Spin-Velocity Metric Tensor

• The spin–velocity metric tensor is a unified tensorial construct that encodes spin–kinematic coupling, integrating intrinsic spin with particle velocity or momentum across diverse physical systems.
• It captures key phenomena ranging from thermal spin magnetization in quantum materials to modified energy–momentum transport in relativistic fluids, underpinning advanced transport models.
• In applications from band geometry to vortex dynamics and gravitational theories, it provides a crucial framework linking quantum transport, spin hydrodynamics, and torsion-based gravity.

The spin-velocity metric tensor is a tensorial construct encoding the interplay between intrinsic spin degrees of freedom and particle or fluid velocity, with manifestations in relativistic field theory, spin hydrodynamics, band geometry, boundary vorticity dynamics, and quantum transport. Its formulation generalizes the standard projection or metric tensors to incorporate spin–kinematic coupling, appearing as a fundamental band-geometric quantity in quantum matter, as well as in classical and continuum mechanics with spin or vorticity. It provides the correct energy-momentum and transport structure for systems where spin cannot be adiabatically separated from momentum or velocity, especially in complex quantum or dissipative environments.

1. Spin–Velocity Metric Tensor in Relativistic Fluids and General Relativity

In classical general relativity, perfect-fluid energy–momentum is encoded via a projection tensor constructed from the fluid's four-velocity $u^\mu$. For spinning matter, as established in the Mathisson–Papapetrou–Dixon (MPD) framework, the dynamical direction of transport is dictated by the momentum $p^\mu$. To maintain compatibility with the Tulczyjew-type supplementary condition $p_\mu S^{\mu\nu}=0$, the projection tensor is modified by replacing $u^\mu$ with

$\pi^\mu = \frac{p^\mu}{\sqrt{-p_\nu p^\nu}}$

so that

$\Pi^{\mu\nu} = g^{\mu\nu} + \pi^\mu \pi^\nu$

serves as the spin-velocity metric tensor (Mohseni, 2010). In the presence of spin, the energy–momentum tensor becomes

$$T^{\mu\nu} = (\rho + P)\pi^\mu \pi^\nu + P\Pi^{\mu\nu} + (\text{spin terms}),$$

where additional corrections are given by Belinfante–Rosenfeld-type tensors arising from the explicit variation of tetrad fields and covariant derivatives. This modified tensor structure ensures accurate transport of energy and momentum in a relativistic spinning fluid, and reduces to the perfect-fluid form in maximally symmetric backgrounds such as Friedmann–Robertson–Walker spacetime, where spin–velocity corrections vanish at leading order.

2. Quantum Geometry: Band-Structure and Spin–Velocity Metric Tensor

The spin–velocity metric tensor occupies a central position as a band-geometric object driving thermal spin magnetization and intrinsic spin currents in quantum systems. It is defined via interband matrix elements: $$\mathcal{Q}^{(\gamma; a)}_{nm} = v^{a}_{nm} s^{\gamma}_{mn} = \mathcal{V}^{(\gamma; a)}_{nm} - \frac{i}{2}\mathcal{U}^{(\gamma; a)}_{nm},$$ where $v^{a}_{nm}$ denotes velocity operator elements and $s^{\gamma}_{mn}$ spin operator elements in the Bloch basis (Sarkar et al., 25 Sep 2025). The real part $\mathcal{V}$ (anomalous spin velocity) and imaginary part $\mathcal{U}$ (anomalous spin polarizability) together produce the complete tensorial response governing thermal spin magnetization (TSM), particularly the intrinsic Fermi-sea contribution, observable even in non-magnetic insulators.

Simultaneously, thermal spin currents are generated by a spin geometric tensor that generalizes the quantum geometric tensor by combining spin Berry curvature and spin quantum metric,

$$\mathcal{T}^{(\gamma; ab)}_{nm} = \frac{\{v^a, s^\gamma\}_{pm} v^b_{nm}}{4\epsilon_{pm}^2} = \mathcal{G}^{(\gamma; ab)}_{nm} - \frac{i}{2} \Omega^{(\gamma; ab)}_{nm},$$

capturing both the quantum metric (real symmetric part) and Berry curvature (imaginary part).

