Spin Inertia: Relativistic and Magnetic Dynamics

• Spin inertia is the delayed response of spin degrees of freedom, reflecting non-instantaneous behavior in diverse physical systems.
• In magnetization dynamics, it appears as a second-order correction to the conventional LLG equation, leading to observable nutational motion alongside precession.
• It provides a precision probe in systems ranging from relativistic bodies to semiconductor spin optics and antiferromagnetic mechanics, linking spin relaxation to modified inertial properties.

Spin inertia denotes a family of inertia-like phenomena associated with spin degrees of freedom rather than a single universally fixed concept. In relativistic continuum mechanics, it modifies the inertial principle itself: for isolated systems with spin, the quantity that moves uniformly is the center of mass corrected by spin, not the ordinary center of mass. In magnetization dynamics, it denotes the appearance of a second-order time derivative in the effective spin equation of motion, producing nutation in addition to precession. In semiconductor spin optics, it denotes the delayed response of longitudinal spin polarization to periodically modulated optical or radio-frequency driving, and functions as a precision probe of slow spin relaxation. In antiferromagnetic spin mechanics, Néel-order dynamics renormalizes the inertia tensor of a rotating body (Medina et al., 2014, Rahman et al., 2021, Smirnov et al., 2018, Cheng et al., 2016).

1. Scope and principal usages

Current literature uses the term in several technically distinct but structurally related ways. The common element is the failure of a spin-related variable to respond instantaneously within a purely first-order description.

Context	Meaning of spin inertia	Representative relation
Relativistic isolated systems	inertial motion of the center of mass and spin	$\frac{d}{dt}\left(U X_\Theta^i\right)=c^2 P^i$
Magnetization dynamics	second-order correction to LLG dynamics, producing nutation	$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$
Optical spin spectroscopy	lag of spin polarization under modulated optical or rf driving	$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$
Antiferromagnetic spin mechanics	spin-induced correction to rigid-body inertia	$$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$

A recurrent misconception is to treat these usages as interchangeable. They are not. The relativistic literature concerns the inertial centroid of an isolated spinning system; the magnetic-dynamics literature concerns nutational corrections to LLG-type equations; the semiconductor literature concerns a modulation-frequency method for extracting $T_1$; and the spin-mechanical literature concerns renormalization of mechanical inertia by ordered spin dynamics (Medina et al., 2014, Ghosh et al., 2024, Belykh et al., 2022, Cheng et al., 2016).

2. Relativistic and kinematic formulations

For an isolated relativistic system with energy-momentum tensor $T^{\mu\nu}$, isolation means $\partial_\mu T^{\mu\nu}=0$, so the four-momentum

$P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$

is conserved. The ordinary center of mass,

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$

moves uniformly only if orbital angular momentum alone is conserved, equivalently when the relevant energy-momentum tensor is symmetric. For systems with intrinsic angular momentum, however, the conserved quantity is the total angular momentum

$$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$

Defining the spin contribution

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$0

one obtains the center of mass and spin

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$1

which satisfies

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$2

The modified Principle of Inertia proposed in this formulation is therefore: there are inertial frames of reference in which the center of mass and spin of isolated systems move with constant velocity (Medina et al., 2014).

This reformulation is not merely formal. The ordinary center of mass can fail to move inertially in an isolated relativistic system when spin and orbital parts exchange angular momentum. The paper emphasizes this for systems with electromagnetic interactions and magnetization, and connects the issue to the Abraham–Minkowski controversy and to the status of the Belinfante–Rosenfeld tensor, which restores an ordinary center-of-mass theorem only after spin has been effectively absorbed into a symmetrized tensor (Medina et al., 2014).

A related but distinct line of work treats spin as an inertial property of a particle on the same structural footing as mass. In this view, mass and spin are the invariants classifying irreducible unitary representations of the inhomogeneous Lorentz group. The inertial character of intrinsic spin is expressed through spin-rotation coupling. In a rotating frame with angular velocity $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$3, intrinsic spin obeys

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$4

with Hamiltonian

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$5

More generally,

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$6

so rotation couples to total angular momentum, not merely orbital angular momentum. For photons this becomes the helicity-rotation effect, $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$7 (Mashhoon, 2015).

