Post-Newtonian N-body dynamics in extended theories of gravity

Abstract We derive the complete first post-Newtonian (1PN) Lagrangian and corresponding equations of motion for the relativistic N -body system in Scalar-Tensor-Fourth-Order Gravity (STFOG), including the Non-Commutative Spectral Geometry (NCSG) sector as a special case. In the regime $$\Phi \sim \Psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>Ψ</mml:mi> </mml:mrow> </mml:math> ( $$\gamma \sim 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), the linearized fourth-order field equations are solved in the Standard Post-Newtonian gauge, and the variational Lagrangian is built directly from the point-particle action. The resulting dynamics is governed by three Yukawa functions $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> , $$\mathcal {W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> and $$\Xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ξ</mml:mi> </mml:math> , which encode the scalar, gravitomagnetic and three-body sectors and depend on the effective masses $$(m_R,m_Y,m_\phi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the additional propagating modes. In this context, we show that the nonlinear metric component (4) $$g_{00}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>00</mml:mn> </mml:msub> </mml:math> plays no role at 1PN level. The 1PN orbital motion of the above extended theories is thus obtained in closed form, and the Einstein–Infeld–Hoffmann equations are recovered in the corresponding general-relativistic limit. The formalism provides a common framework for relativistic celestial mechanics and applies to Solar-System dynamics, binary pulsars such as PSR J0737–3039, Galactic-center stellar orbits, and triple systems.