Dynamical External Scalar Fields in Gravity

• Dynamical External Scalar Field is a scalar degree of freedom that actively shapes geometry and cosmological evolution, rather than acting as a passive background.
• It functions variably as an external potential, an intrinsic clock in gravitational collapse, and an autonomous dark-energy agent, affecting renormalization and stability in diverse models.
• Recent studies employ quantum field techniques, dynamical system analysis, and numerical simulations to elucidate its role in modified gravity, teleparallel cosmologies, and MOND frameworks.

Searching arXiv for the supplied papers to ground the article in current bibliographic records. arXiv search query: (Fulling et al., 2018) OR (Nakonieczna et al., 2015) OR (Faraoni et al., 2012) OR (Azri et al., 2021) OR (Scholz, 20 Oct 2025) OR (Mishra et al., 2024) OR (Kadam et al., 2024) OR (Kadam, 6 Apr 2025) A dynamical external scalar field is a scalar degree of freedom treated not merely as a passive background input but as an active sector that shapes geometry, stress-energy, or cosmological evolution. In the cited literature, the phrase does not denote a single standardized formalism; it covers at least three recurrent constructions: an external scalar potential promoted to a classical dynamical field of its own in quantum field theory, a scalar used as an intrinsic time variable in strong-gravity evolution, and cosmological or modified-gravity scalar sectors whose autonomous dynamics determine radiation, matter, scaling, de Sitter, or bounce phases (Fulling et al., 2018, Nakonieczna et al., 2015, Faraoni et al., 2012).

1. Conceptual range and defining criteria

A common structural distinction in this literature is between a scalar or scalar potential that is prescribed externally and one that is embedded into the dynamical field equations. In the renormalization problem for a quantized scalar field in a background potential, the decisive step is to regard the background potential as “a dynamical object in its own right,” because only then can the ultraviolet-sensitive local terms be absorbed into ordinary couplings of the background-field action (Fulling et al., 2018). In cosmological dynamical-systems language, a minimally coupled scalar field is already dynamical when its evolution closes together with the Friedmann equations; in the flat FLRW case this can be written on the constrained phase space $(H,\phi,\dot\phi)$, with trajectories confined to

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$

and equilibrium points

$$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$

(Faraoni et al., 2012).

Two operational criteria recur when the scalar is used as a relational or effective clock. First, constant-field hypersurfaces must have the correct causal character; second, the field must vary monotonically along the relevant evolution. In gravitational collapse these conditions are tested geometrically, while in cosmology they are encoded in phase-space flow and attractor structure (Nakonieczna et al., 2015, Faraoni et al., 2012). This suggests that “external” becomes physically meaningful only relative to a chosen dynamical closure: the same scalar can function as a prescribed background, a source field, a clock variable, or an autonomous dark-energy degree of freedom depending on how it enters the field equations.

2. External scalar potentials promoted to dynamical variables

The most explicit treatment of an external scalar sector appears in the analysis of a quantized scalar field $\phi$ interacting with an external scalar potential $V$, where the potential is ultimately promoted to a classical dynamical field (Fulling et al., 2018). In the first model, the flat-space Lagrangian density is

$$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$

In the second model, the potential is written as $V=\frac{\lambda}{2}\sigma^2$, with

$$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$

The renormalization analysis combines covariant point splitting with Pauli–Villars regularization. The central result is that the divergences in $\langle \phi^2\rangle$ and $\langle T_{\mu\nu}\rangle$ collapse into a local, covariant functional built from

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$0

where

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$1

In model 1, the local counterterms renormalize $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$2, $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$3, and $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$4, together with a cosmological-constant-type term. In model 2, the corresponding renormalized couplings are $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$5, $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$6, and $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$7 (Fulling et al., 2018).

A distinctive feature is the survival of finite logarithmic terms such as $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$8 and $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$9. These are not removed by renormalization; changing the arbitrary scale $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$0 shifts them by local terms that can be reabsorbed into the renormalized couplings. The paper therefore interprets them as analogous to renormalization-group logarithms. The broader implication is that an “external” scalar potential is renormalizable in a non-ad hoc sense only after it is assigned its own local action and coupling constants (Fulling et al., 2018).

3. Scalar fields as intrinsic time in gravitational collapse

In spherical collapse within general relativity, the scalar field can be tested as an intrinsic time variable in a genuinely dynamical and high-curvature setting. The collapse analysis is performed in double-null coordinates $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$1 with

$$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$2

and the dynamical variables are $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$3, $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$4, and $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$5 in the paper’s 2+2 formalism (Nakonieczna et al., 2015). Although the abstract refers to a self-interacting scalar field, the actual framework is a massless, minimally coupled scalar with

$$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$6

The criterion for using $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$7 as time is geometric. A hypersurface $$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$8 is acceptable only when it is spacelike, and the field must change monotonically in the same region. In the collapse coordinates, the paper determines the causal character of constant-$$(H,\phi,\dot\phi)=\left(\pm\sqrt{\frac{8\pi G}{3}V(\phi_0)},\phi_0,0\right),\qquad V'(\phi_0)=0$$9 lines from the sign of

