Compact-Body Sensitivities and Charges

A compact object does not enter an alternative-gravity binary merely through its asymptotic mass. Its binding energy responds to slowly varying extra fields, and that response becomes an effective worldline charge. This page derives the response for a massless scalar with the kinetic normalization stated below, translates it to the traditional Jordan-frame sensitivity, and explains how an isolated stellar or black-hole solution supplies the numbers used in binary dynamics.

Helpful background. Einstein Equations fixes the field-equation normalization; Gravitational Actions and Boundary Terms supplies the variational principle; Formulations of General Relativity supplies asymptotic mass and constraint data; Relativistic Two-Body Dynamics supplies the effective worldline description; Gravitational Radiation and Waveforms supplies the later multipole expansion; and Gravity Beyond General Relativity supplies the theory-space and matter-metric distinctions used here.

Mass as a response function

Use an Einstein-frame metric $g^*_{\mu\nu}$ and dimensionless scalar $\varphi$ with

$$S=\frac{1}{16\pi G_*}\int d^4x\sqrt{-g_*} \left[R_*-2(\nabla\varphi)^2\right] -\sum_A\int m_A(\varphi)\,ds_A^* .$$

The point-particle replacement is adiabatic: the field varies slowly across the body and on its internal relaxation time. At a fixed asymptotic value $\varphi_0$, define

$$\alpha_A=\left.\frac{\partial\ln m_A}{\partial\varphi_0}\right|_{N_A,S_A,J_A}, \qquad \beta_A=\left.\frac{\partial\alpha_A}{\partial\varphi_0}\right|_{N_A,S_A,J_A}.$$

The held-fixed data—baryon number $N_A$, entropy $S_A$, and angular momentum $J_A$ when relevant—are part of the definition. Damour and Esposito-Farèse use precisely this mass derivative as the strong-field scalar coupling and its second derivative as the susceptibility in Damour and Esposito-Farèse 1996, § II, eqs. (2.3)–(2.4).

Varying the action gives

$$\Box_*\varphi=-4\pi G_*\sum_A\alpha_A(\varphi)T_{*A} .$$

Here $T_{*A}$ is the trace of the effective stress tensor of body $A$; the response must stay inside the sum because strong-field bodies can have different $\alpha_A$. For an isolated static point body, $T_{*A}=-m_A\delta^3(\boldsymbol{x})$, so $\nabla^2\varphi=4\pi G_*m_A\alpha_A\delta^3(\boldsymbol{x})$. Since $\nabla^2(1/r)=-4\pi\delta^3(\boldsymbol{x})$,

$$\varphi=\varphi_0-\frac{G_*q_A}{r}+O(r^{-2}), \qquad q_A\equiv m_A\alpha_A .$$

Thus our source charge $q_A$ has dimensions of mass and the exterior $1/r$ coefficient is $-G_*q_A$. Some references instead call that coefficient the charge; their sign is therefore opposite. In the notation of Damour and Esposito-Farèse 1996, § II, eq. (2.3), $\omega_A=-m_A\alpha_A=-q_A$.

From Jordan sensitivity to scalar charge

In the Jordan description, matter is minimally coupled to the physical metric and the compact mass is written $M_A(\phi)$. Its sensitivity is

$$s_A=\left.\frac{\partial\ln M_A}{\partial\ln\phi}\right|_{\phi_0,N_A}.$$

Writing $\Psi=(\phi-\phi_0)/\phi_0$, $M_{A0}=M_A(\phi_0)$, and $s_A'=\partial s_A/\partial\ln\phi$, an ordinary Taylor expansion gives

$$M_A(\phi)=M_{A0}\left[1+s_A\Psi +\frac12\left(s_A^2+s_A'-s_A\right)\Psi^2+O(\Psi^3)\right].$$

This is the compact-body expansion derived in Mirshekari and Will 2013, § V A, eqs. (5.1)–(5.3), pp. 15–16. The $-s_A$ term is easy to miss: derivatives with respect to $\ln\phi$ are not derivatives with respect to $\Psi$.

