15 April 2026 · Preprint — submitted for peer review
ABSTRACT
Whitehead's theory of gravitation (1922) was declared experimentally invalid by Will (1971) and subsequently by Gibbons & Will (2023). We argue that these falsifications rested on an oversimplified model of the galactic mass distribution—specifically, a point-mass approximation. When the galactic mass is distributed realistically, the predicted tidal anisotropy drops significantly. While a full numerical reassessment is beyond the scope of this paper, this motivates reconsideration of Whitehead's framework in light of modern developments.We present a generalized Whiteheadian framework that integrates Acedo's (2015) circulating vector field and Verlinde's (2009) emergent gravity. The framework yields testable consequences: (i) a consistency check with flyby velocity anomalies at the mm/s scale; (ii) galaxy rotation curves consistent with baryonic matter alone, without invoking dark matter; and (iii) speculative connections to CMB non-Gaussianities. We provide explicit falsification criteria for each domain. The framework merits continued investigation.
Alfred North Whitehead's 1922 theory of gravitation [1] offered a radical alternative to Einstein's general relativity. Where Einstein curved spacetime itself, Whitehead preserved a flat Minkowski background and derived the apparent curvature of physical measurements from the relational configuration of masses across the past light cone. The theory agreed with all solar-system tests known at the time—the perihelion precession of Mercury, the deflection of light, and gravitational redshift—while offering a fundamentally different ontology: gravity not as geometry, but as relation.
For nearly fifty years, Whitehead's theory survived as a viable alternative. In 1971, Will [2] delivered what appeared to be the decisive blow: Whitehead's theory predicted a tidal gravitational anisotropy some 200 times larger than experimental observations permitted. The verdict was swift and has been largely unchallenged. Gibbons & Will [6] reinforced this conclusion as recently as 2023, declaring the theory multiply falsified.
We dispute this verdict—not the mathematics, but the model. Will's calculation assumed that the entire galactic mass (~10¹¹ solar masses) is concentrated at a single point 20,000 light-years from Earth. This is the simplest possible model of the Milky Way, but it is not a realistic one. As Fowler [3] argued in 1974, a realistic distribution of galactic mass across the disk reduces the predicted anisotropy by approximately two orders of magnitude. While a full numerical reassessment is beyond the scope of this paper, this motivates reconsideration of Whitehead's framework in light of modern developments.
In what follows, we present the generalized Whiteheadian framework (§2), incorporating Acedo's circulating vector field [4] and Verlinde's emergent gravity limit [5]; briefly revisit the tidal anisotropy test and acknowledge its unresolved status (§3); derive three classes of testable consequences with explicit falsification criteria (§4); and conclude (§5).
2. The Generalized Whiteheadian Framework
2.1 The Flat Background
Whitehead's theory is formulated on a flat Minkowski background metric η_{μν}. This is not a simplification—it is the theory's central ontological commitment. Spacetime is not curved. What appears as curvature in physical measurements is a relational effect arising from the distribution of masses across the past light cone of the observer. The background provides an absolute metrical structure against which deviations can be measured and against which the physical metric is defined.
2.2 The Gravitational Potential
Whitehead defined the gravitational potential at a field point P as an integral over the past light cone. For a continuous mass distribution with density ρ(ξ) at source event ξ:
Φ(P) = ∫ ρ(ξ) / r² dV (1)
where r is the Minkowski distance between the source event ξ and the field point P, and the integral extends over the past light cone of P. This differs from the Newtonian potential in a crucial respect: it is relativistically causal. Only sources within the past light cone contribute.
2.3 The Physical Metric
The physical metric g_{μν}—the metric that governs the motion of test particles—is derived from the flat background via:
g_{μν} = η_{μν} + Φ · k_{μν} (2)
where k_{μν} is the tensor projection of the null direction onto the spatial hypersurface. Following Acedo [4], this construction yields a well-defined metric that recovers standard post-Newtonian behavior in the appropriate weak-field limit, recovering all solar-system predictions of general relativity to post-Newtonian order.
