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Review of 'Orbital effects of a monochromatic plane gravitational wave with ultra-low frequency incident on a gravitationally bound two-body system'

I failed to understand a couple of key points in the paper.
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Orbital effects of a monochromatic plane gravitational wave with ultra-low frequency incident on a gravitationally bound two-body system

 Lorenzo Iorio (2014)
We analytically compute the long-term orbital variations of a test particle orbiting a central body acted upon by an incident monochromatic plane gravitational wave. We assume that the characteristic size of the perturbed two-body system is much smaller than the wavelength of the wave. Moreover, we also suppose that the wave's frequency νg is much smaller than the particle's orbital one nb. We make neither a priori assumptions about the direction of the wavevector k nor on the orbital configuration of the particle. While the semi-major axis a is left unaffected, the eccentricity e, the inclination I, the longitude of the ascending node Ω, the longitude of pericenter ϖ and the mean anomaly ℳ undergo non-vanishing long-term changes of the form dΨ/dt=F(Kij;e,I,Ω,ω),Ψ=e,I,Ω,ϖ,M, where Kij, i,j=1,2,3 are the coefficients of the tidal matrix K. Thus, in addition to the variations of its orientation in space, the shape of the orbit would be altered as well. Strictly speaking, such effects are not secular trends because of the slow modulation introduced by K and by the orbital elements themselves: they exhibit peculiar long-term temporal patterns which would be potentially of help for their detection in multidecadal analyses of extended data records of planetary observations of various kinds. In particular, they could be useful in performing independent tests of the inflation-driven ultra-low gravitational waves whose imprint may have been indirectly detected in the Cosmic Microwave Background by the Earth-based experiment BICEP2. Our calculation holds, in general, for any gravitationally bound two-body system whose orbital frequency nb is much larger than the frequency νg of the external wave, like, e.g., extrasolar planets and the stars orbiting the Galactic black hole. It is also valid for a generic perturbation of tidal type with constant coefficients over timescales of the order of the orbital period of the perturbed particle.

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    Review text

    The work addresses long-term variations of the Keplerian orbital elements of a test particle
    orbiting a central body acted upon by a plane monochromatic gravitational wave. Both the
    shape and orientation of the orbit turn out to be disturbed by the wave. The calculation holds
    for a two-body system whose mean motion n is much larger than the frequency g of the
    incident gravitational wave.
    I am having mixed feelings about this paper, because I failed to understand a couple of key
    points in it. So I would urge the author to elucidate those points in a more comprehensive version of the paper..

    1. The orbital elements show up as variable “constants” within a perturbed Kepler problem. In that setting, the 00 component of the metric is g_{00} = 2U/c^2 , where U is the Newtonian potential. The author is writing his formulae in the TT gauge, which implies h_{00} = 0 . It therefore would be important to justify accurately the compatibility of these two assertions.

    I am raising this issue because, whenever we introduce a background geometry and impose a gauge on its perturbation, we must prove that the chosen gauge can be propagated on the chosen background – if not globally, then at least over a sufficiently large scale of times and distances and with a precision sufficient for practical applications.
    The TT (transverse, traceless) gauge condition is introduced as
    h_{\mu\nu} U^{\nu} = 0 where U^{\nu ; \mu} = 0
    h{\mu}_{\mu} = 0
    It implies the Lorentz, traceless gauge, but not conversely. Recall that the Lorentz, traceless gauge is given by

    h_{\mu\nu}^{\mu} = 0
    h^{\mu}_{\mu} = 0

    Generally, the TT gauge can be imposed only locally, i.e., over distances much smaller than the typical radius R of curvature of the background. This restriction stems from the fact that the background curvature prevents a vector field from being covariantly constant over regions larger than R, as evident from the expression for the change of a vector over a closed walk. As well known, this expression is: \Delta U_{\mu} = (1/2) \int R^{\kappa}_{\mu\rho\nu} U_{\kappa} d f^{\rho\nu} , where d f^{\rho\nu} denotes an infinitesimal area.

    As a simple example, let us recall how to propagate the Lorentz traceless gauge, when we are dealing with waves on a flat background or with high-frequency waves. From the equation of motion for weak gravitational waves, it possible to show that the trace h satisfies
    h^{; \alpha}_{; \alpha} + 2 R_{\kappa\alpha} h^{\kappa\alpha} = 0
    In vacuum or in the high-frequency limit, this becomes h^{; \alpha}_{; \alpha} = 0 . If the solution to the latter equation is set nil on an initial hypersurface, it remains nil everywhere
    From the equation for weak gravitational waves, it is also possible to prove that the covector field h_{\beta}^{\mu}_{; \mu} satisfies the equation
    h_{\beta}^{\mu}_{; \mu}^{; \alpha}_{; \alpha} + (a linear combination of the components of the Ricci tensor) = 0
    In vacuum or in the high-frequency limit, the above equation becomes: h_{\beta}^{\mu}_{; \mu}^{; \alpha}_{; \alpha} = 0 , so each component of the covector field h_{\beta}^{\mu}_{; \mu} obeys, independently from the other components, the same wave equation. Being set zero on an initial hypersurface, this component remains zero all over the space-time.
    Thus we have proven that, in vacuum or in the high-frequency limit, the equation of motion for the weak gravitational waves is globally compatible with the traceless, Lorentz gauge

    Similarly to this example, the author should prove that the TT gauge, which he has chosen, is globally compatible with the gravitational background of the Kepler problem. (Recall that in the considered problem we are dealing not with the high-frequency limit but with a low-frequency one.)
    If the gauge is not compatible globally, then over what scales should it be compatible, in order for the considered problem to render useful observable quantities?
    2. Very much in line with my previous question, I would also enquire as to what the time t in the author’s formulae is. Given the deviation of g_{00} from unity in the Kepler problem, the authors should comment on how this t will relate to the time of a distant observer. (Be mindful that gravitational waves will influence not only the orbit and clocks, but also the propagation of light rays.)


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