The same rotational logic extends, in the weak-field gravitoelectromagnetic analogy, to spin-gravity coupling. The corresponding Hamiltonian is

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$8

and a position-dependent gravitomagnetic field generates a Stern–Gerlach-like force

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$9

Because this contribution is independent of inertial mass, free fall is violated in principle for spinning particles; for a neutron near Earth the fractional effect is of order $$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$0, far below present experimental reach (Mashhoon, 10 Feb 2025).

3. Inertial magnetization dynamics and nutation

In magnetic systems, spin inertia denotes the appearance of a second-order time derivative in equations that are first order in the conventional Bloch or Landau–Lifshitz–Gilbert description. A representative inertial LLG form is

$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$1

or, in a bath-derived macrospin form,

$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$2

The added term is nondissipative and generates nutation, a fast wobbling motion superposed on ordinary precession (Quarenta et al., 2023, Yuan, 3 Aug 2025).

A central point in this literature is that the inertial correction is not restricted to ultrafast switching protocols. In nanoscale ferromagnets, the modified LLG equation

$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$3

contains an angular-momentum relaxation time

$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$4

The paper states that $$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$5 in ferromagnets can range from a few femtoseconds to about $$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$6 ps. For an elliptical nanomagnet with threshold switching field

$$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$7

using $$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$8, $$\frac{FE(f_m)}{FE(0)}=\frac{1}{\sqrt{1+(2\pi f_m T_s)^2}}$$9, and $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$0, the threshold is $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$1 A/m. Near threshold, inertial nutation in the first few picoseconds changes the long-time fate of trajectories. At $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$2 A/m and room temperature, 25 stochastic trajectories gave a failure count rising from $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$3 without inertia to $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$4 with $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$5 ps; for 1000 trajectories run for 25 ns at each field value, the switching error probability at $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$6 A/m rose from about $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$7 for $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$8 to about $$\mathcal G_{ij} = G_{ij} + \frac{\hbar^2 V}{ZJa^3} g_{ij}$$9 for $T_1$0 ps (Rahman et al., 2021).

This early-time sensitivity is important technologically because it shifts attention from total switching time to the short inertial interval in which the trajectory is selected. The same literature stresses that inertia and damping must not be conflated: Gilbert damping is first order and dissipative, whereas the inertial term is second order and conservative (Rahman et al., 2021, Yuan, 3 Aug 2025).

4. Microscopic origins and tensorial structure

Several microscopic mechanisms have been proposed for magnetic spin inertia. In the Caldeira–Leggett framework, integrating out a bath of harmonic oscillators generates an effective kernel whose low-frequency Ohmic sector yields damping while high-frequency bath modes yield a universal inertial term. The high-frequency contribution satisfies

$T_1$1

The paper’s general claim is that any dissipation channel should also be accompanied by a bath-induced inertia term. In a YIG/GGG phonon-bath example,

$T_1$2

which gives $T_1$3 fs and $T_1$4 fs (Quarenta et al., 2023).

In metallic ferromagnets, integrating out conduction electrons in an $T_1$5 model produces an effective action containing $T_1$6, with

$T_1$7

Here $T_1$8 is the spin polarization of the conduction electrons per lattice site and $T_1$9 is the $T^{\mu\nu}$0 coupling constant. The same derivation modifies the conserved angular momentum density to

$T^{\mu\nu}$1

so spin and angular momentum are no longer strictly collinear in dynamical situations (Kikuchi et al., 2015).