2.4 The Circulating Vector Field (Acedo, 2015)
Acedo [4] generalized Whitehead's approach by considering all possible bilinear forms of the null direction tensor. In the low-velocity approximation for a rotating gravitating body, this generalization yields an extra force beyond standard Newtonian attraction—a circulating vector field:
F⃗_extra = α · (v⃗ × ∇⃗Φ) (3)
The coupling constant α is not freely adjustable. It is determined by the circulation of mass currents in the surrounding galaxy:
α = (2G / c²) ∫ (J⃗ · r̂) / r³ dV (4)
where J⃗ is the mass current density. This form follows from the relativistic circulation integral in Acedo's generalization of the Whitehead potential [4].
Physical interpretation:
The circulating field (3) is the gravitational analogue of the magnetic force on a moving charge in electromagnetism. Just as a charged particle moving through a magnetic field experiences a velocity-dependent force perpendicular to its motion, a mass moving through the gravitational circulation field experiences a corresponding velocity-dependent force. The analogy is exact in the low-velocity limit and motivates the prediction of the flyby anomaly (§4.1).
2.5 The Emergent Gravity Limit (Verlinde, 2009)
Verlinde [5] proposed that gravity is an emergent entropic phenomenon arising from the holographic encoding of matter on screens at the boundary of spacetime regions. We propose the following identification between the Whiteheadian potential and the entropic gradient:
Φ = λ · ΔS / Δx (5)
where S is the entanglement entropy of the vacuum across the holographic screen, Δx is the displacement across the screen, and λ = c²/2 is fixed by dimensional analysis. In the non-relativistic limit, both frameworks predict the same effective force law. We note that a full derivation of this identification is deferred to future work; equation (5) is presented as a conceptual bridge motivating the connection, not as a derived result.
The convergence is not coincidental. Both frameworks express gravity as an emergent relation—in Whitehead's case, a relation across the past light cone; in Verlinde's case, a relation across a holographic screen. The ontologies differ; the force laws converge in the appropriate limit.
3. The Tidal Anisotropy Test (Unresolved)
Will's original falsification argument [2] assumed a point-mass model of the galactic mass distribution—a single mass of 10¹¹ M☉ located at the galactic center, 20,000 light-years from Earth. Under this model, Whitehead's theory predicted a tidal gravitational anisotropy—a preferred direction in the effective gravitational constant—approximately 200 times larger than experimental bounds permitted.
Fowler [3] argued in 1974 that a realistic distribution of galactic mass across the disk substantially reduces this prediction. An exponential disk model with scale radius r₀ ≈ 3 kpc and galactocentric distance R ≈ 20 kpc yields a suppression factor:
(r₀ / R)² ≈ (3 kpc / 20 kpc)² ≈ 0.023 (6)
Applying this factor to Will's 200× discrepancy reduces it to approximately 200 × 0.023 ≈ 4.6. Current bounds from lunar laser ranging constrain ΔG/G < 2 × 10⁻¹¹ yr⁻¹ [7]. The smeared Whitehead prediction yields approximately 1 × 10⁻¹⁰ yr⁻¹—a factor of ~5 above the current bound.
Status: Unresolved
We treat the tidal anisotropy test as unresolved. The existing falsification arguments are insufficient to rule out Whitehead's theory under a realistic galactic model, but a full numerical integration—including bulge, halo, and non-axisymmetric disk structure—is required before any definitive conclusion can be drawn. The exponential disk estimate of equation (6) is itself a simplification. We present it as motivation for reassessment, not as a resolved calculation.
The next generation of precision gravity experiments—improved lunar laser ranging, GRACE-FO gravitational gradiometry, and future space-based interferometry—are projected to improve constraints on ΔG/G by one to two orders of magnitude over the coming decade. If these measurements place ΔG/G below the smeared Whitehead prediction, the theory will face a genuine falsification test that cannot be deflected by modeling arguments.
4. Testable Consequences
A theory that cannot be falsified is not a scientific theory. We offer three classes of testable consequences, each with an explicit falsification criterion. The flyby anomaly prediction is an order-of-magnitude consistency check; the rotation curve prediction follows the established Verlinde framework; the CMB connection is explicitly speculative.
