Environmental Matching and Equivalence-Principle Effects

Screened forces are determined by a boundary-value problem spanning the inside of a body, its transition layer, and its environment. The result can be compressed into an effective scalar charge, but that charge is not generally proportional to inertial mass. Two bodies made of the same microscopic matter can therefore fall differently if their sizes, compactnesses, density profiles, or environments give them different screened charge fractions.

Required background. Ordinary Differential Equations supplies radial boundary-value matching and asymptotic zones.

Helpful background. Einstein Equations supplies the metric acceleration and equivalence-principle comparison; Gravitational Actions and Boundary Terms supplies junction and flux terms; Formulations of General Relativity supplies constraint and boundary-data language; Gravity Beyond General Relativity supplies branch and limit checks; and Relativistic Matter and Compact Objects supplies extended sources whose internal structure affects their scalar charge.

Four radial zones

Use $M_{\rm Pl}^{-2}=8\pi G$ and a nearly constant dimensionless coupling $\beta$. For a chameleon-like scalar around a spherical body, distinguish:

1. an interior zone, where $\phi$ lies near the dense-matter minimum $\phi_c$;
2. a transition shell, beginning at $r_s$, where the field leaves that minimum;
3. an exterior near zone, where the body’s scalar flux is measured; and
4. a far environment, which fixes $\phi_\infty$ and the incoming external gradient.

In the quasistatic weak-field approximation,

$$\frac{1}{r^2}\frac{d}{dr} \left(r^2\phi'\right) =V_{,\phi}+\frac{\beta\rho(r)}{M_{\rm Pl}}.$$

Both $\phi$ and its normal flux are continuous across an ordinary material surface unless the action contains an explicit surface source. Regularity requires $\phi'(0)=0$, while the far condition is $\phi\to\phi_\infty$, not an arbitrarily chosen vacuum value. These conditions select the physical branch.

From the shell to an effective charge

If the net source is negligible inside $r_s$ because the field follows its minimum, and $V_{,\phi}$ is negligible compared with $\beta\rho/M_{\rm Pl}$ within the active shell, radial integration gives

$$\phi'(r) =\frac{\beta}{4\pi M_{\rm Pl}} \frac{M(r)-M(r_s)}{r^2}, \qquad M(r)=4\pi\int_0^r ds\,s^2\rho(s).$$

Outside a body of radius $R$, define

$$Q\equiv\beta\,[M-M(r_s)], \qquad \epsilon\equiv\frac{Q}{\beta M} =1-\frac{M(r_s)}{M}.$$

Here $\epsilon=1$ is fully unscreened and $\epsilon\ll1$ is thin-shell screened. This charge has dimensions of mass. In a linear massive exterior, surface-flux matching gives

$$\phi-\phi_\infty =-\frac{Q}{4\pi M_{\rm Pl}(1+m_\infty R)} \frac{e^{-m_\infty(r-R)}}r.$$

Its gradient includes $(1+m_\infty r)e^{-m_\infty(r-R)}/(1+m_\infty R)$; the massless $1/r^2$ flux is constant only at distances small compared with the ambient Compton wavelength. See the thin-shell derivation. Burrage and Sakstein derive the flux, screening radius, and exterior force in Burrage and Sakstein 2018, § 3.1, equations (3.1)–(3.10).

For a constant-density sphere,

$$\epsilon=1-\left(\frac{r_s}{R}\right)^3 \simeq3\frac{\Delta R}{R}, \qquad \Delta R=R-r_s.$$

This mass fraction is safer than the last approximation when the shell is not thin. If $r_s/R=0.99$, then $\epsilon=1-0.99^3=0.029701$: about three percent of the mass sources the exterior scalar.

