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Bound geodesic orbits around a Kerr black hole can be parametrized by three constants of the motion: the (specific) orbital energy, angular momentum and Carter constant.
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That the mapping { r p , r a , θ min } ↔ { ℰ , ℒ z , 𝒬 } \{r_{\rm p},r_{\rm a},\theta_{\rm min}\}\leftrightarrow\{\mathcal{E},\mathcal{L}_{z},\mathcal{Q}\} is one-to-one can be establishing in the following way. We first note that Schmidt [ 6 ] provides formula for { ℰ , ℒ z , 𝒬 } \{\mathcal{E},\mathcal{L}_{z},\mathcal{Q}\} in terms of ( p , e , θ min ) (p,e,\theta_{\text{min}}) , and that there is a bijection between ( p , e ) ↔ { r p , r a } (p,e)\leftrightarrow\{r_{\rm p},r_{\rm a}\} (straightforward to see from Eqs. ( 2
Cited in the paper.
By “invariant structure” we refer to the fact that the singular surface in the frequency space is invariant under re-parametrization of the orbit, so long as the parameters used are in one-to-one correspondence with { ℰ , ℒ z , 𝒬 } \{\mathcal{E},\mathcal{L}_{z},\mathcal{Q}\}
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T. Damour, Private communication
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L. Blanchet, S. Detweiler, A. Le Tiec, and B. F. Whiting, “High-Accuracy Comparison Between the Post-Newtonian and Self-Force Dynamics of Black-Hole Binaries,” in Mass and Motion in General Relativity , edited by L. Blanchet, A. Spallicci, and B. Whiting (2011) pp. 415–442
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R. Grossman, J. Levin, and G. Perez-Giz, Phys. Rev. D 85
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