Formulation of testing gravitational redshift based on Laser Time link between China Space Station and a ground station

This paper presents a high-precision gravitational redshift test using the China Space Station's Laser Time Transfer system, achieving a verification precision of approximately 10⁻⁷ by leveraging a c⁻³ relativistic model to eliminate ionospheric and first-order Doppler effects, thereby establishing a new benchmark for fundamental physics and geodetic applications.

The Gist

The Big Idea: Testing Einstein's "Slow-Motion" Theory

Imagine you have two identical, super-precise watches. You keep one on your wrist on Earth, and you give the other to an astronaut on the China Space Station (CSS), which is orbiting about 400 kilometers above us.

According to Einstein's theory of gravity (General Relativity), time doesn't tick at the same speed everywhere. Because the space station is higher up, where Earth's gravity is slightly weaker, time should tick faster there than on the ground. This is called Gravitational Redshift.

For decades, scientists have tried to measure this tiny difference. But until now, the tools used to compare the watches (mostly radio waves) weren't precise enough to see the effect clearly without getting confused by other noise.

The New Tool: A Laser "Time-Link"

This paper proposes a new way to compare these watches using a laser beam instead of radio waves. Think of it like this:

• Old Way (Radio): Trying to send a message across a busy, foggy highway where the signal bounces off buildings and gets distorted by the air.
• New Way (Laser): Sending a message through a clear, straight glass tube. The laser beam is so focused that it doesn't get messed up by the atmosphere or the "fog" of the ionosphere that plagues radio signals.

The researchers set up a "two-way" conversation:

1. The ground station shoots a laser pulse up to the space station.
2. The space station catches it, notes the time, and bounces it back.
3. The ground station catches the return pulse and notes the time.

By comparing the "send time," "bounce time," and "return time," they can calculate exactly how much faster the space station's clock is running compared to the Earth clock.

The "Recipe" for Precision

To get a perfect measurement, the scientists had to create a very complex mathematical "recipe" (an observation equation) to account for everything that could mess up the laser's travel time. They went up to the third order of precision (a fancy way of saying they accounted for tiny, tiny details).

Here are the main "ingredients" they had to filter out:

• The Atmosphere: Just like heat haze makes a mirage, the air near the ground bends the laser slightly. They used advanced weather models to correct for this "bending."
• The Earth's Spin: Because the Earth is spinning while the laser is flying, the target moves. They calculated this "Sagnac effect" (like aiming a water hose at a spinning merry-go-round).
• Gravity's Curve: The laser doesn't travel in a perfectly straight line; it curves slightly around the Earth's mass. They corrected for this too.
• Hardware Glitches: The electronics inside the station and on the ground take a tiny fraction of a second to process the signal. They measured and subtracted this delay.

The Simulation: A "Dry Run"

The paper notes that the actual optical clock on the space station is still being debugged (tested and tuned), so they couldn't run the real experiment yet. Instead, they built a super-accurate computer simulation.

They used real data about the space station's orbit and simulated the laser link as if it were happening right now. They fed in all the known errors (like atmospheric turbulence and hardware noise) to see how well their "recipe" worked.

The Results: A Huge Leap Forward

The simulation showed that this laser method is incredibly powerful:

• Precision: They achieved a verification precision of (1.8 ± 47) × 10⁻⁷.
• Comparison: This is about 10 times more precise than previous experiments that used radio waves (microwaves).
• The "Noise" Problem: The biggest remaining "noise" in their measurement comes from the troposphere (the lower layer of the atmosphere) and turbulence (windy air). Even with their advanced models, the air is the hardest thing to predict perfectly. However, by averaging the data over time, these random air fluctuations smooth out.

Why This Matters

The paper concludes that this laser method is a game-changer.

1. For Physics: It offers a new, ultra-precise way to test Einstein's theories. If Einstein was wrong, this method is sensitive enough to catch it.
2. For Mapping (Geodesy): Because time and gravity are linked, measuring the time difference so precisely allows scientists to measure the height difference between two points on Earth with incredible accuracy (down to 0.1 meters squared per second squared). This could help in measuring mountain heights or sea levels across continents without needing physical surveying.

In short: The researchers have designed a "laser time-link" that acts like a super-precise ruler for time. Their simulations prove it can measure the slowing of time due to gravity better than any previous method, paving the way for a new era of testing the laws of the universe from space.