Numerical computations for chiral metals (RhGe) and antiferromagnets (CuMnAs) show sizable band-geometric Fermi-sea contributions to thermal spin transport near band crossings, with symmetry and topology dictating the allowed tensor components (Sarkar et al., 25 Sep 2025).

3. Spin–Velocity Metric Tensor in Nonequilibrium Hydrodynamics and Thermodynamics

In relativistic hydrodynamics with macroscopic spin, the form of the quantum stress–energy tensor (canonical or Belinfante symmetrized) determines the coupling between spin and fluid velocity (Becattini et al., 2018, Fukushima et al., 2020). The local equilibrium density operator includes explicit coupling terms: $$\rho_{LE} = \frac{1}{Z} \exp\left[-\int_\Sigma d\Sigma_\mu \left(T^{\mu\nu} \beta_\nu - \frac{1}{2}\Omega_{\lambda\nu} S^{\mu,\lambda\nu} - \zeta j^{\mu}\right)\right]$$ where $\beta^\nu$ is the four-temperature (proportional to the velocity field) and $\Omega_{\lambda\nu}$ is the spin chemical potential. Derivative couplings such as $$-\frac{1}{2} \varpi_{\lambda\nu} S^{\mu,\lambda\nu}$$ encode spin–vorticity interaction, and the resulting tensor structures mediate the transport properties, spectra, and polarization observables in heavy-ion collisions.

Pseudo-gauge transformations between canonical and Belinfante tensors alter the form of the entropy currents, and the precise spin–velocity tensorial couplings are essential for correct predictions of polarization and spin-current density, especially in the regime of slow spin relaxation (Becattini et al., 2018, Fukushima et al., 2020).

4. Antisymmetric Metric Tensors and Torsion in Gravitation

Extensions of Einsteinian gravity incorporate an antisymmetric part to the metric tensor: $g_{\mu\nu} = \Upsilon_{\mu\nu} + \Phi_{\mu\nu}$ where $\Upsilon_{\mu\nu}$ is symmetric and $\Phi_{\mu\nu}$ is antisymmetric, acting as the potential for the spin or torsion field (Hammond, 2012, Hammond, 2019). The torsion tensor is generated by derivatives of the antisymmetric component,

$$S_{\mu\nu\sigma} = \partial_\sigma \Phi_{\nu\mu} + \partial_\mu \Phi_{\sigma\nu} + \partial_\nu \Phi_{\mu\sigma},$$

which links spin–velocity metric structures to the underlying geometry. To ensure gauge invariance in the spin sector, an explicit addition of an electromagnetic field term $F_{\mu\nu} = \xi_{[\nu,\mu]}$ is required, making the full effective potential $[\Phi]_{\mu\nu} = \Phi_{\mu\nu} + F_{\mu\nu}$ (Hammond, 2019). This establishes a deep connection between spin-velocity metric tensors, torsion, and electromagnetic field theory.

In string theory, the antisymmetric metric part is identified with the Kalb–Ramond field, coupling naturally to world-sheet actions via

$$S = S_{\mathrm{NG}}[\Upsilon_{\mu\nu}] + S_{\mathrm{KR}}[\Phi_{\mu\nu}],$$

thus embedding spin-velocity metric structure directly into both gravitational and quantum geometric frameworks (Hammond, 2012).

5. Velocity Gradient and Spin–Velocity Metric Tensor in Vortex Dynamics

In the context of compressible fluid dynamics and boundary vorticity, the spin–velocity metric tensor arises naturally from decompositions of the velocity gradient tensor at solid walls (Chen et al., 13 Jun 2024). The gradient tensor $$\bm{A}_w = \theta_w \bm{n} \bm{n} + \bm{n} \bm{\xi}$$ (with $\xi = s_w \times n$) separates dilatational and spin contributions, with the spin part originating from the tangential vorticity component.

