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In Newtonian mechanics we perform a multipole expansion on the gravitational potential $V(r)=-GM/r$ by a series expansion of Legendre polynomials. Then the Hamiltonian is given by

\begin{equation} H = T + \tilde{V}, \end{equation}

where $\tilde{V}$ is the expanded gravitational potential. What about in GR?

If we describe the spacetime produced by the Earth with the Schwarzschild metric then the associated Hamiltonian of a test particle in free fall around the Earth is given by

\begin{equation} H = \frac{1}{2} \left(A(r)^{-1} p_t^2 - B(r)^{-1}p_r^2-\frac{p_{\theta}^2}{r^2} - \frac{p_\phi^2}{r^2 \sin^2 \theta} \right) + V(r), \end{equation} where $V(r) = -GM/r.$ To account for the non-spherical geometry of the Earth in GR Is it simply a case of doing the same expansion? I have my doubts because things like this are never as easy as they first appear.

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It is possible to expand the Einstein field equations around their spherical symmetric solutions using so called tensor harmonics. Pioneering work on the topic has been done by T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957), F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970) and many others.

In general the key idea is to expand the metric around the spherical symmetric one using spherical harmonics generalized to tensor and then solving the resulting field equations. In general this is not trivial and I do not know much about the general case.

I am however rather familiar with a special case or application of this approach: the so called Hartle-Thorne formalism: In a series of papers J. B. Hartle, K. S. Thorne and collaborators applied this approach to slowly rotating stars: expanding the metric in the angular velocity around the spherical symmetric one. This is a rather simple application and in its original form (up to second order in the angular velocity) it describes mono- and quadrupole perturbations away from spherical symmetry as well as an $l=1$ frame dragging term. The line element reads:

$$ds^2=-e^{\nu(r)}\left[1+2(h_0(r)+h_2(r)P_2(\cos(\theta)))\right]dt^2+e^{\lambda(r)}\left[1+2(m_0(r)+m_2(r)P_2(\cos(\theta)))/(r-2m(r))\right]dr^2+r^2(1+2k_2(r)P_2(\cos(\theta)))\left[d\theta^2+\sin(\theta)^2(d\phi-\omega(r)dt)^2\right].$$

Treating $h_0$, $h_2$, $m_0$, $m_2$ and $k_2$ as small perturbations and $\omega$ as first order frame dragging term one can solve the Einstein field equations up to second order in the angular velocity. This approach can be used to describe not only deformations due to rotation but also to describe small deformations due to magnetic fields (eg. K. Ioka and M. Sasaki, Astrophys. J. 600:296-316 (2004)).

To conclude a few words on the specific question raised by OP: It is not a "simple" expansion but rather a complicated one and in general the geodesic equations can not be reduced back to a simple Hamiltonian with an effectiv potential in a $V(r)$ form.

There are analytic solutions for the metric perturbation (in case of the Hartle-Thone ansatz) but the geodesic equations become rather lengthy. They include effects like frame dragging as well as the perturbations away form spherical symmetry. The exterior solutions depend only on a few global parameters like, mass and radius of the unperturbed star and mass shift, angular momentum, angular velocity and mass quadrupole moment of the perturbed configuration. In that regard the geodesic equations are not that complicated in this case but that is just the case in this very special ansatz, which has very high symmetry and describes only the lowest order perturbation away from spherical symmetry and only for small perturbations.

For arbitrary large perturbations and higher "multipolarity" things become much more complicated and ultimately a more general treatment in the framework of Numerical Relativity (NR) becomes necessary.

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What you want is knowing the effective gravitational potential given by the Einstein equations when your source isn't spherically symmetric using superposition principle?

I'm not sure if I understanded well your question, that's because I am asking. Maybe my question will get you to the right answer (that kind of superposition principle doesn't work on GR).

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  • $\begingroup$ Not the effective potential. I am interested in whether it is possible to expand the potential given in the Hamiltonian above in a series expansion using Legendre polynomials to account for the slightly non-spherical geometry of the Earth. I feel it is not that simple as it is in the Newtonian case. $\endgroup$ – Rumplestillskin Mar 6 '17 at 5:07
  • $\begingroup$ What I mean is with "effective potential" is not "effective potential because of rotation", I mean "effective potential because of gravitational curvature". I guess it's a good approximation in the regime where the corrections to the corrections given by slightly non-spherical geometry of the Earth to newtonian gravity by GR are small, and that approximation is a lot less bigger than the sum of the first correction plus schwarschild correction. I don't work in GR, so maybe I'm wrong, but it seems reasonable to me that your approximation is good in a lot of regimes. $\endgroup$ – Uli_WH Mar 6 '17 at 5:34
  • $\begingroup$ I probably also don't understand what you are asking, but I fail to see how accounting for a slightly no spherical geometry could accounted for with a spherically symmetric V(r). Seems to me you'd have to take the parametrized post Newtonian (PPN) approximation for a V, and it'll have the mass term, a dipole term if that's enough, quadrupole if not. $\endgroup$ – Bob Bee Mar 6 '17 at 7:35
  • $\begingroup$ This does not really answer the question. If you have a different question, you can ask it by clicking Ask Question. You can also add a bounty to draw more attention to this question once you have enough reputation. - From Review $\endgroup$ – lucas Mar 6 '17 at 8:41
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    $\begingroup$ I don't know if your approximations will be consistent and GR is non linear. To be exact you'd have to start with the Schwartzscild solution, and perturba it and keep the terms consistent with your approximation. I'm sure it's been done, and resulted in dipole and quadrupole moments. But then you have to solve the geodesic equations in that perturbed metric, which would account for the V terms. Remember that even without that there are terms such as perihelion if mercury. Don't know if your approximation would work. Look for PPN formalism or for GR orbits in small perturbations off spherical. $\endgroup$ – Bob Bee Mar 6 '17 at 18:48

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