A different proposal identifies unquenched orbital angular momentum as the missing reversible degree of freedom behind the inertial term. In a two-sublattice spin–orbital model with Russell–Saunders coupling, elimination of the orbital sector yields a single-sublattice equation with an explicit inertia parameter $T^{\mu\nu}$2. For cobalt, using a material-adapted $T^{\mu\nu}$3 from Co$T^{\mu\nu}$4 in CoO, the paper obtains

$T^{\mu\nu}$5

in reasonable agreement with experimentally reported cobalt inertial times of about $T^{\mu\nu}$6, $T^{\mu\nu}$7, and $T^{\mu\nu}$8 fs. The same work emphasizes a diagnostic distinction between genuine OAM-driven nutation and a spurious optical mode in a multisublattice ferromagnet: the field slope should correspond to an effective $T^{\mu\nu}$9-factor near $\partial_\mu T^{\mu\nu}=0$0 in the OAM case and near $\partial_\mu T^{\mu\nu}=0$1 for the conventional optical magnon mode (Moussa et al., 31 Mar 2026).

The scalar picture is not exhaustive. In normal metal–ferromagnet heterostructures, dynamical RKKY interactions generate a frequency expansion

$\partial_\mu T^{\mu\nu}=0$2

where the $\partial_\mu T^{\mu\nu}=0$3 term is the nutational contribution. Because the ferromagnetic state already breaks time-reversal symmetry, the resulting inertia tensor is anisotropic,

$\partial_\mu T^{\mu\nu}=0$4

rather than isotropic as in the paramagnetic case (Johnsen et al., 2024).

More generally, tensorial magnetic inertia may be decomposed as

$\partial_\mu T^{\mu\nu}=0$5

This separates scalar isotropic inertia $\partial_\mu T^{\mu\nu}=0$6, symmetric anisotropic inertia $\partial_\mu T^{\mu\nu}=0$7, and chiral antisymmetric inertia $\partial_\mu T^{\mu\nu}=0$8. The reported consequence is that precession resonances remain largely unaffected, while nutation resonances are shifted and their effective damping increases strongly with the chiral part (Ghosh et al., 2024).

Spin inertia also appears as a correction to ordinary mechanical inertia in antiferromagnets. After elimination of the induced magnetization, Néel dynamics acquires the kinetic term

$\partial_\mu T^{\mu\nu}=0$9

and, in the adiabatic strong-anisotropy limit, the rigid-body action becomes

$P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$0

The correction is tensorial, generally non-diagonal, scales relatively as $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$1, and is reduced by zero-point and thermal magnons through

$P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$2

The effect is therefore predicted to be most visible in small antiferromagnetic objects at low temperature (Cheng et al., 2016).

5. Collective dynamics, topology, and detection

Once the inertial term is present, the excitation spectrum is doubled. In metallic ferromagnets, linearization of

$P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$3

yields

$P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$4

For small $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$5, one branch is the usual gapless magnon $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$6, while the second is a new gapped inertial mode $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$7. The same inertial scale $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$8 appears in domain-wall dynamics, where the collective-coordinate Lagrangian acquires a wall mass $P^\nu=\frac{1}{c}\int T^{0\nu}\, dV$9 (Kikuchi et al., 2015).

In a normal metal–ferromagnet heterostructure with anisotropic nutation, the generalized LLG equation predicts not only the ordinary ferromagnetic resonance and a nutation-related high-frequency peak, but also an additional third resonance. This third peak is field-tunable, and a suitably chosen field can eliminate it, which is proposed as a specific fingerprint of anisotropic inertial dynamics in the ordered ferromagnetic state (Johnsen et al., 2024).

Domain-wall dynamics acquires a genuine effective mass when the inertial relaxation time $$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$0 is included. In the collective-coordinate description,

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$1

Without Gilbert damping, the domain-wall dynamics become chaotic, resembling that of an electron in a two-dimensional crystal. For finite damping, field-like driving can significantly increase the wall velocity compared to conventional massless dynamics, with pronounced velocity maxima that are sometimes nearly twice the non-inertial value for representative parameters. In the low-driving regime, the wall width contracts because the inertial correction enters the effective stiffness as

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$2

(Bassant et al., 11 Mar 2026).

Spin inertia also creates new routes to topological magnon bands. In a honeycomb ferromagnet, the inertial LLG equation produces low-frequency precessional bands and high-frequency nutational bands. Hybridization between these sectors requires interactions that break angular-momentum conservation; in the example studied, the crucial term is a pseudodipolar interaction. The resulting avoided crossings open topological gaps with nonzero Chern numbers and chiral edge states in slab geometries. The paper emphasizes that the inertial mechanism differs from the effect of the Dzyaloshinsky–Moriya interaction considered there, because the latter does not hybridize opposite-rotation inertial modes in the same way (Ghosh et al., 6 Mar 2026).