$\phi$0

with positive slope identified numerically as spacelike and negative slope as timelike (Nakonieczna et al., 2015). The monotonicity claim is not presented as a formal theorem; it is inferred from the geometry of the plotted level sets. The paper states that in regions where the lines are spacelike, the changes are monotonic along constant $\phi$1 and $\phi$2, and therefore also along

$\phi$3

For all nine families of initial data that form black holes, the spacetime is of Schwarzschild type, with a spacelike central singularity at $\phi$4 and a single apparent horizon. The key result is restricted but precise: in the dynamical part of the high-curvature region near the singularity, the hypersurfaces $\phi$5 are spacelike for all studied profiles, and $\phi$6 varies monotonically there (Nakonieczna et al., 2015). The numerical evolution reaches

$\phi$7

so the conclusion pertains to regions extremely close to the singularity. No new critical scaling law of Choptuik type is reported.

4. Autonomous cosmological dynamics and late-time asymptotics

In flat FLRW cosmology with a minimally coupled scalar field, the phase-space description can be formulated directly in $\phi$8, and the Hamiltonian constraint confines the dynamics to a two-sheeted surface. Because the actual motion is restricted to a two-dimensional constrained manifold, the cited analysis emphasizes that chaos is excluded in this minimally coupled, spatially flat setting (Faraoni et al., 2012). Stability of equilibrium points is governed by the sign of $\phi$9 and the curvature of the potential: for expanding de Sitter solutions with $V$0, local minima of $V$1 are asymptotically stable, local maxima are unstable, and Minkowski is stable through third order if $V$2 (Faraoni et al., 2012).

The normalized dynamical-system literature rewrites the same problem in variables such as

$V$3

and, when needed, slope or curvature variables. For a canonical scalar field with exponential potential in flat FRW, the standard fixed points are a matter point $V$4, kinetic points $V$5, a scaling point $V$6, and a scalar-field dominated point $V$7; $V$8 is stable whenever it exists, while $V$9 is stable for $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$0 and accelerated for $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$1 (Tamanini, 2014). The same paper emphasizes a persistent phenomenological tension: one cannot simultaneously obtain an early-time nucleosynthesis-compatible scaling regime and a late-time accelerated attractor from the same canonical exponential potential.

When spatial curvature is included, the phase space becomes three-dimensional and two additional fixed points appear: a curvature-dominated point $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$2 and a curvature-scalar scaling point $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$3. In that setting, the scalar-field dominated point

$$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$4

remains the full three-dimensional attractor only for $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$5, while open trajectories with $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$6 are drawn toward the curvature-scalar scaling solution $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$7 with $$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$8 (Gosenca et al., 2015). This is a concrete instance in which an apparently external geometric ingredient—curvature—changes the late-time scalar-field fate.

Several noncanonical extensions sharpen the viability question. For the dilatonic model with

$$\mathcal L_1 = \frac12\Big[(\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\lambda \phi^2 V +(\partial_t V)^2-(\nabla V)^2-M^2V^2-2JV\Big].$$9

the background admits mathematically stable nodes $V=\frac{\lambda}{2}\sigma^2$0 and $V=\frac{\lambda}{2}\sigma^2$1, but both lie in the regime $V=\frac{\lambda}{2}\sigma^2$2, where

$V=\frac{\lambda}{2}\sigma^2$3

so they are classically and quantum mechanically unstable (Mahata et al., 2013). For the square-root kinetic correction

$V=\frac{\lambda}{2}\sigma^2$4

the background dynamics allows phantom acceleration and dynamical crossing of the phantom barrier, but the sound speed

$V=\frac{\lambda}{2}\sigma^2$5

can diverge or become negative, and the model is therefore not perturbatively viable (Tamanini, 2014). A complementary integrability result is obtained through Wheeler–DeWitt symmetries: if the current cosmological model is Liouville integrable, then there is a unique stable point describing the de Sitter phase of the universe (Paliathanasis et al., 2015).

5. Scalar-torsion and aether-coupled cosmologies

Teleparallel models replace curvature by torsion and allow the scalar sector to couple directly to the torsion scalar. In one formulation,

$V=\frac{\lambda}{2}\sigma^2$6

with the specialization

$V=\frac{\lambda}{2}\sigma^2$7

the effective dark-energy density and pressure are

$V=\frac{\lambda}{2}\sigma^2$8

$V=\frac{\lambda}{2}\sigma^2$9

and the bounce profile

$$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$0

is used to study five potentials: $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$1, $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$2, $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$3, a generalized Starobinsky potential, and an $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$4-attractor $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$5 (Mishra et al., 2024). The paper finds that all potentials show broadly universal evolution away from the bounce, but near the bounce the detailed profile of $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$6 is potential dependent.