The normal–nilpotent decomposition splits $\bm{A} = \bm{N} + \bm{S}$, isolating pure spin dynamics from orbital motion. Boundary fluxes of tensor invariants such as $Q$ (the second principal invariant) and $R$ (the third) involve combinations of spin–curvature and spin–pressure gradient couplings, and precise formulae are derived for wall-normal derivatives, e.g.

$$\left[\partial_n Q\right]_w = (\nabla_p \cdot \xi) \theta_w - \xi \cdot (\nabla_p \theta_w) - K \theta_w^2 - \xi \cdot (K \cdot \xi)$$

where $K$ is mean curvature and $\nabla_p$ the surface gradient. This formalism provides a unified metric for vortex generation and flow noise at boundaries, essential for understanding coherent near-wall structures (Chen et al., 13 Jun 2024).

6. Spin Quantum Metric and Spin Hall Effects in Quantum Transport

In quantum transport, the spin–velocity metric tensor is realized as the real, symmetric part of the spin quantum geometric tensor, defined by

$$T^{(\alpha \beta;\gamma)}_{nm} = \frac{v^{(\alpha;\gamma)}_{nm} v^{\beta}_{mn}}{\epsilon_{nm}^2} = g^{(\alpha\beta;\gamma)}_{nm} - \frac{i}{2} \Omega^{(\alpha\beta;\gamma)}_{nm}$$

where $v^{(\alpha;\gamma)}$ incorporates spin-resolved velocity (Xiang et al., 4 Jun 2024). Fundamental symmetry analysis reveals:

• The spin quantum metric $g^{(\alpha\beta;\gamma)}$ is $\mathcal{T}$-odd (reverses under time reversal).
• The spin Berry curvature $\Omega^{(\alpha\beta;\gamma)}$ is $\mathcal{T}$-even.

In systems subject to high-frequency AC electric fields, the $\mathcal{T}$-odd spin quantum metric can drive an intrinsic spin Hall effect (ISHE) comparable in magnitude to the conventional Berry curvature-driven ISHE, especially near band crossings or anticrossings. Calculations in magnetically tilted surface Dirac cones and ferromagnetic monolayer MnBi$_2$Te$_4$ demonstrate pronounced spin Hall responses initiated by the spin quantum metric under THz/IR fields (Xiang et al., 4 Jun 2024). This establishes the metric as a new fundamental object in quantum geometry and ultrafast spintronics.

7. Spin–Velocity Tensor in Generally Covariant Mechanics

The concept of the velocity tensor provides a generally covariant foundation by defining particle motion via $V^\mu(dx^\mu) = 0$ (Kapuścik et al., 2011). The extension toward a spin–velocity metric tensor is anticipated by combining velocity tensor structures with antisymmetric spin tensors, yielding a generalized tensorial object that governs direction and dynamics in systems with intrinsic angular momentum. Covariant transformation properties insured by $V'(x') = S(x) V(x) S^{-1}(x)$ guarantee proper behavior under changes of inertial frame, setting the stage for a unified tensorial formalism for both translation and spin in non-relativistic and relativistic mechanics. Although explicit construction is not provided, the suggested methodology points toward future developments integrating spin into covariant tensor frameworks.

Conclusion

The spin–velocity metric tensor generalizes the conventional metric or projection tensors, integrating spin degrees of freedom with velocity or momentum in a unified tensorial construct. Its manifestations range from relativistic fluid dynamics and continuously deforming quantum states, to band geometry controlling thermal spin currents and magnetization, to boundary vorticity sources in turbulent flows, and to advanced quantum transport effects. Its theoretical development and physical repercussions are deeply linked to fundamental principles of covariance, symmetry, and geometric topology, with practical relevance for spintronics, caloritronics, quantum hydrodynamics, and gravitational theories embracing torsion and gauge invariance.