In this case, the system is still described by the EKG Eqs. (27 – 32
), with the the additional
simplification that the scalar field is strictly real,
. In order to find equilibrium configurations, one
expands both metric components
and the scalar field
as a
truncated Fourier series
A careful analysis of the high frequency components of this construction reveals difficulties in
avoiding infinite total energy while maintaining the asymptotically flat boundary condition [172].
Therefore, the truncated solutions constructed above are not exactly time periodic. Indeed, very
accurate numerical work has shown that the oscillatons radiate scalar field on extremely long time
scales while their frequency increases [84, 97]. This work finds a mass loss rate of just one part
in
per oscillation period, much too small for most numerical simulations to observe.
The solutions are, therefore, only near-equilibrium solutions and can be extremely long-lived.
Although the geometry is oscillatory in nature, these oscillatons behave similar to BSs. In
particular, they similarly transition from long-lived solutions to a dynamically unstable branch
separated at the maximum mass . Figure 6
displays the total mass curve,
which shows the mass as a function of central value. Compact solutions can be found in the
Newtonian framework when the weak field limit is performed appropriately, reducing to the so-called
Newtonian oscillations [216]. The dynamics produced by perturbations are also qualitatively similar,
including gravitational cooling, migration to more dilute stars, and collapse to black holes [5]. More
recently, these studies have been extended by considering the evolution in 3D of excited states [13]
and by including a quartic self-interaction potential [217]. In [129], a variational approach is
used to construct oscillatons in a reduced system similar to that of the sine-Gordon breather
solution.
Closely related, are oscillons that exist in flatspace and that were first mentioned as “pulsons” in
1977 [33]. There is an extensive literature on such solutions, many of which appear in [79]. A series of
papers establishes that oscillons similarly radiate on very long time scales [79, 80, 81, 82]. An interesting
numerical approach to evolving oscillons adopts coordinates that blueshift and damp outgoing radiation of
the massive scalar field [115, 117]. A detailed look at the long term dynamics of these solutions suggests
the existence of a fractal boundary in parameter space between oscillatons that lead to expansion of a
true-vacuum bubble and those that disperse [116].
http://www.livingreviews.org/lrr-2012-6 |
Living Rev. Relativity 15, (2012), 6
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