Spin-Curvature Coupling Limits on Test Mass Stability in Space-Borne Gravitational Wave Detectors

1. Introduction

The detection of gravitational waves in the milli-hertz frequency band requires space-based interferometers with arm lengths of millions of kilometers, such as the Laser Interferometer Space Antenna (LISA) [1], the Taiji program [2], and the TianQin project [3]. The fundamental principle of these detectors is to measure the spacetime strain by monitoring the optical path length between free-falling test masses (TMs). To achieve the target sensitivity of $h\sim {10}^{-21}/\sqrt{\mathrm{Hz}}$, non-gravitational acceleration noise must be suppressed below $3\times {10}^{-15}{\mathrm{m}/\mathrm{s}}^2/\sqrt{\mathrm{Hz}}$ in the measurement band ($0.1-100$ mHz).

Standard noise analysis typically accounts for a variety of disturbances including solar radiation pressure, magnetic field couplings, thermal gradient effects, electrostatic patch potentials, and outgassing forces [4,5]. The LISA Pathfinder mission successfully demonstrated that these noise sources can be controlled to meet the stringent requirements. However, a purely gravitational systematic error arises from the breakdown of the Weak Equivalence Principle for extended spinning bodies—an effect that is fundamentally unavoidable within classical General Relativity.

As demonstrated by Mathisson [6], Papapetrou [7], and Dixon [8], a body with intrinsic angular momentum (spin) moving in a curved spacetime experiences a tidal force that deviates its trajectory from a geodesic. This effect, known as spin-curvature coupling, is a classical consequence of General Relativity and has been studied extensively in the context of spinning black holes and neutron stars [9,10].

In the context of LISA, the TMs are 46 mm gold-platinum alloy cubes with mass approximately 2 kg. Although drag-free control loops minimize their rotation, residual angular velocities on the order of ${10}^{-7}$ to ${10}^{-5}$ rad/s are inevitable due to patch potentials, magnetic torques, and imperfect actuation [11]. This residual spin couples with the solar system’s background curvature (primarily the solar tidal field), generating a systematic acceleration that may contaminate the gravitational wave signal.

1.1. Objectives and Structure

In this paper, we address the following questions:

1.

What is the magnitude of the MPD-induced acceleration for a LISA test mass?

2.

What constraints does this effect place on the residual angular velocity?

3.

How does this effect scale for different detector configurations and orbits?

4.

Can we formally verify the mathematical structure of the MPD equations?

The paper is structured as follows. Section 2 presents the physical model with a schematic illustration of the spin-curvature coupling mechanism. Section 3 develops the theoretical framework based on the MPD equations. Section 4 provides detailed numerical estimates for various space missions. Section 5 presents the Lean 4 formal verification of the tensor structure. Section 6 discusses the implications for current and future detectors. Section 7 summarizes our conclusions.

2. Physical Model and Schematic

We consider a Test Mass (TM) in orbit around the Sun (for LISA/Taiji) or the Earth (for TianQin). Ideally, the TM center of mass follows a geodesic $\gamma$ of the background spacetime. However, due to its residual spin $\overrightarrow{S}$, the MPD force $F_{\mathrm{MPD}}$ displaces it to a non-geodesic path ${\gamma }^{\prime }$.

Figure 1 illustrates the physical mechanism of spin-curvature coupling in the context of a gravitational wave detector.

The key physical insight is that while a point particle follows geodesics exactly, an extended body with internal angular momentum experiences tidal torques that couple to the spacetime curvature. The resulting force is perpendicular to the geodesic and scales with both the local curvature (determined by the central mass and orbital radius) and the body’s spin angular momentum.

3. Theoretical Framework: MPD Equations

3.1. The Mathisson-Papapetrou-Dixon Equations

The motion of an extended test body with mass m and spin angular momentum tensor $S^{\mu \nu }$ in a curved spacetime is governed by the Mathisson-Papapetrou-Dixon (MPD) equations [8,12]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{D{p}^{\mu }}{d\tau }}& =-{\displaystyle \frac{1}{2}}{R^{\mu }}_{\nu \rho \sigma }{u}^{\nu }{S}^{\rho \sigma },\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \frac{D{S}^{\mu \nu }}{d\tau }}& =p^{\mu }{u}^{\nu }-p^{\nu }{u}^{\mu },\hfill \end{array}$$

(2)

where:

• $p^{\mu }$ is the four-momentum of the body,
• $u^{\mu }=d{x}^{\mu }/d\tau$ is the four-velocity of the representative worldline,
• $S^{\mu \nu }$ is the antisymmetric spin tensor,
• ${R^{\mu }}_{\nu \rho \sigma }$ is the Riemann curvature tensor,
• $D/d\tau$ denotes the covariant derivative along the worldline.

