Gravitational Lorentz Force

• Gravitational Lorentz force is defined as the gravitational analogue of the electromagnetic Lorentz force, where SME-induced Lorentz violations replace electric currents with mass currents.
• It reveals novel gravitomagnetic fields even for static sources by mixing gravitoelectric and gravitomagnetic contributions, leading to unique test-body accelerations.
• SME modifications predict measurable effects like spin precession and oscillatory corrections in planetary motions, offering pathways to test Lorentz symmetry breaking.

The gravitational Lorentz force is a concept arising from the formal analogy between gravitation—particularly in the weak-field, slow-motion (post-Newtonian) regime—and electromagnetism, extended to include effects from Lorentz symmetry violation. In this context, the equation of motion for a test mass in a gravitational field exhibits terms directly analogous to the electric and magnetic components of the classical Lorentz force, albeit with mass currents replacing electric currents and with additional contributions governed by Lorentz-violating coefficients. The theoretical framework underpinning this analogy is supplied by the Standard-Model Extension (SME), which parameterizes possible departures from exact local Lorentz invariance in both gravity and electromagnetism. The gravitational Lorentz force, especially in the presence of SME-induced Lorentz-violating effects, reveals novel "magnetic-type" behaviors of gravity, such as the appearance of gravitomagnetic fields from static sources and an enriched structure of test-body accelerations, with implications for experimental gravity.

1. Gravitoelectromagnetic Analogy and SME Framework

The analogy between electromagnetism and gravity, commonly formalized as gravitoelectromagnetism (GEM), becomes particularly robust in the weak-field, post-Newtonian expansion. In Lorentz-invariant General Relativity (GR), the metric perturbations (notably $h_{00}$ and $h_{0i}$) correspond respectively to gravitoelectric (Newtonian) and gravitomagnetic potentials. In the SME, both the electromagnetic and gravitational sectors acquire Lorentz-violating corrections: the electromagnetic sector via $19$ CPT-even coefficients, nine of which, $(c_F)^{\mu\nu}$, are directly analogous to the nine traceless coefficients $s^{\mu\nu}$ in linearized gravity. This numerical and structural parallel enables a mapping in the stationary, slow-motion regime: $(c_F)^{\mu\nu} \longleftrightarrow s^{\mu\nu}$ Under this correspondence, the modified Maxwell equations and the linearized Einstein equations of the SME admit analogous solution strategies, with the gravitational metric perturbations $h_{0\mu}$ constructed from SME-modified electromagnetic potentials $A_\mu$ via prescribed replacement rules including $\rho \to$ mass density and $J^j \to$ mass current.

2. Explicit Structure of the Gravitational Lorentz Force

The post-Newtonian approximation yields explicit expressions for the metric perturbations up to $\mathcal{O}(3)$ in velocity: $$\begin{aligned} h_{00} &= (2 + 3 s^{00}) U + s^{jk} U^{jk} - 4 s^{0j} V^j \ h_{0j} &= -s^{0j} U - s^{0k} U^{jk} - 4(1 + \frac{1}{2} s^{00}) V^j + 2 s^{jk} V^k + 2 s^{kl}(X^{klj} - X^{jkl}) \end{aligned}$$ where integrals $U$, $U^{jk}$, $V^j$, and $X^{jkl}$ are sourced by the mass density and mass current distributions. In the Lorentz-violating SME, these components yield gravitoelectric and gravitomagnetic fields

$$E_G^j = \frac{1}{2} \partial_j\left[(h_{00}) + (h_J)_{00}\right], \quad B_G^j = \epsilon^{jkl} \partial_k (h_{0l})$$

leading to the equation of motion for a test particle

$a^j \simeq E_G^j + (\vec{v} \times \vec{B}_G)^j$

This mirrors the structure of the electromagnetic Lorentz force but with "currents" and fields defined via the gravitational analogs and with SME-induced augmentations to both $E_G$ and $B_G$.