A direct spectroscopic proposal uses graphene plasmons to detect nutation spin waves. In a graphene$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$3magnet heterostructure, the inertial branch

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$4

hybridizes with the plasmon, and in the thick-film limit the hybrid frequency becomes

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$5

The experimentally proposed signature is a dip in the reflection spectrum of an Otto-geometry heterostructure, with the dip position providing a quantitative measure of $$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$6 (Yuan, 3 Aug 2025).

6. Spin inertia as a spectroscopy of slow spin relaxation

In semiconductor spin optics, spin inertia is not a second-order equation of motion for the spin vector. It is the lag of longitudinal spin polarization under periodic reversal of the drive. For helicity-modulated circular pumping, the signal is the Fourier component of $$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$7 at the modulation frequency. In the general theory for singly charged quantum dots,

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$8

and, when the trion relaxes much faster than the resident spin dynamics,

$$X^i=\frac{1}{U}\int x^i T^{00}\, dV, \qquad U=cP^0,$$9

Here $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$0 is the spin-flip probability during the trion lifetime and $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$1 is the effective longitudinal relaxation time. The theory incorporates frozen and finite-correlation-time Overhauser fields, trion dynamics, longitudinal $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$2-factors, hyperfine anisotropy, and pump-induced saturation (Smirnov et al., 2018).

This framework explains why polarization-recovery curves can be V-like or M-like. In singly charged $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$3 quantum dots, the measured Faraday ellipticity obeys, in the simple regime,

$$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$4

For $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$5-type dots the polarization-recovery curve is V-like, while for $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$6-type dots it is M-like, because the field dependence of the trion spin-flip probability differs for electron-dominated and hole-dominated trions. The reported extrapolated equilibrium lifetimes are $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$7 for $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$8-type dots and $$J^{\mu\nu}=L^{\mu\nu}+S^{\mu\nu}, \qquad \dot J^{\mu\nu}=0.$$9 for $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$00-type dots (Zhukov et al., 2018).

The same language of spin inertia is used for other localized-carrier systems. In fluorine-doped ZnSe, helicity-modulated pump–probe Kerr rotation yields

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$01

from which the donor-bound electron longitudinal relaxation time was extracted as $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$02 for one sample. The notable result is that this $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$03 remains nearly constant from zero field to $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$04 T in Faraday geometry and from $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$05 K to $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$06 K, strongly constraining candidate relaxation mechanisms (Heisterkamp et al., 2015).

A selective extension, termed resonant spin inertia, uses rf-driven depolarization instead of helicity reversal. In Ce$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$07:YAG, amplitude-modulated resonant rf addresses a chosen Larmor frequency and therefore a chosen $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$08-selected subensemble. The low-amplitude response obeys

$$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$09

The measured $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$10 is on the order of milliseconds, extrapolation to zero optical power gives an intrinsic $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$11 of about $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$12 ms, and the field-orientation dependence shows that $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$13 varies by more than a factor of $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$14. The same experiment indicates that the spread of Larmor frequencies is not frozen but caused by fluctuating internal magnetic fields with correlation time $$\partial_t \mathbf{m} = -\gamma \mu_0\,\mathbf{m}\times \mathbf{H}_\mathrm{eff} +\mathbf{m}\times\left(\alpha\,\partial_t\mathbf{m}+\eta\,\partial_{tt}\mathbf{m}\right)$$15 (Belykh et al., 2022).

The semiconductor and resonant-rf literature therefore uses spin inertia as a frequency-domain relaxation probe rather than as a nutational correction to LLG. That distinction is essential. The former measures how a long-lived longitudinal spin polarization fails to keep up with a periodically varying drive; the latter describes how magnetization acquires an additional dynamical degree of freedom and a high-frequency nutation mode (Schering et al., 2019, Belykh et al., 2022).