Table of Links Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon


Clock Rate Differences Between Earth and Moon


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


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 APPENDIX 1: FERMI COORDINATES WITH ORIGIN AT THE CENTER OF THE MOON We give the transformation equations between barycentric coordinates and Fermi normal coordinates with the center at the Moon as follows:[6] Here, the notation (m) as in V(m) represents quantities evaluated at the Moon’s center of mass. The quantity V (m) is the magnitude of the Moon’s velocity. Transformation coefficients can be derived and are: Transformation of the metric tensor is accomplished with the usual formula: where the summation convention for repeated indices applies. Thus, for the time-time component of the metric tensor in the freely falling frame, APPENDIX 2: CONSTRUCTION OF FREELY FALLING CENTER OF MASS FRAME The transformation coefficients are easily obtained from the above coordinate transformations and are Transformation of the metric tensor using Eq. (72): the metric component g00 in the center of mass frame, Summarizing, the scalar invariant in the center of mass system is Authors:
(1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu);
(2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). This paper is available on arxiv under CC0 1.0 license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon 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 Conclusions Clock in Orbit 2.1 Coordinate Time 2.2 Local Frame for the Moon Clock in Orbit Clock in Orbit 2.1 Coordinate Time 2.1 Coordinate Time 2.2 Local Frame for the Moon 2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon 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 Clocks at Earth-Moon Lagrance Points 4.1 Clock at Lagrange point L1 4.1 Clock at Lagrange point L1 4.2. Clock at Lagrange point L2 4.2. Clock at Lagrange point L2 4.3. Clock at Lagrange point L4 or L5 4.3. Clock at Lagrange point L4 or L5 Conclusions Conclusions Conclusions Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 3: Equations of Motion of Earth and Moon Appendix 3: Equations of Motion of Earth and Moon Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Acknowledgments and References Acknowledgments and References APPENDIX 1: FERMI COORDINATES WITH ORIGIN AT THE CENTER OF THE MOON We give the transformation equations between barycentric coordinates and Fermi normal coordinates with the center at the Moon as follows:[6] Here, the notation (m) as in V (m) represents quantities evaluated at the Moon’s center of mass. The quantity V (m) is the magnitude of the Moon’s velocity. Transformation coefficients can be derived and are: V Transformation of the metric tensor is accomplished with the usual formula: where the summation convention for repeated indices applies. Thus, for the time-time component of the metric tensor in the freely falling frame, APPENDIX 2: CONSTRUCTION OF FREELY FALLING CENTER OF MASS FRAME The transformation coefficients are easily obtained from the above coordinate transformations and are Transformation of the metric tensor using Eq. (72): the metric component g00 in the center of mass frame, Summarizing, the scalar invariant in the center of mass system is Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu); (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). Authors: Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 ( Neil.Ashby@Colorado.Edu ); Neil.Ashby@Colorado.Edu (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 ( bijunath.patla@nist.gov ). bijunath.patla@nist.gov This paper is available on arxiv under CC0 1.0 license. This paper is available on arxiv under CC0 1.0 license. available on arxiv available on arxiv

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Abstract

Orbit

Origin

Equations of Motion in the Earth-Moon System

How Keplerian Orbits Help Define Time on the Moon

Exoplanetology Tech

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Keeping Time on the Moon: A Relativistic Approach to Lunar Clocks

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