A related scalar-torsion system with

$$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$7

admits autonomous variables $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$8, $$\mathcal L_2 = \frac12\Big[ (\partial_t\phi)^2-(\nabla\phi)^2-m^2\phi^2-\frac{\lambda}{2}\phi^2\sigma^2 +(\partial_t\sigma)^2-(\nabla\sigma)^2-\frac{\Lambda}{12}\sigma^4-M^2\sigma^2 \Big].$$9, $\langle \phi^2\rangle$0, and $\langle \phi^2\rangle$1, with

$\langle \phi^2\rangle$2

Its principal late-time accelerating points are $\langle \phi^2\rangle$3 or $\langle \phi^2\rangle$4, for which

$\langle \phi^2\rangle$5

accelerated when $\langle \phi^2\rangle$6, and exact de Sitter points $\langle \phi^2\rangle$7 or $\langle \phi^2\rangle$8 with $\langle \phi^2\rangle$9 and $\langle T_{\mu\nu}\rangle$0 (Kadam et al., 2024). In the more general scalar–torsion model with Sorkin–Schutz matter couplings and power-law torsion coupling $\langle T_{\mu\nu}\rangle$1, the autonomous system contains radiation points $\langle T_{\mu\nu}\rangle$2, matter points $\langle T_{\mu\nu}\rangle$3, stiff points $\langle T_{\mu\nu}\rangle$4, and dark-energy attractors $\langle T_{\mu\nu}\rangle$5, so the sequence radiation $\langle T_{\mu\nu}\rangle$6 matter $\langle T_{\mu\nu}\rangle$7 dark energy is explicitly available (Kadam, 6 Apr 2025).

Einstein–Aether cosmology introduces a different external structure: a timelike unit aether field. For the potential

$\langle T_{\mu\nu}\rangle$8

the effective scalar density retains the GR form

$\langle T_{\mu\nu}\rangle$9

while the pressure is modified to

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$00

The dynamical-system analysis yields kinetic, scaling, matter-like, and de Sitter critical points, and the paper emphasizes that exact de Sitter attractors appear even for exponential $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$01, with the GR limit recovered when $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$02 (Paliathanasis et al., 2019). This suggests that coupling an otherwise standard scalar to additional background structure can enlarge the attractor set without changing the Friedmann density equation in the same way.

6. Affine and scale-covariant generalizations

In asymmetric Eddington gravity, the fundamental variable is a general affine connection $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$03, with two independent Ricci-type tensors,

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$04

External scalar fields $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$05 enter a determinant action through the tensor

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$06

or, in the simpler case, $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$07 (Azri et al., 2021). The crucial outcome of the connection variation is an emergent metric density

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$08

and because $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$09 is not symmetric, the emergent metric is generically nonsymmetric: $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$10

The scalar equations are correspondingly modified. In the two-coupling model they take the form

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$11

so the second Ricci tensor directly influences scalar evolution (Azri et al., 2021). This is a qualitatively different notion of “external scalar field”: the scalars are external to the purely affine gravitational sector at the level of field content, but their dynamics and the affine connection become mutually entangled once the variational problem is solved.

A different extension is formulated in integrable Weyl geometry. There the scalar field $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$12 has Weyl weight $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$13, the Einstein-gauge variable is $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$14, and the Milgrom-regime scalar Lagrangian combines an ordinary quadratic term, a Bekenstein-type aquadratic term, a second-order mass-generating term, and a quartic potential (Scholz, 20 Oct 2025). In Einstein gauge, the Bekenstein sector becomes a cubic $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$15 term, and the weak-field scalar equation reduces to

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$16

which is the deep-MOND equation. The same model assigns the scalar field real energy density and pressure, with

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$17

so the scalar sector modifies both free fall and light deflection (Scholz, 20 Oct 2025).

7. External field effect and hierarchical systems

The Weyl-geometric MOND model gives a precise reinterpretation of the external field effect. Its nonstandard scalar terms are switched on only if two conditions hold simultaneously: the scalar gradient is spacelike,

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$18

and its norm lies in the low-gradient Milgrom sector, implemented by the switching function

$$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$19

(Scholz, 20 Oct 2025). In that regime the scalar sector produces MOND-like dynamics and a modified light-cone structure.

The paper rejects the standard strong external field effect according to which a subsystem loses internal MOND behavior whenever the ambient external acceleration satisfies $$H^2=\frac{8\pi G}{3}\left(\frac{\dot\phi^2}{2}+V(\phi)\right),$$20. Its argument is explicitly relativistic: in a local freely falling frame, an approximately homogeneous external field is transformed away, so the subsystem should not respond to the homogeneous field itself. The proposed replacement is a weak or relativistic external field effect: only sufficiently strong tidal forces, rather than homogeneous external acceleration, can inhibit entry into the Milgrom regime (Scholz, 20 Oct 2025).

This change has direct consequences for hierarchical systems. A star cluster moving in a galactic field, or a galaxy moving inside a cluster, can still carry its own scalar halo if the relevant tidal field over the subsystem is weak enough. The paper argues that galaxies inside clusters can retain internal scalar-field halos and that, together with the scalar halo of the hot gas, these contributions may significantly reduce or perhaps remove the need for additional dark matter in clusters (Scholz, 20 Oct 2025). In that sense, the “external” scalar field is no longer merely a background modifier of local dynamics; it becomes a scale-dependent mediator whose activation depends on geometry, gradients, and environment.