itself which preserve the metric tensor [6].
The Lie and Noether symmetries of the geodesic equations have been discussed in detail
in [7], the more interesting case being the latter since these lead to conservation laws via
Noether’s theorem [8]. We showed that we totally recover the information regarding the
isometries of the metric from a study of the Noether symmetries associated with the corresponding natural Lagrangian, L. The Noether symmetries of a system are a one-parameter
DOI: 10.1007/s12043-011-0175-3; ePublication: 27 August 2011
555
R Narain and A H Kara
Lie groups of transformations that preserve the action L = L and are determined by a
Killing-type equation given later. That is, a larger algebra of generators of symmetry are
obtained and, hence, more conservation laws are classified. The Lie symmetries, on the
other hand, leave the system invariant under the transformation and contain the Noether
symmetries [9].
The standard wave equation in (1+3) dimensions has been extensively studied in the
literature from the point of view of its Lie point symmetries. A detailed symmetry analysis
of this equation is discussed by Ibragimov [10]. It is well known that in three-dimensional
Euclidean space, the linear wave equation admits a maximal 16-dimensional Lie algebra of
point symmetries excluding the infinite symmetry. In this work, we use a purely geometric
consideration to construct the wave equation in a curved background geometry in such a
way that the wave equation inherits nonlinearities of the respective geometry in a natural
way. Keeping in mind that the wave equation in four-dimensional spacetime may be of
more physical significance, we use the Vaidya manifold for our purposes.
We present some salient features of an Euler–Lagrange system of differential equations
[8,9]. Consider an r th-order system of partial differential equations of n independent and
m dependent variables,
E β (x, u, u (1) , . . . , u (r ) ) = 0,
β = 1, . . . , m.
(1)
A conservation law of (1) is a closed form of the equation given by
Di T i = 0,
(2)
on the solutions of (1), where Di is the total derivative operator given by
Di =
∂
∂
∂
+ u iα α + u iαj α + · · · ,
∂xi
∂u
∂u j
i = 1, . . . , n,
where u iα represents the first derivative of u α with respect to the ith independent variable
x i and u iαj represents all second derivatives of u α and so on and T = (T 1 , . . . , T n ) is a
conserved vector of (1). Suppose A is the universal space of differential functions. A
Lie–Bäcklund operator (vector field) is given by
X = ξi
∂
∂
∂
∂
+ ηα α + ζiα α + ζiα1 i2 α + · · · ,
∂xi
∂u
∂u i
∂u i1 i2
(3)
where ξ i , ηα ∈ A and the additional coefficients are
ζiα = Di (W α ) + ξ j u iαj ,
ζiα1 i2 = Di1 Di2 (W α ) + ξ j u αji1 i2 ,
(4)
and so forth, where W α is the Lie characteristic function defined by
W α = ηα − ξ j u αj .
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(5)
Wave equation on Vaidya manifolds
In this paper, we shall assume that X is a Lie point symmetry operator, that is, ξ and
η are functions of x and u and are independent of derivatives of u. The Euler–Lagrange
equations, if they exist, associated with (1) is the system,
δL
= 0,
δu α
α = 1, . . . , m,
(6)
where δ/δu α is the Euler–Lagrange operator given by
∂
∂
δ
=
+
(−1)s Di1 · · · Dis
,
α
δu α
∂u α
∂u
i 1 ···i s
s≥1
α = 1, . . . , m.
(7)
L is referred to as a Lagrangian and a Noether symmetry operator X of L arises from a
study of the invariance properties of the associated functional,
L=
L(x, u, u (1) , . . . , u (r ) )dx,
(8)
defined over . If we include point-dependent gauge terms g1 , . . . , gn , then the Noether
symmetries X are given by a Killing-type equation,
X L + L Di ξi = Di gi .
(9)
Corresponding to each X , a conserved vector T = (T 1 , . . . , T n ) is obtained via Noether’s
theorem.
The paper is organized in the following form. In §2 we discuss symmetries of classes
of wave equations that arise as a consequence of some Vaidya metrics. The objective is
to show how the respective geometry affects the Lie point symmetry algebras of the corresponding wave equation. Then, in §3, the Noether point symmetries of the corresponding
wave equations are determined via a Lagrangian. It will be shown that the algebras of
Noether point symmetries are also reduced when compared with the algebra of the standard wave equation (radically, in some cases). Via Noether’s theorem, some conserved
flows are constructed. Finally, in §4, we pursue the existence of higher-order variational
symmetries of wave equations on the respective manifolds.
