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Scalar

scalar

Table of Links Acknowledgements 1 Introduction to thesis 1.1 History and Evidence 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.5 Summary 5 Summary Appendices A Standard model B Friedmann equations C Type I seasaw mechanism D Feynman diagrams in two-component DM model Bibliography 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM In this chapter, we study a pseudo-scalar DM in a generic U(1)𝑋 extension of SM. The results are based on the work: Shivam Gola, "Pseudo scalar dark matter in a generic U(1)𝑋 model, Phys.Lett.B 842 (2023) 137982” 4.1 Introduction WIMP-type dark matter is heavily constrained by direct experimental searches [32, 184]; therefore, to continue with WIMP, one has to find a way to evade the stringent direct detection bounds. A pseudo-scalar DM can naturally evade the strong direct detection constraints as it has derivative couplings that imply a momentum suppression in the tree-level DM-nucleon scattering matrix, which vanishes in the non-relativistic limit [185–192]. Hence, it is interesting to seek a pseudo-scalar particle as a WIMP dark matter. In this chapter, we consider a generic U(1)𝑋 model for pseudo-scalar DM and discuss its phenomenological implications in this study. The interesting aspect of the 𝑈(1)𝑋 models is that the three generations of right-handed neutrinos (RHNs) are required to eliminate the gauge and mixed gauge-gravity anomalies [144–147]. The RHNs mix with active neutrinos of SM via type I seesaw mechanism [71–74] to generate the required light neutrino masses and flavor mixing. A model that explains both the neutrino mass problem and the nature of dark matter would be a major step forward in high-energy physics [148–153]. The pseudo-scalar is not protected by any symmetry; therefore, for it to be a DM candidate, it must have a lifetime much greater than the age of the universe. We study the feasible parameter space allowed by the lifetime constraint on our pseudo-scalar particle, then we scan for the allowed parameter space by relic and direct detection bounds while respecting several other theoretical and experimental constraints. The chapter is organised as follows: In sec. 4.2, we introduce the model and discuss the details of the new fields and their interactions. In sec. 4.3, we discuss some of the relevant theoretical and experimental constraints. In sec. 4.4, we discuss relic density, direct detection, and other phenomenologically relevant studies. Finally, in sec. 4.5, we summarize the chapter. This paper is available on arxiv under CC BY 4.0 DEED license. Author:
(1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Table of Links Acknowledgements Acknowledgements 1 Introduction to thesis 1 Introduction to thesis 1.1 History and Evidence 1.1 History and Evidence 1.2 Facts on dark matter 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.4 Dark matter detection 1.5 Outline of the thesis 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.2 Model 2.3 Existing constraints on ALP parameter space 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.4 Dark matter analysis 2.5 Summary 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.2 Model 3.3 Theoretical and experimental constraints 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.2 Model 4.3 Theoretical and experimental constraints 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.4 Dark Matter analysis 4.5 Summary 4.5 Summary 5 Summary 5 Summary Appendices Appendices A Standard model A Standard model B Friedmann equations B Friedmann equations C Type I seasaw mechanism C Type I seasaw mechanism D Feynman diagrams in two-component DM model D Feynman diagrams in two-component DM model Bibliography Bibliography 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM In this chapter, we study a pseudo-scalar DM in a generic U(1)𝑋 extension of SM. The results are based on the work: Shivam Gola, "Pseudo scalar dark matter in a generic U(1)𝑋 model, Phys.Lett.B 842 (2023) 137982” 4.1 Introduction WIMP-type dark matter is heavily constrained by direct experimental searches [32, 184]; therefore, to continue with WIMP, one has to find a way to evade the stringent direct detection bounds. A pseudo-scalar DM can naturally evade the strong direct detection constraints as it has derivative couplings that imply a momentum suppression in the tree-level DM-nucleon scattering matrix, which vanishes in the non-relativistic limit [185–192]. Hence, it is interesting to seek a pseudo-scalar particle as a WIMP dark matter. In this chapter, we consider a generic U(1)𝑋 model for pseudo-scalar DM and discuss its phenomenological implications in this study. The interesting aspect of the 𝑈(1)𝑋 models is that the three generations of right-handed neutrinos (RHNs) are required to eliminate the gauge and mixed gauge-gravity anomalies [144–147]. The RHNs mix with active neutrinos of SM via type I seesaw mechanism [71–74] to generate the required light neutrino masses and flavor mixing. A model that explains both the neutrino mass problem and the nature of dark matter would be a major step forward in high-energy physics [148–153]. The pseudo-scalar is not protected by any symmetry; therefore, for it to be a DM candidate, it must have a lifetime much greater than the age of the universe. We study the feasible parameter space allowed by the lifetime constraint on our pseudo-scalar particle, then we scan for the allowed parameter space by relic and direct detection bounds while respecting several other theoretical and experimental constraints. The chapter is organised as follows: In sec. 4.2, we introduce the model and discuss the details of the new fields and their interactions. In sec. 4.3, we discuss some of the relevant theoretical and experimental constraints. In sec. 4.4, we discuss relic density, direct detection, and other phenomenologically relevant studies. Finally, in sec. 4.5, we summarize the chapter. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Author: Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai.

