Use of Approximate Use of Approximate Symmetry Methods for Symmetry Methods for

Differential Equations in Differential Equations in General RelativityGeneral Relativity

IbrarIbrar HussainHussainSchool of Electrical Engineering and Computer School of Electrical Engineering and Computer

Science, National University of Sciences and Science, National University of Sciences and Technology, Technology, IslamabadIslamabad

PlanPlanGeneral Relativity: Brief Introduction and the Problem of General Relativity: Brief Introduction and the Problem of EnergyEnergy

Approximate Lie Symmetries and Approximate Noether Approximate Lie Symmetries and Approximate Noether SymmetriesSymmetries

Approximate Symmetries and Energy of Approximate Symmetries and Energy of ReissnerReissner--Nordstrom (RN), Kerr, KerrNordstrom (RN), Kerr, Kerr--Newman (KN) and KNNewman (KN) and KN--AdSAdSSpacetimesSpacetimes

Approximate Symmetries and Energy in Gravitational Wave Approximate Symmetries and Energy in Gravitational Wave SpacetimesSpacetimes

Summary Summary

(1 2) ,ab ab abR Rg Tκ− =Einstein field equations (Einstein field equations (EFEsEFEs) relate geometry to matter, ) relate geometry to matter,

=metric tensor, = gravitational coupling, =matter=metric tensor, = gravitational coupling, =matter tensor, tensor, = Ricci tensor= Ricci tensor

where, Riemann curvature tensor given bywhere, Riemann curvature tensor given by

and and ChristoffelChristoffel symbols given bysymbols given by

κ

, , ,a a a a e a ebcd bd c bc d ec bd ed bcR = Γ −Γ +Γ Γ −Γ Γ

Brief Introduction to GRBrief Introduction to GR

, , ,(1 2) ( ).a adbc bd c cd b bc dg g g gΓ = + −

abTabga

bd badR R=

Some well known static solutions: Schwarzschild solution, RN Some well known static solutions: Schwarzschild solution, RN solution, Kerr solution.solution, Kerr solution.

NonNon--static vacuum solutions which represent Gravitational Waves static vacuum solutions which represent Gravitational Waves ((GWsGWs). ).

EinsteinEinstein--Rosen, cylindrically symmetric solution (1937).Rosen, cylindrically symmetric solution (1937).

BondiBondi--Robinson, plane symmetric solution (1957). Robinson, plane symmetric solution (1957).

These are fluctuations in spacetime. In Maxwell's theory, acceleThese are fluctuations in spacetime. In Maxwell's theory, accelerated rated charges, electromagnetic waves. In a similar way, Einstein's thecharges, electromagnetic waves. In a similar way, Einstein's theory, ory, accelerated masses accelerated masses GWsGWs..

Problem of energy in GR Problem of energy in GR NonNon--static spacetime no timestatic spacetime no time--like Killing vector (KV).like Killing vector (KV).

Energy conservation in a spacetime only using timeEnergy conservation in a spacetime only using time--like KV to define like KV to define the time direction. No timethe time direction. No time--like KV energy is not conservedlike KV energy is not conserved. .

The problem of defining the energy content of The problem of defining the energy content of GWsGWs is particularly is particularly severe.severe.

There was a debate whether these waves really exist.There was a debate whether these waves really exist.

Weber and Wheeler (Weber Weber and Wheeler (Weber et. alet. al 1957) for the cylindrical waves and 1957) for the cylindrical waves and Ehlers and Ehlers and KundtKundt (Ehlers (Ehlers et. alet. al 1962) for plane waves showed they 1962) for plane waves showed they impart energy and momentum to the test particles in their pathimpart energy and momentum to the test particles in their path..

Using the pseudoUsing the pseudo--Newtonian formalism a general formula for the Newtonian formalism a general formula for the momentum imparted to the test particles in arbitrary spacetimes momentum imparted to the test particles in arbitrary spacetimes was was obtained (obtained (QadirQadir et. alet. al 1992). 1992).

Energy and momentum conservation are described by the requiremenEnergy and momentum conservation are described by the requirement t that the divergence of the stressthat the divergence of the stress--energy tensor is zero.energy tensor is zero.

The represents the energy and momentum of matter and allThe represents the energy and momentum of matter and all nonnon--gravitational fields and no longer satisfies .gravitational fields and no longer satisfies .

A contribution from the gravitational field must be added to obtA contribution from the gravitational field must be added to obtain an ain an energyenergy--momentum expression with zero divergence.momentum expression with zero divergence.

Einstein, LandauEinstein, Landau--LifshitzLifshitz, , PapapetrouPapapetrou and Weinberg gave energyand Weinberg gave energy--momentum complexes. They can be expressed as a combination of momentum complexes. They can be expressed as a combination of and a pseudoand a pseudo--tensor, which is interpreted to represent the energy and tensor, which is interpreted to represent the energy and momentum of the gravitational field (momentum of the gravitational field (SzabadosSzabados 2004).2004).

T μν

, 0T μν μ =

T μν

They are coordinate dependent and hence nonThey are coordinate dependent and hence non--tensorialtensorial..

Due to the coordinate dependence many others, including Due to the coordinate dependence many others, including MMøøllerller, , BondiBondi, , KomarKomar, , AshtekarAshtekar--Hansen, Penrose, Isaacson, Christodoulou Hansen, Penrose, Isaacson, Christodoulou have proposed coordinate independent definitions (have proposed coordinate independent definitions (SharifSharif 2003, 2003, HussainHussain et. al et. al 2009).2009).

However, each of these, has its own drawbacks (However, each of these, has its own drawbacks (BergqvistBergqvist 1992). 1992).

Different Different approximate symmetryapproximate symmetry approaches (approaches (HussainHussain et. alet. al 2007). 2007).

