Oct. 2016 / JGRG26 at Osaka City Univ.

T. Mishima (Nihon Univ.)

S. Tomizawa (Tokyo Univ. of Tech.)

P24

Generation of non-solitonic cylindrically symmetric

gravitational waves with mode-mixing

by the harmonic mapping method

In the previous work[T&M(16’)], we constructed new cylindrically symmetric

gravitational solitonic waves which behave like regular wave packets in the

space of radial and time coordinates to clarify nonlinear properties of strong

gravitational fields.

This time, to advance the study further, using the harmonic mapping method,

we generate a different type of solutions from non-solitonic seed

solutions( generalized WWB sol. ).

After the method and the solutions are presented, we clarify the nonlinear

properties of the solutions by paying attention to the way of conversion or

mixing between plus and cross modes.

I. Introduction

< Purpose >

Investigation of nonlinear / non-perturbed effects

of Strong gravitational fields

Using the new exact solutions corresponding to

cylindrically symmetric gravitational waves

Through the scattering or reflection of the waves

near the axis

Having nonlinearly interacting two modes: (＋) and (×)

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< schematic pictures >

3

「 Schematic spacetime diagram

(z = const. , φ = const.) 」

Regular packet-like waves on the space of radial-time coordinates

Coming into the symmetric axis and reflecting off

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( Kompaneets – Jordan_Ehlers metric for cylindrically symmetric spacetimes )

( The metric depends only on ρ and t )

< Preparation－(1)：metric, amplitudes and basic equations >

( Vacuum Einstein equations for the above metric )

Following Piran, Safier and Stark[‘85]

Solving first two nonlinear equations (i) and (ii) is crucial.

(i)

(iv)

(iii)

(ii)

0

1

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( nonlinear term )

( Amplitudes used in the rest )

( Basic equations for the amplitudes converted from the vacuum Einstein equation )

( ingoing + mode ) ( outgoing + mode ) ( ingoing mode ) ( outgoing mode )

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C-energy:

< Preparation－(2)：C-energy based energy density and fluxes >

Thorne[‘65]

：outward null flux

( Energy density and fluxes )

：inward null flux

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II. Construction of new solutions and the metric form

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< Application of the harmonic mapping method to gravitational waves>

（1） Introducing the twist potential from the twist

Following Halilsoy[‘88]

（2） Introducing the Ernst potentials and

（3） The equations (i) and (ii) is described with .

(Ernst equation)

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6

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（4） Ernst potential as a harmonic map

The above Ernst equation is derived from the ‘energy’ functional below:

(Ernst equation)

(on )

(on :SU(1,1)/S(U(1)×U(1)) )

Solutions of the Ernst eq. are, so called, Harmonic maps

from a 2-dim. manifold to a 2-dim. manifold (target space)

with the following metrics, respectively.

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( ～ Nonlinear σ model)

(Ernst equation)

（5） Composite harmonic maps using harmonic potentials

Introducing 1-dimensional manifold : ,

is considered as a composite map :

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10

geodesic equation

on N

Only the solutions of the geodesic equation on N are need.

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< New Solutions generated by harmonic mapping method >

The manifold N is just a 2-dim. hyperbolic space, so that

the geodesic equation can be solved completely.

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6

9

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From any harmonic potential the solution can be derived.

(Formal solution: A const. )

(1) Original Weber-Wheeler[’57]・Bonnor[’57] solution （WWB sol.）

Regular and packet-like wave solution : ＋ mode only

Even function for time reflection

＜ Generalized WWB solution with mixed modes＞

(2) Generalized WWB solution

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(odd function )

Substituting for the formulas , we obtain the explicit form of the solution.

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(3) Generalized WWB solution with mixed modes

(Integration )

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( Full metric coefficients : and are essential parameters. )

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III. The behavior of the new waves

< Shape of the C-energy density corresponding to gravitational waves >

From , we know the corresponding to the C-energy does not depend on the

mode-mixing parameter A, so that the behavior of the energy density coincides with

WWB wave’s.

We can expect the new solutions are also regular ‘localized’ waves.

Very Similar to the soliton solution previously obtained !

(e.g.1) Distribution of C energy at t = -15, 0, 14

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< How the nonlinearity appears in C-energy >

(e.g.2) Time dependence of the ratio of + mode contribution to C-energy

A = 1/10

A = 1/3

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A = 2

A = 1

A = 7

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IV. Summary

time

constructed as composite harmonic maps with geodesics

on SU(1,1)/S(U(1)×U(1)) ,

from non-solitonic seed solutions (generalized WWB solutions)

seem to be regular packet-like waves like seed solutions.

have similar behavior to solitonic waves constructed by ISM.

mode conversion/mixing occurs near the rotational axis.

As further investigations,

(nonlinear interaction）

「 Systematic analysis of scattering and collision of

cylindrically symmetric waves 」

Deep understanding of the nonlinearity of gravity

The cylindrically symmetric gravitational solutions we constructed here: