Lie point symmetry reductions of Bondi's radiating metric

Lie point symmetry reductions of Bondi’s radiating

metric

S. Dimas, D. Tsoubelis and P. XenitidisUniversity of Patras, Department of MathematicsRio 26 500 - Greece

E-mail: [email protected], [email protected], [email protected]

Abstract. The Lie point symmetries of the Einstein vacuum equations corresponding to theBondi form of the line element are presented. Using these symmetries, we study reductionsof the field equations, which might lead to new asymptotically flat solutions, representinggravitational waves emitted by an isolated source.

1. IntroductionMost of the exact solutions of the Einstein field equations constructed so far depend on twoindependent variables, only. This is the case, in particular, with the whole class of stationaryand axially symmetric metrics, which cover the physically most realistic solutions. However,in order to arrive at a general relativistic description of dynamic gravitational phenomena, wemust free ourselves from ... the two-dimensional chains.

The evolution of a compact object, such as a star, is the prime example of the above kindof phenomena. It is a process expected to be accompanied by the emission of gravitationalradiation and, as a result, it is of special interest in connection of the intense current effort, onan international scale, to detect gravitational waves.

The foundations for constructing models of gravitational waves emitted by a compact sourcewere laid down by the pioneering work of Sir Herman Bondi and his collaborators in the 1960s[1]: In a nutshell, the results of their investigations can be described by the following set ofrequirements:

(i) In the axially symmetric case, the space-time region outside the radiating object isdescribed by the line element

ds2 =(

e2βV

r− r2e2γU2

)du2 + 2e2βdudr + 2r2e2γUdudθ − r2

(e2γdθ2 + sin2θe−2γdφ2

).

This is referred to as Bondi’s radiating metric, and β, γ, U and V represent functionsdepending only on u, r, and θ.

12th Conference on Recent Developments in Gravity (NEB XII) IOP PublishingJournal of Physics: Conference Series 68 (2007) 012018 doi:10.1088/1742-6596/68/1/012018

c© 2007 IOP Publishing Ltd 1

(ii) The latter must comply to the boundary conditions

β → 0, γ → 0, r U → 0,V

r→ 1, (r →∞)

which insure that the space-time is asymptotically flat. In other words, as r → ∞, theline element must approach

ds2 = du2 + 2dudr − r2(dθ2 + sin2θdφ2

),

which is but the Minkowski line element

ds2 = dt2 − dx2 − dy2 − dz2,

in the retarded, spherical coordinates (u, r, θ, φ) defined by

t = u + r, z = r cos θ, x + y = r sin θeφ.

(iii) For u0 ≤ u ≤ u1, r0 ≤ r ≤ ∞, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π, one can expand all metriccomponents in powers of r−1 with at most a finite pole at r = ∞, and the resulting seriescan be added, multiplied, differentiated, etc., freely.

(iv) As sin θ → 0, β, V, U/ sin θ and γ/ sin2 θ should be regular functions of cos θ. Thisregularity at the axis condition is imposed by the fact that the axis of rotational symmetryis free of matter.

Unfortunately, the vacuum field equations corresponding to Bondi’s radiating metric are socomplicated that no exact solution of really dymamic character has been obtained yet. Hopingthat symmetry analysis could help in breaking out of this impasse, we have started a systematicinvestigation of the symmetries admitted by the above system. In the following sections,we present the first results of our effort: The Lie point symmetries of the Einstein vacuumequations, the corresponding reductions, and some known solutions that can be obtained fromthe reduced system.

2. Einstein’s vacuum field equationsIn the case of the Bondi metric, the vacuum field equations split into the following twosubsystems:

(a) The four main equations

2β,r − rγ2,r = 0

r2U,r(4− 2rβ,r + 2rγ,r) + r3U,rr + e2(β−γ){2β,θ + r[2(cot θ − γ,θ)γ,r − β,rθ + γ,rθ]} = 0

4e2(β−γ)[− 1 + β2,θ + β,θ(cot θ − 2γ,θ)− 3 cot θγ,θ + 2γ2

,θ + β,θθ − γ,θθ] +

e−2(β−γ)r4U2,r + 4V,r − 2r[4 cot θU + 4U,θ + r(cot θU,r + U,rθ] = 0

e2(β−γ)[− 1 + 2β,θ(cot θ − γ,θ)− 3 cot θγ,θ + 2γ2,θ − γ,θθ] + V,r − V γ,r +

rU,θ(−1 + rγ,r) + rU [2γ,θ + cot θ(−3 + rγ,r) + 2rγ,rθ]− r [r(cot θ − γ,θ)U,r

+V,rγ,r + V γ,rr − 2(γ,u + rγ,ur)] = 0


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(b) The two supplementary equations