Firstly, we note that the Vaidya metric [3] is given by
2m(t)
2
dt 2 − 2 dt dr + r 2 dθ 2 + sin2 θ dφ 2 ,
(10)
ds = − 1 −
r
for which a special case, m(t) = t, is known as the Papapetrou model [4].
A wave equation on a Lorentzian manifold endowed with a metric gi j is given by the
expression
2
∂
1 i j 00
2
00 2
k
u
x,
¯
t)
=
g
g
g
u (x,
¯ t) = 0,
−
∇
∂
+
g
+
∂
∂
(
(∂
)∂
i
00
j
i
j
k
00
ij
∂t 2
2
where u(x,
¯ t) is some given wave function,
ikj =
1 km g ∂ j gim + ∂i g jm − ∂m gi j ,
2
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R Narain and A H Kara
Table 1. The Lie point symmetries of the wave equation for various cases of m(t).
m=0
∂t
t∂t + r ∂r
t 2 ∂t + 2r (r + t)∂r − 2u(r + t)∂u
u∂u
∂φ
m=k
m=t
m = et
m = m(t)
∂t
u∂u
∂φ
t∂t + r ∂r
u∂u
∂φ
u∂u
∂φ
u∂u
∂φ
represents the Christoffel symbols, g i j is the inverse of the metric gi j with spherical
variables r, θ, φ, and
⎛
⎞
−(1 − 2m(t)
) −1 0
0
r
⎜
⎟
−1
0 0
0
⎟.
gi j = ⎜
⎝
⎠
0
0
0 r2
0
0 0 r 2 sin2 θ
Consequently, the wave equation on a curved Vaidya background takes the form
2m(t)
− r 2 sin θ u r t − 2r sin θ u t + 2r sin θ 1 −
u r + 2 sin θ m(t)u r
r
2m(t)
u rr + cos θ u θ + sin θ u θθ + u φφ = 0.
+ r 2 sin θ 1 −
r
(11)
We shall classify the Lie and Noether point symmetries of this equation and show the
effect of the curved background on the respective symmetry algebras.
2. Lie symmetries
We determine the Lie point symmetry generators of the wave equation (11) and split
these into various cases (table 1). The principle Lie algebra is stated below. The most
significant result we note is the reduction in the dimension of the algebra of Lie point
symmetries when compared with the algebra of the wave equation on a flat manifold.
This, as will be seen later, also has consequences on the number of standard conservation
laws (usually first-order) of (11). Furthermore, the number of exact or invariant solutions
are reduced drastically. For illustration, we perform a reduction corresponding to some
two-dimensional subalgebras.
It is well known that the case m = 0 is isomorphic to a flat manifold and one would
expect a maximal 16-dimensional algebra. It is clear that the wave equation is somewhat
distorted even in this case and the number conservation laws will be reduced (see §2.1 for
a confirmation of this).
2.1 Reduction of order – some illustrations
We demonstrate two reductions of the wave equation using two-dimensional subalgebras
for the case m = t. In both cases, this leads to a partial differential equation in just two
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Wave equation on Vaidya manifolds
independent variables which can be further analysed using another Lie symmetry reduction
or an appropriate, alternative method.
(i) If X¯ 1 = ∂φ + u∂u , [ X¯ 1 , X 2 ] = 0 so that reducing may begin with either X¯ 1 or X 2 .
The respective wave equation,
2t
u r + 2t sin θ u r
−2r 2 sin θ u r t − 2r sin θ u t + 2r sin θ 1 −
r
2t
u rr + cos θ u θ + sin θ u θθ + u φφ = 0,
+r 2 sin θ 1 −
(12)
r
becomes
2t
−2r 2 sin θ wr t − 2r sin θ wt + 2r sin θ 1 −
wr + 2t sin θ wr
r
2t
wrr + cos θ wθ + sin θ wθθ + w = 0
+r 2 sin θ 1 −
r
(13)
via the generator X¯ 1 since
dr
dθ
dφ
du
dt
=
=
=
=
0
0
0
1
u
leads to the new dependent variable w defined by u = w(t, r, θ )eφ. From X 2 =
t∂t + r ∂r we get
dr
dθ
dw
dt
=
=
=
,
t
r
0
0
so that w = W (α, θ ) leads to the partial differential equation,
(2α 3 + α 2 − α) sin θ Wαα + (4α 2 + 2α − 2) sin θ Wα
+ cos θ Wθ + sin θ Wθθ + W = 0,
(14)
where α = r/t.