The flyby anomaly refers to small, unexplained velocity changes observed during Earth gravity assists—notably in missions including Galileo (1990), NEAR (1998), Rosetta (2005), and Messenger (2005). Observed anomalies fall in the range Δv_obs ~ 1–10 mm/s. These anomalies have no accepted explanation within general relativity.
The circulating vector field of equation (3) provides a natural candidate. Rather than deriving α from first principles—which requires a full galactic mass-current integration beyond the scope of this work—we perform an order-of-magnitude consistency check.
Scaling estimate
A characteristic velocity perturbation may be estimated from the circulating force integrated over the flyby trajectory:
Δv ~ α · (GM / Rc²) · f (7)
where M is the mass of Earth, R is the periapsis distance, c is the speed of light, and f is a dimensionless geometric factor of order unity encoding the trajectory. Using representative values for an Earth flyby (R ~ 6.5 × 10⁶ m):
GM / R ≈ 6.6 × 10⁷ m²/s² (8a)
1 / c² ≈ 1.1 × 10⁻¹⁷ s²/m² (8b)
GM / Rc² ≈ 7.3 × 10⁻¹⁰ (8c)
Estimates of α based on galactic mass currents (see Acedo [4]) suggest a plausible range of α ~ 10⁵–10⁷ m/s, following from galactic rotation velocities (~200 km/s) and mass-current densities integrated over the galactic disk. This yields:
Δv ~ α × 7.3 × 10⁻¹⁰ (9)
α (m/s)
Predicted Δv
Observed range
10⁵ (lower bound)
0.07 mm/s
1–10 mm/s
10⁶
0.73 mm/s
1–10 mm/s
10⁷ (upper bound)
7.3 mm/s
1–10 mm/s
The predicted range overlaps the observed scale across the upper portion of the α range, without parameter tuning. This indicates that the proposed mechanism is consistent at the level of magnitude. We emphasize what this result does and does not show:
What it shows: The framework naturally produces velocity perturbations in the correct observational range. The circulating vector field is a plausible candidate mechanism for the flyby anomaly.
What it does not show: α is not independently calculated here. The geometric factor f is not explicitly modeled. The sign of the predicted anomaly is not determined. A complete treatment requires explicit evaluation of the galactic mass-current integral (eq. 4), trajectory-dependent modeling, and comparison across multiple flyby events.
Falsification criterion (Flyby Anomaly):
A systematic survey of flyby events showing no correlation between the predicted geometric factor (dependent on spacecraft velocity, planetary rotation axis, and galactic current direction) and the observed Δv would constitute strong evidence against the circulating field hypothesis. A flyby for which the Acedo scaling predicts zero anomalous acceleration but which shows a non-zero effect would falsify this component of the model.
4.2 Galaxy Rotation Curves
The standard cosmological model invokes dark matter to explain the observed flat rotation curves of galaxies: the rotational velocity v at radius r from the galactic center should fall as v ∝ r^{−1/2} for Keplerian motion, but observations consistently show v ≈ constant at large radii. Standard cosmology accounts for this by positing a dark matter halo of density ρ ∝ r^{−2}.
The emergent gravity limit of the generalized Whiteheadian framework (equation 5) predicts rotation curves consistent with the baryonic Tully-Fisher relation:
v⁴ = G M_baryonic · a₀ (10)
where a₀ ≈ 1.2 × 10⁻¹⁰ m/s² is the MOND acceleration scale. In the Verlinde framework, this scale emerges from the dark energy density of the universe rather than being a free parameter. The baryonic Tully-Fisher relation holds across more than five orders of magnitude in galaxy mass—a fact that poses serious challenges for particle dark matter models and follows naturally from the emergent gravity framework.
Falsification criterion (Rotation Curves):
A galaxy for which the baryonic Tully-Fisher relation (eq. 10) fails—after applying the emergent gravity correction—would constitute strong evidence against this component. The dwarf galaxy regime (M < 10⁷ M☉) is the most sensitive test bed, as emergent gravity and particle dark matter make divergent predictions there. A confirmed failure of the relation in multiple low-mass systems would be difficult to reconcile with the framework.