UU2,rr

6 − e2(β−γ) csc θ{r cos θU [rU,r + 2U(rγ,r + 1)] + 4U2(γ,θ + rγ,rθ)r +

sin θ{[2U,r(U,θ − β,u + γ,u) + U,ur]r2 + V [2U,r(rβ,r − rγ,r − 2)− rU,rr] +U{r[−2r(β,θ − 2γ,θ)U,r − 2V,rγ,r + 2U,θ(rγ,r + 3) + 3rU,rθ − 2V γ,rr + 4γ,u + 4rγ,ur]−2(V,r + V γ,r)}}}r2 + e4(β−γ){V,θ(2rβ,r − 2rγ,r − 1) +r{V,rθ + 2r{U [β,θ(cot θ + 2β,θ − 2γ,θ) + β,θθ] + 2(cot θ − γ,θ)γ,u − β,uθ + γ,uθ}}} = 0

−e2(γ−β)V U,r2r4 + 2e2(γ−β)U3[(rγ,r + 1) cot θ + 2γ,θ + 2rγ,rθ]r4 +

U2{ − e4(γ−β)U,r2r4 − 2[2β,θ(β,θ − γ,θ) + cot θγ,θ + γ,θθ] +

2e2(γ−β){−V,r − V γ,r + r[r(cot θ − 2β,θ + 3γ,θ)U,r −V,rγ,r + U,θ(rγ,r + 3)− V γ,rr + 2γ,u + 2r(U,rθ + γ,ur)]}}r3 +

e2(γ−β)U{2r2{[2U,r(U,θ − β,u + γ,u) + U,ur]r2 + V [2U,r(rβ,r − rγ,r − 2)− rU,rr]} −e2(β−γ){−4V,θ + V [4β,θ + cot θ(2rβ,r − 1) + 4rβ,rθ] +r{2rU,θ(cot θ − 2β,θ + 2γ,θ) + cot θV,r +4V,θ(β,r − γ,r) + 2V,rθ + 2r{U,θθ + 2[cot θ(γ,u − β,u)− β,uθ + γ,uθ]}}}}r +

e2(β−γ)[(V,θ + 2V β,θ)(cot θ + 2β,θ − 2γ,θ) + V,θθ + 2V β,θθ]−− csc θ{2 cos θU,ur3 + sin θ{ − 2(β,r + rβ,rr)V 2 +r[2rβ,θU,r − 2V,rβ,r + U,θ(2rβ,r − 1)− V,rr + 4β,u + 4rβ,ur]V +

r{r[2rU,θ2 + (V,r − 4rβ,u + 4rγ,u)U,θ + 4rγ,u

2 − V,θU,r + 2rU,uθ]− 2V,u}}} = 0

The first objective of the symmetry analysis is to determine the point transformations

(xa, ga b) → (xa, ga b)

that leave the above set of equations invariant. A short outline of the way this group oftransformations can be determined and exploited in constructing exact solutions is presentedin the following section.

3. Symmetries of systems of PDEsLet (x, u) denote an arbitrary point of the space Rn ×Rm. In other words, let

(x, u) := (x1, . . . , xi, . . . , xn, u1, . . . , uµ, . . . , um) .

Then the vector field

X = ξi(x, u)∂xi + ηµ(x, u)∂uµ , i = 1, 2, . . . , n, µ = 1, 2, . . . ,m ,

where∂xi :=

∂

∂xi, ∂uµ :=

∂

∂uµ,

determines an infinitesimal transformation of the following form:

(x)i = xi + ε ξi(x, u) + · · · , (u)µ = uµ + ε ηµ(x, u) + · · · .


3

When the uµ are considered to be functions of the coordinates xi, the above transformationin the space Rn × Rm (of the independent and dependent variables) can be extended to thepartial derivatives of the uµ with respect to the xi. Specifically,

uµ,i = uµ

,i + ε ηµi (x, u, ∂u) + · · ·

whereηµ

i (x, u, ∂u) = Di ηµ − uµ

,jDi ξj .

In the last expression Di denotes the total derivative operator. Thus, from the fact thatηµ = ηµ(x, u) it follows that

Di ηµ ≡ D ηµ

D xi= ∂xiηµ + (∂uν ηµ)uν

,j .