(ii) If we first reduce using X 1 = ∂φ we get
2t
u r + 2t sin θ u r
−2r 2 sin θ u r t − 2r sin θ u t + 2r sin θ 1 −
r
2t
+r 2 sin θ 1 −
u rr + cos θ u θ + sin θ u θθ = 0,
r
(15)
and then X¯ 2 = u∂u + t∂t + r ∂r leads to
dt
dr
dθ
du
=
=
=
,
t
r
0
u
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R Narain and A H Kara
so that with invariants α = r/t and w(θ, α) = tu we get the reduced partial
differential equation
(2α 3 + α 2 − 2α) sin θ wαα + (2α 2 + 2α − 2) sin θ wα
− 2α sin θ w + cos θ wθ + sin θ wθθ = 0.
(16)
3. Noether symmetries and conservation laws
Since (11) is variational, we determine the Noether symmetries using (9) and the corresponding conserved flows (T t , T r , T θ , T φ ) via Noether’s theorem. First, it can be shown
that a Lagrangian of (11) is given by
1
2m (t)
1
1
u r2 + sin θ u 2θ + u 2φ .
(17)
L = −r 2 sin θ u t u r + r 2 sin θ 1 −
2
r
2
2
(i) The case m = 0.
(a)
X 11 = ∂t
Tt =
1 2
(r sin θ u r u t − u(u φφ + cos θ u θ
2
+ sin θ (u θθ + r (2u r + r u rr − 2u t − r u tr )))),
1
T r = − r 2 sin θ (u r u t − u 2t + u(−u tr + u tt )),
2
1
T θ = − sin θ (u θ u t − uu tθ ),
2
Tφ =
(b)
1
(−u φ u t + uu tφ ).
2
X 21 = ∂φ
1
T t = r 2 sin θ (u φ u r − uu r φ ),
2
1
T r = − r 2 sin θ (u φ (u r − u t ) + u(−u r φ + u tφ )),
2
1
T θ = − sin θ (u φ u θ − uu θφ ),
2
Tφ =
560
1
(−u φ 2 − u(cos θ u θ + sin θ (u θθ
2
+ r (2u r + r u rr − 2(u t + r u tr ))))).
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Wave equation on Vaidya manifolds
(c)
X 31 = u∂u + t∂t + r ∂r
Tt =
1 2
(r sin θ u r (r u r + tu t ) − u(tu φφ + t cos θ u θ + sin θ (tu θθ
2
+ r ((r + 2t)u r + r (r + t)u rr − t (2u t + r u tr ))))),
1
T r = − r (r sin θ (u r − u t )(r u r + tu t ) + u(u φφ + cos θ u θ
2
+ sin θ (u θθ − r (−u r + u t + r u tr + tu tr − tu tt )))),
1
T θ = − sin θ (u θ (r u r + tu t ) − u(r u r θ + tu tθ )),
2
Tφ =
(d)
1
(−u φ (r u r + tu t ) + u(r u r φ + tu tφ )).
2
X 41 = t 2 ∂t + 2r (r + t)∂r − 2u(r + t)∂u
Tt =
1 2
(2r sin θ u 2 − r 2 sin θ u r (2r (r + t)u r + t 2 u t ) + u(t 2 u φφ
2
+ t 2 cos θ u θ + sin θ (t 2 u θθ + r (2(2r 2 + r t + t 2 )u r + r (2r 2
+ 2r t + t 2 )u rr − t 2 (2u t + r u tr ))))),
1
T r = r (r sin θ (u r − u t )(2r (r + t)u r + t 2 u t ) + u(2(r + t)u φφ
2
+ 2(r + t) cos θ u θ + sin θ (2(r + t)u θθ − r (−2(r + t)u r
+ 2(2r + t)u t + 2r 2 u tr + 2r tu tr + t 2 u tr − t 2 u tt )))),
(e)
Tθ =
1
sin θ (u θ (2r (r + t)u r + t 2 u t ) − u(2r (r + t)u r θ + t 2 u tθ )),
2
Tφ =
1
(u φ (2r (r + t)u r + t 2 u t ) − u(2r (r + t)u r φ + t 2 u tφ )).