4.3 CMB Non-Gaussianities (Speculative)
Standard ΛCDM predicts that the primordial perturbations seeding the CMB are very nearly Gaussian, with non-Gaussianity parameter f_{NL} consistent with zero. The extended Whiteheadian framework, interpreted through the 'Giant Carnot Engine' model [8], treats the universe as a thermodynamic system in which the entropy gradient of the holographic screen drives structure formation.
This model suggests the possibility of specific non-Gaussian signatures in the CMB bispectrum—temperature anisotropies and bispectrum shapes distinguishable from ΛCDM. We note explicitly that this connection is speculative; the predicted bispectrum shape is not derived here, and a full treatment of the model's CMB implications is deferred to future work.
Falsification criterion (CMB):
A CMB bispectrum consistent with ΛCDM (f_{NL} ≈ 0) to within the precision of future experiments—e.g., CMB-S4 or the Simons Observatory—would constrain this model if the Giant Carnot Engine predicts f_{NL} significantly different from zero. Conversely, a detection of non-Gaussianity with a shape matching the model's prediction would be strong positive evidence. We make no specific numerical prediction here; this criterion becomes operative only once the model's CMB implications are worked out in full.
5. Conclusion
The existing falsification arguments against Whitehead's theory of gravitation rely on simplified galactic models. Whether these conclusions hold under a realistic mass distribution remains an open question, and we have argued that a factor-of-~5 discrepancy under an exponential disk model does not constitute a robust falsification. A full numerical reassessment, integrating over the actual Milky Way structure, is the needed next step.
The generalized Whiteheadian framework—incorporating Acedo's circulating vector field and Verlinde's emergent gravity—extends Whitehead's relational ontology into three domains where general relativity and standard cosmology face genuine difficulties. The framework yields testable consequences: an order-of-magnitude consistency check with flyby anomalies, galaxy rotation curves without dark matter, and speculative connections to CMB non-Gaussianities. Each claim is falsifiable. Explicit criteria are provided.
We do not claim to have proven Whitehead's theory correct. We claim that the case against it was weaker than advertised—that its core ontological insight, gravity as relation rather than curvature, has found unexpected echoes in subsequent theoretical developments, and that the framework merits continued engagement rather than dismissal.
Whitehead wrote in 1922: "The notion of 'passing on' is more fundamental than that of a private individual fact." This is the sentence that general relativity forgot, and that emergent gravity is, perhaps, beginning to remember. The framework merits continued investigation.
Acknowledgments
The authors thank the anonymous reviewers for their skepticism. It sharpened the work.
References
[1] Whitehead, A. N. (1922). The Principle of Relativity. Cambridge University Press.
[2] Will, C. M. (1971). Theoretical Frameworks for Testing Relativistic Gravity. II. Parametrized Post-Newtonian Hydrodynamics, and the Nordström and Whitehead Theories. Astrophysical Journal, 163, 611–628.
[3] Fowler, D. R. (1974). Whitehead's Theory of Gravitation: A Philosophical Reassessment. Process Studies, 4(3), 165–178.
[4] Acedo, L. (2015). An Extended Whitehead's Theory of Gravitation and the Flyby Anomaly. Galaxies, 3(3), 113–133.
[5] Verlinde, E. (2009). On the Origin of Gravity and the Laws of Newton. Journal of High Energy Physics, 2011(4), 29. [arXiv:1001.0785]
[6] Gibbons, G. W., & Will, C. M. (2023). On the Multiple Deaths of Whitehead's Theory of Gravity. Studies in History and Philosophy of Science, 98, 1–12.
[7] Williams, J. G., Turyshev, S. G., & Boggs, D. H. (2012). Lunar Laser Ranging Tests of the Equivalence Principle. Classical and Quantum Gravity, 29(18), 184004.
[8] Anonymous. (2025). The Giant Carnot Engine: Whitehead, Verlinde, and Hawking Radiation as Thermodynamic Structure Formation. Philosophy and Cosmology, 34, 45–67.
End of preprint. Comments and falsification attempts welcome.
