Terrestrial Test of Shapiro Time Delay: Forth test of Einstein General Theory of Relativity

This paper proposes a novel fiber-based Sagnac interferometer to perform a precision terrestrial measurement of the Shapiro time delay, aiming to determine the PPN parameter gamma with a sensitivity of 10⁻⁹ and thereby provide an independent laboratory-scale test of General Relativity.

The Gist

Imagine you are trying to weigh a feather, but you don't have a scale sensitive enough to see it. Now, imagine that feather is actually a tiny delay in time caused by the Earth's gravity slowing down light. That is the challenge this paper tackles.

Here is a simple breakdown of what the authors, Farhad and Hosain Hakimi, are proposing:

The Big Idea: Catching a "Time Ghost"

In the 1960s, a physicist named Irwin Shapiro discovered a weird rule of the universe: Gravity slows down light. It's not that light gets tired; it's that gravity stretches the "fabric" of space and time, making the light take a slightly longer path to get somewhere. This is called the Shapiro Time Delay.

Usually, we only see this effect with massive objects like the Sun and planets. It's like shouting across a canyon; the echo takes a tiny bit longer because the air is thick. But on Earth, the gravity is so weak that this "echo delay" is incredibly small—smaller than the time it takes for light to cross a single atom.

The authors say: "Let's try to measure this tiny delay right here in a laboratory, using a tabletop machine."

The Machine: A "Time-Looping" Race Track

To catch this ghost, they designed a special device called a Fiber-Optic Sagnac Interferometer. Think of it like a very sophisticated race track for light.

1. The Track: Instead of a physical track, they use a coil of glass fiber optic cable (the kind used for internet) that is 100 kilometers long (about 62 miles), but it is wound up tightly so it fits on a table.
2. The Racers: They send two beams of light racing around this loop in opposite directions—one clockwise, one counter-clockwise.
3. The Twist: In a normal race, if the track is flat, both racers finish at the exact same time. But the authors propose stacking two of these loops: one sitting on the floor and one lifted up on a table (about 1 meter higher).
4. The Gravity Effect: Because the top loop is higher up, it is in a slightly weaker gravitational field than the bottom loop. According to Einstein, light travels slightly faster (or rather, takes less time) in the weaker gravity.
5. The Result: The light in the top loop finishes its race a tiny, tiny fraction of a second before the light in the bottom loop. The machine is designed to detect this microscopic difference.

The Challenge: Finding a Needle in a Haystack

The delay they are looking for is about 0.00000000000000000007 seconds (7.2 x 10⁻²⁰ seconds). That is so small it's hard to imagine. To put it in perspective, if that delay were a second, a single second would last longer than the entire history of the universe.

How do they find it?
The paper suggests using a "noise-canceling" strategy, similar to how noise-canceling headphones work:

• The Problem: The machine is noisy. The light source flickers, the temperature changes, and the electronics buzz. These noises are like a loud rock concert drowning out a whisper.
• The Solution: They use a special trick called modulation. They make the light pulse very fast (billions of times a second) and use a "lock-in" technique. Imagine trying to hear a specific person in a crowded room. If you ask them to speak only in a specific rhythm, you can tune out everyone else. This machine does that with light, filtering out the "rock concert" noise to hear the "whisper" of the time delay.

The Promise: A New Way to Test Einstein

The paper claims that with current, off-the-shelf technology (like high-speed computers and lasers you can buy today), this machine could measure this delay with incredible precision.

If they succeed, they could test Einstein's General Theory of Relativity right in a university lab, rather than needing to send rockets to other planets. They aim to measure a specific number (called the PPN parameter $\gamma$) that tells us how much gravity bends space. If their machine measures it differently than Einstein predicted, it would mean our understanding of gravity is wrong. If it matches, it proves Einstein was right again, but this time, on a tabletop.

Summary

• The Goal: Measure how Earth's gravity slows down light in a lab.
• The Tool: A 100km-long fiber-optic cable coiled up, with one part lifted higher than the other.
• The Trick: Using high-speed electronics to filter out noise and hear the tiny "time difference" between the high and low loops.
• The Outcome: A potential new, compact way to prove Einstein's theories without needing a telescope or a spaceship.

The authors are essentially saying: "We have the tools to weigh the weight of a feather using a scale we can build on a desk."

Technical Summary

Technical Summary: Terrestrial Test of Shapiro Time Delay

Problem Statement
The Shapiro time delay, the fourth classical test of Einstein's General Theory of Relativity, describes the additional time required for light to traverse a region of curved spacetime near a massive body. Historically, this effect has been measured exclusively on astronomical scales (e.g., radar ranging to Venus or signals from the Cassini spacecraft), where the delay reaches hundreds of microseconds. While these measurements have confirmed General Relativity with high precision, constraining the parametrized post-Newtonian (PPN) parameter $\gamma$ to approximately $10^{-5}$, there has been no successful terrestrial laboratory measurement of this effect. Previous proposals to detect Shapiro delays on meter-scale distances using massive rotating units or gravitational wave detectors (like LIGO) have suggested signals on the order of $10^{-32}$ seconds, which remain extremely challenging to detect. The central problem addressed by this work is the feasibility of measuring the minute Shapiro time delay induced by Earth's own gravitational field within a controlled, tabletop laboratory environment.

