Muon Storage-Ring Time Dilation and g-2 Experiments

The ring is a clock, and everybody knows it. Because $\tau_{\text{lab}} = \gamma\tau_0$ is verified to $10^{-3}$ at $10^{18}\,g$, storage rings are now used as time-dilation instruments rather than tested as such — and the same ledger is spent on other quantities. A muon electric dipole moment would tip the precession plane out of the horizontal, a vertical oscillation at the same $\omega_a$ and $90^\circ$ out of phase; the arithmetic is identical, so the same $\gamma$ that buys $14.7$ cycles for $a_\mu$ buys them for $d_\mu$, and the EDM search rides free on the $g\!-\!2$ data.

The frontier now is a ring that does not need the magic momentum at all. J-PARC's E34 will store ultra-cold muons at $p = 300$ MeV/$c$ ($\gamma = 3$) in a compact solenoid with no electric focusing — no $\vec E$ term to cancel, no magic condition to satisfy. The price is paid in exactly the currency this article counts: $\gamma = 3$ instead of $29.3$ is ten times fewer precession cycles per muon, bought back with intensity and a far colder phase space. The design trade is a line item in the cycle ledger.

Three things in this ring happen at some rate, and $\gamma$ treats each differently. Getting them straight is the special relativity of the muon anomaly.

Clock one: the muon dies. Proper lifetime $\tau_0 = 2.1969811\,\mu$s; in the lab, $\tau_{\text{lab}} = \gamma\tau_0$. Slower by $\gamma$.

Clock two: the muon goes round. The Lorentz force bends momentum $p = \gamma m v$, and heavier momentum bends more reluctantly: $\omega_c = eB/\gamma m$. Slower by $\gamma$. Two clocks, both dilated, no surprises.

Clock three: the spin walks ahead of the momentum. If $g$ were exactly $2$ the spin would track the velocity forever and this clock would not tick. It is not — QED, the strong interaction, and anything else that couples to a muon see to that — and the spin gains on the momentum at

Look at what is missing. There is no $\gamma$. The individual frequencies $\omega_s$ and $\omega_c$ are both riddled with $\gamma$; their difference is clean. A muon at rest and a muon at $\gamma = 29$ have their spins walking away from their momenta at the same number of radians per second of laboratory time. Why, in one line: the field boosts up by $\gamma$ and the clock slows down by $\gamma$. In its own frame the muon does not see $B$; it sees $\gamma B$. So the anomalous cycles a muon lives to show you — frequency $\times$ lifetime — is

Linear in $\gamma$, with nothing cancelling. Every factor of $\gamma$ bought with the accelerator is a factor of $\gamma$ more signal — and at $\gamma = 1$, a muon sitting still in the Fermilab magnet, $N_{\text{cyc}} = 0.5033$. Half a cycle. It decays before its spin has finished the first turn.

Equation (7) is the whole apparatus in one line. The exponent has a $\gamma$ in it and the cosine does not. Every other equation here exists to explain that asymmetry. Two numbers pin the ring. The magic momentum $p = mc/\sqrt{a_\mu} = 3.0943$ GeV/$c$ follows from (6) because $\gamma\beta = \sqrt{\gamma^2-1} = 1/\sqrt{a_\mu}$ exactly. Feed it into magnetic rigidity, $p[\mathrm{GeV}/c] = 0.299792\,B[\mathrm{T}]\,\rho[\mathrm{m}]$, with the design radius $\rho = 7.112$ m, and the field falls out: $B = 1.4513$ T — the field the Fermilab magnet is actually set to. The anomaly picked the momentum; the momentum and the radius picked the field.

:::svg[THE NEWTONIAN COUNTERFACTUAL. Both panels plot the same physical signal — the g−2 wiggle, N(t) = N₀·exp(−t/τ)·[1 + A·cos(ω_a t)] — over exactly ONE mean lifetime, with the same anomalous frequency f_a = 229.08 kHz in both. The ONLY difference is γ. LEFT, at rest (γ = 1, τ = τ₀ = 2.1969811 µs): the muon is dead before the cosine has completed half a turn. N = 0.5033 cycles. There is nothing here to fit — no period, no phase, no frequency, just a single sagging curve. This is what the muon anomaly looks like in a universe without time dilation, and it is why the measurement would simply not exist. RIGHT, at the magic momentum (γ = 29.3034, τ_lab = 64.379 µs): the SAME frequency, in the SAME field, but now the exponential envelope has been stretched by 29.3 while the cosine has not been stretched at all — because ω_a = a(e/m)B carries no γ. N = 14.748 cycles per muon. Note carefully what did and did not change: the envelope dilated, the wiggle did not. That asymmetry is the entire experiment, and it is written into the fit function itself — the γ sits in the exponent and is absent from the cosine]

1 · Build the entire storage ring out of one number

Start with the anomaly and nothing else: $a_\mu = 1.1659207\times10^{-3}$ (the measured value, PDG/Muon $g\!-\!2$).

The Lorentz factor. The magic condition $a_\mu = 1/(\gamma^2-1)$ gives $\gamma = \sqrt{1 + 1/a_\mu} = \sqrt{1 + 857.69} = \mathbf{29.3034}$, hence $\beta = 0.9994175$.

The momentum. $\gamma\beta = \sqrt{\gamma^2-1} = 1/\sqrt{a_\mu} = 29.286$, so $p = \gamma\beta\, m c = 29.286 \times 105.6583755\ \mathrm{MeV}/c = \mathbf{3.0943\ GeV}/c$. The muon's magnetic anomaly has just chosen the beam energy of a $\$700$-million experiment.

