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2014-05-24

Michael Efroimsky4

I failed to understand a couple of key points in the paper.

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Lorenzo Iorio (2014)

10.14293/S2199-1006.1.SOR-ASTRO.AWXWIL.v1.RPBUWJ

This work has been published open access under Creative Commons Attribution License **CC BY 4.0**, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at **www.scienceopen.com**.

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.)

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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