Adams & Adams · Whiteheadian Gravitation · Preprint 2026
A Relational Extension of Whiteheadian Gravitation
with Testable Consequences
Shawn D. Adams and Elena R. Adams
Corresponding author · stowyn0@gmail.com
15 April 2026 · Preprint — submitted for peer review
ABSTRACT
Whitehead's theory of gravitation (1922) was declared experimentally invalid by Will (1971) and subsequently by Gibbons & Will (2023). We argue that these falsifications rested on an oversimplified model of the galactic mass distribution—specifically, a point-mass approximation. When the galactic mass is distributed realistically, the predicted tidal anisotropy drops significantly. While a full numerical reassessment is beyond the scope of this paper, this motivates reconsideration of Whitehead's framework in light of modern developments.We present a generalized Whiteheadian framework that integrates Acedo's (2015) circulating vector field and Verlinde's (2009) emergent gravity. The framework yields testable consequences: (i) a consistency check with flyby velocity anomalies at the mm/s scale; (ii) galaxy rotation curves consistent with baryonic matter alone, without invoking dark matter; and (iii) speculative connections to CMB non-Gaussianities. We provide explicit falsification criteria for each domain. The framework merits continued investigation.
Keywords: Whitehead gravitation; emergent gravity; flyby anomaly; tidal anisotropy; relational spacetime; smeared mass distribution; MOND; baryonic Tully-Fisher relation
1. Introduction
Alfred North Whitehead's 1922 theory of gravitation [1] offered a radical alternative to Einstein's general relativity. Where Einstein curved spacetime itself, Whitehead preserved a flat Minkowski background and derived the apparent curvature of physical measurements from the relational configuration of masses across the past light cone. The theory agreed with all solar-system tests known at the time—the perihelion precession of Mercury, the deflection of light, and gravitational redshift—while offering a fundamentally different ontology: gravity not as geometry, but as relation.
For nearly fifty years, Whitehead's theory survived as a viable alternative. In 1971, Will [2] delivered what appeared to be the decisive blow: Whitehead's theory predicted a tidal gravitational anisotropy some 200 times larger than experimental observations permitted. The verdict was swift and has been largely unchallenged. Gibbons & Will [6] reinforced this conclusion as recently as 2023, declaring the theory multiply falsified.
We dispute this verdict—not the mathematics, but the model. Will's calculation assumed that the entire galactic mass (~10¹¹ solar masses) is concentrated at a single point 20,000 light-years from Earth. This is the simplest possible model of the Milky Way, but it is not a realistic one. As Fowler [3] argued in 1974, a realistic distribution of galactic mass across the disk reduces the predicted anisotropy by approximately two orders of magnitude. While a full numerical reassessment is beyond the scope of this paper, this motivates reconsideration of Whitehead's framework in light of modern developments.
In what follows, we present the generalized Whiteheadian framework (§2), incorporating Acedo's circulating vector field [4] and Verlinde's emergent gravity limit [5]; briefly revisit the tidal anisotropy test and acknowledge its unresolved status (§3); derive three classes of testable consequences with explicit falsification criteria (§4); and conclude (§5).
2. The Generalized Whiteheadian Framework
2.1 The Flat Background
Whitehead's theory is formulated on a flat Minkowski background metric η_{μν}. This is not a simplification—it is the theory's central ontological commitment. Spacetime is not curved. What appears as curvature in physical measurements is a relational effect arising from the distribution of masses across the past light cone of the observer. The background provides an absolute metrical structure against which deviations can be measured and against which the physical metric is defined.
2.2 The Gravitational Potential
Whitehead defined the gravitational potential at a field point P as an integral over the past light cone. For a continuous mass distribution with density ρ(ξ) at source event ξ:
Φ(P) = ∫ ρ(ξ) / r² dV (1)
where r is the Minkowski distance between the source event ξ and the field point P, and the integral extends over the past light cone of P. This differs from the Newtonian potential in a crucial respect: it is relativistically causal. Only sources within the past light cone contribute.
