Nordstrom's scalar theory of gravity and the equivalence principle

TL;DR

Deruelle shows that Nordström’s scalar theory of gravity, like General Relativity, can satisfy the strong equivalence principle beyond post-Newtonian order, with gravitational mass equal to inertial mass when \(F(\Phi)=\Phi\). The paper supports this via explicit examples and a proposed Nordström-Katz superpotential, while noting the general proof remains incomplete.

Key findings

• Nordström’s theory is observationally ruled out: it gives PPN parameter \(\gamma=-1\), perihelion advance \(\Delta\omega = -\frac{1+a_2}{6}\Delta\omega_{\rm GR}\), and \(\beta=\frac{1+a_2}{2}\).
• For a two-body system at lowest order, the paper derives \(M_g/M_{\rm in}=1+\eta E_g/(Mc^2)\) with \(\eta=2a_2\), so \(M_g=M_{\rm in}\) if \(a_2=0\).
• For a constant-density perfect-fluid star, it finds \(M_g/M_{\rm in}=1+\eta E_g/(Mc^2)\) with \(\eta=2a_2\) and \(E_g=-3GM^2/(5R)\); a Mathematica check shows \(M_{\rm in}=M_g\) exactly if \(F(\Phi)=\Phi\).
• In the Jordan-frame superpotential approach, the inertial mass is \(M_{\rm in}=\frac{M_g}{3}(1-\alpha)\); fixing \(\alpha=-2\) from the Einstein-frame constant-density result gives \(M_{\rm in}=M_g\).
• Appendix result: for a radiating binary, the leading energy-loss term is \(1/6\) the general relativistic value, with an orbit-averaged expression given in the paper.

Methods

• Theoretical study of Nordström’s scalar gravity in both the Einstein frame (scalar field on flat spacetime) and Jordan frame (conformally flat metric).
• Uses analytic calculations for two point-like particles at lowest order and for a static, spherically symmetric constant-density perfect-fluid star, including a numerical/symbolic check of junction conditions.
• Constructs a Katz-type superpotential and conserved charge for Nordström’s Jordan-frame action, and proposes a Palatini-like action for future work.

Limitations

• The paper states that its argument that the strong equivalence principle always holds in Nordström’s final theory is “a bit weak.”
• The Jordan-frame Lagrange multiplier \(\lambda^{\mu\nu\rho\sigma}\) has ten independent components while the traceless equations give only nine, so the system is underdetermined and the inertial mass is not fully fixed; the constant \(\alpha\) is fixed by returning to the Einstein-frame constant-density example.
• The explicit star calculation uses the “unrealistic but simple” constant-density equation of state; more realistic barotropic fluids are cited as studied numerically elsewhere but are not used to prove the general case.
• The proposed Palatini-like action and construction of a “Nordström-Katz” superpotential are left to further work.