This extension process can be generalized easily to the derivatives of all higher orders. Forexample,

uµ,i j = uµ

,i j + ε ηµi j(x, u, ∂u, ∂2u) + · ,

whereηµ

i j(x, u, ∂u, ∂2u) = Dj ηµi − uµ

,i kDj ξk .

Now ηµi = ηµ

i (x, u, ∂u), i.e. the coefficients ηµi depend, in general, on the first derivatives of the

functions uν . Therefore,

Dj ηµi ≡

D ηµi

D xj=

∂ ηµi

∂ xj+

∂ ηµi

∂ uν

∂ uν

∂ xj+

∂ ηµi

∂ uν,k

∂ uν,k

∂ xj.

Equivalently,Dj ηµ

i = ∂xjηµi + (∂uν ηµ

i )uν,j + (∂uν

,kηµ

i )uν,k j .

A system of S partial differential equations (PDEs) for the functions uµ can be written as

HA(x, u, ∂u, ∂2u, · · · , ∂pu) = 0 , A = 1, 2, ... , S .

It is said to be invariant under the one-parameter group of transformations

(x, u) → (x, u) = G(x, u, ε) , G(x, u, 0) = (x, u)

ifHA(x, u, ∂u, ∂2u, · · · , ∂pu) = 0 .

Then, the above transformation is called a Lie point symmetry of the given system of PDEs.At the infinitesimal level, the condition

X HA = 0 (modHA = 0)

is necessary and sufficient for the transformation (x, u) → (x, u) to be a symmetry of the systemHA = 0. Here, X denotes the extension of the vector field determined by (x, u) → (x, u) to theorder p of the system (the order of the highest derivative of uµ that appears in the system), i.e.

X = ξi(x, u)∂xi + ηµ(x, u)∂uµ + ηµj (x, u, ∂u)∂uµ

,j+

ηµi j(x, u, ∂u, ∂2u)∂uµ

,i j+ · · ·+ ηµ

j1j2··· jp(x, u, ∂u, ∂2u, · · · , ∂pu)∂u,j1j2···jp


4

In the case of the Einstein vacuum equations,

HA = 0 ⇔ Ri j = 0 ,

where Ri j denotes the Ricci tensor. Now the unknown functions uµ are the components gi j ofthe metric tensor, so that the generator of a Lie point symmetry of Einstein’s equations shouldbe written as

X = ξi(x, g)∂xi + ηi j(x, g)∂gi j .

It turns out [3] that the above symmetry generator is restricted to the following form, where ais an arbitrary real number:

X = ξi(x)∂xi + (gi kξk,j + gj kξ

k,i − 2a gi j)∂gi j

Once (some of) the symmetries of a given system have been determined, one can exploitthem in one of the following ways:

(i) To generate new solutions from known, simpler ones.(ii) To find solutions that remain invariant under the action of the symmetry group.

The latter are called similarity solutions and the method of their construction is referred to assimilarity reduction of the system.

If one represents the solution of the system HA = 0 as the n-dimensional submanifold

uµ = Φµ(x) , µ = 1, 2, . . . ,m .

of the Rn ×Rm space, then the condition of its invariance under the action of the vector field

X = ξi(x, u)∂xi + ηµ(x, u)∂uµ

readsηµ(x,Φν(x)) = ξi(x,Φν(x))∂xiΦµ(x) .

This set of first order qausilinear equations for the functions Φµ can, in principle, be solved.The substitution of the result into the original system of PDEs leads to the latter’s similarityor group invariant solutions.

In the case of the Einstein vacuum equations the above invariant surface condition becomes

Lξgi j := gi j, kξk + gi kξ

k,j + gj kξ

k,i = 2a gi j

where Lξgi j denotes the Lie derivative of the metric tensor. This relation shows that thesimilarity solutions of Einstein’s equations are those which admit a Killing vector (a = 0) ora homothetic vector (a 6= 0). “This connection with the symmetries explains the outstandingrole Killing vectors and homothetic vectors play compared with other vector fields.” [2], p.132.

4. Symmetries Einstein’s vacuum equations for the Bondi metricIn order to find the symmetries of a given system of PDEs, one has to set up and solve thesystem’s determining equations. The latter is a (usually, very large) system of linear PDEs forthe n + m components ξi(x, u), ηµ(x, u) of the generator

X = ξi(x, u) ∂xi + ηµ(x, u) ∂uµ .


5

In the case of Einstein’s vacuum equations, the determining equations concern the four functionsξi(x).

