[Paper Review] Post-Newtonian effects of Dirac particle in curved spacetime - I : magnetic moment in curved spacetime

Takahiro Morishima, Toshifumi Futamase|arXiv (Cornell University)|Jan 30, 2018

Quantum and Classical Electrodynamics1 references5 citations

TL;DR

This paper investigates the post-Newtonian correction to the magnetic moment of Dirac fermions in curved spacetime using the Schwarzschild metric. It shows that gravity universally modifies the effective magnetic moment and gyro-magnetic ratio via a factor of $(1 + 3\phi/c^2)$, implying a gravitational anomaly of $2.1 \times 10^{-9}$ in the anomalous magnetic moment on Earth's surface, independent of fermion composition or electromagnetic coupling.

ABSTRACT

The magnetic moment of free fermions in the curved spacetime has been studied based on the general relativity. Adopting the Schwarzschild metric for the background spacetime, the effective value of the magnetic moment has been calculated up to the post-Newtonian order $O(1/c^2)$ for three cases (A) Dirac particles with g=2, (B) neutral fermions with g$ e$2 and e=0 and (C) charged fermions with g$ e$2 and e$ e$0. The result shows their gravity dependence is given as $\mu_{ m m}^{ m eff}= (1\!+\!3\phi/c^2) \,\mu_{ m m} $ for all of these cases in which the coupling between fermions and the electromagnetic field is essentially different. It implies that the magnetic moment is influenced by the spacetime curvature on the basis of the general relativity commonly for point-like fermions, composite fermions and spread fermions dressed with the vacuum fluctuation. The gravitational effect affects the gyro-magnetic ratio and the anomalous magnetic moment as ${ m g}^{ m eff} \!\simeq\! (1 \!+\! 3\phi/c^2)\,{ m g} $, ${a}^{ m eff} \!\simeq\! a \!+\! 3(1\!+\!a)\,\phi/c^2 $. Consequently, the anomalous magnetic moment of fermions with g$\simeq$2 measured on the Earth's surface contains the gravitational effect as $|a^{ m eff}| \simeq 3|\phi|/c^2 \simeq 2.1\! imes\! 10^{-9}$, which implies that the gravitational anomaly of $2.1\! imes\! 10^{-9}$ is induced by the curvature of the spacetime on the basis of the general relativity in addition to the quantum radiative corrections for all fermions including electrons and muons.

Motivation & Objective

• To investigate how spacetime curvature in general relativity influences the magnetic moment of fermions at post-Newtonian order $O(1/c^2)$.
• To determine whether the gravitational correction to the magnetic moment depends on the fermion's charge, g-factor, or internal structure (point-like, composite, or vacuum-dressed).
• To assess whether the anomalous magnetic moment of fermions on Earth's surface contains a gravitational contribution beyond quantum corrections.
• To derive a universal expression for the effective magnetic moment and gyro-magnetic ratio in curved spacetime.

Proposed method

• Adopting the Schwarzschild metric as the background spacetime geometry to model weak gravitational fields.
• Solving the Dirac equation for fermions in curved spacetime up to $O(1/c^2)$ post-Newtonian order.
• Calculating the effective magnetic moment $\mu^{\text{eff}}_{\text{mm}}$ for three distinct cases: g=2 Dirac particles, neutral fermions with g≠2, and charged fermions with g≠2 and e≠0.
• Deriving the effective gyro-magnetic ratio $g^{\text{eff}}$ and anomalous magnetic moment $a^{\text{eff}}$ from the effective magnetic moment.
• Using the post-Newtonian expansion to isolate the gravitational contribution to the magnetic moment, expressed as $\mu^{\text{eff}}_{\text{mm}} = (1 + 3\phi/c^2)\mu_{\text{mm}}$.
• Applying the derived expressions to the Earth's gravitational potential $\phi \simeq -GM/r$ to estimate the observable gravitational anomaly on the surface.

Experimental results

Research questions

• RQ1 How does spacetime curvature in the post-Newtonian regime affect the magnetic moment of Dirac fermions?
• RQ2 Does the gravitational correction to the magnetic moment depend on the fermion's charge or g-factor?
• RQ3 What is the magnitude of the gravitational contribution to the anomalous magnetic moment of fermions on Earth's surface?
• RQ4 Can a universal gravitational correction be derived for all fermions, regardless of their internal structure or electromagnetic coupling?

Key findings

• The effective magnetic moment of fermions in curved spacetime is universally enhanced by a factor of $(1 + 3\phi/c^2)$, regardless of their charge or g-factor.
• The effective gyro-magnetic ratio is corrected as $g^{\text{eff}} \simeq (1 + 3\phi/c^2)g$, indicating a gravitational influence on the g-factor.
• The anomalous magnetic moment is modified as $a^{\text{eff}} \simeq a + 3(1+a)\phi/c^2$, showing a gravitational contribution even for $a \simeq 2$.
• On Earth's surface, the gravitational anomaly in the anomalous magnetic moment is estimated at $|a^{\text{eff}}| \simeq 2.1 \times 10^{-9}$, arising purely from spacetime curvature.
• This gravitational correction is universal and applies to all fermions, including electrons, muons, and composite fermions, independent of quantum radiative corrections.