Methodology
The authors propose a compact fiber-optic Sagnac interferometer designed to convert tiny relativistic propagation delays into measurable optical phase shifts. The core methodology relies on the following architectural and analytical components:

1. Sensor Architecture: The instrument utilizes a pair of fiber-optic loops, each containing approximately 100 km of single-mode optical fiber wound on spools with zero net enclosed geometric area (to suppress rotation-induced Sagnac phase shifts). One loop serves as a horizontal reference, while the second is vertically displaced by a height $d$ (e.g., 1 meter).
2. Physical Principle: Based on the effective refractive index of spacetime in a gravitational field ($n \approx 1 - 2\Phi/c^2$), light propagating at different gravitational potentials experiences different propagation speeds. The differential Shapiro time delay ($\Delta t_{Shapiro}$) between the two coils is derived as a function of Earth's mass, the fiber length, and the vertical separation.
3. Signal Detection: The system operates as a Sagnac interferometer biased at the quadrature point ($90^\circ$) to maximize sensitivity. It employs a broadband light source (ASE from an EDFA) passed through a Semiconductor Optical Amplifier (SOA) driven into saturation to suppress Relative Intensity Noise (RIN) by up to 22 dB.
4. Noise Mitigation and Processing: To address dominant noise sources (RIN, thermo-optic noise, and shot noise), the system operates at high frequencies (GHz range) using modulation and demodulation techniques. The authors analyze the impact of cryogenic cooling (down to ~2 K) on reducing fiber phase noise. Furthermore, they propose using digital post-processing with oversampling (using a 14-bit ADC at 6 GSa/s) and digital filtering to enhance the Signal-to-Noise Ratio (SNR) by approximately 75 dB.

Key Contributions

• Novel Sensor Design: The design of a compact, fiber-based Sagnac interferometer specifically configured to measure differential Shapiro delays between vertically separated optical paths.
• Terrestrial Feasibility Analysis: A comprehensive derivation showing that Earth's mass, combined with long fiber lengths (100 km) and differential measurement techniques, can produce a detectable signal, unlike previous proposals relying on smaller rotating masses.
• Performance Framework: A detailed signal-to-noise analysis that identifies RIN and thermo-optic noise as primary limitations and outlines specific engineering solutions (SOA saturation, cryogenic cooling, and high-frequency operation) to overcome them.
• Post-Processing Strategy: The demonstration that modern digital signal processing (oversampling and lock-in amplification) can theoretically boost the sensitivity of the raw detector by orders of magnitude.

Results and Performance Estimates

• Signal Magnitude: For a representative configuration with 100 km of fiber and a 1-meter vertical separation, the expected one-way differential Shapiro delay is calculated to be $\Delta t_{Shapiro} \approx 7.2 \times 10^{-20}$ seconds (equivalent to an optical path difference of $\approx 2.2 \times 10^{-11}$ meters).
• Raw Sensitivity: The analysis indicates that, prior to post-processing, the system can achieve a delay detection sensitivity of approximately $\Delta t \sim 10^{-23}$ seconds, limited primarily by RIN.
• Enhanced Sensitivity: By applying digital post-processing (oversampling and filtering), the authors estimate the effective sensitivity could reach $\Delta t \approx 10^{-31}$ seconds.
• PPN Parameter Constraint: This enhanced sensitivity corresponds to a potential constraint on the PPN parameter $\gamma$ at the level of $\sim 10^{-11}$, which would significantly exceed current astrophysical bounds of $\sim 10^{-5}$.
• Technological Readiness: The paper asserts that all necessary components (high-speed modulators, photodetectors, SOAs, cryogenic systems, and GHz lock-in amplifiers) are commercially available and mature.

Significance and Claims
The paper claims that this work establishes a practical platform for "tabletop" tests of General Relativity, moving the measurement of the Shapiro delay from astronomical scales to the laboratory. The authors argue that this approach offers a complementary method to astronomical observations, providing a controlled environment for systematic studies and independent cross-checks.

The significance of the work lies in its potential to:

1. Rival Astrophysical Constraints: Achieve a precision in determining $\gamma$ that rivals or surpasses current limits derived from space missions like Cassini.
2. Demonstrate Scalability: Show that fiber-based interferometry, originally developed for gyroscopes, can be adapted for fundamental physics tests with unprecedented sensitivity.
3. Enable Future Research: Provide a framework for probing other relativistic effects, such as gravitational waves, using compact, modular instruments.

The authors conclude that while the intrinsic sensitivity of the sensor is high, the ultimate precision of the $\gamma$ measurement in this specific setup is currently limited by the uncertainty in Earth's gravitational parameter ($GM_{\oplus}$), suggesting that improvements in geophysical measurements would further enhance the utility of this terrestrial test.