Einstein Was Right (Again): Why Moon Clocks Run Faster Than Earth’s | HackerNoon

Table of Links

Abstract and 1. Introduction

2. Clock in Orbit

   2.1 Coordinate Time

   2.2 Local Frame for the Moon

3. Clock Rate Differences Between Earth and Moon

4. Clocks at Earth-Moon Lagrance Points

   4.1 Clock at Lagrange point L1

   4.2. Clock at Lagrange point L2

   4.3. Clock at Lagrange point L4 or L5

5. Conclusions

Appendix 1: Fermi Coordinates with Origin at the Center of the Moon

Appendix 2: Construction of Freely Falling Center of Mass Frame

Appendix 3: Equations of Motion of Earth and Moon

Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems

Acknowledgments and References

As humanity aspires to explore the solar system and investigate distant worlds such as the Moon, Mars, and beyond, there is a growing need to establish and broaden coordinate time references that depend on the rate of standard clocks. According to Einstein’s theory of relativity, the rate of a standard clock is influenced by the gravitational potential at the location of the clock and the relative motion of the clock. A coordinate time reference is established by a grid of synchronized clocks traceable to an ideal clock at a predetermined point in space. This allows for the comparison of local time variations of clocks due to gravitational and kinematic effects. We present a relativistic framework to introduce a coordinate time for the Moon. This framework also establishes a relationship between the coordinate times for the Moon and the Earth as determined by standard clocks located on the Earth’s geoid and the Moon’s equator. A clock near the Moon’s equator ticks faster than one near the Earth’s equator, accumulating an extra 56.02 microseconds per day over the duration of a lunar orbit. This formalism is then used to compute the clock rates at Earth-Moon Lagrange points. Accurate estimation of the rate differences of coordinate times across celestial bodies and their inter-comparisons using clocks onboard orbiters at relatively stable Lagrange points as time transfer links is crucial for establishing reliable communications infrastructure. This understanding also underpins precise navigation in cislunar space and on celestial bodies’ surfaces, thus playing a pivotal role in ensuring the interoperability of various position, navigation, and timing (PNT) systems spanning from Earth to the Moon and to the farthest regions of the inner solar system.

1. INTRODUCTION

More than 50 years after the first lunar landing, a multinational consortium, which includes NASA, is working towards a return to the Moon under the Artemis Accords [1]. Our ability to explore distant worlds will require the design and development of a communication and navigation infrastructure within and beyond cislunar space. With the expectation of a significant increase in assets on the lunar surface and in cislunar space in the near future, developing a robust architecture for accurate position, navigation, and timing (PNT) applications has become a matter of paramount interest.

Communication and navigation systems rely on a network of clocks that are synchronized to each other within a few tens of nanoseconds. As the number of assets on the lunar surface grows, synchronizing local clocks with higher precision using remote clocks on Earth becomes challenging and inefficient. An optimal solution would be to draw from the heritage of global navigation satellite systems (GNSS) by envisioning a system or constellation time common to all assets and then relating this time to clocks on Earth.

The relativistic framework presented here enables us to compare clock rates on the Moon and cislunar Lagrange points with respect to clocks on Earth by using a metric appropriate for a locally freely falling frame. The time measured by a clock at any given location is known as the proper time. Relativity of simultaneity implies that no two observers will agree on a given sequence of events if they are in different reference frames [2]. In other words, clocks in different reference frames tick at different rates. The gravitational and motional effects affect the ticking rate of clocks when compared with “ideal” clocks that are at rest and sufficiently far away from any gravitating mass. For example, clocks farther away from Earth tick faster, and clocks in uniform motion will tick slower with respect to “ideal” clocks, and vice-versa. Therefore, choosing an appropriate reference frame becomes essential for obtaining self-consistent results when comparing clocks on two celestial bodies.

In this paper, mainly we seek answers to the following questions: What is a good choice for the coordinate system that can be used to relate the proper times on the Earth and the Moon? What is an appropriate choice for the locations of ideal clocks on the surfaces of the Earth and Moon that makes it easier to compare their proper times? What is the proper time difference between clocks on the Moon and the Earth? What are the proper time differences between clocks located at the Earth-Moon Lagrange points and the Earth? The stability offered by Lagrange points provides a low acceleration noise environment for spacecraft with clocks. The relativistic corrections for such clocks can be precisely estimated as their positions and velocities are well-determined and can be used to compare the proper times of clocks on Earth, Moon, and in cislunar orbits.

In Section 1, we use the global positioning system (GPS) as an example to illustrate the relativistic effects on clocks if the Moon is treated just like an artificial satellite of the Earth and obtain a rough estimate for the clock rates on the Moon with respect to clocks on the geoid. Section 2 introduces a freely falling coordinate system with its center coinciding with the center of mass of the Earth and Moon. Section 3 compares the rate offset of a clock on the lunar surface to clocks on the geoid using this freely falling coordinate system, assuming the Moon is in a Keplerian orbit around the Earth. The results are compared with precise orbits for the Moon obtained using the latest planetary ephemerides DE440 [3]. Section 4 discusses the time rate offsets at Earth-Moon Lagrange points L1, L2, and L4/L5. Conclusions and future outlook are presented in Section 5. Appendices 1 and 2 introduce the framework for developing the metric used in all calculations. Appendix 3 justifies our assumptions of using a Keplerian model ignoring tidal effects, and a discussion in Appendix 4 establishes general covariance, meaning that the results are coordinate-independent.