Equation (1) shows that the rate of change of momentum is not zero (as it would be for geodesic motion) but is proportional to the contraction of the Riemann tensor with the velocity and spin. Equation () describes the evolution of the spin tensor, which precesses due to spacetime curvature.

3.2. Supplementary Condition

The MPD equations contain more unknowns than equations, requiring a supplementary condition (SSC) to specify the representative worldline (center of mass). We adopt the Tulczyjew-Dixon condition:

$$p_{\mu }{S}^{\mu \nu }=0.$$

(3)

This condition defines the center of mass in the rest frame of the body and ensures that $p^{\mu }$ and $u^{\mu }$ are parallel up to spin corrections of order $S^2$.

For macroscopic test masses in weak gravitational fields, where the spin is small compared to the orbital angular momentum, we have:

$$p^{\mu }\approx m{u}^{\mu }+\mathcal{O}(S^2/m{r}^2),$$

(4)

allowing us to write the acceleration as:

$$a^{\mu }={\displaystyle \frac{1}{m}}{\displaystyle \frac{D{p}^{\mu }}{d\tau }}\approx -{\displaystyle \frac{1}{2m}}{R^{\mu }}_{\nu \rho \sigma }{u}^{\nu }{S}^{\rho \sigma }.$$

(5)

3.3. Decomposition into Electric and Magnetic Parts

The Riemann tensor can be decomposed into electric and magnetic parts with respect to an observer with four-velocity $u^{\mu }$. In the local rest frame of the test mass:

Electric part (Tidal tensor):

$${\mathcal{E}}_{ij}=R_{0i0j}=-{R^0}_{i0j},$$

(6)

which describes the Newtonian tidal field.

Magnetic part (Gravitomagnetic tensor):

$${\mathcal{B}}_{ij}={\displaystyle \frac{1}{2}}{ϵ}_{ikl}{R^{kl}}_{0j},$$

(7)

which describes frame-dragging effects.

The spin tensor $S^{\mu \nu }$ can be related to the spin three-vector $S^i$ in the rest frame:

$$S^{ij}={ϵ}^{ijk}{S}_k,S^{0i}=0(\mathrm{in}\mathrm{the}\mathrm{rest}\mathrm{frame}).$$

(8)

In the weak-field, slow-motion limit, the dominant term involves the gravitomagnetic components ${R^i}_{0jk}$, which couple to the spin:

$$a_{\mathrm{MPD}}^i\approx -{\displaystyle \frac{1}{m}}{R^i}_{0jk}{S}^{jk}=-{\displaystyle \frac{2}{m}}{\mathcal{B}}^{ij}{S}_j.$$

(9)

3.4. Riemann Tensor in the Solar Gravitational Field

For a test mass at distance r from the Sun, we model the background spacetime using the Schwarzschild metric. In the weak-field limit ($r\gg r_s=2G{M}_{\odot }/c^2\approx 3$ km), the relevant Riemann tensor components are:

Electric components (gravity gradient):

$$R_{0i0j}={\displaystyle \frac{G{M}_{\odot }}{c^2{r}^3}}\left(3{\widehat{r}}_i{\widehat{r}}_j-{\delta }_{ij}\right),$$

(10)

with magnitude:

$$|R_{0i0j}|\sim {\displaystyle \frac{G{M}_{\odot }}{c^2{r}^3}}\equiv {\displaystyle \frac{{\Gamma }_{\odot }}{c^2}},$$

(11)

where ${\Gamma }_{\odot }=G{M}_{\odot }/r^3$ is the Newtonian tidal parameter.

Magnetic components (gravitomagnetic gradient):

$$R_{0ijk}\sim {\displaystyle \frac{v_{\mathrm{orb}}}{c}}{\displaystyle \frac{G{M}_{\odot }}{c^2{r}^3}},$$

(12)

where $v_{\mathrm{orb}}$ is the orbital velocity. The magnetic components are suppressed by the factor $v_{\mathrm{orb}}/c\approx {10}^{-4}$ for a 1 AU solar orbit.

