Post-Newtonian Dynamics Beyond GR

Post-Newtonian dynamics beyond general relativity is not obtained by inserting one arbitrary coefficient into a GR acceleration. The field content determines new long-range exchanges; compact-body response fixes their charges; and the PN expansion then orders velocity, nonlinear-field, preferred-frame, and radiation-reaction operators. A massless scalar supplies the threaded example, while a separate operator table shows what changes when a preferred vector is present.

Helpful background. Einstein Equations supplies the GR limit; Gravitational Actions and Boundary Terms fixes exchange normalizations; Formulations of General Relativity supplies gauge and constraint language; Relativistic Two-Body Dynamics supplies PN counting and invariant orbital variables; Gravitational Radiation and Waveforms supplies the balance-law interface; and Gravity Beyond General Relativity supplies the field-content and cutoff record.

Newtonian exchange with body-dependent charges

Continue with the Einstein-frame point-body action and charges $q_A=m_A\alpha_A$ defined on the preceding page. One-graviton and one-scalar exchange give

$$L_N=\frac12m_1v_1^2+\frac12m_2v_2^2 +\frac{G_{12}m_1m_2}{r}, \qquad G_{12}=G_*\left(1+\alpha_1\alpha_2\right).$$

The potential energy is therefore $V_N=-G_{12}m_1m_2/r$. Equal-sign scalar charges strengthen attraction; opposite-sign charges weaken it, subject to $G_{12}\gt0$ on the admitted branch. In relative variables $\boldsymbol{x}=\boldsymbol{x}_1-\boldsymbol{x}_2$, $M=m_1+m_2$, and $\boldsymbol{n}=\boldsymbol{x}/r$,

$$\boldsymbol{a}_N=-\frac{G_{12}M}{r^2}\boldsymbol{n}.$$

In Jordan variables Mirshekari and Will write the same coefficient as $G\alpha$, where their $\alpha$ contains $(1-2s_1)(1-2s_2)$; see Mirshekari and Will 2013, § I, eqs. (1.3)–(1.5) and Table I, pp. 3–4. Their symbol $\alpha$ is a two-body potential coefficient, not either body’s $\alpha_A$ here.

This already violates the strong equivalence principle: $G_{12}$ depends on the bodies’ internal binding energies through $\alpha_A$. It also shows why a universal rescaling of Newton’s constant is generally insufficient. Three bodies carry three pairwise products $\alpha_A\alpha_B$, which cannot all be represented by one constant unless their charges are universal.

Which operators enter at each PN order?

Let $\epsilon\sim v^2\sim G_{12}M/r$. After eliminating redundant accelerations by a coordinate or contact transformation, a nonspinning, parity-even two-body Lagrangian can be organized using $v_1^2$, $v_2^2$, $\boldsymbol{v}_1\!\cdot\!\boldsymbol{v}_2$, $\boldsymbol{n}\!\cdot\!\boldsymbol{v}_A$, and powers of $G_{12}M/r$. The coefficients, not the kinematic basis alone, carry the theory.

Relative order	Massless scalar–tensor content	Additional preferred-vector content	Status
Newtonian (0PN)	$G_{12}=G_*(1+\alpha_1\alpha_2)$	body/æther sensitivities change the relation between rest, inertial, and active gravitational masses	conservative
1PN	velocity potentials and nonlinear scalar response through $\alpha_A,\beta_A$	scalars built from the binary velocity relative to the æther, such as $w^2$, $(\boldsymbol{n}\!\cdot\!\boldsymbol{w})^2$, and $\boldsymbol{v}\!\cdot\!\boldsymbol{w}$	conservative
1.5PN	leading scalar dipole reaction when $\alpha_1\ne\alpha_2$	spin-0/spin-1 dipole and monopole channels when allowed by the vector constraints	dissipative
2PN	higher response derivatives and nonlinear two-field potentials	further æther-velocity and sensitivity operators	conservative
2.5PN	tensor quadrupole reaction plus PN corrections to scalar multipoles	tensor, vector, and scalar quadrupole reaction with mode-speed factors	dissipative

For the massless scalar theory, the explicit 1PN acceleration contains $\bar\gamma$ and two body-dependent $\bar\beta_A$ coefficients, while 2PN introduces $\bar\delta_A$ and $\bar\chi_A$. Their definitions in terms of $\omega(\phi)$ and $s_A,s_A',s_A''$ are tabulated in Mirshekari and Will 2013, Table I and eqs. (1.4)–(1.8), pp. 3–5. The full conservative body accelerations and Lagrangian are given in Mirshekari and Will 2013, § VI A, eqs. (6.1)–(6.4), pp. 21–24.