The system of determining equations is specified in an algorithmic way. However, this canbe done by hand only in very simple cases, involving scalar PDEs of low order. In all other casesone has to turn to algebraic computing packages which have been specifically developed forthat purpose. One such package, operating in the context of Mathematica, has been developedby our group recently. It is called SYM [4] and is freely available upon request. So far, it hasbeen proven very effective in both obtaining and solving systems of the determining equationsof very complicated systems of linear and nonlinear PDEs.

This includes the sixty (60) determining equations corresponding to the Einstein vacuumequations for the Bondi metric:

ξ2,u = 0, ξ3

,r = 0, ξ1,r = 0, η4,r = 0, η3,r = 0, η2,r = 0, η1,r = 0, ξ2

,θ = 0, ξ1,θ =, η3,θ

= 0

η2,θ= 0, η1,UU = 0, ξ3

,U = 0, ξ2,U = 0, ξ1

,U = 0, η4,U = 0, η3,U = 0, η2,U = 0, ξ3,V = 0

ξ2,V = 0, ξ1

,V = 0, η4,V = 0, η3,V = 0, η1,V = 0, ξ3,β = 0, ξ2

,β = 0, ξ1,β = 0, η4,β

= 0

η3,β= 0, η2,β

= 0, η1,β= 0, ξ3

,γ = 0, ξ2,γ = 0, ξ1

,γ = 0, η4,γ = 0, η3,γ = 0, η2,γ = 0, η1,γ = 0

rξ2,r − ξ2 = 0, V η2,V − η2 = 0, rη2 − 2V ξ2 + rV ξ1

,u = 0

− csc θ sec θξ3 − tan θη4,θ + ξ3,θ = 0, 2 cot θη4,u + ξ3

,u + η4,uθ = 0

4 csc(2θ)ξ3 + (cos(2θ) + 5) csc(2θ)η4,θ + η4,θθ = 0, V (2rη3,θ + ξ2,θ)− rη2,θ = 0

rη2 − V (2rη3 − 2rη4 + ξ2 − 2rξ3,θ) = 0, V (− 2rη3 + 2rη4 + 3ξ2 + 2rη1,U )− rη2 = 0

− cot θη1 + U cot θη1,U − η1,θ + Uη1,θU = 0

2 tan θη3,u − 2 tan θη4,u + ξ3,u − tan θη1,uU = 0

rV cot θη1 − rU cot θη2 + V (− rUξ3csc2θ + 2U cot θξ2 + rη1,θ) = 0

−2rV η1 + rUη2 + V (2rUη3 − 2rUη4 − 3Uξ2 + 2rξ3,u) = 0

r cot θη2 + V (2rξ3csc2θ − 2r cot θη3 + 2r cot θη4 − cot θξ2 + 2rη4,θ) = 0

−2rV cot θη1 + rU cot θη2 +V (2rU cot θη3 − 2rU cot θη4 − 3U cot θξ2 + 2rη4,u) = 0

cot θη2 + V (3ξ3csc2θ − 4 cot θη3 + 4 cot θη4 + cot θη1,U + 2η4,θ − η1,θU ) = 0

2rV cot θη1 − rU cot θη2 +V (− 2rU cot θη3 + 2rU cot θη4 + 3U cot θξ2 − r cot θξ3

,u + rξ3,uθ) = 0

rη1 − U(2rη3 + 2rη4 + ξ2 + r csc θ sec θξ3 + r tan θη4,θ) + 2r cot θη4,u + rη4,uθ = 0

2e2γrU cot θη2 + V (− 4e2γrU cot θη3 + 4e2γrU cot θη4 − 2e2γU cot θξ2 +3e2β cot θη4,θ + 2e2βξ3

,θ + 4e2γrU cot θξ3,θ + e2βη4,θθ) = 0

2rV cot θη1 − rU cot θη2 − 2rUV cot θη3 + 2rUV cot θη4 + 3UV cot θξ2 −3rUη2,θ − 2rUV η3,θ + 7UV ξ2

,θ − 2rη2,u + 4V ξ2,u + 2rV η1,uU = 0


6

−3e2γrUξ3csc2θ + 3e2γr cot θη1 − 6e2γrU cot θη3 +6e2γrU cot θη4 + 3e2γU cot θξ2 + e2γrη1,θ − 2e2γrUη4,θ +

