generation of non-solitonic cylindrically symmetric gravitational …€¦ · gravitational...
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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
4
( 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 )
3
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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
4
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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(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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geodesic equation
on N
Only the solutions of the geodesic equation on N are need.
7
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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
16
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:
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