The preferred-vector column is an operator inventory, not a universal formula. Which coefficients survive depends on the vector constraint, matter metric, mode speeds, and sensitivities. Einstein–æther strong-field point-body dynamics illustrates this dependence in Foster 2007, § II.A, Eqs. (4)–(6), and § III. A calculation that writes preferred-frame terms without specifying the æther rest frame and its coupling has not defined an observable theory.

Gauge-dependent acceleration, invariant orbit

Individual PN acceleration coefficients depend on coordinates. For a circular orbit the invariant frequency parameter

$$x=\left(G_{12}M\Omega\right)^{2/3}$$

removes the coordinate radius. Newtonian force balance gives

$$\Omega^2=\frac{G_{12}M}{r^3}, \qquad x=\frac{G_{12}M}{r},$$

and the conservative binding energy takes the form

$$E(x)=-\frac12\mu x\left[1+e_1(\eta,\alpha_A,\beta_A)x +e_2(\eta,\alpha_A,\beta_A,\ldots)x^2+\cdots\right],$$

where $\mu=m_1m_2/M$ and $\eta=\mu/M$. This invariant relation, rather than a coordinate coefficient in $\boldsymbol a$, is the portable conservative output for a waveform. Mirshekari and Will construct the relative equation and the conserved energy before adding radiation reaction in Mirshekari and Will 2013, §§ VI A and VI C, eqs. (6.3)–(6.13), pp. 22–26.

Two limits discriminate different physics. If $\alpha_A,\beta_A\to0$, then $G_{12}\to G_*$ and $e_n$ approach their GR values. If instead $\alpha_1=\alpha_2\ne0$, the dipole channel vanishes but $G_{12}\ne G_*$ and scalar nonlinearities can remain. “No dipole” is therefore not the GR limit.

Consistency checks and regime

Every term in $\boldsymbol a$ must scale as $1/L$: the Newtonian term is $G_{12}M/r^2$, while a 1PN correction carries one extra dimensionless factor $v^2$ or $G_{12}M/r$. Conservative terms must follow from a time-reversal-even Lagrangian and admit conserved $E$ and $\boldsymbol J$. Dissipative half-integer terms must instead make the mechanical energy decrease after total time derivatives are removed. The 2.5PN calculation explicitly separates these pieces and checks the GR limit in Mirshekari and Will 2013, § VII A, pp. 27–28.

The expansion fails near merger when $x$ is no longer small, near a nonadiabatic charge transition, when a massive field crosses its radiation threshold, or when gradients approach the theory cutoff. Strong-field charges must be matched on the same isolated-object branch used to interpret the binary; inserting weak-field PPN coefficients for a scalarized star is inconsistent.

Exercises

Invariant Newtonian diagnostic. Starting from $L_N$ above, derive the circular-orbit relation and $E(x)$. Then decide what changes when $\alpha_1=\alpha_2=a\ne0$.

Solution

In the center-of-mass frame $L_N=\mu v^2/2+G_{12}\mu M/r$. Radial force balance for $v=r\Omega$ gives $\mu r\Omega^2=G_{12}\mu M/r^2$, hence $\Omega^2=G_{12}M/r^3$ and $x=G_{12}M/r$. The mechanical energy is $E=\mu v^2/2-G_{12}\mu M/r=-G_{12}\mu M/(2r)=-\mu x/2$. If both charges equal $a$, then $G_{12}=G_*(1+a^2)$ and this conservative relation uses the modified coupling. The charge difference is zero, so leading scalar dipole radiation vanishes, but the system has not reached GR unless $a\to0$ and the higher responses also decouple.

References

• Foster, Brendan Z. “Strong Field Effects on Binary Systems in Einstein-Æther Theory.” Physical Review D 76 (2007): 084033. DOI; Open PDF.
• Mirshekari, Saeed, and Clifford M. Will. “Compact Binary Systems in Scalar-Tensor Gravity: Equations of Motion to 2.5 Post-Newtonian Order.” Physical Review D 87 (2013): 084070. DOI; Open PDF.