Conservation of energy, asymptotically and examine whether it woConservation of energy, asymptotically and examine whether it would uld work for work for GWsGWs ((KomarKomar 1962). 1962).

Almost symmetric space and the corresponding vector field an almAlmost symmetric space and the corresponding vector field an almost ost KV (York 1974).KV (York 1974).

Was applied to the Was applied to the TaubTaub cosmological solution (cosmological solution (TaubTaub 1951) and to study 1951) and to study gravitational radiation. gravitational radiation.

Provides a choice of gauge making calculation simpler and was usProvides a choice of gauge making calculation simpler and was used for ed for this purpose (Bona this purpose (Bona et. al et. al 2005).2005).

The approach of slightly broken symmetry, promising but merely The approach of slightly broken symmetry, promising but merely providing simplicity of calculation is not physically convincingproviding simplicity of calculation is not physically convincing. .

Other approaches need to be tried to find one that significantlyOther approaches need to be tried to find one that significantlybetter than the others.better than the others.

Approximate Lie Symmetries and Approximate Lie Symmetries and Approximate Noether SymmetriesApproximate Noether Symmetries

A vector field A vector field ,,is called secondis called second--order approximate symmetry of the system of perturbed order approximate symmetry of the system of perturbed ODEsODEs

if the following condition holdsif the following condition holds

wherewhere. .

2 30 1 2 ( )Oε ε ε= + + +X X X X

2 30 1 2 ( ) 0,Oε ε ε= + + + =E E E E

2 2 30 1 2 0 1 2 0( )( ) ( ),Oε ε ε ε ε=+ + + + =EX X X E E E

( 0,1, 2, 0,1, 2, 3)ij j j i j i

s xξ η∂ ∂

= + = =∂ ∂

X

For a secondFor a second--order perturbed system of order perturbed system of DEsDEs, ,

Corresponding to a firstCorresponding to a first--order Lagrangianorder Lagrangian, ,

the functional is invariant under one parameter apthe functional is invariant under one parameter approximate groupproximate group

of transformation with symmetry generator of transformation with symmetry generator ,,

up to gauge , if up to gauge , if ,,

, ,

,,

2 30 1 2 ( )Oε ε ε= + + +E E E E

2 30 1 2( , , ) ( , , ) ( , , ) ( , , ) ( )i i i i i i i iL s x x L s x x L s x x L s x x Oε ε ε= + + +& & & &

V

Lds∫

2 20 1 2 ( )Oε ε ε= + + +X X X X

20 1 2A A A Aε ε= + + 0 0 0 0 0( )L D L DAξ+ =X

0 1 1 0 0 1 1 0 1( ) ( )L L D L D L DAξ ξ+ + + =X X

2 0 1 1 0 2 2 0 1 1 0 2 2( ) ( ) ( )L L L D L D L D L DAξ ξ ξ+ + + + + =X X X

where . where .

The secondThe second--order approximate symmetry is called nonorder approximate symmetry is called non--trivial if any trivial if any one of is nonone of is non--zero. In the case of trivial symmetries it is also zero. In the case of trivial symmetries it is also possible that lower order symmetries cancel out in the determinipossible that lower order symmetries cancel out in the determining ng equations.equations.

The approximation involves a small parameter whose powers, higheThe approximation involves a small parameter whose powers, higher r then some chosen value, is neglected. The scaling factors (discuthen some chosen value, is neglected. The scaling factors (discussed ssed later), are independent of the small parameter. This is reminisclater), are independent of the small parameter. This is reminiscent of ent of the d' the d' AlembertAlembert principle . This is an extension of the principle of principle . This is an extension of the principle of virtual work from virtual work from staticsstatics to dynamics.to dynamics.

iiD x

s x∂ ∂

= +∂ ∂

&

0 1, ,X X

Approximate Symmetries and Approximate Symmetries and Energy in the RN spacetimeEnergy in the RN spacetime

RN spacetime RN spacetime

wherewhere

For the approximate symmetries For the approximate symmetries

2 ( ) 2 ( ) 2 2 2 2 2( sin ),r rds e dt e dr r d dν ν θ θ φ−= − − +

2( )

2

21 , ( 1, 1).r m Qe G cr r

ν = − + = =

2 2 12 , , (0 ).4

m Q k kε ε= = < ≤

SecondSecond--order perturbed geodesic equationsorder perturbed geodesic equations

22 3

2 2 2 2 2 2 2 22

2 2 2 2 2 23

2

1 2( ) 0,

1( sin ) [ ( ) sin ]2

1 [(1 2 ) (1 2 ) 2 ( sin )] 0,22 sin cos 0,

2 2cot 0.

tr kt trr r

r r t r rr

k t k r rkr

rr

rr

ε ε

θ θφ ε θ θφ

ε θ θφ

θ θ θ θφ

φ φ θθφ

−+ + =

− + + − + +

− + + − + + =

+ − =

+ + =

&&&& &&

& & & &&&& & &&

& && &

&& & &&

&& & & &&

The exact case ( ), 35 symmetry generators. In the first oThe exact case ( ), 35 symmetry generators. In the first order and rder and secondsecond--order approximate cases we use the 10 KVs andorder approximate cases we use the 10 KVs andwith only. No nonwith only. No non--trivial approximate symmetry. trivial approximate symmetry.

The exact case, include generators of corresponding to The exact case, include generators of corresponding to

In the determining equations for the firstIn the determining equations for the first--order approximate symm. for order approximate symm. for the Schwarzschild spacetime the terms involving , canthe Schwarzschild spacetime the terms involving , cancel out. cel out.

For the secondFor the second--order approximate symm. the terms involving do order approximate symm. the terms involving do not automatically cancel out but collect a scaling factor of not automatically cancel out but collect a scaling factor of so as to cancel out.so as to cancel out.