2
X 51 = g(t, r, θ, φ)∂u where the function g(t, r, θ, φ) satisfies the equation
gφφ + cos θgθ + sin θ (gθθ + r (2gr + rgrr − 2(gt + rgtr ))).
T t = r 2 sin θ (ugr − gu r ),
T r = r 2 sin θ (u(−gr + gt ) + g(u r − u t )),
T θ = sin θ (−ugθ + gu θ ),
T φ = −ugφ + gu φ .
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R Narain and A H Kara
(ii) The case m = k where k is an arbitrary constant.
(a)
X 12 = ∂t
Tt =
1 2
(r sin θ u r u t − u(u φφ + cos θ u θ + sin θ (u θθ − 2(k − r )u r
2
+ r ((−2k + r )u rr − 2u t − r u tr )))),
1
T r = − r sin θ ((−2k + r )u r u t − r u 2t + u((2k − r )u tr + r u tt )),
2
1
T θ = − sin θ (u θ u t − uu tθ ),
2
Tφ =
(b)
1
(−u φ u t + uu tφ ).
2
X 22 = ∂φ
1
T t = r 2 sin θ (u φ u r − uu r φ ),
2
1
T r = − r sin θ (u φ ((−2k + r )u r − r u t ) + u((2k − r )u r φ + r u tφ )),
2
1
T θ = − sin θ (u φ u θ − uu θφ ),
2
Tφ =
1
(−u 2φ − u(cos θ u θ + sin θ (u θθ − 2(k − r )u r + r ((−2k + r )u rr
2
− 2(u t + r u tr ))))).
(c)
X 32 = g(t, r, θ, φ)∂u where the function g(t, r, θ, φ) satisfies the equation
gφφ + cos θgθ + sin θ (gθθ − 2(k − r )gr + r ((−2k + r )grr − 2(gt + rgtr )))
T t = r 2 sin θ (ugr − gu r ),
T r = r sin θ (u((2k − r )gr + rgt ) + g((−2k + r )u r − r u t )),
T θ = sin θ (−ugθ + gu θ ),
T φ = −ugφ + gu φ .
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(iii) The case m = t.
(a)
X 13 = ∂φ
1
T t = r 2 sin θ (u φ u r − uu r φ ),
2
1
T r = − r sin θ (u φ ((r − 2t)u r − r u t ) + u(−(r − 2t)u r φ + r u tφ )),
2
1
T θ = − sin θ (u φ u θ − uu θφ ),
2
Tφ =
1
(−u 2φ − u(cos θ u θ + sin θ (u θθ + 2(r − t)u r + r ((r − 2t)u rr
2
− 2(u t + r u tr ))))).
(b)
X 23 = u∂u + t∂t + r ∂r
Tt =
1 2
(r sin θ u r (r u r + tu t ) + u(−tu φφ − t cos θ u θ + sin θ (−tu θθ
2
− (r 2 + 2r t − 2t 2 )u r + (2r t 2 − r 3 − r 2 t)u rr + t (2u t + r u tr ))))),
1
T r = r (sin θ (r (2t − r )u r2 + (r 2 − r t + 2t 2 )u r u t + r tu 2t )
2
− u(u φφ + cos θ u θ + sin θ (u θθ + r (u r − u t )
− r 2 u tr − r tu tr + 2t 2 u tr + r tu tt ))),
1
T θ = − sin θ (u θ (r u r + tu t ) − u(r u r θ + tu tθ )),
2
Tφ =
(c)
1
(−u φ (r u r + tu t ) + u(r u r φ + tu tφ )).
2
X 33 = g(t, r, θ, φ)∂u where the function g(t, r, θ, φ) satisfies the equation
gφφ + cos θgθ + sin θ (gθθ + 2(r − t)gr + r ((r − 2t)grr − 2(gt + rgtr )))
T t = r 2 sin θ (ugr − gu r ),
T r = r sin θ (u(−(r − 2t)gr + rgt ) + g((r − 2t)u r − r u t )),
T θ = sin θ (−ugθ + gu θ ),
T φ = −ugφ + gu φ .