3. Emergence of Gravitomagnetic Fields in Lorentz-Violating Gravity

In the conventional (Lorentz-invariant) theory, gravitomagnetic fields are produced by rotating mass currents (frame-dragging). The SME framework introduces striking new behaviors:

• Static Sources: Even in the absence of rotation or mass current, static masses generate a nonzero gravitomagnetic field when SME coefficients $s^{0j}$ are present:

$$B_G^j = -\frac{2 G_N m}{r^2} \epsilon^{jkl} s^{0k} \hat{x}^l$$

• Mixing of Gravitoelectric and Gravitomagnetic Sectors: The clean separation between fields sourced by densities and by currents is lost. The scalar (gravitoelectric) potential $h_{00}$ gains contributions from mass currents, and the vector (gravitomagnetic) potential $h_{0j}$ is influenced by mass density.

A tabulated summary:

Source Type	Gravitomagnetic Field $B_G$	Dependency
Static point mass	Nonzero if $s^{0j}\neq 0$	Proportional to $s^{0j}$
Rotating sphere	Enhanced, includes SME-induced dipole term	Relates to $s^{j}$ and angular momentum

4. Unconventional Effects: Mixings and Experimental Manifestations

The coupling of mass current into $h_{00}$ and mass density into $h_{0j}$—termed gravitoelectromagnetostatics (GEMS)—produces unconventional, Lorentz-violation-specific phenomena:

• New Test-Body Accelerations: Even for a mass at rest, the presence of nonzero SME parameters allows velocity-dependent forces ("magnetic-type") to act on other masses.
• Spin Precession: SME modifications introduce corrections to the rate and orientation of spin precession in analogy to the interaction of magnetic moments with electromagnetic fields.
• Oscillatory Corrections in Planetary Motions: SME-induced gravitomagnetic components (notably via $s^{TJ}$) yield periodic corrections, observable in high-precision experiments such as lunar laser ranging.

These effects provide unique experimental windows into Planck-scale Lorentz symmetry breaking, with testable consequences for atom interferometry and spin-dependent free-fall tests.

5. Examples: Static Point Mass and Rotating Sphere

Two explicit SME-modified cases demonstrate the methodology:

• Static Point Mass: The metric perturbations acquire terms proportional to $s^{00}$ and $s^{jk}$ in $h_{00}$ and to $s^{0j}$ in $h_{0j}$. The result is a spherically asymmetric gravitoelectric field and a nonzero gravitomagnetic field realigned with SME parameters.
• Rotating Mass: In this scenario, the gravitomagnetic dipole moment, originally proportional to angular momentum, is modified to include cross-products with $s^j = -s^{0j}$, inducing an effective gravitoelectric dipole deformation.

The mapped solutions rigorously extend known electromagnetic analogs, ensuring that any Lorentz-violating electromagnetic solution (with $c_F^{\mu\nu}$ replaced by $s^{\mu\nu}$) directly yields a gravitational counterpart.

6. Impact on the Principle of Equivalence and Tests of Lorentz Symmetry

Within the SME framework, the gravitational Lorentz force law maintains the weak equivalence principle at leading order (mass cancels in the test-body acceleration), but Lorentz violation introduces spin-dependent and potentially trajectory-dependent corrections. These new effects:

• Result in possible violations of universality of free fall at subleading order, especially for polarized or structured test bodies.
• Signal explicit breaking of local Lorentz invariance, providing motivation to design and interpret ultra-sensitive gravitational experiments with directional and compositional discrimination.
• Offer a unified theoretical infrastructure for analyzing diverse gravitational phenomena in both astrophysical systems (e.g., binary precession) and laboratory contexts.

7. Conclusions and Outlook

The gravitational Lorentz force, when analyzed through the lens of Lorentz-violating extensions, exhibits an enriched phenomenology. The SME mapping formalism enables direct translation of Lorentz-violating electromagnetic solutions into the gravitational domain, illuminating a host of unconventional effects—particularly gravitomagnetic phenomena arising from static sources and new mixings between the gravitoelectric and gravitomagnetic sectors. Experimental consequences include predicted oscillatory corrections to celestial motions, spin-precession effects, and other measurable departures from General Relativity, tightly constrained by (but also ideally targeted for) contemporary high-precision tests. This theoretical scaffolding elevates the gravitational Lorentz force from a merely formal analogy to a framework for searching for new fundamental physics.