2.3 The Physical Metric
The physical metric g_{μν}—the metric that governs the motion of test particles—is derived from the flat background via:
g_{μν} = η_{μν} + Φ · k_{μν} (2)
where k_{μν} is the tensor projection of the null direction onto the spatial hypersurface. Following Acedo [4], this construction yields a well-defined metric that recovers standard post-Newtonian behavior in the appropriate weak-field limit, recovering all solar-system predictions of general relativity to post-Newtonian order.
2.4 The Circulating Vector Field (Acedo, 2015)
Acedo [4] generalized Whitehead's approach by considering all possible bilinear forms of the null direction tensor. In the low-velocity approximation for a rotating gravitating body, this generalization yields an extra force beyond standard Newtonian attraction—a circulating vector field:
F⃗_extra = α · (v⃗ × ∇⃗Φ) (3)
The coupling constant α is not freely adjustable. It is determined by the circulation of mass currents in the surrounding galaxy:
α = (2G / c²) ∫ (J⃗ · r̂) / r³ dV (4)
where J⃗ is the mass current density. This form follows from the relativistic circulation integral in Acedo's generalization of the Whitehead potential [4].
Physical interpretation:
The circulating field (3) is the gravitational analogue of the magnetic force on a moving charge in electromagnetism. Just as a charged particle moving through a magnetic field experiences a velocity-dependent force perpendicular to its motion, a mass moving through the gravitational circulation field experiences a corresponding velocity-dependent force. The analogy is exact in the low-velocity limit and motivates the prediction of the flyby anomaly (§4.1).
2.5 The Emergent Gravity Limit (Verlinde, 2009)
Verlinde [5] proposed that gravity is an emergent entropic phenomenon arising from the holographic encoding of matter on screens at the boundary of spacetime regions. We propose the following identification between the Whiteheadian potential and the entropic gradient:
Φ = λ · ΔS / Δx (5)
where S is the entanglement entropy of the vacuum across the holographic screen, Δx is the displacement across the screen, and λ = c²/2 is fixed by dimensional analysis. In the non-relativistic limit, both frameworks predict the same effective force law. We note that a full derivation of this identification is deferred to future work; equation (5) is presented as a conceptual bridge motivating the connection, not as a derived result.
The convergence is not coincidental. Both frameworks express gravity as an emergent relation—in Whitehead's case, a relation across the past light cone; in Verlinde's case, a relation across a holographic screen. The ontologies differ; the force laws converge in the appropriate limit.
3. The Tidal Anisotropy Test (Unresolved)
Will's original falsification argument [2] assumed a point-mass model of the galactic mass distribution—a single mass of 10¹¹ M☉ located at the galactic center, 20,000 light-years from Earth. Under this model, Whitehead's theory predicted a tidal gravitational anisotropy—a preferred direction in the effective gravitational constant—approximately 200 times larger than experimental bounds permitted.
Fowler [3] argued in 1974 that a realistic distribution of galactic mass across the disk substantially reduces this prediction. An exponential disk model with scale radius r₀ ≈ 3 kpc and galactocentric distance R ≈ 20 kpc yields a suppression factor:
(r₀ / R)² ≈ (3 kpc / 20 kpc)² ≈ 0.023 (6)
Applying this factor to Will's 200× discrepancy reduces it to approximately 200 × 0.023 ≈ 4.6. Current bounds from lunar laser ranging constrain ΔG/G < 2 × 10⁻¹¹ yr⁻¹ [7]. The smeared Whitehead prediction yields approximately 1 × 10⁻¹⁰ yr⁻¹—a factor of ~5 above the current bound.
Status: Unresolved
We treat the tidal anisotropy test as unresolved. The existing falsification arguments are insufficient to rule out Whitehead's theory under a realistic galactic model, but a full numerical integration—including bulge, halo, and non-axisymmetric disk structure—is required before any definitive conclusion can be drawn. The exponential disk estimate of equation (6) is itself a simplification. We present it as motivation for reassessment, not as a resolved calculation.
The next generation of precision gravity experiments—improved lunar laser ranging, GRACE-FO gravitational gradiometry, and future space-based interferometry—are projected to improve constraints on ΔG/G by one to two orders of magnitude over the coming decade. If these measurements place ΔG/G below the smeared Whitehead prediction, the theory will face a genuine falsification test that cannot be deflected by modeling arguments.
