Modified-gravity proposals and their predictions

An altered response should follow from a stated model

A modified-gravity proposal changes the dynamical account of gravitational interactions. An arbitrary fitted force curve alone does not specify its fields, conservation, initial conditions or applicable regimes.

Additional gravitational fields can affect matter motion and light differently. Comparing those responses is useful only after the physical metric, coupling and approximations are defined.

Use a canonical massive scalar with universal conformal matter coupling as one explicit example. Its weak static exterior is a controlled specialization, not a universal description of gravity alternatives.

The inputs below are hypothetical. No observed force anomaly, current exclusion, actual object mass or complete expansion history is inferred from their numerical values.

Distinguish the action convention from SI observables

For optional mathematical depth, use c = ħ = 1 in the Einstein-frame action S = ∫d⁴x√−g[M_P²R/2−(∂φ)²/2−m²φ²/2]+S_m[e^(2βφ/M_P)g,χ], with M_P = (8πG_*)^(−1/2).

Here g is the metric determinant, R the Ricci scalar, χ the matter fields and S_m their action; (∂φ)² = g^μν∂_μφ∂_νφ. The parameter β is a dimensionless coupling and m is the scalar mass in the action’s natural units. The scalar has a positive canonical kinetic term, m²>0 and background φ = 0. Matter follows the conformally related Jordan metric; at background its conformal factor is 1. These checks concern this linear model, not full ultraviolet or cosmological consistency.

Write the physical weak static metric in SI coordinates as ds² = −(1+2Ψ)c²dt²+(1−2Φ)d𝐱². The potentials Ψ and Φ are dimensionless, and G_* is the bare tensor coupling.

Take an ideal point source of mass M, unscreened nonrelativistic test matter and a positive range λ in metres. Neglect scalar-stress backreaction and higher weak-field orders; define x = r/λ over 1/4≤x≤8, G_*M/(λc²) = 10⁻⁶ and β² = 1/6.

Differentiating the range factor changes the force

Let U_N = −G_*M/(rc²), α = 2β² = 1/3 and q = αexp(−x). The decaying exterior Helmholtz solution and linear conformal expansion give Ψ = U_N(1+q) and Φ = U_N(1−q).

The inward nonrelativistic acceleration is g_r = −c²∂Ψ/∂r. Its magnitude relative to G_*M/r² is 1+α(1+x)exp(−x).

The factor 1+x comes from differentiating both 1/r and exp(−r/λ). Using the potential factor 1+αexp(−x) as a force ratio would omit part of the derivative.

At x = 1, the potential correction is 1/(3e), while the force correction is 2/(3e). At large x both decay, within the finite range where this exterior approximation is being evaluated.

The leading light response involves both potentials

The Weyl mean is (Φ+Ψ)/2 = U_N in this leading approximation because the conformal scalar terms cancel in the sum. The potential ratio is η = (1−q)/(1+q), generally different from 1.

A conditional GR interpretation using the fixed bare G_* would assign motion-response mass M[1+α(1+x)exp(−x)] but Weyl-response mass M. This is a comparison within the supplied model and convention.

G_* is not automatically the Newton constant inferred from a laboratory experiment in a specified environment. Changing the calibration convention can change how an inferred mass comparison is expressed.

The first-order cancellation is not a universal theorem that modified gravity leaves lensing unchanged. Actual lensing also requires light-path geometry, source selection and a mass-reconstruction model, none of which is supplied here.

A finite exterior result leaves further consistency tests

The chosen radii keep the potentials weak and exclude the singular ideal point-source origin. The calculation supplies an exterior response, not an interior material solution or an extrapolation to r = 0.

The quadratic massive scalar here is different from an inverse-power chameleon with environmental screening. A finite range, source screening and test-body screening are different effects and need their own equations.

A full gravity proposal must address propagating degrees of freedom, stable applicable dynamics, conservation, boundary conditions and agreement with tests in other regimes. A static force calculation does not complete those tasks.

The useful prediction is conditional and specific: this scalar changes motion through a Yukawa derivative while the leading Weyl mean retains U_N. Those linked responses supply a question to test, not a completed explanation of cosmological observations.

Worked reasoning

Try the reasoning

1. What are the extra-force ratio and Weyl mean at x = 2?

   Show the answer and reasoning

   The extra-force ratio is e⁻²; the total is 1+e⁻², and the Weyl mean is U_N.

   With α = 1/3, α(1+2)e⁻² = e⁻², while the conformal terms cancel in the potential sum.

2. Does the conditional motion–lensing mass mismatch prove extra material or a complete alternative cosmology?

   Show the answer and reasoning

   Neither.

   It follows from a specified weak scalar response and bare-coupling convention; other physical and observational ingredients remain to be tested.

3. May the potential correction αe⁻ˣ be used as the complete force correction?

   Show the answer and reasoning

   No.

   Differentiation supplies an additional factor 1+x in the acceleration response.

What to carry forward

• A modified force should follow from stated fields and approximations.
• The potential and acceleration range factors are different.
• Linked motion and light predictions need a consistent coupling convention and tests across regimes.