Majdi Amr
Amman, Jordan
Correspondence to: Majdi Amr , Amman, Jordan.
Email: | |
Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
In this work I will discuss the possibility of creating particles due to the interaction of a weakly incident gravitational perturbation with a spin-zero weakly interacting static one-dimensional domain wall in the harmonic coordinates and aim to calculate the number of particles created due to the interaction with a thin static example in the one-dimensional case without any resort to the real mechanism underlying the creation process.
Keywords:
Gravitational waves, Particle creation, Domain walls
Cite this paper: Majdi Amr , Particle Creation by Gravitational Waves in Domain Walls, International Journal of Theoretical and Mathematical Physics, Vol. 9 No. 1, 2019, pp. 20-24. doi: 10.5923/j.ijtmp.20190901.04.
1. Introduction
The idea of particle creation by gravitational waves has been considered extensively by many particle physicists where it has been shown in a previous work by F. Sorge [1] that particles can indeed be created by weakly incident gravitational waves in the TT and harmonic gauges [2] due to its interaction with a bounded scalar field after considering methods of quantum filed theory in curved spacetimes (namely considering the Klein-Gordon equation in a curved background and a calculation of some coefficients involved in the analysis) where the number of particles created has also been calculated in contrary to the unbounded case where no particle production was shown to exist due to the absence of the β Bogoliubov coefficient (a parameter related to the number of particles created after a time evolution of the filed from one vacuum state to another). Although the physics of domain walls is more involved due to its gravitational nature it still can be shown that particles can be produced due to its interaction with a weakly incident gravitational perturbation in the de Donder gauge [3] where the number of particles created has been calculated here for a thin one-dimensional static example after the calculation of the Bogoliubov coefficient involved in the general solution of the filed perturbation of the wall long-after the interaction with the incident gravitational wave.
2. Particle Production in Non-Stationary Space-Times
Starting with the Lagrangian density | (1) |
for a scalar field of particles with mass m in a curved background (where an extra scalar-gravitational coupling term with being the Ricci scalar has been added) and assuming that | (2) |
(i.e. asymptotic flatness so that in these limits our field equation reduces to the usual Minkowski Klein-Gordon equation), we can imagine that the space-time considered is a ‘sandwiched’ space-time domain as shown below:Therefore, it is possible to write the scalar quantum field solution in of the Klein-Gordon equation as | (3) |
The continuation of this solution through will be | (4) |
As both sets and are complete we can set | (5) |
Thus in we have | (6) |
where | (7) |
which is called a Bogoliubov transformation with called the Bogoliubov coefficients respectively. Here the choice of depends on the choice of the basis, where the particle number operator for the mode is | (8) |
| (9) |
The expected number of particles in the mode in in the in- vacuum is then | (10) |
Therefore, the expected total number of particles is given by and will vanish if [4,5].
3. Domain Walls
In simple words a planar domain wall is a sheet-like distribution of matter that is invariant under translations in the two directions in the plane, rotations around the normal, and reflection through the midplane of the sheet at Under these conditions the line element can be written as | (11) |
where are some functions of t and Domain walls appear through the breaking of a discrete symmetry of the vacuum which is usually described in terms of a scalar field. In the simplest realization the vacuum has two degenerate states, in which the scalar field can take values and say. When on one side of two neighboring regions it takes the value and on the other side then a domain wall occurs in the separation layer, with the field interpolating between these two values. These are the Z_{2} domain walls, which arise due to a discrete symmetry breaking of two possible states [6].
4. Particle Creation by Gravitational Waves in a 1-D Domain Wall
We aim now to the study of the interaction of a gravitational wave with a static 1-D domain wall for an gravitational wave using harmonic coordinates.