Energy conservation comes from time translational invariance, Energy conservation comes from time translational invariance, is the is the coefficient of in the point transformations , is thcoefficient of in the point transformations , is the proper time, e proper time, the scaling factor corresponds to a rescaling of energy.the scaling factor corresponds to a rescaling of energy.

0ε =4(1,3) sso ⊕

/ , ( / )s s s∂ ∂ ∂ ∂2d

2d 0 1( ) .s c s cξ = +

0s cξ =

sξ2 2(1 2 )Q m−

ξ/ s∂ ∂ s

Approximate Noether Symmetries of Approximate Noether Symmetries of the Kerr spacetimethe Kerr spacetime

The Kerr spacetimeThe Kerr spacetime

WhereWhere

Kerr spacetime as a first perturbation of the Schwarzschild spacKerr spacetime as a first perturbation of the Schwarzschild spacetime etime take take LagrangianLagrangian

2 2 22 2 2 2 2 2

2 2 2

2 sin 2 sin(1 ) ,mr mrads dt dr d d dtdρ θ θρ θ φ φρ ρ ρ

Λ= − − − − +

Δ

2 2 2 2 2 2 2 2 2 2 2cos , ( ) sin , 2 .r a r a a r a mrρ θ θ= + Λ = + − Δ Δ = + −

a ε=

22 1 2 2 2 2 2

2 2

2 2 2 sin(1 ) (1 ) ( sin ) .mr mr mL t r r tr

θθ θφ ε φρ ρ

−= − − − − + +& & && &&

For , the above Lagrangian reduces to that of the SchwarFor , the above Lagrangian reduces to that of the Schwarzschild zschild spacetime. For the Schwarzschild spacetime the Noether symmetry spacetime. For the Schwarzschild spacetime the Noether symmetry algebra is 5 dimensional, given by witalgebra is 5 dimensional, given by with generators,h generators,

N nonN non--trivial approximate symmetry. We recover the exact 5 trivial approximate symmetry. We recover the exact 5 symmetries.symmetries.

0ε =

1(3)so d⊕ ⊕

0 1

2 3 0

, cos cot sin ,

sin cot cos , , .

t

s

φ θ φθ φ

φ θ φθ φ φ

∂ ∂ ∂= = −∂ ∂ ∂

∂ ∂ ∂ ∂= + = =

∂ ∂ ∂ ∂

Y Y

Y Y Z

KeerKeer spacetime as a second perturbation of the Minkowski spacetime.spacetime as a second perturbation of the Minkowski spacetime.For this we use the same small parameter defined for thFor this we use the same small parameter defined for the RN e RN spacetime and spacetime and

LagrangianLagrangian

For this Lagrangian reduces to that of the MinkowskiFor this Lagrangian reduces to that of the Minkowskispacetime.spacetime.

The Noether symmetries form a 17 dim Lie algebra. The generatorsThe Noether symmetries form a 17 dim Lie algebra. The generators are are the 10 KVs and the 10 KVs and

,ε2 2

1 1, (0 1/ 4).a k kε= < ≤

22 2 2 2 2 2 2 2 2 2 21

2

12 2 2 2 2 2 31

2 1( sin ) ( ) [ (1 sin )4

(cos sin ) sin ] ( ).

kL t r r t r rr r

kk tr O

r

θ θφ ε ε θ

θθ θφ θ ε

= − − − − + − −

+ + − +

& && && & &

& & &&

0,ε =

The is translation in , is scaling symmetry in The is translation in , is scaling symmetry in and and . Using we can write or. Using we can write or and and

0 1 2

23

4

5

6

1, ( ), ,2

1 [ ( )],2

cos cos csc sin[ sin cos (sin cos )],

cos sin csc cos[ sin sin (sin sin )],

[ cos (c

s t r ss s t r t

s s t rs t r

s r tt r r r

s r tt r r r

s r tt

θ φ θ φθ φ θ φθ φ

θ φ θ φθ φ θ φθ φ

θ

∂ ∂ ∂ ∂ ∂= = + + =∂ ∂ ∂ ∂ ∂

∂ ∂ ∂= + +

∂ ∂ ∂∂ ∂ ∂ ∂

= + + −∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= + + −∂ ∂ ∂ ∂

∂= +

∂

Z Z Z

Z

Z

Z

Z sinos ].r r

φθθ

∂ ∂−

∂ ∂0Z s 1Z , ,s t r

0 3 1[ , ] =Z Z Z1Z 2s t= 2s r=

22 2

31[ ( 2 ) 3 ].

4r r t r

t t r∂ ∂

= + +∂ ∂

Z

The Lie algebra of the Conformal Killing Vectors (CKVs) for a The Lie algebra of the Conformal Killing Vectors (CKVs) for a conformally flat spacetime is 15 dimensional. Therefore for the conformally flat spacetime is 15 dimensional. Therefore for the Minkowski spacetime there are 15 CKVs. The 5 symmetry generatorsMinkowski spacetime there are 15 CKVs. The 5 symmetry generators, , i.e. , are proper CKVs with conformal factoi.e. , are proper CKVs with conformal factor r

Not only the KVs but also the CKVs form a subalgebra of the Not only the KVs but also the CKVs form a subalgebra of the symmetries of the Lagrangian for the Minkowski symmetries of the Lagrangian for the Minkowski spacetime.spacetime.

( 2...6)i i =Z2

0 11 ( ).2

c t cψ = +

Retaining terms of firstRetaining terms of first--order in the secondorder in the second--order perturbed order perturbed Lagrangian becomes a firstLagrangian becomes a first--order perturbed Lagrangian for the order perturbed Lagrangian for the Schwarzschild spacetime Schwarzschild spacetime

In the set of determining equations only 12 of the 17 constants In the set of determining equations only 12 of the 17 constants appear. No nonappear. No non--trivial approximate symmetry. trivial approximate symmetry.