Technical Summary

Technical Summary: Formulation of Testing Gravitational Redshift Based on Laser Time Link between China Space Station and a Ground Station

Problem Statement
Testing Einstein's General Relativity Theory (GRT) via the Gravitational Redshift (GRS) effect requires comparing clocks at different gravitational potentials with extreme precision. While previous experiments using microwave links (e.g., Gravity Probe A, Galileo satellites) and ground-based optical clocks have achieved significant milestones, they face limitations. Microwave-based methods are susceptible to ionospheric delays and first-order Doppler shifts, often requiring complex multi-frequency combinations to mitigate these errors. Furthermore, existing high-precision tests have lacked detailed systematic modeling and elimination of link and atmospheric error effects. The China Space Station (CSS) offers a unique platform equipped with a Strontium (Sr) optical lattice clock and a Laser Time Transfer (CLT) system, yet a comprehensive observation model specifically tailored to $c^{-3}$ order relativistic effects for GRS verification using this specific space-ground link had not been fully developed or simulated.

Methodology
The authors developed a high-precision observation model for GRS testing based on the two-way Laser Time Transfer (CLT) link between the CSS and a ground station (Xi'an Laser Station).

1. Relativistic Modeling: The study derives a comprehensive observation equation accurate to the $c^{-3}$ order. This model integrates the Geocentric Celestial Reference System (GCRS) coordinate time (TCG) with the proper time (PT) of physical clocks. It accounts for:

   • Special Relativistic Effects: Time dilation due to velocity (Transverse Doppler Effect).
   • General Relativistic Effects: Gravitational time dilation due to Earth's potential and tidal potentials.
   • Propagation Delays: Shapiro time delay, Sagnac effects (both $c^{-2}$ and $c^{-3}$ orders), and tropospheric delays.
   • Systematic Errors: Hardware delays, position delays between detectors and reflectors, and atmospheric turbulence.

2. Error Mitigation Strategy: Unlike microwave approaches, the laser link inherently avoids ionospheric refraction. The model employs a two-way measurement configuration to cancel common-mode errors. The authors explicitly model the asymmetry between uplink and downlink paths caused by Earth's rotation and atmospheric refraction, introducing correction terms for Sagnac effects induced by tropospheric delays.

3. Simulation Framework: Since the CSS optical clock is currently in the orbital debugging phase, the authors conducted a simulation using actual CSS orbit data and published system parameters.

   • Clocks: Simulated a space-based Sr optical clock ($2 \times 10^{-15}/\sqrt{\tau}$) and a ground-based optical clock ($1 \times 10^{-15}/\sqrt{\tau}$).
   • Error Sources: Systematic modeling of tropospheric delays (using GPT3/VMF3 models), Solid Earth Tides (SET), orbit errors, and atmospheric turbulence.
   • Data Processing: A weighted averaging method was applied to 142 observation arcs, excluding outliers and correcting for geometric, relativistic, and atmospheric effects before calculating the GRS parameter $\alpha$.

Key Contributions

• First Application of CLT for GRS: This work presents the first formulation and simulation of using the CSS CLT system specifically for gravitational redshift verification.
• $c^{-3}$ Order Observation Equation: The authors derived a rigorous observation equation that incorporates $c^{-3}$ order terms, including higher-order Sagnac effects and Shapiro delays, providing a more complete relativistic model than previous microwave-based studies.
• Atmospheric Error Characterization: The study identifies tropospheric delay variations and atmospheric turbulence as the primary remaining uncertainty contributors, quantifying their impact on link stability through Allan deviation analysis.
• Precision Benchmark: The simulation demonstrates that the laser approach avoids ionospheric effects and first-order Doppler shifts, offering a pathway to significantly higher precision than microwave methods.

Results
The simulation results indicate a gravitational redshift verification precision of $(1.8 \pm 47) \times 10^{-7}$.

• Improvement: This represents approximately a one-order-of-magnitude improvement over previous experiments (e.g., the microwave-based tri-frequency combination method which achieved $\sim 10^{-6}$).
• Residual Analysis: After applying the comprehensive error model, residual stabilities for most effects (including Sagnac, Shapiro, and system delays) were reduced to levels below $10^{-17}$ after 1000 seconds of averaging.
• Dominant Errors: The residual analysis revealed that tropospheric delay ($\sim 2 \times 10^{-7}$ uncertainty) and atmospheric turbulence ($\sim 1 \times 10^{-7}$ uncertainty) remain the dominant sources of error, limiting the current precision.
• Geodetic Capability: The achieved precision enables gravitational potential difference measurements with 0.1 m²/s² precision, suggesting utility for intercontinental height transfer.

Significance
The paper claims that this work establishes a new benchmark for high-precision tests of relativistic physics. By demonstrating the transformative potential of space-based optical time transfer, the study validates the feasibility of using the CSS CLT system to test fundamental aspects of General Relativity with unprecedented accuracy. The authors assert that their methodology provides a robust framework for future experiments, potentially allowing for the detection of subtle deviations from Einstein's theory once the CSS optical clock system enters full operation and more advanced tropospheric models are implemented. The transition from microwave to laser-based time transfer is highlighted as a critical step in eliminating ionospheric errors and achieving the stability required for next-generation fundamental physics investigations.