4. Testable Consequences
A theory that cannot be falsified is not a scientific theory. We offer three classes of testable consequences, each with an explicit falsification criterion. The flyby anomaly prediction is an order-of-magnitude consistency check; the rotation curve prediction follows the established Verlinde framework; the CMB connection is explicitly speculative.
4.1 Flyby Anomaly: Order-of-Magnitude Consistency Check
The flyby anomaly refers to small, unexplained velocity changes observed during Earth gravity assists—notably in missions including Galileo (1990), NEAR (1998), Rosetta (2005), and Messenger (2005). Observed anomalies fall in the range Δv_obs ~ 1–10 mm/s. These anomalies have no accepted explanation within general relativity.
The circulating vector field of equation (3) provides a natural candidate. Rather than deriving α from first principles—which requires a full galactic mass-current integration beyond the scope of this work—we perform an order-of-magnitude consistency check.
Scaling estimate
A characteristic velocity perturbation may be estimated from the circulating force integrated over the flyby trajectory:
Δv ~ α · (GM / Rc²) · f (7)
where M is the mass of Earth, R is the periapsis distance, c is the speed of light, and f is a dimensionless geometric factor of order unity encoding the trajectory. Using representative values for an Earth flyby (R ~ 6.5 × 10⁶ m):
GM / R ≈ 6.6 × 10⁷ m²/s² (8a)
1 / c² ≈ 1.1 × 10⁻¹⁷ s²/m² (8b)
GM / Rc² ≈ 7.3 × 10⁻¹⁰ (8c)
Estimates of α based on galactic mass currents (see Acedo [4]) suggest a plausible range of α ~ 10⁵–10⁷ m/s, following from galactic rotation velocities (~200 km/s) and mass-current densities integrated over the galactic disk. This yields:
Δv ~ α × 7.3 × 10⁻¹⁰ (9)
α (m/s)
Predicted Δv
Observed range
10⁵ (lower bound)
0.07 mm/s
1–10 mm/s
10⁶
0.73 mm/s
1–10 mm/s
10⁷ (upper bound)
7.3 mm/s
1–10 mm/s
The predicted range overlaps the observed scale across the upper portion of the α range, without parameter tuning. This indicates that the proposed mechanism is consistent at the level of magnitude. We emphasize what this result does and does not show:
What it shows: The framework naturally produces velocity perturbations in the correct observational range. The circulating vector field is a plausible candidate mechanism for the flyby anomaly.
What it does not show: α is not independently calculated here. The geometric factor f is not explicitly modeled. The sign of the predicted anomaly is not determined. A complete treatment requires explicit evaluation of the galactic mass-current integral (eq. 4), trajectory-dependent modeling, and comparison across multiple flyby events.
Falsification criterion (Flyby Anomaly):
A systematic survey of flyby events showing no correlation between the predicted geometric factor (dependent on spacecraft velocity, planetary rotation axis, and galactic current direction) and the observed Δv would constitute strong evidence against the circulating field hypothesis. A flyby for which the Acedo scaling predicts zero anomalous acceleration but which shows a non-zero effect would falsify this component of the model.
4.2 Galaxy Rotation Curves
The standard cosmological model invokes dark matter to explain the observed flat rotation curves of galaxies: the rotational velocity v at radius r from the galactic center should fall as v ∝ r^{−1/2} for Keplerian motion, but observations consistently show v ≈ constant at large radii. Standard cosmology accounts for this by positing a dark matter halo of density ρ ∝ r^{−2}.
The emergent gravity limit of the generalized Whiteheadian framework (equation 5) predicts rotation curves consistent with the baryonic Tully-Fisher relation:
v⁴ = G M_baryonic · a₀ (10)
where a₀ ≈ 1.2 × 10⁻¹⁰ m/s² is the MOND acceleration scale. In the Verlinde framework, this scale emerges from the dark energy density of the universe rather than being a free parameter. The baryonic Tully-Fisher relation holds across more than five orders of magnitude in galaxy mass—a fact that poses serious challenges for particle dark matter models and follows naturally from the emergent gravity framework.