The second approximation, in the new set of determining equationThe second approximation, in the new set of determining equations s 14 of the 17 constants appear. Again there is no non14 of the 17 constants appear. Again there is no non--trivial trivial symmetry.symmetry.

,ε

2 2 2 2 2 2 2 2 22( sin ) ( ) ( ).L t r r t r Or

θ θφ ε ε= − − − − + +& && && &

Approximate Lie Symmetries and Approximate Lie Symmetries and Energy in the ChargedEnergy in the Charged--Kerr Kerr

SpacetimeSpacetimeProblem arises in the search for a scaling factor. In the RNProblem arises in the search for a scaling factor. In the RN--case the case the rescaling was by , there is a simple multiplrescaling was by , there is a simple multiplicative factor for icative factor for the Kerr spacetime. In the absence of the constant, it is not clthe Kerr spacetime. In the absence of the constant, it is not clear what ear what significance to attach to the rescaling. To relate that factor tsignificance to attach to the rescaling. To relate that factor to the factor o the factor arising in the RNarising in the RN--case, we investigate secondcase, we investigate second--order approximate order approximate symmetries for the chargedsymmetries for the charged--Kerr spacetime. Kerr spacetime.

Line elementLine element

where where

2 2(1 / 2 )Q m−

2 2 2 22 2 2 2 2 2 2

2 2 2

2 sin sin(1 ) (2 ) ,mr Q ads dt dr d d mr Q dtdρ θ θρ θ φ φρ ρ ρ− Λ

= − − − − + −Δ

2 2 22 .a r mr QΔ = + − +

Using the same defined for the RN spacetime and settingUsing the same defined for the RN spacetime and settingwe have we have

2 2 ,Q kε=

12 2 32 3 2

2 2 2 2 2 2 2 22

212 2 2

13 2 3

2 2 2 2 2112

21 1[ (1 2 ) s in ] ( ) ,

1( s in ) [ ( ) s in ]2

1[ (1 2 ) s in { 2 ( s in ) 1}2

1s in ( s in ) ( s in ) ] (

kt t r k t r r O

r r r

r r t rr

k rk t t k kr r r

k r k k Or r

ε ε θ φ ε

θ θ φ ε θ θ φ

ε θ φ θ

θ θ θ θ θ φ ε

+ + − − +

− + + − + + −

+ + − + −

+ + + + +

&&& & && & &

& & & &&&& &

&&& &

& & && 3

12 2 213 4

2 2 2 2 31 13 2

123

313

) ,

2 s in c o s [ ( ) s in 22

2 c o s s in 2 ( s in ) ] ( ) ,2

2 2 c o t c o s [ ( 2 c o t )

2 ] ( ) .

k kr t rr r r

k kr Or r

k trr tr r r

k r Or

θ θ θ θ φ ε φ θ

θ θ θ θ θ φ ε

φ φ θ θ θ φ ε θ θ

φ ε

+ − + − −

− + +

+ + + −

− +

&& & & &&& &

& & &&

&&&& & & & &&&

&&

ε

The scaling factor The scaling factor

For this reduces to mFor this reduces to m--times of the expression of the RN metric. times of the expression of the RN metric.

Here the reHere the re--scaling consists of two parts scaling consists of two parts -- one is due to charge and the one is due to charge and the other is due to spin which depends on other is due to spin which depends on

The charge comes in quadratically compared to unity in one term.The charge comes in quadratically compared to unity in one term. The The spin comes in linearly. It does not come with a constant term tospin comes in linearly. It does not come with a constant term tocompare. However, taken as a whole, the spin has an effectively compare. However, taken as a whole, the spin has an effectively lower lower order effect.order effect.

2

.2c KQ maM mm r− = − +

0,a =

.r

Using the Using the KomarKomar integral Cohen and de integral Cohen and de FeliceFelice obtained a formula for obtained a formula for the mass (energy) for the chargedthe mass (energy) for the charged--Kerr spacetime (Cohen Kerr spacetime (Cohen et. alet. al 1984)1984)

Here does not appear explicitly and only appears in a productHere does not appear explicitly and only appears in a product with with . When the effects of rotation also disappear. Th. When the effects of rotation also disappear. This does is does

not seem reasonable.not seem reasonable.

ChellathuraiChellathurai et. alet. al modified the modified the KomarKomar integral. Obtained an expression integral. Obtained an expression for the mass of the chargedfor the mass of the charged--Kerr black hole (Kerr black hole (ChellathuraiChellathurai et. al et. al 1990)1990)

Using Using --formalism formalism QadirQadir and and QuamarQuamar obtained an expression for obtained an expression for the approximate mass of the chargedthe approximate mass of the charged--Kerr spacetime (Kerr spacetime (QadirQadir et. al et. al 1986)1986)

2 2 2 2 2 2

3 4

(12 ) 14 ... .3 3c K

Q m Q a mQ aM mr r r−

+= − − + +

Nψ

2 2 2 2 2 2

2 3

cos cos ... .2 2c KQ ma Q aM m

r r rθ θ

− = − − + +

2 2 2 21

2

( ) tan ( ).c KQ Q r a aM mr ar r

−−

+= − −

aQ 0Q →

In all three expressions the charge and spin appear at the same In all three expressions the charge and spin appear at the same order. order. The last one comes with a The last one comes with a --dependent part. In our expression in the dependent part. In our expression in the absence of charge, the effect is to enhance the mass. absence of charge, the effect is to enhance the mass.