Environmental matching

The screening radius follows from matching to the ambient field. In the commonly used constant-coupling parametrization,

$$\chi\equiv\frac{\phi_\infty}{2\beta M_{\rm Pl}} =4\pi G\int_{r_s}^{\infty}dr\,r\,\delta\rho(r),$$

after the small interior field is neglected. This radial formula assumes a spherically symmetric density contrast, a nearly constant coupling, and a range long enough that the Yukawa term can be neglected in the matching region. It is not an exact rule for an arbitrarily placed host. A nearby host can make $\phi_\infty$ small or contribute a large external potential, screening an object that would be unscreened in isolation.

This is why density at one point is not a screening criterion. One must state the host, separation, ambient solution, and scalar Compton wavelength. Nonlinearity also prevents superposing isolated profiles. A practical calculation solves the host environment first, uses its field and gradient as outer data for the body, then checks overlap between the body, transition, and far-zone expansions.

Effective charges and free fall

Let body $A$ move in a slowly varying external scalar field. Its center-of-mass acceleration is

$$\boldsymbol a_A =-\nabla\Phi_N -\frac{Q_A}{M_A M_{\rm Pl}}\nabla\phi_{\rm ext} =-\nabla\Phi_N -\frac{\beta\epsilon_A}{M_{\rm Pl}}\nabla\phi_{\rm ext}.$$

For bodies $A$ and $B$ at the same event,

$$\Delta\boldsymbol a =-\frac{\beta(\epsilon_A-\epsilon_B)}{M_{\rm Pl}} \nabla\phi_{\rm ext}.$$

A universal microscopic conformal coupling can therefore coexist with different accelerations of extended, self-gravitating bodies. The distinction is important: fully unscreened test particles with the same microscopic coupling obey universality of free fall, while partially screened bodies carry structure-dependent scalar charges. The charge formula and its equivalence-principle consequence are given in Burrage and Sakstein 2018, § 3.3, especially Eq. (3.34).

Composition can enter if microscopic couplings are nonuniversal, but size and compactness effects arise even for universal coupling. Calling every differential acceleration “composition dependence” would obscure that separation.

Matching workflow and checks

A controlled calculation should:

1. specify the action, conformal frame, density convention, and branch;
2. solve the environmental background and record $\phi_\infty$, $\nabla\phi_\infty$, and the local mass;
3. integrate the interior with regular central data;
4. locate the transition without assuming a thin shell;
5. match field and flux at every interface;
6. extract $Q$ in an overlap zone and vary that zone;
7. check EFT control, nonsphericity, and time dependence.

Gauss integration provides an independent normalization check: the exterior flux at the surface, or in an effectively massless overlap zone, must equal the active shell source. Farther out a massive exterior contributes to the volume source, so $r^2\phi'$ is no longer constant. Two limiting checks are equally important. Taking $r_s\to0$ gives $Q\to\beta M$ and the unscreened force; taking $r_s\to R$ gives $Q\to0$. Failure of either limit signals a matching or density-normalization error.

Exercise

Body A is fully unscreened, while a constant-density body B has $r_s/R=0.99$. They have equal inertial mass and experience the same external scalar gradient. Compute their charge fractions and the scalar part of their differential acceleration.

Solution

For A, $\epsilon_A=1$. For B, $\epsilon_B=1-0.99^3=0.029701$. Hence

$$\Delta\boldsymbol a_\phi =-\frac{\beta(1-0.029701)}{M_{\rm Pl}} \nabla\phi_{\rm ext} =-\frac{0.970299\,\beta}{M_{\rm Pl}} \nabla\phi_{\rm ext}.$$

Equal inertial masses do not imply equal scalar charges because most of B’s interior does not source the exterior field. The result assumes a slowly varying external gradient, negligible Yukawa variation across each body, a universal microscopic coupling, and a valid constant-density screening solution. It cannot be inferred from mass alone.

References

• Burrage, Clare, and Jeremy Sakstein (2018), “Tests of Chameleon Gravity,” Living Reviews in Relativity 21, 1. DOI. Open PDF.
• Hui, Lam, Alberto Nicolis, and Christopher Stubbs (2009), “Equivalence Principle Implications of Modified Gravity Models,” Physical Review D 80, 104002. DOI. Open PDF.