Falsification criterion (Rotation Curves):
A galaxy for which the baryonic Tully-Fisher relation (eq. 10) fails—after applying the emergent gravity correction—would constitute strong evidence against this component. The dwarf galaxy regime (M < 10⁷ M☉) is the most sensitive test bed, as emergent gravity and particle dark matter make divergent predictions there. A confirmed failure of the relation in multiple low-mass systems would be difficult to reconcile with the framework.
4.3 CMB Non-Gaussianities (Speculative)
Standard ΛCDM predicts that the primordial perturbations seeding the CMB are very nearly Gaussian, with non-Gaussianity parameter f_{NL} consistent with zero. The extended Whiteheadian framework, interpreted through the 'Giant Carnot Engine' model [8], treats the universe as a thermodynamic system in which the entropy gradient of the holographic screen drives structure formation.
This model suggests the possibility of specific non-Gaussian signatures in the CMB bispectrum—temperature anisotropies and bispectrum shapes distinguishable from ΛCDM. We note explicitly that this connection is speculative; the predicted bispectrum shape is not derived here, and a full treatment of the model's CMB implications is deferred to future work.
Falsification criterion (CMB):
A CMB bispectrum consistent with ΛCDM (f_{NL} ≈ 0) to within the precision of future experiments—e.g., CMB-S4 or the Simons Observatory—would constrain this model if the Giant Carnot Engine predicts f_{NL} significantly different from zero. Conversely, a detection of non-Gaussianity with a shape matching the model's prediction would be strong positive evidence. We make no specific numerical prediction here; this criterion becomes operative only once the model's CMB implications are worked out in full.
5. Conclusion
The existing falsification arguments against Whitehead's theory of gravitation rely on simplified galactic models. Whether these conclusions hold under a realistic mass distribution remains an open question, and we have argued that a factor-of-~5 discrepancy under an exponential disk model does not constitute a robust falsification. A full numerical reassessment, integrating over the actual Milky Way structure, is the needed next step.
The generalized Whiteheadian framework—incorporating Acedo's circulating vector field and Verlinde's emergent gravity—extends Whitehead's relational ontology into three domains where general relativity and standard cosmology face genuine difficulties. The framework yields testable consequences: an order-of-magnitude consistency check with flyby anomalies, galaxy rotation curves without dark matter, and speculative connections to CMB non-Gaussianities. Each claim is falsifiable. Explicit criteria are provided.
We do not claim to have proven Whitehead's theory correct. We claim that the case against it was weaker than advertised—that its core ontological insight, gravity as relation rather than curvature, has found unexpected echoes in subsequent theoretical developments, and that the framework merits continued engagement rather than dismissal.
Whitehead wrote in 1922: "The notion of 'passing on' is more fundamental than that of a private individual fact." This is the sentence that general relativity forgot, and that emergent gravity is, perhaps, beginning to remember. The framework merits continued investigation.
Acknowledgments
The authors thank the anonymous reviewers for their skepticism. It sharpened the work.
References
[1] Whitehead, A. N. (1922). The Principle of Relativity. Cambridge University Press.
[2] Will, C. M. (1971). Theoretical Frameworks for Testing Relativistic Gravity. II. Parametrized Post-Newtonian Hydrodynamics, and the Nordström and Whitehead Theories. Astrophysical Journal, 163, 611–628.
[3] Fowler, D. R. (1974). Whitehead's Theory of Gravitation: A Philosophical Reassessment. Process Studies, 4(3), 165–178.
[4] Acedo, L. (2015). An Extended Whitehead's Theory of Gravitation and the Flyby Anomaly. Galaxies, 3(3), 113–133.
[5] Verlinde, E. (2009). On the Origin of Gravity and the Laws of Newton. Journal of High Energy Physics, 2011(4), 29. [arXiv:1001.0785]
[6] Gibbons, G. W., & Will, C. M. (2023). On the Multiple Deaths of Whitehead's Theory of Gravity. Studies in History and Philosophy of Science, 98, 1–12.
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End of preprint. Comments and falsification attempts welcome.