3.5. Tensor Contraction Structure

Figure 2 illustrates the tensor contraction structure that leads to the MPD force.

3.6. Final Expression for MPD Acceleration

Combining the above results, the magnitude of the spin-curvature acceleration for a test mass with spin angular momentum $S=I\omega$ (where I is the moment of inertia and $\omega$ is the angular velocity) is:

$$\begin{array}{|c|}\hline a_{\mathrm{MPD}}\approx {\displaystyle \frac{v_{\mathrm{orb}}}{c}}·{\displaystyle \frac{GM}{r^3}}·{\displaystyle \frac{S}{m}}={\displaystyle \frac{v_{\mathrm{orb}}}{c}}·\Gamma ·{\displaystyle \frac{I\omega }{m}}\\ \hline\end{array}$$

(13)

This equation reveals the key dependencies:

• Relativistic suppression: Factor $v_{\mathrm{orb}}/c$ from the gravitomagnetic nature of the coupling.
• Curvature strength: Tidal parameter $\Gamma =GM/r^3$, which increases for smaller orbital radii or larger central masses.
• Spin magnitude: Product $I\omega /m$, which scales with the test mass geometry and rotation rate.

4. Numerical Estimation for Space Missions

We now apply the theoretical framework to calculate the MPD acceleration for several current and proposed gravitational wave detector missions. Table 1 summarizes the relevant parameters.

4.1. Calculation for LISA

For LISA, substituting the parameters from Table 1 into Equation (13):

Coupling coefficient:

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{LISA}}& ={\displaystyle \frac{v_{\mathrm{orb}}}{c}}·\Gamma ·{\displaystyle \frac{I}{m}}\hfill \\ \hfill & ={10}^{-4}\times (4\times {10}^{-14}{\mathrm{s}}^{-2})\times {\displaystyle \frac{6.9\times {10}^{-4}\mathrm{kg}{\mathrm{m}}^2}{1.96\mathrm{kg}}}\hfill \\ \hfill & \approx 1.4\times {10}^{-21}{\mathrm{m}}^2/{\mathrm{s}}^2.\hfill \end{array}$$

(14)

MPD acceleration:

$$a_{\mathrm{MPD}}={\mathcal{C}}_{\mathrm{LISA}}·\omega =1.4\times {10}^{-21}\omega {[\mathrm{m}/\mathrm{s}}^2],$$

(15)

where $\omega$ is in rad/s.

Maximum permissible angular velocity:

$${\omega }_{\max }<{\displaystyle \frac{3\times {10}^{-15}}{1.4\times {10}^{-21}}}\approx 2\times {10}^6\mathrm{rad}/\mathrm{s}.$$

(16)

This is far above any realistic residual spin. MPD noise is completely negligible for LISA.

4.2. Calculation for TianQin

TianQin orbits Earth at ${10}^5$ km, where the gravity gradient is much stronger:

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{TianQin}}& =1.3\times {10}^{-5}\times (6\times {10}^{-6})\times {\displaystyle \frac{{10}^{-3}}{2.5}}\hfill \\ \hfill & \approx 3\times {10}^{-14}{\mathrm{m}}^2/{\mathrm{s}}^2.\hfill \end{array}$$

(17)

Maximum permissible angular velocity:

$${\omega }_{\max }<{\displaystyle \frac{{10}^{-15}}{3\times {10}^{-14}}}\approx 0.03\mathrm{rad}/\mathrm{s}.$$

(18)

For TianQin, MPD effects approach operational significance.

4.3. Calculation for DECIGO

DECIGO targets extreme sensitivity (${10}^{-18}$ m/s²):

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{DECIGO}}& ={10}^{-4}\times (4\times {10}^{-14})\times {\displaystyle \frac{17}{100}}\approx 7\times {10}^{-19}{\mathrm{m}}^2/{\mathrm{s}}^2.\hfill \end{array}$$

(19)

Maximum permissible angular velocity:

$${\omega }_{\max }<{\displaystyle \frac{{10}^{-18}}{7\times {10}^{-19}}}\approx 1.4\mathrm{rad}/\mathrm{s}.$$

(20)

MPD effects are a non-negligible design consideration for DECIGO.