One can extract rotational energy from a rotating black hole, heOne can extract rotational energy from a rotating black hole, hence the nce the rotation should add into the mass. As would be expected, this efrotation should add into the mass. As would be expected, this effect fect decreases with . The other three expressions give a reduction decreases with . The other three expressions give a reduction of the of the rotating mass. rotating mass.

Our expression gives change in mass due to charge that is positiOur expression gives change in mass due to charge that is position on independent. The force experience by a particle in the field of independent. The force experience by a particle in the field of a a charged source would be position dependent, but this does not sacharged source would be position dependent, but this does not say that y that mass should be modified by a position dependent expression. It mmass should be modified by a position dependent expression. It might ight be that in our expression modification is due to the electromagnbe that in our expression modification is due to the electromagnetic etic selfself--energy to the gravitational selfenergy to the gravitational self--energy. energy.

The other expressions have drawbacks of which our seems to be frThe other expressions have drawbacks of which our seems to be free.ee.

θ

r

Approximate Symmetries and Approximate Symmetries and Energy in WaveEnergy in Wave--like Spacetimeslike Spacetimes

Plane symmetric static spacetime Plane symmetric static spacetime

where a constant of dimwhere a constant of dimensions of .ensions of .

Lagrangian:Lagrangian:

Noether symmetry generators:Noether symmetry generators:

2 2 ( ) 2 2 2 ( ) 2 2( ),x xds e dt dx e dy dzν μ= − − +2

( ) , ( ) ,x xx xX X

ν μ ⎛ ⎞= = ⎜ ⎟⎝ ⎠

X x

22( / ) 2 2 2( / ) 2 2( ).x X x XL e t x e y z= − − +& & & &

0 1 2 3 0, , , , ,X X X X y z Y A ct y z z y s∂ ∂ ∂ ∂ ∂ ∂

= = = = − = =∂ ∂ ∂ ∂ ∂ ∂

For the approximate symmetries, For the approximate symmetries,

a constant of dimensions of .a constant of dimensions of .FirstFirst--order perturbed Lagrangian:order perturbed Lagrangian:

NonNon--trivial firsttrivial first--order approximate symmetry:order approximate symmetry:

Contract the energy momentum vector with this Noether symmetry, Contract the energy momentum vector with this Noether symmetry, we we getget

Energy imparted to the test particle, does not give energy in thEnergy imparted to the test particle, does not give energy in the field.e field.

2

2( ) 2 , ( ) 2 .x t x tx xX T X T

ν ε μ ε⎛ ⎞⎛ ⎞= + = +⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

T t

2 22( / ) 2 2 2( / ) 2 2 2( / ) 2 2( / ) 2 2 2( ) 2 [ ( )] ( ).x X x X x X x XtL e t x e y z e t e y z OT

ε ε= − − + + − + +& && & & & &

1 ( ).a t y zt T t y z

ε∂ ∂ ∂ ∂= − + +∂ ∂ ∂ ∂

X

( ),y zQ E Et yp zpTε

= − + +

Cylindrically symmetric static spacetime,Cylindrically symmetric static spacetime,

and where is constaand where is constant of dimensions of . nt of dimensions of .

Lagrangian: Lagrangian:

Noether symmetry generators areNoether symmetry generators are

2 ( ) 2 2 ( ) 2 2 2( ),ds e dt d e a d dzν ρ μ ρρ φ= − − +2

( )Rρν ρ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

3

( )Rρμ ρ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

2 3( / ) 2 2 ( / ) 2 2 2( ).R RL e t e a zρ ρρ φ= − − +&& & &

R ρ

20 1 2 3

0

, , , ,

, .

X X X X z at z z

Y A cs

φφ φ

∂ ∂ ∂ ∂ ∂= = = = −∂ ∂ ∂ ∂ ∂∂

= =∂

To obtain approximate symmetries we take, To obtain approximate symmetries we take,

FirstFirst--order perturbed Lagrangian: order perturbed Lagrangian:

NonNon--trivial firsttrivial first--order approximate symmetry existorder approximate symmetry exist

Conserved quantity:Conserved quantity:

2 3

( ) 2 , ( ) 2 .t txR T R Tρ ρν ρ ε μ ε⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 3 2 3( / ) 2 2 ( / ) 2 2 2 ( / ) 2 ( / ) 2 2 2 2( ) 2 [ ( )] ( ).R R R RtL e t e a z e t e a z OT

ρ ρ ρ ρρ φ ε φ ε= − − + + − + +& && && & &

1 ( ).a t zt T t z

ε φφ

∂ ∂ ∂ ∂= − + +∂ ∂ ∂ ∂

X

( ).zQ E Et p zpT φε φ= − + +

Energy in KN-AdS SpacetimeThe line element

where

Reduces to KN spacetime when Admit the same two KVs .

22 2 2 22 2 2 2

2 2

2 2 22

sinsin( ) (

sin ) ,

r

r

ads dt d dr d adt

r a d

θ

θ

θθ ρ ρφ θρ ρ

θ φ

ΔΔ= − − − −

Ξ Δ Δ

+−

Ξ

2 2 22 2 2 2

2 2 21 , 1 cos , ( )(1 ) 2 .ra a rr a mr Ql l lθ θΞ = − Δ = − Δ = + + − +

l

2

3l−

Λ =

.l →∞

Introduce the small parameter

From the second-order approximate symmetries of the perturbed geodesic equations, the energy scaling factor

For this reduces to that of KN spacetime.

