Higher order analysis of the gravitational wave velocity memory effect between two free-falling gyroscopes in the plane wave spacetime (2501.15745v1)

Abstract: In the plane wave spacetime, when gravitational waves pass by, an angular deviation exists between two free-falling gyroscopes, which naturally corresponds to the velocity memory effect. In the shackwave spacetime background, the angular deviation between two free-falling gyroscopes is calculated, which is also found to correspond to the velocity memory effect. In the plane wave spacetime, with linear polarization taken into account, no contribution is made by the first-order terms of the initial separation distance (L) (or the initial separation velocity (v_0)), (\bm{P}) (or (\bm{M})) of two free-falling gyroscopes to the velocity memory effect, while contributions are initiated from the second-order terms. Under certain circumstances, the second-order contribution of the initial separation distance (L) is of the same order of magnitude as the first-order contribution of the initial separation velocity (v_0). When both + polarization and (\times) polarization are taken into account, in the context of the merger of supermassive black holes and with the initial separation velocity approaching the speed of light, the order of magnitude of the angle is (10^{-16}) rads.

• The paper utilizes Brinkmann coordinates and a Dyson series approach to analyze the angular deviation between gyroscopes at higher orders.
• Angular deviation and the velocity memory effect appear only at higher orders, not first-order linear approximations, confirmed by second-order results.
• The results enhance understanding of non-linear gravitational waves and have implications for future gravitational wave observatory missions.

Higher Order Analysis of the Gravitational Wave Velocity Memory Effect

The paper "Higher Order Analysis of the Gravitational Wave Velocity Memory Effect between Two Free-Falling Gyroscopes in the Plane Wave Spacetime" introduces a comprehensive paper of the velocity memory effect (VME) as a result of the passage of gravitational waves in plane wave spacetime. The authors delve into the angular deviation between two free-falling gyroscopes and evaluate its relation to the VME, which arises from non-oscillatory components of gravitational waves.

Overview

The concept of memory effects in gravitational waves is pivotal, addressing permanent changes incurred in physical systems post wave interaction. The paper focuses on VME, contrasting the relatively more familiar displacement memory effect. The authors' methodology distinguishes between the separation velocity and distance in calculating the angular deviation, thereby disentangling complex interactions intrinsic to higher-order terms of the interaction.

Methodology

Utilizing Brinkmann coordinates, the research elaborates on the equation of motion governing gyroscopic precession within plane wave spacetimes. The solutions to the geodesic equations are expressed iteratively, leveraging a Dyson series approach to overcome the limitations associated with direct numerical solutions. Higher-order terms in the expansion are particularly emphasized, as they are critical for capturing the non-linear and memory-inducing aspects of the gravitational wave interactions.

Key Findings

• The angular deviation between the gyroscopes is shown not to manifest at first-order terms regarding the initial separation distance or velocity. The implications of this property affirm that VME contributions emerge solely at the higher orders and are inherently linked to non-linear gravitational wave effects.
• Strong numerical results report no discernible final angular deviation under simplified models when relying on linear approximations. In contrast, significant deviations appear upon incorporating second-order corrections, substantiating the necessity for higher-order considerations.
• Analytical formulations quantify the angular deviations as a combination of memory effects from both the plus (+) and cross (×) polarizations of gravitational waves, with specific magnitudes pertaining to stellar and black hole merger events evaluated.

Implications

The theoretical importance lies foremost in enhancing comprehension of gravitational wave characteristics beyond linear approximations, suggesting potential observational tests employing advanced detectors or pulsar timing arrays. The practical application is evident in future gravitational wave observatory missions, where understanding of VME can refine signal calibration and improve the interpretability of complex astrophysical data.

Future Work

Research derivations suggest further empirical exploration into the multi-polarization effects and the accompanying observational signatures. There remains notable scope to extend kernel methods into non-plane wave scenarios, potentially illuminating yet unexplored mechanisms within astrophysical events.

In conclusion, this paper underscores the intricate nature of VME, necessitating higher-order analysis to unravel the subtle, but significant, contributions of non-linear gravitational wave dynamics. The methodologies and resulting insights pave the way for refined future research and experimental validation within the field of gravitational wave astronomy.