3e2β cot θη4,θ + 2e2βξ3,θ + 6e2γrU cot θξ3

,θ + e2βη4,θθ − 2e2γrη4,u = 0

−2UV η1r2 + 2U2η2r

2 + U2 sin(2θ)η2,θr2 + 2U2V ξ3

,θr2 + U sin(2θ)η2,ur2 −

4UV cos θ sin θη3,ur2 − 4UV cot θη4,ur2 − 2UV η4,uθr2 − 4U2V ξ2r −

2U2V sin(2θ)ξ2,θr − 2UV sin(2θ)ξ2

,ur − V η2,u + V cos(2θ)η2,u + 4V 2sin2θη3,u = 0

Here, ξi and ηj are functions of (u, r, θ, β, γ, U, V ).The solution of the above system determines the Lie point symmetries of the Einstein

vacuum equations for the Bondi metric. In terms of the generators of the correspondingsymmetry algebra it reads

X1 = ∂β + ∂γ ,

X2 = ∂β + 2r∂r + 4V ∂V ,

X3 = ∂β − cot θ∂θ − cot2 θ∂γ + U csc2 θ∂U ,

X4 = F1(u)∂u −F ′1(u)[

12∂β + U∂U + V ∂V ],

X5 = csc θF2(u)∂θ + cot θ csc θF2(u)∂γ + csc θ[F ′2(u)− U cot θF2(u)]∂U ,

where Fi(u) are arbitrary functions of u.

5. Similarity reductionsThe combination of the above symmetries that respects the boundary conditions at infinity,

β → 0 , γ → 0 , r U → 0 ,V

r→ 1 ,

is given byX = ∂u + λ (r∂r + u∂u − U∂U + V ∂V )

where λ is an arbitrary real parameter.Using this symmetry, the field equations reduce to the following system

The main equations

ζF1,ζ2 − 2F2,ζ = 0

e2F1F4,ζζζ3 − 2e2F1(ζF2,ζ − 2)F4,ζζ

2 + 2F1,ζ(e2F1F4,ζζ2 +

2e2F2( cot θ −F1,θ))ζ + 2e2F2F1,θζζ − 2e2F2F2,θζζ + 4e2F2F2,θ = 0

e4F1 sin θF4,ζ2ζ4 − 2e2(F1+F2) cos θF4,ζζ

2 − 2e2(F1+F2) sin θF4,θζζ2 −

8e2(F1+F2) cos θF4ζ − 8e2(F1+F2) sin θF4,θζ + 8e4F2 sin θF1,θ2 + 4e4F2 sin θF2,θ

2 −e4F2 sin θ + 4e2(F1+F2) sin θF3,ζ − 12e4F2 cos θF1,θ + 4e4F2 cos θF2,θ −8e4F2 sin θF1,θF2,θ − 4e4F2 sin θF1,θθ + 4e4F2 sin θF2,θθ = 0

2e2F1λF1,ζζζ3 + e2F1 cot θF4,ζζ

2 − e2F1F4,ζF1,θζ2 + e2F1F3F1,ζζζ + e2F1F4,θζ −


7

e2F1F4(ζF1,ζ cot θ − 3 cot θ + 2F1,θ + 2ζF1,θζ)ζ + e2F2 − 2e2F2F1,θ2 − e2F1F3,ζ +

3e2F2 cot θF1,θ − 2e2F2 cot θF2,θ + 2e2F2F1,θF2,θ + e2F1F1,ζ(F3 + ζ(4ζλ + F3,ζ −ζF4,θ)) + e2F2F1,θθ = 0