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(iv) The case m = et .
(a)
X 14 = ∂φ
1
T t = r 2 sin θ (u φ u r − uu r φ ),
2
1
T r = − r sin θ (u φ ((−2et + r )u r − r u t ) + u((2et − r )u r φ + r u tφ )),
2
1
T θ = − sin θ (u φ u θ − uu θφ ),
2
1
φ
T = (−u 2φ − u(cos θ u θ + sin θ (u θθ − 2(et − r )u r
2
+ r ((−2et + r )u rr − 2(u t + r u tr ))))).
(b)
X 24 = g(t, r, θ, φ)∂u where the function g(t, r, θ, φ) satisfies the equation
gφφ + cos θgθ + sin θ (gθθ − 2(et − r )gr + r ((−2et + r )grr − 2(gt + rgtr )))
T t = r 2 sin θ (ugr − gu r ),
T r = r sin θ (u((2et − r )gr + rgt ) − g((2et − r )u r + r u t )),
T θ = sin θ (−ugθ + gu θ ),
T φ = −ugφ + gu φ .
(v) The case m = m(t) where m(t) is an arbitrary function of t.
(a)
X 15 = ∂φ
1
T t = r 2 sin θ (u φ u r − uu r φ ),
2
1
T r = − r sin θ (u φ ((r − 2m(t))u r − r u t ) + u((2m(t) − r )u r φ + r u tφ )),
2
1
T θ = − sin θ (u φ u θ − uu θφ ),
2
1
T φ = (−u 2φ − u(cos θ u θ + sin θ (u θθ + 2(r − m(t))u r
2
(b)
+ r ((r − 2m(t))u rr − 2(u t + r u tr ))))).
X 25 = g(t, r, θ, φ)∂u where the function g(t, r, θ, φ) satisfies the equation
gφφ + cos θgθ + sin θ (gθθ + 2(r − m(t))gr + r ((r − 2m(t))grr − 2(gt + rgtr )))
T t = r 2 sin θ (ugr − gu r ),
T r = r sin θ (u(−(r − 2m(t))gr + rgt ) + g((r − 2m(t))u r − r u t )),
T θ = sin θ (−ugθ + gu θ ),
T φ = −ugφ + gu φ .
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4. Higher-order symmetries and conserved densities
In the standard (1+3) wave equation there exist higher-order variational symmetries
which lead to nontrivial conserved flows. Thus, even though there is a radical reduction in the number of variational point symmetries and conservation laws, one could
analyse the curved wave equation on a knowledge of the higher-order symmetries and
conservation laws. In this section we list some of these variational symmetries X =
η(x, u, u (1) , u (2) , u (3) )∂u and nontrivial conserved flows (T t , T r , T θ , T φ ) wherein the
conserved density is given by T t (see [9] for a discussion on recursion operators and
generalized symmetries).
(i) The case m = 0.
