and the distribution of digital products.
Keeping Time on the Moon: A Relativistic Approach to Lunar Clocks
Clocks at Earth-Moon Lagrance Points
4.1 Clock at Lagrange point L1
\ 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
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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]
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\ 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:
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\ Transformation of the metric tensor is accomplished with the usual formula:
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\ where the summation convention for repeated indices applies. Thus, for the time-time component of the metric tensor in the freely falling frame,
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\ \ The transformation coefficients are easily obtained from the above coordinate transformations and are
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\ \ Transformation of the metric tensor using Eq. (72): the metric component g00 in the center of mass frame,
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\ \ Summarizing, the scalar invariant in the center of mass system is
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:::info Authors:
(1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 ([email protected]);
(2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 ([email protected]).
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:::info This paper is available on arxiv under CC0 1.0 license.
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