4.1. Background Configuration for a Domain Wall
The action for a scalar field in a curved background is: | (12) |
where is the scalar field Lagrangian, and is the determinant of the metric involved. The action for the gravitational field is | (13) |
where is the Ricci scalar. The equations of motion, of the full action for the scalar and gravitational fields can be obtained by considering the total Lagrangian density of the joint fields | (14) |
from which we get by the use of the Euler-Lagrange equations for both fields the two field equations | (15) |
and certainly the Einstein's equation | (16) |
(where considered to be the generalized d'Alembertian) with being the Ricci tensor, and is the energy-momentum tensor given by | (17) |
The domain wall we are considering is specified by the scalar and gravitational field solutions respectively, and it is only assumed that depends on only, and that the gravitational field depends on both and with The wall creates a gravitational field with a line element of the form | (18) |
where and is the wall tension [7]. Therefore, the metric of the wall is given by | (19) |
with the inverse metric | (20) |
A somewhat lengthy calculation of the Ricci tensor components (a Mathematica program in appendix C of [8] would be very handy) gives with and the following three useful equations for our domain wall | (21) |
| (22) |
| (23) |
where some extra assumptions on the fields and parameters involved are assumed as in [7] which we will not consider by any way here in this work..Turning off gravity, i.e. using in all the previously given equations, the scalar field equation of motion admits a solution satisfying: | (24) |
The gravitational field produced by the wall back reacts producing the scalar field solution instead of
4.2. Gravitational Perturbation Equations
The fields of the domain wall as we have previously mentioned is specified by and and we will consider perturbations around this background configuration, | (25) |
| (26) |
where and obey the Klein-Gordon and Einstein equations (15) and (16) respectively. We know proceed to derive a linearized equation satisfied by Substituting equation (25) into the field equation (15) and assuming that:1- (as given in (19)) differs slightly from and as a consequence2- differs also slightly from 3- Terms with products of small contributions (such as and ) are neglected4- (i.e a use of harmonic coordinates [9])We end up with an equation for the field perturbation | (27) |
which can be treated as a unique field by its own (a massless Klein-Gordon field with a source term as will be shown in what follows) and assumed to vanish at both sides of the wall [1].Before the gravitational perturbation is switched on we trivially have which corresponds to the no-particle production ground state which is expressed in the notation of (3) of the region with the (properly normalized) complete set of basis solutions | (28) |
of a free massless Klein-Gordon field with a vanishing boundary condition at both sides of the wall (but indeed after a justification for proving that the term on the left of (27) can be dropped as a first approximation to the formal solution to as will be shown in what follows).All what we need to show for particle creation to take place is a solution of the field perturbation as a new basis solution that mixes both frequency modes of the old basis solutions, i.e. we need to reach the condition | (29) |
The solution of (27) now proceeds as follows:First we can rewrite (27) in the form | (30) |
which can be considered as an equation to a massless scalar field with a source given by the term on the right hand side of (30), and therefore, the formal solution to is | (31) |
where is the Green’s function for a z-bounded wave equation given by | (32) |
[1] where is given by | (33) |
If we define | (34) |
then (31) can be written as | (35) |
which is a Fredholm integral equation of the second kind with a convergent Neumann solution provided that | (36) |
The last condition (36) given above (with the fact that the unknown function of the integral equation is just a small perturbation) gives us a good justification to only consider the zeroth-order approximation in the Neumann series solution | (37) |
What we need now is to look at the time dependence of the solution (37) to see if we have a two frequency-mode solution for particles to exist after the interaction and calculate the number of particles created if it does indeed exist.The solution to (37) under the assumption that the incident gravitational perturbation depends only on and can now be written as | (38) |
writing this in terms of the basis solutions of the original free-filed equation (28) we get | (39) |
Comparing (39) with (4), and (5) above (and by assuming that exists in a flat (or a semi-flat) background for a weakly interacting domain wall after the perturbation has been switched off) we can write | (40) |
with | (41) |
from which we can write | (42) |
and finally consider (to at least zero-order approximation) that the coefficient is given by | (43) |
which indeed should prove that particles should be produced after such an interaction of the wall with an incident gravitational perturbation to zero order approximation of the expansion of the field perturbation under all the assumptions given above.
4.3. A Simple Calculational Example for a Thin Domain Wall Case
Now at and with the approximation that the wavelength of the gravitational perturbation is large compared to (which is suitable for the case of a thin domain wall) we can assume to lowest order that is a function of time only (where a similar assumption has been made in [1]) from which we get with the further assumption that to this order a convenient choice of the gravitational perturbation working well in the limits is a Gaussian-like perturbation, i.e. with | (44) |
we get | (45) |
which gives | (46) |
(where represents the typical wavelength of the gravitational perturbation).Now according to (10) we have for the total number of particles created after the interaction of a thin domain wall with such a gravitational perturbation | (47) |
5. Conclusions
As we can see the proof of particle production in a general static one-dimensional domain wall has been shown to a zero order approximation and the number of particles created for the case of a thin domain wall and a Gaussian like perturbation has been calculated which may stimulate the idea for a similar calculation to other systems other the one at hand and give some sort of encouragement to the applicability of the combination of the methods of quantum field theory to classical theory of general relativity (i.e the subject of quantum field theory in curved backgrounds). It's worth mentioning here that the true mechanism of the creation process though not discussed in this work is fairly discussed in [1] if such considerations are sought after.
ACKNOWLEDGEMENTS
Special and cordial thanks go to my former teacher and supervisor Dr. F. Paul Esposito at the physics department of the University of Cincinnati for ingraining me with the seeds of general relativity and its applications in quantum field theory and stimulating the idea of particle creation by gravity waves in different systems. I would also like to thank my respectful supervisor Dr. Noor Chair at the physics department of Jordan University for his supportive efforts during the writings of this work. Wish them all the best.
References
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