Regimes, Scales, and Environmental Dependence

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“Weak” and “strong” gravity are not single labels. A system can have a small potential but a long propagation distance, weak curvature but relativistic motion, or strong self-gravity inside bodies while the orbital field remains post-Newtonian. A gravity constraint is portable only after the dimensionless control parameters, boundary conditions, and environment of both calculations are matched.

Required background. The Einstein Equation supplies the Einstein equations and their dimensional scales. Probability and Inference for Gravity supplies the conditional reasoning needed to compare constraints defined on different parameter domains.

Helpful background. Weak Metric Perturbations and Gauge Freedom develops the perturbative metric expansion. The Tests of Gravity overview maps the regimes used by later chapters.

Dimensionless coordinates on a test regime

For a gravitating body of mass $M$ , characteristic radius or separation $r$ , internal size $R$ , and speed $v$ , useful controls include

\epsilon_\Phi=\frac{GM}{r}, \qquad \epsilon_v=v^2, \qquad \mathcal C=\frac{GM}{R}, \qquad \epsilon_{\mathcal R}=\frac{GM}{r^3}L^2.

The site uses $c=1$ ; restoring units replaces $GM/r$ by $GM/(rc^2)$ and $v^2$ by $v^2/c^2$ . The potential parameter $\epsilon_\Phi$ controls many weak-field metric corrections, $\epsilon_v$ organizes slow-motion expansions, and compactness $\mathcal C$ measures self-gravity. Curvature has dimensions of inverse length squared, so it becomes a comparison parameter only after choosing the physical probe scale $L$ .

These controls need not be comparable. A neutron star in a wide binary has large internal compactness but a small orbital potential. A gravitational wave can propagate through extremely weak local curvature for a cosmological distance, accumulating a small dispersive or frictional effect. A laboratory experiment has negligible compactness yet can probe a mediator whose Compton wavelength matches a submillimetre gap.

Additional fields introduce their own ratios. For a mediator mass $m$ , $mr=r/\lambda$ distinguishes long- and short-range response. A time-dependent probe compares $\omega r$ , $\omega/m$ , and the observation time with any relaxation scale. An effective field theory also requires $E/\Lambda\ll1$ and field gradients below its cutoff; a tight fit outside this domain is not a controlled constraint.

Boundary conditions and environments

Two calculations make the distinction between field strength and curvature tangible. A $1.4M_\odot$ neutron star with radius $12\,\mathrm{km}$ has compactness about $0.17$ , using $GM_\odot/c^2\simeq1.477\,\mathrm{km}$ . The orbital potential of a $2.6M_\odot$ binary at separation $10^6\,\mathrm{km}$ is only $3.8\times10^{-6}$ . Its orbit can therefore be weak-field while its constituents have strong self-gravity.

For a Schwarzschild black hole, define $r_g=GM/c^2$ . The curvature invariant

K=R_{abcd}R^{abcd}=\frac{48r_g^2}{r^6}

has dimension length $^{-4}$ . At the horizon $r=2r_g$ , the potential ratio is $r_g/r=1/2$ for every mass, but $\sqrt K=\sqrt3/(2r_g^2)$ decreases as $M^{-2}$ . A more massive black hole has the same horizon compactness and smaller local curvature. Thus a potential bound, a curvature bound, and a bound at a specified acceleration probe different physical directions.

The same local Lagrangian can yield different observable responses because nonlinear equations select environment-dependent branches. A screened scalar illustrates the point. Schematically,

\nabla^2\phi=V_{,\phi}(\phi)+\frac{\beta(\phi)}{M_{\rm Pl}}\rho,

Here $M_{\rm Pl}^{-2}=8\pi G$ defines the reduced Planck mass. The field profile depends on the density $\rho$ , chamber or halo geometry, material boundaries, and the ambient value supplied by a larger environment. A vacuum chamber, Solar-System body, dwarf galaxy, and neutron star cannot be compared by inserting one universal constant into a common straight line. The field equation must be solved or matched in each regime, including the possibility of a thin shell, scalarization, or no stable branch Burrage and Sakstein 2018, §§2–5.

Environmental dependence is broader than screening. Galactic acceleration contaminates pulsar orbital decay; plasma and source structure enter horizon-scale images; host-galaxy catalogs enter standard-siren redshifts; and cosmological boundary values can set local effective couplings. These are not optional “astrophysical details.” They are part of the map from a gravitational hypothesis to an observable.

Matching an approximation to an observable

A controlled comparison should answer four questions.

Which quantity is expanded? A post-Newtonian expansion orders powers of velocity and potential; a waveform expansion may instead order frequency-dependent phase terms.
Where is it valid? State radial, frequency, compactness, curvature, and redshift intervals, together with boundary or initial conditions.
Which neglected term is next? An error estimate should scale with a known omitted order or be calibrated against a more complete calculation.
Does the measurement weight the valid region? A formally valid low-frequency expression can still bias inference if the likelihood is dominated near merger.

The familiar PPN construction is deliberately limited to slowly moving sources and weak, quasistatic fields; compact-body strong-field effects require additional body-dependent parameters or a different framework Will 2014, §3.2 and Box 2. Similarly, a quasistatic cosmological parameterization does not automatically describe horizon-scale evolution, and a geometric-optics lens equation does not describe wave-optics fringes when the wavelength resolves the lens.

A regime table is part of the result

Before combining constraints, record at least the probe scale, potential, compactness, curvature scale, characteristic frequency, environment, and approximation/cutoff. The table should distinguish directly measured inputs from model-derived ones. If a parameter maps differently in two rows, the constraints cannot be multiplied until a theory supplies that mapping and its covariance.

Limit checks are especially effective. Taking $v\to0$ and $GM/r\to0$ should recover the Newtonian observable. Taking $mr\to0$ and $mr\to\infty$ should recover long-range and decoupled mediator limits when those limits exist. Varying the outer boundary in a screened calculation should expose whether the alleged laboratory constraint is actually controlled by an assumed ambient field.

Check your understanding

A proposed combination treats a Solar-System bound on a scalar coupling as an unchanged prior for a neutron-star binary. Identify the missing regime information and give a valid route to a combined constraint.

Solution

The Solar-System result probes weak self-gravity, a particular ambient density and scalar boundary value, and slow quasistatic motion. A neutron star has large compactness and may develop a body-dependent sensitivity or a nonperturbative scalarized branch. Therefore the same microscopic coupling need not produce the same effective charge. A valid combination begins with one specified scalar theory, solves or matches the field in both environments, checks stability and the effective-theory cutoff, and maps the common theory parameters to each observable. The two likelihoods can then be multiplied with shared nuisance and boundary parameters represented only once.

References

Burrage, C., and Sakstein, J. (2018), “Tests of chameleon gravity,” Living Reviews in Relativity 21, 1. Burrage and Sakstein 2018, §§2–5.
Will, C. M. (2014), “The confrontation between general relativity and experiment,” Living Reviews in Relativity 17, 4. Will 2014, §3.2, Box 2, and Table 2.