2 2 2 21 22 , , .m a k Q kε ε ε= = =

2 4 2 2 2 2 2

2 4 2 2 2 2 2

2(1 ...)[1 (1 )( ) { (1 )}].2 2 2KNA

r r r Q a r Q rM ml l l m r l m l

= − + − − + − + − −

l →∞

Energy in Exact Gravitational Wave Energy in Exact Gravitational Wave SpacetimesSpacetimes

Plane fronted waves with parallel rays (ppPlane fronted waves with parallel rays (pp--waves),waves),

is the arbitrary amplitude, is the frequency.is the arbitrary amplitude, is the frequency.Lagrangian:Lagrangian:

Noether symmetry generators:Noether symmetry generators:Remove the tRemove the t--dependent part, put , get static Spacetime dependent part, put , get static Spacetime

Lagrangian:Lagrangian:

Noether symmetry generators:Noether symmetry generators:

2 2 2 2 2 2 2 2 2 2[( ) sin ( ) 2 cos ( )]( 2 ) .ds h x y t z xy t z dt dz dtdz dt dx dy dzω ω ω= − − + − + − + − − −h ω

2 2 2 2

2 2 2 2 2

[( )sin ( ) 2 cos ( )](2 ) .

L h x y t z xy t z tz tz t x y z

ω ω ω= − − + −

+ − + − − −

&

& && & & & &

0 0, , .A ct z s∂ ∂ ∂

= + = =∂ ∂ ∂

X Y

1h =2 2 2 2 2 2 2 2 2 2[( ) 2 ]( 2 ) .ds x y xy dt dz dtdz dt dx dy dzω= − + + − + − − −

2 2 2 2 2 2 2 2 2[( ) 2 ]( 2 ) .L x y xy t z tz t x y zω= − + + − + − − −& & && & & & &

0 1 0, , , .A ct z s∂ ∂ ∂

= = = =∂ ∂ ∂

X X Y

For the approximate symmetries, consider the exact ppFor the approximate symmetries, consider the exact pp--waves as a waves as a perturbation on the above static spacetime and put . perturbation on the above static spacetime and put .

Lagrangian:Lagrangian:

No nonNo non--trivial approximate symmetry found. trivial approximate symmetry found.

Only the exact symmetries are recovered as trivial firstOnly the exact symmetries are recovered as trivial first--order order approximate symmetries.approximate symmetries.

h ε=

2 2 2 2 2 2

2 2 2 2 2

[( ) 2 {( )sin ( ) 2 cos ( )}](2 ) .

L x y xy x y t z xy t z tz tz t x y z

ω ε ω ω= − + + − − + −

+ − + − − −

&

& && & & & &

Cylindrically symmetric gravitational waves,Cylindrically symmetric gravitational waves,

and are arbitrary functions of and subject to the vand are arbitrary functions of and subject to the vacuum acuum EFEsEFEs

Solution of these equations:Solution of these equations:

Lagrangian: Lagrangian:

Noether symmetries:Noether symmetries:

2 2( ) 2 2 2 2 2 2 2( ) .ds e dt d e d e dzγ ψ ψ ψρ ρ φ− −= − − −

2 21 0, ( ), 2 .ψ ψ ψ γ ρ ψ ψ γ ρψψρ

′′ ′ ′ ′ ′+ − = = + =&& & & &

0 0

2 20 0 0 0 0 0 0 0 0 0 0 0

( ) cos ( )sin ,

[( ) cos(2 ) {( )sin(2 ) 2( ) }]2

AJ t BY t

A J J B Y Y t AB J Y J Y t J Y J Y t

ψ ρω ω ρω ωρωγ ω ω ω

= +

′ ′ ′ ′ ′ ′= + − + − −

2( ) 2 2 2 2 2 2 2( ) .L e t e e zγ ψ ψ ψρ ρ φ− −= − − −&& & &

0 1 0, , , .A cz sφ

∂ ∂ ∂= = = =∂ ∂ ∂

X X Y

γψ t ρ

Remove the tRemove the t--dependent part and put the strength of the wave A=1 and dependent part and put the strength of the wave A=1 and B=0. B=0.

Lagrangian:Lagrangian:

Noether symmetries:Noether symmetries:

For the approximate symmetries, put and take the eFor the approximate symmetries, put and take the exact wave xact wave as a perturbation on the above static spacetime. as a perturbation on the above static spacetime.

0 0 0 02( ) 2 22 2 2 2 2 2( ) .ds e dt d e d e dzγ ψ ψ ψρ ρ φ− −= − − −

0 0 0 0 0( ), ( ) ( ).2

J J Jρωψ ρω γ ρω ρω′= =

0 0 0 02( ) 2 22 2 2 2 2( ) .L e t e e zγ ψ ψ ψρ ρ φ− −= − − −&& & &

0 1 2 0, , , , .A ct z sφ∂ ∂ ∂ ∂

= = = = =∂ ∂ ∂ ∂

X X X Y

A ε=

0 0 1

2 20 0 0 1

( )(1 cos( )) ,

( ) ( )(1 sin(2 )) .2

J t

J J t

ψ ρω ε ω ψ εψρωγ ρω ρω ε ω γ ε γ

= + = +

′= + = +

FirstFirst--order perturbed Lagrangian: order perturbed Lagrangian:

No nonNo non--trivial approximate symmetry found. trivial approximate symmetry found.

Only the exact symmetries are recovered as trivial firstOnly the exact symmetries are recovered as trivial first--order order approximate symmetries.approximate symmetries.

0 0 0 0 0 0

0 0

2( ) 2 2 2( )2 2 2 2 2 2 21

2 22 2 2 2

( ) 2 ( ( )

) ( ).