X11 = (2u tt + tu ttt + r u r tt )∂u ,
Tt =
1
(2 sin θ u θθ u t + 2r sin θ u r u t − r u r φφ u t − r cos θ u r θ u t
6
− r sin θ u r θθ u t − 2r 2 sin θ u rr u t − r 3 sin θ u rrr u t
− 2r sin θ u 2t + 2uu tφφ − r u r u tφφ − 3tu t u tφφ
+ 2 cos θ uu tθ − r cos θ u r u tθ − 3t cos θ u t u tθ
+ 2 sin θ uu tθθ − r sin θ u r u tθθ − 3t sin θ u t u tθθ
+ 8r sin θ uu tr + 2r sin θ u θθ u tr + 2r 2 sin θ u r u tr
+ 2r 3 sin θ u rr u tr − 2r 2 sin θ u t u tr − 6r t sin θ u t u tr
− 4r 3 sin θ u 2tr + 2r uu tr φφ + 2r cos θ uu tr θ
+ 2r sin θ uu tr θθ + 10r 2 sin θ uu trr − r 3 sin θ u r u trr
+ 2r 3 sin θ u t u trr − 3r 2 t sin θ u t u trr + 2r 3 sin θ uu trrr
− 8r sin θ uu tt + 3t sin θ u θθ u tt − 4r 2 sin θ u r u tt
+ 6r t sin θ u r u tt + 3r 2 t sin θ u rr u tt − 6r 2 t sin θ u tr u tt
+ u φφ (2u t + 2r u tr + 3tu tt ) + cos θ u θ (2u t + 2r u tr + 3tu tt )
+ 3tuu ttφφ + 3t cos θ uu ttθ + 3t sin θ uu ttθθ − 7r 2 sin θ uu ttr
+ 6r t sin θ uu ttr − r 3 sin θ u r u ttr + 6r 2 t sin θ u t u ttr
− r 3 sin θ uu ttrr + 3r 2 t sin θ uu ttrr − 6r t sin θ uu ttt
− 3r 2 t sin θ u r u ttt − 3r 2 t sin θ uu tttr ),
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1
T r = r (u φφ u tt + cos θ u θ u tt + sin θ u θθ u tt + 8r sin θ u r u tt
6
+ r 2 sin θ u rr u tt − 2r 2 sin θ u tr u tt + uu ttφφ
+ cos θ uu ttθ + sin θ uu ttθθ − 7r sin θ uu ttr
+ 3r 2 sin θ u r u ttr − 2r 2 sin θ uu ttrr + 7r sin θ uu ttt
+ 3r t sin θ u r u ttt − u t (u tφφ + cos θ u tθ
+ sin θ (u tθθ + r (2u tr + r u trr + 6u tt + r u ttr + 3tu ttt )))
+ r 2 sin θ uu tttr − 3r t sin θ uu tttr + 3r t sin θ uu tttt ),
Tθ =
1
sin θ (u θ (2u tt + r u ttr + tu ttt ) − u(2u ttθ + r u ttr θ + tu tttθ )),
2
Tφ =
1
(u φ (2u tt + r u ttr + tu ttt ) − u(2u ttφ + r u ttr φ + tu tttφ )).
2
(ii) The case m = k where k is an arbitrary constant.
X12 = u φtt ∂u ,
Tt =
1
(−u φφφ u t − cos θ u θφ u t − sin θ u θθφ u t + 2k sin θ u r φ u t − 2r sin θ u r φ
6
+ 2kr sin θ u rr φ u t − r 2 sin θ u rr φ u t + 2 cos θ u θ u tφ + 2r sin θ u φ u tt
+ 2 sin θ u θθ u tφ − 4k sin θ u r u tφ + 4r sin θ u r u tφ − 4kr sin θ u rr u tφ
+ 2r 2 sin θ u rr u tφ −2r sin θ u t u tφ − u φ u tφφ + 2uu tφφφ −2r sin θ u φ u tr
+ 2 cos θ uu tθφ −sin θ u φ u tθθ +2 sin θ uu tθθφ +2k sin θ u φ u tr −cos θ u φ
− 4r 2 sin θ u tφ u tr − 4k sin θ uu tr φ + 4r sin θ uu tr φ + 2r 2 sin θ u t u tr φ
+ 2kr sin θ u φ u trr −r 2 sin θ u φ u trr −4kr sin θ uu trr φ +2r 2 sin θ uu trr φ
− 4r sin θ uu ttφ − 3r 2 sin θ u r u ttφ + 2r 2 sin θ u φ u ttr − r 2 sin θ uu ttr φ ),
1
T r = r sin θ ((−2k + r )u r u ttφ − r u t u ttφ + u((2k − r )u ttr φ + r u tttφ )),
2
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Tθ =
1
sin θ (u θ u ttφ − uu ttθφ ),
2
Tφ =
1
(u φφ u tt + cos θ u θ u tt + sin θ u θθ u tt − 2k sin θ u r u tt + 2r sin θ u r u tt
6
− 2kr sin θ u rr u tt + r 2 sin θ u rr u tt − 2r 2 sin θ u tr u tt + 3u φ u ttφ
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+ cos θ uu ttθ + sin θ uu ttθθ − 2k sin θ uu ttr + 2r sin θ uu ttr − u t (u tφφ
+ cos θ u tθ + sin θ (u tθθ − 2(k − r )u tr + r ((−2k + r )u trr − 2r u ttr )))
− 2kr sin θ uu ttrr + r 2 sin θ uu ttrr − 2r sin θ uu ttt − 2r 2 sin θ uu tttr ).
