Document Type
Article
Publication Date
1985
Abstract
We analyse the equation of dispersion for the gravitational radiation in the matter assuming the space is filled homogeneously and isotropically by the viscose fluid. In presence of the matter the refraction index is shown to differ from 1 yielding an opportunity to focuse the gravitational waves. Due to the cosmological effects the magnitude of wave damping may become sign reversed causing the instability of the homogeneous isotropic model.
1. Hawking (1966) demonstrated a technique which allows to calculate perturbations of the homogeneous and isotropic universe. Based on some results of Jordan et al. (1960, 1961, 1962) this technique does not employ any restricting approximation. The arising equations have some analogy to the Maxwell equations and are applicable to the problem of interaction of the gravitational radiation with the matter.
In the work presented the Hawking set of equations is re-derived properly and generalized for the case or viscose matter filling the universe. The analyses of this set yields the conclusions: (a) the refraction index of the gravitational waves is not 1 even in the case of a perfect fluid -contrary to some previous opinions; (b) under the influence of the cosmological effects the damping of gravitational radiation may change its sign causing an exponential increase of the perturbations -this may lead to the instability of isotrpic homogeneous model of the universe; (c) There is a good analogy between the behaviour of the electromagnetic radiation in a plasma and of the gravitational radiation in a fluid -in the latter case the plasma frequency is replaced by the (imaginary) Jeans (1929) frequency.
2. For the viscose fluid case the generalized Hawking equations for the "electric" (2.6) and "magnetic" (2.7) components of the Weyl tensor (2.5) take the form (2.I6)(2.I9). Scalars μ, p, λ are respectively the density, the pressure and the (dynamic) viscosity of the fluid. The velocity gradient ua;b is decomposed according (2.II) into the acceleration u'a , the expansion θ, the shear σab, and the rotation ωab of the flow lines ua.
Here hab is the projection operator: habub = 0, ηabcd is the alternating tensor; ωa is the rotation vector; Ʇ means the projection by hab. Other notations are ordinary.
To derive the final results the equations of "constraint" and of "motion" (2.20)-(2.25) are needed. The full set of equations (2.I6)-(2.25) is obtained using the Einstein equation (2.2), the Bianchi identity and the hydrodynamic form of energy-momentum tensor (2.3) (Einstein's gravitational constant and the speed of light are 1). The metric tensor gab is of signature (+2), Riemann and Ricci tensors are defined by the relations (2.1) and (2.4).
At vanishing viscosity (λ = 0) the present set of equations have to coincide with the one obtained by Hawking (1966). Nevertheless, there are some discrepancies, among which the most important is the reversed sign of the E-term in (2.22). This term reveals itself still in linear approximation*).
3. In linear approximation after exclusion of H-terms the equations (3.5)-(3.11) for the transverse perturbations (3.12) yield the dispersion equation (4.1). The first two terms in (4.1) correspond to the D'Alambertian in the synchronous frame of reference. Other ones correspond to the interaction of radiation with the matter.
4. Assuming the gravitational wave be of the form (4.3) in synchronous frame of reference, the expression (5.2) for the refraction index n may be found. In some cases (e.g. for the case of vanishing viscosity and of negligible cosmological effects) n > 1. This may result to the focusing of gravitational radiation when the wave propagates through the dense astrophysical object (e.g. through a star). This opportunity will increase the wave intensity in certain spatial regions, being useful for the experimental detection of gravitational radiation.
5. The damping γ of the gravitational waves (6.6) may become negative (according to (2.14) at γ=0 θ< 0 if μ'>0) due to the cosmological effects. In the case of viscose fluid γ reverses its sign also because of negative effective viscosity (7.10). Negative damping means the (gravitational wave) instability of the model supposed. The gravitational wave instability may "be interpreted from the quantum mechanical view-point taking into account the boson nature of transverse gravitational fields. The instability of such a kind is principal for the usual laser action.
6. In this paper we restrict the consideration by the high frequency limit of the dispersion equation. The case of longer wavelength should be investigated in further reports.
*) Another spurious result is contained in the Eq.(26} of Hawking's paper (cf. (7.2) ) compensating the mentioned drawback and yielding the correct result γ=λ/2 for the damping.
Recommended Citation
Abramian, P. B. and Gulian, Armen, "Gravitational Waves in an Evolutionary Universe" (1985). Mathematics, Physics, and Computer Science Faculty Articles and Research. 1211.
https://digitalcommons.chapman.edu/scs_articles/1211
Copyright
The authors
Comments
This article is a preprint and has not undergone peer review.
This article is in Russian, although the extended abstract is in English.