Do not identify $M_{A0}$ silently with the Einstein-frame mass $m_A$. If $\widetilde g_{\mu\nu}=A^2(\varphi)g^*_{\mu\nu}$, equality of the worldline action gives $m_A=A M_A$. For the simplest massless one-scalar theory, $A\propto\phi^{-1/2}$, $\alpha_0=d\ln A/d\varphi|_0$, and $\alpha_0^2=(2\omega_0+3)^{-1}$. Therefore

$$\alpha_A=\alpha_0(1-2s_A).$$

This relation is not a definition valid for every scalar theory. Nonlinear scalarization can make $\alpha_A$ a nonperturbative function of mass and $\varphi_0$ even when the weak-field coupling $\alpha_0$ is small. Likewise, a vector or æther theory uses a different response variable. If $u^\mu$ is a preferred unit timelike field and $\gamma_A=-u_\mu v_A^\mu$, a compact-body action may contain

$$S_A=-m_A\int ds_A\left[1-\sigma_A(\gamma_A-1) +\frac12\sigma_A'(\gamma_A-1)^2+\cdots\right].$$

Here $\sigma_A=-d\ln m_A(\gamma)/d\ln\gamma|_1$, while $\sigma_A'$ names the quadratic expansion coefficient; it is not simply $d\sigma_A/d\gamma$. If $\bar\sigma_A=d^2\ln m_A/d(\ln\gamma)^2|_1$, Taylor expansion gives $\sigma_A'=\sigma_A+\sigma_A^2+\bar\sigma_A$. These definitions and the action are given in Foster 2007, § II.A, Eqs. (4)–(6). As a sign check, in the local æther frame $ds=dt\sqrt{1-v^2}$ and $\gamma-1=v^2/2+\cdots$, so the coordinate-time Lagrangian is $-m_A+m_A(1+\sigma_A)v^2/2+\cdots$. Its inertial mass is $m_A(1+\sigma_A)$. These responses are not scalar charges; their source multipoles and preferred-frame forces depend on the vector action.

Matching an isolated body

A sensitivity must be calculated on the same solution branch later used for the binary. For a star, choose an equation of state and central data, integrate the interior equations with a regular center, match at the surface, and impose $\varphi\to\varphi_0$ at infinity. Read the ADM mass $m_A$ and the $1/r$ coefficient, then vary $\varphi_0$ while holding baryon number fixed. Damour and Esposito-Farèse give the exterior solution and surface matching that extract $m_A$ and $\alpha_A$ in Damour and Esposito-Farèse 1996, § III, eqs. (3.8)–(3.13).

Two independent checks are available:

1. compare $q_A=m_A\alpha_A$ obtained by differentiating the mass curve with $-1/G_*$ times the fitted exterior coefficient;
2. repeat the derivative at fixed baryon number with two step sizes and verify convergence.

For weakly self-gravitating matter, $s_A$ is proportional to minus the binding energy per unit mass; therefore bodies of different compactness fall differently in an external scalar gradient. Mirshekari and Will exhibit this strong-equivalence-principle violation in Mirshekari and Will 2013, § VII B, eqs. (7.4)–(7.7), pp. 28–29. In the regular stationary black-hole branch of massless Brans–Dicke theory, $s_{\rm BH}=1/2$ and hence $\alpha_{\rm BH}=0$; this conclusion requires the no-hair hypotheses and does not transfer automatically to scalar–curvature or time-dependent hairy branches.

Checks and limitations

The dimensions close: $[G_*]=L/M$, $[q_A]=M$, so $G_*q_A/r$ is dimensionless. The sign follows directly from the Green function and the declared scalar equation. A sufficient decoupling limit to GR sends $\alpha_A,\beta_A\to0$, whereas equality $\alpha_1=\alpha_2$ alone is insufficient: equal nonzero charges remove the leading dipole but leave a conservative scalar force. An uncharged branch with $\alpha_A=0$ can have a GR background while retaining $\beta_A\ne0$ and a susceptibility to scalar perturbations; agreement of that solution with GR is a weaker statement than decoupling the extra field. The point-particle model fails when the external field varies on the object scale, excites internal modes, crosses a scalarization bifurcation nonadiabatically, or probes a field-theory cutoff.

Exercises

Extract a sensitivity and verify its charge. Suppose an isolated sequence at fixed baryon number has

$$m_A(\varphi_0)=M\left[1+a\varphi_0+\frac12b\varphi_0^2+O(\varphi_0^3)\right].$$

Find $\alpha_A(0)$, $\beta_A(0)$, the exterior field through first order, and the condition for this body to have no scalar charge at the background.

Solution

Taking logarithms gives $\ln m_A=\ln M+a\varphi_0+(b-a^2)\varphi_0^2/2+\cdots$. Hence $\alpha_A(0)=a$ and $\beta_A(0)=b-a^2$. The source charge is $q_A=Ma$ at this order, so $\varphi=\varphi_0-G_*Ma/r+O(r^{-2})$. The body is uncharged at this background precisely when $a=0$; $b$ may remain nonzero, describing a susceptibility that can generate charge when the ambient field changes.

References

• Damour, Thibault, and Gilles Esposito-Farèse. “Tensor-Scalar Gravity and Binary-Pulsar Experiments.” Physical Review D 54 (1996): 1474–1491. DOI; Open PDF.
• 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.