The role of Wigner rotation in estimating the specific angular momentum of a Kerr spacetime

This paper proposes a geodesic interferometer scheme using a single photon to estimate the specific angular momentum of a Kerr spacetime by analyzing the interferometric visibility resulting from the combined effects of gravitational time delay and Wigner-induced polarization rotation.

The Gist

The Big Idea: A Cosmic Spin Doctor

Imagine the Earth (or a black hole) isn't just a heavy ball sitting in space; it's a spinning top. According to Einstein's theory of General Relativity, when a massive object spins, it doesn't just sit there—it actually "drags" the fabric of space and time around with it, like a spoon stirring honey. This is called frame-dragging.

This paper asks a tricky question: Can we measure exactly how fast this cosmic top is spinning by watching a single photon (a particle of light) travel through it?

The authors propose a method to estimate this "spin speed" (called specific angular momentum) by using a very sensitive tool: a quantum interferometer.

The Setup: The Cosmic Maze

To do this, the scientists imagine a machine called a Mach-Zehnder interferometer.

• The Analogy: Think of a race track with two lanes. A runner (the photon) starts at the beginning and splits into two versions of themselves. One version runs down the "inner lane" (closer to the spinning Earth), and the other runs down the "outer lane" (farther away).
• The Twist: In normal space, these lanes are straight. But in the space around a spinning Earth (Kerr spacetime), space itself is twisted. The "inner lane" gets dragged along by the spin, while the "outer lane" feels less of this drag.
• The Reunion: The two versions of the photon eventually meet back at the finish line. Because they traveled through slightly different "twisted" spaces, they arrive with a tiny difference in their internal state.

The Two Effects: The Clock and the Compass

When the two photon paths meet, the paper says two things have happened to them:

1. The Time Delay (The Clock): Because space is curved and moving, one path takes a tiny bit longer to travel than the other. It's like one runner had to run through thick mud while the other ran on pavement. This creates a "time difference."
2. The Wigner Rotation (The Compass): This is the star of the show. As the photon travels through the spinning space, its "polarization" (which you can think of as the direction its internal compass is pointing) gets rotated.
   • The Analogy: Imagine the photon is a spinning arrow. As it flies through the "honey" of the spinning Earth, the honey twists the arrow slightly. By the time it reaches the finish line, the arrow isn't pointing in the exact same direction it started. This twist is called Wigner rotation.

The Measurement: Reading the Result

The machine detects the photon at the end. The probability of finding the photon in one detector versus the other depends on how much the two paths differed.

• The paper shows that the detection probability is a mix of the "Time Delay" and the "Compass Twist."
• The "Time Delay" is actually quite large (relatively speaking) and easy to see.
• The "Compass Twist" (Wigner rotation) is incredibly tiny—so small it's hard to imagine. The authors calculate that for an experiment near Earth, this twist is about $10^{-30}$ (a decimal point followed by 29 zeros).

The Goal: Cracking the Code

The main point of the paper is to show that if you can measure the final result (where the photon lands) with extreme precision, you can work backward to figure out the spin speed of the Earth (or the black hole).

• The Math: They created a formula. If you know the probability of the photon landing in a specific spot, you can plug that number into their equation to solve for the spin speed ($a$).
• The Uncertainty: They also calculated how much error would be in their answer. They found that if you build a very large interferometer (with mirrors separated by hundreds of kilometers) and can measure the photon's landing spot with high precision, you could estimate the Earth's spin speed with an error of only about one part in a million.

Summary in a Nutshell

The paper proposes a theoretical experiment where a single photon is sent through a "twisted" space created by a spinning planet. By measuring how the photon's internal "compass" (polarization) gets rotated by the spin of the planet, scientists could theoretically calculate exactly how fast the planet is spinning. While the effect is incredibly tiny, the math proves it is possible to extract this information from the photon's behavior.

Technical Summary

Technical Summary: The Role of Wigner Rotation in Estimating the Specific Angular Momentum of a Kerr Spacetime

Problem Statement
The paper addresses the challenge of reconciling Quantum Mechanics and General Relativity by investigating the interaction between quantum systems and gravitational fields. Specifically, it focuses on the propagation of photons in the Kerr spacetime, which describes the geometry around a rotating mass. While relativistic effects such as the Skrotskii effect (gravitational Faraday rotation) and Wigner rotation are theoretically predicted to modify a photon's polarization state, direct experimental measurements of these effects along astrophysical paths remain limited. The authors aim to study the rotation of photon polarization in the Kerr metric and determine if this phenomenon can be utilized to estimate the specific angular momentum ($a$) that parameterizes the metric.