L e t e e z e t

e e z O

γ ψ ψ ψ γ ψ

ψ ψ

ρ ρ φ εψ ρ

ρ φ ε

− − −

−

= − − − − −

− + +

&& && &&

& &

SecondSecond--order approximate geodesic equations for plane order approximate geodesic equations for plane symmetric wavesymmetric wave--like spacetime:like spacetime:

2 2

2 2

22 2 2 2(( / ) / ) 2 2 2 2(( / ) / ) 3

2

2 22( / ) 2 2 2( / ) 2 2( / ) 2 2 2( / )

2

2 22 2( / ) 2 2 2(

2

2 [ ( ) ] [ ( ) ] ( ) 0,

2 2 2( ) [ ( ) ]

2 2[ ( )

x X x X x X x X

x X x X x X x X

x X x

tt tx t x y e t x y e OX T Tt x t xx e y z e t e y z eX X TX X

t xt e y z eT X X

ε ε ε

ε

ε

− −+ − − + + + + + =

+ − + + − + +

− +

&& & & && & & & &

&&&& & & & &

& & &2/ ) 3

23

2 2

23

2 2

] ( ) 0,

4 2 2 ( ) 0,

4 2 2 ( ) 0.

X O

x ty xy ty ty OX T Tx tz xz tz tz O

X T T

ε

ε ε ε

ε ε ε

+ =

+ + − + =

+ + − + =

& &&& & & & &

& &&& & & & &

scaling factor: scaling factor: 4 / 2( / )( / 1)[ 2 ].4

x X x X x Xt e eT

− − ++

SecondSecond--order approximate geodesic equations for cylindrically order approximate geodesic equations for cylindrically symmetric wavesymmetric wave--like spacetime:like spacetime:

Scaling factor:Scaling factor:The plots for this case are similar to those given in the casThe plots for this case are similar to those given in the case of plane e of plane wavewave--like case, with replaced by . like case, with replaced by .

2 3 2 3

2 3 2 3

2

22 2 2 2 (( / ) ( / ) ) 2 2 2 2 (( / ) ( / ) ) 3

2

2( / ) 2 2 2 ( / ) 2 ( / ) 2 2 2 ( / )

2

2 22 ( / ) 2

2

2 [ ( ) ] [ ( ) ] ( ) 0,

2 2 2( ) [ ( ) ]

2 2[ (

R R R R

R R R R

R

tt t t a z e t a z e OR T Tt te a z e t e a z eR R TR R

t t e aT R R

ρ ρ ρ ρ

ρ ρ ρ ρ

ρ

ε ερ φ φ ε

ρ ε ρρ φ φ

ε ρ

− −+ − − + + + + + =

+ − + + − + +

−

& &&& & & && & &

&& &&&& & &

&& 32 2 ( / ) 3

23

2 2

23

2 2

) ] ( ) 0,

4 2 2 ( ) 0,

4 2 2 ( ) 0.

Rz e O

tt t OR T T

tz tz tz OR T T

ρφ ε

ρ ε εφ ρφ φ φ ε

ρ ε ερφ ε

+ + =

+ + − + =

+ + − + =

&

&& & & && &&

& & &&&& & &

2 22( / ) ( / ) ( / 1)[ 2 ].4

R R Rt e eT

ρ ρ ρ− − ++

x ρ

Perturbed ppPerturbed pp--waves, system of firstwaves, system of first--order approximate geodesic order approximate geodesic equations:equations:

32 2 2

2 2 2

2

2 2 2 2

( ){( ) ( ) } [ {( )cos ( )2

2 sin ( )}( 2 ) ( ){ sin ( )cos ( )} { sin ( ) cos ( )} ] 0,

[ ( ) { sin ( ) cos ( )}]( 2

t t z x y x x y y x y z t

xy z t t z tz t z x z ty z t x y z t x z t yx x y x z t y z t t z

ωω ε ω

ω ω ω

ω ω ω ω

ω εω ω ω

+ − + + − + − −

+ − + − − − − −

− + − + − =

+ + − − − − + −

&& & & & &

& & && & &

& &

& &&& &2 2 2 2

32 2 2

2 2 2

2

) 0,[ ( ) { cos ( ) sin ( )}]( 2 ) 0,

( ){( ) ( ) } [ {( )cos ( )2

2 sin ( )}( 2 ) ( ){ sin ( )cos ( )} { sin ( ) co

tzy x y x z t y z t t z tz

z t z x y x x y y x y z t

xy z t t z tz t z x z ty z t x y z t x

ω εω ω ω

ωω ε ω

ω ω ω

ω ω ω

=

+ + − − − − + − =

+ − + + − + − −

+ − + − − − − −

− + − +

&

& &&& & &

&&& & & &

& & && & &

& s ( )} ] 0.z t yω − =&

There is no appear in the above geodesic equations. There is no appear in the above geodesic equations.

We can not apply the definition of secondWe can not apply the definition of second--order approximate order approximate symmetries of symmetries of ODEsODEs..

The approximation do not alter the behavior of waves and go lineThe approximation do not alter the behavior of waves and go linearly. arly. This is consistent with the wave geometry. The wave front moves This is consistent with the wave geometry. The wave front moves as as parallel planes. The curvature is absolutely zero before the pulparallel planes. The curvature is absolutely zero before the pulse se arrives and after it has passed. No region where there is a sligarrives and after it has passed. No region where there is a slight shift ht shift from the flat geometry for obtaining an approximate symmetry.from the flat geometry for obtaining an approximate symmetry.

Hence there is no energy reHence there is no energy re--scaling.scaling.