The field. Rigidity: $B = p/(0.299792\,\rho) = 3.0943/(0.299792 \times 7.112) = \mathbf{1.4513\ T}$ at $\rho = 7.112$ m. The Fermilab magnet runs at $1.45$ T.

The clocks. $\tau_{\text{lab}} = \gamma\tau_0 = \mathbf{64.379\ \mu s}$. Orbit period $T = 2\pi\rho/\beta c = \mathbf{149.14\ ns}$, so $f_c = \mathbf{6.705\ MHz}$. Anomalous frequency $f_a = a_\mu eB/2\pi m = \mathbf{229.08\ kHz}$.

The yield. $N_{\text{cyc}} = f_a\tau_{\text{lab}} = \mathbf{14.75}$ cycles and $\tau_{\text{lab}}/T = \mathbf{431.7}$ laps per muon. The spin gains a full turn on the momentum every $\omega_c/\omega_a = \mathbf{29.27}$ laps.

Everything above came from $a_\mu$, $m_\mu$, $\tau_0$ and the ring radius. No fitting, no lookup. That closure is the fastest way to know the physics is right — and the fastest way to see the base's $\omega_a$ cannot be, since feeding its formula into the last step returns $-189.77$ MHz.

2 · The clock hypothesis at a billion billion $g$

Special relativity says the dilation factor is $\gamma$, and $\gamma$ is built from speed alone — nothing in $\mathrm{d}\tau = \mathrm{d}t/\gamma$ knows about acceleration. That is an assumption, the clock hypothesis, logically independent of the postulates, and it has to be tested. The stored muon is the test. Its proper acceleration on the $\rho = 7.112$ m orbit is

If proper acceleration entered the dilation at all, at any order and with any coefficient of reasonable size, a number that large would show. It does not. The CERN Muon Storage Ring measured the stored lifetimes directly: $\tau_+ = 64.419(58)\ \mu$s and $\tau_- = 64.368(29)\ \mu$s (Bailey et al., Nature 268, 301 (1977)), against the prediction $\gamma\tau_0 = \mathbf{64.379\ \mu s}$ at $\gamma = 29.33$. Agreement at the $10^{-3}$ level, on a clock swung around at $10^{18}$ times Earth's gravity.

The clock hypothesis holds — and the experiment that verified it was not designed to. The lifetime measurement was a by-product of a magnetic-moment measurement; the CERN final report is titled, in part, "…and a Direct Test of Relativistic Time Dilation" (Bailey et al., Nucl. Phys. B150, 1 (1979)). It remains among the sharpest tests of the clock hypothesis ever made, and it is why a muon ring can be trusted as a clock at all.

Texts and papers. Jackson, Classical Electrodynamics, 3rd ed., §11.11 · Bargmann, Michel & Telegdi, Phys. Rev. Lett. 2, 435 (1959) · Thomas, Nature 117, 514 (1926) · Bailey et al., Nature 268, 301 (1977) — the direct time-dilation test · Bailey et al., Nucl. Phys. B150, 1 (1979) — the CERN final report · Muon $g\!-\!2$ Collaboration, Phys. Rev. Lett. 126, 141801 (2021) and the 2025 final result · Muon $g\!-\!2$ Theory Initiative, White Paper 2025 · Particle Data Group, Review of Particle Physics.

Provenance. Signature $(+,-,-,-)$. Every number here is recomputed at 50 decimal digits in verify_aq-sr-muon-g2-time-dilation.py, which reads this file, extracts the numbers from the prose, and asserts them. Inputs: $m_\mu c^2 = 105.6583755$ MeV, $\tau_0 = 2.1969811\ \mu$s, $a_\mu = 1.1659207\times10^{-3}$, $\rho = 7.112$ m (PDG / Muon $g\!-\!2$). $B = 1.4513$ T is derived from $p_{\text{magic}}$ and $\rho$ by magnetic rigidity.

Base defect repaired. The base's eq[2] gives $\omega_s = \tfrac{g}{2}\tfrac{eB}{\gamma m} + (1-\gamma)\tfrac{eB}{\gamma m}$. The Larmor term must not carry $1/\gamma$ — the rest-frame field is $\gamma B$, and the boost cancels the dilation. Corrected, $\omega_s = \tfrac{eB}{m}\left[\tfrac{g}{2} - 1 + \tfrac1\gamma\right]$ and $\omega_a = a_\mu eB/m$ exactly. The base's form yields $-189.77$ MHz instead of $+229.08$ kHz, is $\gamma$-dependent against its own eq[5], and fails to vanish at $g = 2$ against its own limits card. Its eq[3], eq[5] and eq[7] are correct and used as written — eq[3] with the caveat that it assumes $\vec\beta\cdot\vec B = 0$, true in this ring and false in general.

Scope, and the fence. The covariant spin-transport law is the topic of `aq-sr-bmt-equation`; the kinematic frame rotation is the topic of `aq-sr-thomas-precession`. This article does not re-derive either and makes no claim about what either contains. The Thomas rate $\omega_T = (1-\gamma)\omega_c$ is taken from the primary literature (Thomas 1926; Jackson §11.11) and used, not proved.

Builds on. Time Dilation · The Electromagnetic Field Tensor · Relativistic Beam Dynamics. Leads to. The BMT Equation · Thomas Precession · GPS Relativistic Corrections.