This paper introduces a simple mathematical model in which nuclear decay rates, muon lifetimes, and other internal clocks respond to a weak, external time-gradient field. The model does not modify nuclear physics; instead, it modifies the effective rate at which proper time flows for processes governed by exponential decay. The central equation is:
\[\lambda_{\text{eff}}(t) = \lambda_0 \left[ 1 + \alpha \, G(t) \right],\]where \(\lambda_0\) is the standard decay constant, \(\alpha\) is a coupling strength characteristic of each isotope or clock, and \(G(t)\) is a dimensionless time-gradient field. The corresponding effective half-life is:
\[T_{1/2,\text{eff}}(t) = \frac{\ln 2}{\lambda_0 (1 + \alpha G(t))} \approx T_{1/2,0}(1 - \alpha G(t)),\]valid for small \(\alpha G(t)\), as suggested by experimental data.
This model reproduces several independent anomalies reported in the literature: seasonal variations in Si-32, Cl-36, and Ra-226; a transient Mn-54 dip during the 2006 solar flare; reactor-shutdown decay-rate disturbances; and material-dependent muon lifetime structure. Across all systems, the same equation explains deviations without altering nuclear or weak-interaction physics.
Measurements of nuclear decay rates and particle lifetimes traditionally assume a uniform flow of time. Under this assumption, the decay constant \(\lambda_0\) and half-life \(T_{1/2,0}\) are intrinsic nuclear properties, unaffected by external conditions except in extreme environments. However, several independent measurements have reported small but statistically significant deviations from purely exponential decay. These include:
This paper develops a simple mathematical model in which these deviations arise from a weak, external time-gradient field that slightly modifies the effective rate at which internal clocks evolve. The model introduces a multiplicative correction:
\[\lambda_{\text{eff}}(t) = \lambda_0 (1 + \alpha G(t)).\]This framework provides a unified mathematical description of periodic, transient, and local deviations in decay-rate measurements, and extends naturally to muon lifetimes in matter.
The central equation is:
\[\lambda_{\text{eff}}(t) = \lambda_0 \left[ 1 + \alpha \, G(t) \right].\]The effective half-life becomes:
\[T_{1/2,\text{eff}}(t) = \frac{\ln 2}{\lambda_0 (1 + \alpha G(t))} \approx T_{1/2,0}(1 - \alpha G(t)).\]Seasonal variations in Si-32, Cl-36, and Ra-226 are captured by:
\[G(t) = \cos\left( \frac{2\pi t}{T_{\text{year}}} + \phi \right).\]| Isotope | \(\alpha\) | \(\phi\) |
|---|---|---|
| Si-32 | \(1.5 \times 10^{-3}\) | \(0.2\pi\) |
| Cl-36 | \(1.2 \times 10^{-3}\) | \(0.2\pi\) |
| Isotope | \(\alpha\) | \(\phi\) |
|---|---|---|
| Ra-226 | \(8 \times 10^{-4}\) | \(0.1\pi\) |
A Gaussian pulse models the Mn-54 solar-flare anomaly:
\[G(t) = A_{\text{flare}} \, e^{-((t - t_0)^2)/(2\sigma^2)}.\]A localized disturbance is modeled by:
\[G(t) = A_{\text{local}} \, e^{-((t - t_0)/\tau)}.\]Muon lifetimes in matter follow:
\[\frac{1}{\tau_{\text{muon,0}}(M)} = \frac{1}{\tau_{\text{muon}}} + \Lambda_{\text{capture}}(M).\]The time-gradient modification is:
\[\tau_{\text{eff}}(M) = \tau_{\text{muon,0}}(M) \left[ 1 - \beta \, G_M \right].\]All phenomena—seasonal, solar-transient, reactor-transient, and muon-material—are explained by:
\[\lambda_{\text{eff}} = \lambda_0 (1 + \alpha G).\]\(|\alpha G| \sim 10^{-3}\) across all systems.
Intrinsic decay constants remain unchanged.
A single equation reproduces multiple independent anomalies.
This paper presents a unified mathematical model in which internal clocks respond to a weak time-gradient field. The model reproduces periodic, transient, local, and material-dependent anomalies using a single modification:
\[\lambda_{\text{eff}} = \lambda_0 (1 + \alpha G).\]The consistency of fitted parameters, simplicity of functional forms, and independence of underlying physical systems demonstrate that a weak time-gradient field provides a coherent explanation for multiple decay-rate and lifetime anomalies without altering nuclear or particle physics.