2ε

Cylindrically symmetric approximate gravitational Wave secondCylindrically symmetric approximate gravitational Wave second--order order geodesic equations: geodesic equations:

Scaling factor:Scaling factor:

0 0 0

0 0

0 0 0

2 2(2 )2 2 20 0 1 1 1 1

2(2 )2 2 2 2 31 1 1

2 2(2 )2 2 2 2 20 0 0 1

2

2( ) [ ( ) 2 ]

[ ( ) 4 2 ] ( ) 0,

( )( ) ( 1) [ (

)

t t t e e z t

t e z t O

t t e e z t

γ ψ γ

ψ γ

γ ψ γ

γ ψ ρ ε ψ ρ ρ ψ φ ψ ψ ρ

ε γ ρ ψ γ ρ ε

ρ γ ψ ρ ρ ρ ρψ φ ψ ε ψ

ρ

− −

−

− −

′ ′ ′+ − − + + − +

′+ + + − + =

′ ′ ′ ′ ′+ − + + + − + −

+

&&& & & && & & & & &&

& && & &&

&& & &&& & & &

& 0 0 0

0 0 0

2 2(2 )2 2 2 2 2 21 0 1 1 1

2 2(2 )2 2 2 31 0 0 1 1 0 1 1 1

23

0 1 1 0

(4 ) ] [ ( )

( 1) 2 (4 ) ] ( ) 0,

1 (1 ) ( ) (1 ) ( ) 0,

e e z t t

e e z t O

t O

γ ψ γ

γ ψ γ

ρ ψ φ ψ ψ ψ ψ ρ ε γ ρ

ργ ρψ φ ψ ψ γ ψ ψ ψ γ ρ ε

εφ ρψ ρφ ε ψ ρ ψ φ ρψ ρφ ερ ρ

− −

− −

′ ′ ′ ′ ′− − − + + + −

′ ′ ′ ′ ′− + − − + + =

′ ′ ′+ − − + − − + =

& & && &&

& & &&

&& & & &&& & & &

&& 2 2 30 1 1 0 1( ) 2 ( ) 0.z z t z z Oψ ρ ε ψ ρ ψ ε ψ ψ ρ ε′ ′ ′+ + + − + =&& & & && & &

0 0 0 0 0 0 0 0 0 02( ) 2( ) 3( ) 3( ) 3( ) 1/ 21 1( 1) 2 ( 1) .e e e e eψ γ ψ γ ψ γ ψ γ ψ γγ γ− − − − −′+ − + −&

This scaling factor involves the Bessel function of the first kiThis scaling factor involves the Bessel function of the first kind and its derivatives. nd and its derivatives. Using that asymptotic representation of the Bessel function we oUsing that asymptotic representation of the Bessel function we obtain an btain an asymptotic representation of the scaling factor asymptotic representation of the scaling factor

113 1 342 2 2

32

3 2 [( cos( ) ) sin(2 )]( ) ([ ] ).t Oωρ ω ωρ ωρπ

− −×+

Whether there is the analogue of LandauWhether there is the analogue of Landau--damping of electromagnetic damping of electromagnetic waves for waves for GWsGWs. .

With the use of approximate Lie symmetries the question seems toWith the use of approximate Lie symmetries the question seems to be be answerable. answerable.

Classically the energy density in cylindrical waves reduces by tClassically the energy density in cylindrical waves reduces by the he factor . From the factor energy density decreases by factor . From the factor energy density decreases by a further a further factor of . factor of .

Hence for sufficiently large the scaling factor Hence for sufficiently large the scaling factor is a is a significant selfsignificant self--damping of the waves.damping of the waves.

1/ 2πρ11/ 4 33 2 π ωρ× ×

ρ 31/ ωρ

SummarySummaryThe RN spacetime. Importantly from the consistency of the The RN spacetime. Importantly from the consistency of the trivial approx. symm. of the geodesic equations the scaling trivial approx. symm. of the geodesic equations the scaling factor obtained. This gives the refactor obtained. This gives the re--scaling of energy.scaling of energy.

Kerr spacetime, no nonKerr spacetime, no non--trivial approx. Noether symm.trivial approx. Noether symm.

For the Minkowski spacetime, the CKVs form a proper For the Minkowski spacetime, the CKVs form a proper subalgebra of the 17 Noether symm. subalgebra of the 17 Noether symm.

Scaling factor for the KN spacetime, gives energy in the field. Scaling factor for the KN spacetime, gives energy in the field. Compared with the other and found more reasonable.Compared with the other and found more reasonable.

Due to the cosmological term an extra contribution in the Due to the cosmological term an extra contribution in the energy expression for the KNenergy expression for the KN--AdSAdS spacetime. Since is a spacetime. Since is a candidate for the dark energy therefore the relation of the candidate for the dark energy therefore the relation of the --dependent terms with the dark energy is worth exploring.dependent terms with the dark energy is worth exploring.

lΛ

A timeA time--like nonlike non--trivial approx. Noether symm. obtained. Give trivial approx. Noether symm. obtained. Give conserved quantities. The energy imparted to the test particles.conserved quantities. The energy imparted to the test particles.

No nonNo non--trivial approximate Noether symm. for the pp and cylindrically trivial approximate Noether symm. for the pp and cylindrically symmetric waves. symmetric waves.

Scaling factor were obtained, plotted for different values of tiScaling factor were obtained, plotted for different values of time and me and space variables.space variables.

For ppFor pp--waves, no , appeared in the geodesic equations. No scaling waves, no , appeared in the geodesic equations. No scaling factor.factor.

Resemblance with Resemblance with QadirQadir and and SharifSharif formula, momentum imparted to formula, momentum imparted to test particles by cylindrical waves. For pptest particles by cylindrical waves. For pp--waves no such formula.waves no such formula.

2ε

For cylindrical waves, scaling factor. Plotted for different valFor cylindrical waves, scaling factor. Plotted for different values of , ues of , and .and .

SelfSelf--damping of cylindrical damping of cylindrical GWsGWs was seen.was seen.

It would be of interest to apply this analysis to the KhanIt would be of interest to apply this analysis to the Khan--Penrose and Penrose and SzekeresSzekeres solutions. Whether they suffer selfsolutions. Whether they suffer self--damping or enhancement.damping or enhancement.

The analysis should be applied to spherical solutions like thoseThe analysis should be applied to spherical solutions like those of of NutkuNutku..

tρ

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