Methodology
The authors propose a theoretical framework based on a "geodesic interferometer," defined as a Mach-Zehnder interferometer where the arms are strictly defined by null geodesics. The methodology proceeds through the following steps:

1. Geometric Setup: The study utilizes the Kerr metric in Boyer-Lindquist coordinates. Unlike the Schwarzschild metric, the Kerr metric lacks spherical symmetry, meaning purely radial null geodesics do not exist due to frame-dragging. Consequently, the authors focus on "principal null geodesics" confined to a cone of constant polar angle ($\theta = \text{const}$). They derive analytical solutions for these trajectories, parameterizing them by the radial coordinate $r$.
2. Tetrad Formalism and Polarization: To analyze the photon's polarization, the authors employ a tetrad (orthonormal basis) formalism aligned with the principal null geodesics. They adopt the "adapted-tetrad" method, aligning the basis vectors with the direction of propagation to isolate the polarization degree of freedom. Within the Wentzel-Kramers-Brillouin (WKB) approximation, the evolution of the polarization vector is governed by parallel transport, leading to coupled differential equations involving the Wigner rotation matrix.
3. Quantum State Modeling: A single photon is modeled as a localized qubit defined by its two-dimensional polarization (Jones vector) and frequency dispersion. The quantum state is described as a superposition of momentum-helicity eigenstates. The evolution of this state through the interferometer is calculated using unitary operators that incorporate both the gravitational time delay (transport phase) and the Wigner phase.
4. Interferometer Configuration: The authors define a specific Mach-Zehnder configuration where a photon emitted at radial coordinate $r_2$ is split into two paths: one traveling inward to $r_1$ and the other outward to $r_3$. Both paths reflect and recombine at a coordinate $r_4$. The authors analytically solve for $r_4$ by equating the azimuthal angles of the incoming and outgoing geodesics, demonstrating that such an interferometer is feasible in Kerr spacetime despite the absence of purely radial paths.
5. Approximations: The calculations for phase differences and detection probabilities are performed under the "slow rotation" ($a \ll r$) and "weak field" ($m \ll r$) approximations, expanding terms up to the fourth order in $a/r$ and $m/r$.

Key Contributions and Results

• Analytical Derivation of Geodesics: The paper provides exact analytical solutions for principal null geodesics in Kerr spacetime, establishing the feasibility of defining a Mach-Zehnder interferometer with arms strictly following these geodesics.
• Phase Decomposition: The detection probability at the output ports is shown to be a function of two distinct phase differences:
  1. $\Delta \vartheta_\tau$ (Transport Phase): Arising from the gravitational time delay (arrival time difference) between the two arms.
  2. $\Delta \vartheta$ (Wigner Phase): Arising from the rotation of the polarization vector due to the spacetime curvature and rotation (frame-dragging).
• Magnitude of Effects: Under Earth-like conditions (radius $\approx 7 \times 10^6$ m, rotation parameter $a \approx 3.97$ m) and an interferometer area of $\sim 1 \text{ m}^2$ with light frequency $10^{12}$ Hz:
  • The phase difference due to arrival time delay ($\Delta \vartheta_\tau$) is of the order of $10^{-16}$.
  • The Wigner phase difference ($\Delta \vartheta$) is significantly smaller, of the order of $10^{-30}$.
  • The authors note that increasing the mirror separation distance to $\sim 10^3$ m can increase the Wigner phase to $\sim 10^{-23}$.
• Detection Probability and Visibility: The probability of detecting the photon at the output ports is derived as an armonic function of both phase differences. The interferometric visibility is shown to be modulated by the Wigner phase ($\cos(\Delta \vartheta)$) and exponentially suppressed by the transport phase ($e^{-(\sigma/\omega \Delta \vartheta_\tau)^2}$).
• Estimation of Specific Angular Momentum: By expanding the detection probability for small phases, the authors derive an algebraic expression to estimate the specific angular momentum $a$ based on the measured detection probability. They further calculate the relative uncertainty ($\delta a/a$) of this estimate. For a geodesic interferometer near Earth with a mirror separation of $\sim 800$ km and a detection probability error of $10^{-14}$, the relative error in estimating $a$ is on the order of $10^{-6}$.

Significance and Claims
The paper claims that the detection probability in such a geodesic interferometer serves as a signature of two purely relativistic effects acting on a quantum system: the gravitational time delay and the polarization rotation (Wigner rotation). The authors assert that their approach allows for the characterization of the specific angular momentum of the Kerr metric using quantum interference.

The significance of the work lies in demonstrating a theoretical pathway to estimate the spin parameter of a rotating mass (specifically Earth-like parameters in their examples) using quantum optical techniques. The authors emphasize that while the Wigner phase is extremely small compared to the time-delay phase, the interferometric visibility retains a dependence on the Wigner phase, coupling the metric elements ($a$ and $m$) to the photon's helicity. This provides a theoretical basis for using quantum interferometry to probe frame-dragging effects and test fundamental aspects of General Relativity, although the paper does not propose a specific immediate experimental realization beyond the theoretical estimation of uncertainties.