The supplementary equations

8e2(F1+F2) cot θ(F4)2ζ3 −F4(e4F1F4,ζ

2ζ4 −4e2(F1+F2)F4,ζ( cot θ + F1,θ −F2,θ)ζ2 + 4e2(F1+F2)F3,ζ +

4e2F2(e2F2F2,θ2 + 2e2F2( cot θ −F1,θ)F2,θ − e2F1ζ(2F4,θ + ζF4,θζ)))ζ2 +

2e2F2(− e2F1λF4,ζζζ5 + 2e2F1F4,ζ(ζF2,ζλ− λ + F4,θ)ζ4 +

2e2F2λF1,θζζ3 − 2e2F2λF2,θζζ

3 − e2F2F1,ζ2F3,θζ

2 + 2e2F2F3F1,θζζ −2F1,ζ(e2F1λF4,ζζ

4 − e2F2(2λ cot θζ2 − 2λF1,θζ2 + F3,θ)−

e2F2F3(2 cot θ − 2F1,θ − ζF1,θζ))ζ − e2F2F3,θζζ + 4e2F2F3F2,θ + e2F2F3,θ) = 0

−4e2(F1+F2)(ζF1,ζ − 1)(F1,ζ + ζF1,ζζ)F32 − (− 3e4F1F4,ζ

2ζ4 − 8e2(F1+F2)λF1,ζζζ3 +

8e2(F1+F2)λF2,ζζ2 − 2e2(F1+F2) cot θF4,ζζ

2 + 4e2(F1+F2)F4,ζF1,θζ2 − 4e2(F1+F2)F4,ζF2,θζ

2 +

2e2(F1+F2)F1,ζ2(4ζλ + F3,ζ − ζF4,θ)ζ2 + 2e2(F1+F2)F4,θζζ

2 + 4e2(F1+F2)λζ +

2e2(F1+F2)F3,ζζζ + 6e2(F1+F2)F4,θζ + 4e2(F1+F2)F1,ζ

(2λF1,ζζ

ζ3 − 4λζ + F4,θζ −F3,ζ

)ζ −

2e2(F1+F2)F4(ζ2 cot θF1,ζ2 + 2ζ(2ζF1,θζ − cot θ)F1,ζ + cot θ − 4F1,θ + 4F2,θ −

4ζF1,θζ)ζ + 4e4F2F2,θ2 + 8e4F2 cot θF2,θ − 8e4F2F1,θF2,θ)F3 − 2e2F2 csc θ

(− 4e2F1λ2 sin θF1,ζ2ζ5 + 2e2F1λ cos θF4,ζζ

4 + 4e2F1λ sin θF1,ζF4,θζ4 −

4e2F1λ sin θF2,ζF4,θζ4 + 2e2F1λ sin θF4,θζζ

4 − 2e2F1 sin θF4,θ2ζ3 −

2e2F1F42(2 sin θF1,θ

2 − 2 cos θF1,θ − sin θ − 2 cos θF2,θ)ζ3 + 2e2F1λ sin θF4,θζ3

−2e2F1λ sin θF3,ζζ2 + e2F1 sin θF4,ζF3,θζ

2 − e2F1 sin θF3,ζF4,θζ2 −

e2F1F4(4λ cos θF2,ζζ3 + 4λF1,ζ( cos θ − 2 sin θF1,θ)ζ3 − 2λ cos θζ2 + 2 cos θF4,θζ

2 +

4 sin θF1,θF4,θζ2 − 4 sin θF2,θF4,θζ

2 + 2 sin θF4,θθζ2 + cos θF3,ζζ − 2 sin θF3,θ)ζ +

e2F2 cos θF3,θ − 2e2F2 sin θF1,θF3,θ + 2e2F2 sin θF2,θF3,θ + e2F2 sin θF3,θθ) = 0

Here, Fi = Fi(ζ, θ) and ζ = r1+λ u .

6. ConclusionsIn the process of analyzing further the above system, we were led to the following results.

(i) The Schwarzschild solution is contained in the solution set of the above reduced system,and can be obtained by demanding that the functions involved are independent of theangular variable θ.

(ii) The solution based on the Bondi form of the metric, which was given by Hobill [5], can beobtained from our reduced system above via the ansatz

F3(ζ, θ) = e2(F2−F1)f3(ζ, θ) + f(θ) , F4(ζ, θ) = e2(F2−F1)f4(ζ, θ)

together with the assumption that, in the resulting equations the coefficients of differentpowers of exp(F2 −F1) are identically zero.


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The above results strengthen our hope that an asymptotically flat and regular on the symmetryaxis solution can be obtained from the Lie point symmetry reduction presented above. Ifthis hope does not materialize, one would have to turn for help to other kind of symmetries(potential, conditional, generalized or Backlund, etc.) that are probably admitted by the Bondimetric.

AcknowledgmentsWe thank European Social Fund (ESF), Operational Program for Educational and VocationalTraining II (EPEAEK II) and particularly the program Irakleitos and, the University of Patrasand particularly the program Karatheodory 2001 for funding the above work.

References[1] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. A 269 (1962) 21[2] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of Einstein’s Field

Equations, 2nd ed., Cambridge University Press (2003)[3] N. H. Ibragimov, Transformation groups applied to mathematical physics, Reidel (1985)[4] S. Dimas and D. Tsoubelis, Proceedings of The 10th International Conference in MOdern GRoup ANalysis,

edited by N. H. Ibragimov et al, University of Cyprus (2004), pp. 64 - 70[5] D. W. Hobill, Gen. Relativity Gravitation 19 (1987) 121


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Lie point symmetry reductions of Bondi's radiating metric

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