Kerr metric, geodesic motion, and Flyby Anomaly in fourth-order Conformal Gravity

Abstract

In this paper we analyze the Kerr geometry in the context of Conformal Gravity, an alternative theory of gravitation, which is a direct extension of General Relativity (GR). Following previous studies in the literature, we introduce an explicit expression of the Kerr metric in Conformal Gravity, which naturally reduces to the standard GR Kerr geometry in the absence of Conformal Gravity effects. As in the standard case, we show that the Hamilton–Jacobi equation governing geodesic motion in a space-time based on this geometry is indeed separable and that a fourth constant of motion—similar to Carter’s constant—can also be introduced in Conformal Gravity. Consequently, we derive the fundamental equations of geodesic motion and show that the problem of solving these equations can be reduced to one of quadratures. In particular, we study the resulting time-like geodesics in Conformal Gravity Kerr geometry by numerically integrating the equations of motion for Earth flyby trajectories of spacecraft. We then compare our results with the existing data of the Flyby Anomaly in order to ascertain whether Conformal Gravity corrections are possibly the origin of this gravitational anomaly. Although Conformal Gravity slightly affects the trajectories of geodesic motion around a rotating spherical object, we show that these corrections are minimal and are not expected to be the origin of the Flyby Anomaly, unless conformal parameters are drastically different from current estimates. Therefore, our results confirm previous analyses, showing that modifications due to Conformal Gravity are not likely to be detected at the Solar System level, but might affect gravity at the galactic or cosmological scale.

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Fig. 1
Fig. 2

Notes

  1. 1.

    We follow here the convention [7] of introducing the energy-momentum tensor \(T_{\mu \nu }\) so that the quantity \(cT_{00}\) has dimensions of an energy density.

  2. 2.

    The function \(a(x)\) in Eq. (16) should not be confused with the angular momentum parameter \(a\).

  3. 3.

    The parameter \(r^{\prime }\) used here should not be confused with the radial coordinate \(r\).

  4. 4.

    Here we are improperly naming the fourth-order CG solutions as “fourth-order Kerr,” and “fourth-order Schwarzschild” solutions. All these solutions were in fact introduced by Mannheim and Kazanas. By using these names we simply mean the equivalent CG fourth-order solutions for the Kerr and Schwarzschild geometries.

  5. 5.

    Following the discussion in Sect. V of Ref. [20], this requires us to set as integration constant \(c=\gamma /(2-3\beta \gamma )\), which defines the transformation \(r=\rho /(1-\rho c)\) between the radial coordinates, \(r\) and \(\rho \), of the two equivalent metrics. This choice also sets the function \(p(r)=r/(1+cr)\), which determines the conformal factor \(\Omega (r)=p(r)/r\) connecting the two metrics.

  6. 6.

    Due to the numerical values of \(\gamma \) and \(\kappa \) in Eq. (12), or in Eq. (13), we have \(k\simeq \kappa \) in Eq. (23). Therefore, the parameters \(k\) and \(\kappa \) are practically equivalent but different in principle and should not be confused.

  7. 7.

    In the form of Eq. (29) our metric looks similar to the well-known Kerr-AdS\(_{4}\) black hole metric [38] for an asymptotically anti-de Sitter space. However, our solution is different and fully satisfies Eq. (4) of Conformal Gravity with \(T_{\mu \nu }=0\). It should also be noted that, since our fourth-order Kerr metric is not asymptotically flat, it might be also superradiantly unstable (as in the case of standard GR, see [39] or [40] for a review).

  8. 8.

    In the metric of Eq. (25) we used the quantity \(u=-\beta (2-3\beta \gamma )\), also defined in Eq. (23) with \(\beta =GM/c^{2}\). Replacing \(\beta \) with the geometrized mass \(M\) yields: \(u=-M(2-3M\gamma )=-2M(1-\frac{3}{2}M\gamma )=-2\widetilde{M}\), which explains the definition of \(\widetilde{M}\) in Eq. (30).

  9. 9.

    The negative sign appearing in Eq. (32), after the first equality sign, is due to our choice of the metric signature (different from the one adopted by Chandrasekhar), so that all the following equations can be compared directly with those in Ref. [37]. It should also be noted that the Lagrangian used to study geodesics is not conformally invariant, even in the non-rotating black hole case [41].

  10. 10.

    As for the other quantities, in the following we will denote the fourth-order constants \(\widetilde{\mathcal {K}}\) (or \(\widetilde{\mathcal {Q}}\)) with a tilde (\(^{\sim }\)) superscript to distinguish them from their second-order equivalents, \(\mathcal {K}\) (or\(\mathcal {\ Q}\)).

  11. 11.

    The problem of finding a rotating perfect-fluid interior solution, which can be matched to a Kerr exterior solution, has not been solved yet.

  12. 12.

    This is actually true for the case of unbound orbits with \(\eta >0\) (or \(\widetilde{\eta }>0\)), which is valid for the spacecraft motion related to the FA. See Ref. [37] (Chapter 7, §63–64) for a full discussion of all other possible cases.

  13. 13.

    JPL/NASA website at: http://ssd.jpl.nasa.gov/horizons.cgi.

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Acknowledgments

The author would like to thank Loyola Marymount University and the Seaver College of Science and Engineering for continued support and for granting a sabbatical leave of absence to the author, during which this work was completed. The author is indebted to Ms. Z. Burstein for helpful comments and for proofreading the original manuscript. Finally, the author also thanks the anonymous reviewers for their useful comments and suggestions, which helped improve the final version of this paper.

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Correspondence to Gabriele U. Varieschi.

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Varieschi, G.U. Kerr metric, geodesic motion, and Flyby Anomaly in fourth-order Conformal Gravity. Gen Relativ Gravit 46, 1741 (2014). https://doi.org/10.1007/s10714-014-1741-z

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Keywords

  • Conformal gravity
  • Kerr metric
  • Geodesics
  • Flyby Anomaly