(iii) The case m = t.
X13 = (tu tφφ + r u r φφ + u φφ )∂u ,
Tt =
1 2
(tu + tuu φφφφ + t cos θ uu θφφ + t sin θ uu θθφφ + 6r 2 sin θ uu r φφ
6 φφ
+2r t sin θ uu r φφ −2t 2 sin θ uu r φφ −3r 3 sin θ u r u r φφ +3r 3 sin θ uu rr φφ
+r 2 t sin θ uu rr φφ −2r t 2 sin θ uu rr φφ −2r t sin θ uu tφφ −3r 2 t sin θ u tφφ
+u φφ (t cos θ u θ +sin θ (tu θθ +(−3r 2 +2r t −2t 2 )u r + r t ((r −2t)u rr
−2(u t +r u tr ))))+tu φ (−u φφφ −cos θ u θφ −sin θ (u θθφ +2(r −t)u r φ
+r ((r − 2t)u rr φ − 2(u tφ +r u tr φ ))))+r 2 t sin θ uu tr φφ ),
1
T r = r (u 2φφ + uu φφφφ + cos θ uu θφφ + sin θ uu θθφφ − 4r sin θ uu r φφ
6
+10t sin θ uu r φφ +3r 2 sin θ u r u r φφ −6r t sin θ u r u r φφ − 2r 2 sin θ u r φφ
+4r t sin θ uu rr φφ −3r 2 sin θ u r φφ u t +4r sin θ uu tφφ +3r t sin θ u r u tφφ
−6t 2 sin θ u r u tφφ − 3r t sin θ u t u tφφ + u φφ (cos θ u θ + sin θ (u θθ + (5r
−8t)u r + r ((r − 2t)u rr − 5u t − 2r u tr ))) − u φ (u φφφ + cos θ u θφ
+ sin θ (u θθφ + 2(r − t)u r φ + r ((r − 2t)u rr φ − 2(u tφ + r u tr φ ))))
+r 2 sin θ uu tr φφ −3r t sin θ uu tr φφ +6t 2 sin θ uu tr φφ +3r t sin θ uu ttφφ ),
Tθ =
1
sin θ (u φφ u θ + u θ (r u r φφ + tu tφφ ) − u(u θφφ + r u r θφφ + tu tθφφ )),
2
Tφ =
1
(−r u φφφ u r − r cos θ u θφ u r − r sin θ u θθφ u r + 2r u φφ u r φ − tu φφφ u t
6
+2r sin θ u θθ u r φ + 2r 2 sin θ u r u r φ − 2r t sin θ u r u r φ + 2r 3 sin θ u r φ u rr
−4r 2 t sin θ u r φ u rr − r 3 sin θ u r u rr φ + 2r 2 t sin θ u r u rr φ + 2tu φφ u tφ
−t cos θ u θφ u t − t sin θ u θθφ u t − 4r 2 sin θ u r φ u t − 2r t sin θ u r φ u t
+2t 2 sin θ u r φ u t − r 2 t sin θ u rr φ u t +2r t 2 sin θ u rr φ u t +2r cos θ u θ u r φ
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+ 2t cos θ u θ u tφ + 2t sin θ u θθ u tφ + 2r 2 sin θ u r u tφ + 4r t sin θ u r u tφ
− 4t 2 sin θ u r u tφ + 2r 2 t sin θ u rr u tφ − 4r t 2 sin θ u rr u tφ
− 2r t sin θ u t u tφ−4r 3 sin θ u r φ u tr − 4r 2 t sin θ u tφ u tr + 2r 3 sin θ u tr φ
+ 2r 2 t sin θ u t u tr φ + u φ (4u φφ + cos θ u θ + sin θ u θθ + 2r u r φφ
− r cos θ u r θ − r sin θ u r θθ − 3r 2 sin θ u rr + 4r t sin θ u rr
− r 3 sin θ u rrr + 2r 2 t sin θ u rrr + 2tu tφφ − t cos θ u tθ − t sin θ u tθθ
+ 4r 2 sin θ u tr − 2r t sin θ u tr + 2t 2 sin θ u tr + 2r 3 sin θ u trr
+ (2r t 2 − r 2 t) sin θ u trr + 2r t sin θ u tt + 2r 2 t sin θ u ttr ) − u(2u φφφ
− cos θ u θφ − sin θ u θθφ − 6r sin θ u r φ + 6t sin θ u r φ + r u r φφφ
− 2r cos θ u r θφ − 2r sin θ u r θθφ − 9r 2 sin θ u rr φ + 14r t sin θ u rr φ
− 2r 3 sin θ u rrr φ + 4r 2 t sin θ u rrr φ + 6r sin θ u tφ + tu tφφφ
− 2t cos θ u tθφ − 2t sin θ u tθθφ + 14r 2 sin θ u tr φ + (4t 2 − 4r t)
× sin θ u tr φ + 4r 3 sin θ u trr φ − 2r 2 t sin θ u trr φ + 4r t 2 sin θ u trr φ
+ 4r t sin θ u ttφ + 4r 2 t sin θ u ttr φ )).