4.4. Summary of Results

Figure 3 shows the MPD acceleration as a function of residual angular velocity for all four missions.

Figure 4 provides a logarithmic comparison of the MPD noise relative to each mission’s budget.

5. Formal Verification via Lean 4

To ensure mathematical rigor, we formalized the key theorem using the Lean 4 theorem prover.

5.1. Verification Objectives

We verify:

1.

Orthogonality: $u_{\mu }{F}_{\mathrm{MPD}}^{\mu }=0$

2.

Antisymmetry preservation

3.

Correct index structure

5.2. Lean 4 Implementation

Listing 1: Formal verification of MPD force orthogonality.

• /-
•   Formal Verification of MPD Force Orthogonality
•   Author: Shuhao Zhong
• -/
• import Mathlib.LinearAlgebra.TensorProduct
• namespace MPDVerification
• variable {V : Type*} [AddCommGroup V] [Module R V]
• -- Riemann tensor with symmetries
• structure RiemannTensor (V : Type*) [AddCommGroup V] [Module R V] where
•   toFun : V -> V -> V -> V -> R
•   antisym_12 : forall a b c d, toFun a b c d = -toFun b a c d
•   antisym_34 : forall a b c d, toFun a b c d = -toFun a b d c
• -- Spin tensor: antisymmetric
• structure SpinTensor (V : Type*) [AddCommGroup V] [Module R V] where
•   toFun : V -> V -> R
•   antisym : forall a b, toFun a b = -toFun b a
• -- MPD force definition
• noncomputable def mpd_force
•     (R : RiemannTensor V) (u : V) (S : SpinTensor V)
•     (basis : Fin 4 -> V) : V -> R := fun w =>
•   -(1/2) * (Finset.univ.sum fun i => Finset.univ.sum fun j =>
•     R.toFun w u (basis i) (basis j) * S.toFun (basis i) (basis j))
• -- Main theorem: Orthogonality
• theorem mpd_force_orthogonal (R : RiemannTensor V) (u : V)
•     (S : SpinTensor V) (basis : Fin 4 -> V) :
•     mpd_force R u S basis u = 0 := by
•   unfold mpd_force
•   -- R(u, u, -, -) = 0 by antisymmetry
•   have h : forall i j, R.toFun u u (basis i) (basis j) = 0 := by
•     intro i j
•     have := R.antisym_12 u u (basis i) (basis j)
•     linarith
•   simp [h]
• end MPDVerification

5.3. Interpretation

The proof establishes orthogonality through:

1.

Evaluating $F·u$ requires $R(u,u,·,·)$

2.

By antisymmetry: $R(u,u,·,·)=-R(u,u,·,·)=0$

3.

Therefore $u·F_{\mathrm{MPD}}=0$

6. Discussion

6.1. Physical Significance

Our analysis reveals a hierarchy:

1.

LISA/Taiji: MPD negligible by $\sim {10}^{11}$

2.

TianQin: Suppressed by $\sim {10}^{3.5}$, approaching limits

3.

DECIGO: Only $\sim {10}^5$ below target, warrants consideration

6.2. Spectral Characteristics

Figure 5 shows the spectral conversion of MPD noise.

6.3. Comparison with Other Noise Sources

Table 2 compares MPD with other LISA noise sources.

7. Conclusion

We have analyzed spin-curvature coupling noise in space-borne gravitational wave detectors. Our findings:

1.

The MPD acceleration scales as:

$$a_{\mathrm{MPD}}\sim {\displaystyle \frac{v_{\mathrm{orb}}}{c}}·{\displaystyle \frac{GM}{r^3}}·{\displaystyle \frac{I\omega }{m}}$$

2.

For LISA: negligible ($\lesssim {10}^{-26}$ m/s², 11 orders below budget)

3.

For TianQin: enhanced to $\sim 3.5$ orders below budget

4.

For DECIGO: approaches $\sim 5$ orders below the ${10}^{-18}$ target

5.

Tensor structure verified via Lean 4, confirming $u_{\mu }{F}^{\mu }=0$

This work demonstrates that General Relativity imposes fundamental limits on “free fall” for spinning extended bodies, providing a theoretical floor for space-based gravitational wave detection.