(iv) The case m = et .
X14 = u φφφ ∂u ,
1
T t = − r 2 sin θ (u φφφ u r − uu r φφφ ),
2
1
T r = r sin θ (u φφφ ((−2et + r )u r − r u t ) + u((2et − r )u r φφφ + r u tφφφ )),
2
1
T θ = sin θ (u φφφ u θ − uu θφφφ ),
2
1
T φ = (u φφ 2 + u φφ (cos θ u θ + sin θ (u θθ − 2(et − r )u r + r ((−2et + r )u rr
2
− 2(u t + r u tr )))) − u φ (cos θ u θφ + sin θ (u θθφ − 2(et − r )u r φ
+ r ((−2et + r )u rr φ − 2(u tφ + r u tr φ )))) + u(cos θ u θφφ
+ sin θ (u θθφφ − 2(et − r )u r φφ + r ((−2et + r )u rr φφ
− 2(u tφφ + r u tr φφ ))))).
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Wave equation on Vaidya manifolds
(v) The case m = m(t) where m(t) is an arbitrary function of t.
X15 = u φφφ ∂u ,
1
T t = − r 2 sin θ (u φφφ u r − uu r φφφ ),
2
1
T r = r sin θ (u φφφ ((r − 2m(t))u r − r u t ) + (2m(t) − r )uu r φφφ + r u tφφφ )),
2
Tθ =
1
sin θ (u φφφ u θ − uu θφφφ ),
2
Tφ =
1
(u φφ 2 + u φφ (cos θ u θ + sin θ (u θθ + 2(r − m(t))u r
2
+ r ((r − 2m(t))u rr − 2(u t + r u tr ))))
− u φ (cos θ u θφ + sin θ (u θθφ + 2(r − m(t))u r φ
+ r ((r − 2m(t))u rr φ − 2(u tφ + r u tr φ ))))
+ u(cos θ u θφφ + sin θ (u θθφφ + 2(r − m(t))u r φφ
+ r ((r − 2m(t))u rr φφ − 2(u tφφ + r u tr φφ ))))).
5. Discussion and conclusion
We have considered the classical wave equation in some Lorentzian spacetime backgrounds
with a point in mind that the wave equation there may naturally inherit nonlinearities from
the geometry. In this connection, we have considered the Vaidya metric for which a special
case is the Papapetrou metric. We have given some symmetry reductions to show how
the wave equation there can be either solved or reduced to ordinary differential equations
using the method of invariants. Also, some conservation laws were constructed. In his
book, Ibragimov [10] suggests that in three-dimensional flat space the linear wave equation
admits a 16-dimensional Lie algebra of point symmetries excluding the infinite symmetry.
In this study we show that the wave equations admit fewer symmetries when it is solved on
a curved manifold. A special case to note is m = 0 when the Vaidya manifold is supposedly
flat. Manifolds that are flat need not to lead to the wave equation admitting the maximal
16-dimensional Lie algebra of point symmetries. Finally, some higher-order symmetries
and associated conservation laws were presented. Solving or analysing the nonlinear wave
equation in curved spacetime background using the invariance approach may provide some
insight in geometry or relativity for different manifolds.
Acknowledgement
AHK thanks the NRF for support under programme FA2007041200006.
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