A Pseudo-Scalar Dark Matter Case: An Introduction to Get You Up to Speed

Table of Links Acknowledgements 1 Introduction to thesis 1.1 History and Evidence 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.5 Summary 5 Summary Appendices A Standard model B Friedmann equations C Type I seasaw mechanism D Feynman diagrams in two-component DM model Bibliography 3.2 Model In the next subsections, we discuss various parts of the lagrangian of the model, 3.2.1 Scalar Sector where 𝛼 is the mixing angle. The rotation matrix satisfies The real and imaginary components of 𝜒 have the following masses 3.2.2 Gauge sector To determine the gauge boson spectrum, we have to expand the scalar kinetic terms and replace 3.2.3 Yukawa sector The Yukawa sector of the model can be written in a gauge-invariant way as This paper is available on arxiv under CC BY 4.0 DEED license. Author:
(1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Table of Links Acknowledgements Acknowledgements 1 Introduction to thesis 1 Introduction to thesis 1.1 History and Evidence 1.1 History and Evidence 1.2 Facts on dark matter 1.2 Facts on dark matter 1.3 Candidates to dark matter 1.3 Candidates to dark matter 1.4 Dark matter detection 1.4 Dark matter detection 1.5 Outline of the thesis 1.5 Outline of the thesis 2 Dark matter through ALP portal and 2.1 Introduction 2 Dark matter through ALP portal and 2.1 Introduction 2.2 Model 2.2 Model 2.3 Existing constraints on ALP parameter space 2.3 Existing constraints on ALP parameter space 2.4 Dark matter analysis 2.4 Dark matter analysis 2.5 Summary 2.5 Summary 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3 A two component dark matter model in a generic 𝑈(1)𝑋 extension of SM and 3.1 Introduction 3.2 Model 3.2 Model 3.3 Theoretical and experimental constraints 3.3 Theoretical and experimental constraints 3.4 Phenomenology of dark matter 3.4 Phenomenology of dark matter 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.5 Relic density dependence on 𝑈(1)𝑋 charge 𝑥𝐻 3.6 Summary 3.6 Summary 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4 A pseudo-scalar dark matter case in 𝑈(1)𝑋 extension of SM and 4.1 Introduction 4.2 Model 4.2 Model 4.3 Theoretical and experimental constraints 4.3 Theoretical and experimental constraints 4.4 Dark Matter analysis 4.4 Dark Matter analysis 4.5 Summary 4.5 Summary 5 Summary 5 Summary Appendices Appendices A Standard model A Standard model B Friedmann equations B Friedmann equations C Type I seasaw mechanism C Type I seasaw mechanism D Feynman diagrams in two-component DM model D Feynman diagrams in two-component DM model Bibliography Bibliography 3.2 Model In the next subsections, we discuss various parts of the lagrangian of the model, 3.2.1 Scalar Sector where 𝛼 is the mixing angle. The rotation matrix satisfies The real and imaginary components of 𝜒 have the following masses 3.2.2 Gauge sector To determine the gauge boson spectrum, we have to expand the scalar kinetic terms and replace 3.2.3 Yukawa sector The Yukawa sector of the model can be written in a gauge-invariant way as This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai. Author: Author: (1) Shivam Gola, The Institute of Mathematical Sciences, Chennai.

A Phenomenological Study of WIMP Models: Scalar Sector, Gauge Sector, and More

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the random signal in frequency space (c.f. Eq. (2.9)) takes the following form where (and following the notation of [45]) Proceeding similarly as in Appendix B, the marginalized likelihood is which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We arrive at the following: where This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis

Let be a cyclic group, is the element in the group, is the scalar, and the Multi-Scalar Multiplication (MSM) is the algorithm for calculating , the sum of multiple scalar multiplications. As the group operation is more complicated than the addition and multiplication of elements in the finite field, the MSM algorithm is employed to reduce the number of times of group operation as much as possible. Usually, is a cyclic group defined on the elliptic curve of on the finite field of , with the order being bits, and . If the group operation is conducted by the plain algorithm of fast exponentiation, the number of group operations needed for every is times on average, and for all is times in total. Next, we will introduce some optimization methods corresponding to larger . G P_i ∈ G a_i ∈ Z ∑_{i=1}^{n} a_{i} P_{i} G y^2=x^3+ax+b F_p | | G b ≈ 256 0 ≤ a_i< 2^b a_i.P_i 1.5bn a_i.P_i 1.5b n Optimization Based on the Windowing Technique We can split the bit scalar into c width windows: reduce the scalar to base system. b 2^c Here, } , so we reduce the original algorithm issue of -bit MSM into a smaller -bit issue. We can put the same scalars from into a bucket and write them in another form: Q_j=∑_{i=1}^{n a_{i j} P_{i} b c ∑_{i=1}^{n} a_{i j} P_{i} For example, when , it is calculated as: c = 3 When calculating , set the partial sum T_l=S_l+⋯+S_k, then ∑kS_k However, the operation results of can be obtained through the recurrence sequence . It means that only one group operation is needed for the operation for each . Under the MSM algorithm with cc-bit scalar, all requires times of group operation. All requires times and the sum of these requires times of group operation. Therefore, each window with the value of c requires times of group operations. The operation of } only requires the normal method of fast exponentiation by times of group operation. T_l T_l=T_l+1+S_l T_l S_k n−2^c+1 T_k 2^ 1 c− T_k ^c−2 2 n+2^c−2 ∑_{j=0}^{b / c-1} Q_{j} 2^{j c (b/c−1)(c+1)=b − c + b/c−1 In summary, we can conclude that the total number of group operations required by the method of the Windowing Technique of a width of is c Set , then the number of group operations is . When , the number of group operations reduces to of its original value. c = logn 2bn/logn n≈10^5 1/12 Optimization Based on Group Endomorphism For the cyclic group on the elliptic curve on the finite field F_p, if the following group endomorphism of φ can be found: there exists , satisfying holds for all the points on , then it is easy to prove that such an endomorphism is a multiplication map, namely there is a making hold for all the points of on , which means that whenever we know the coordinates of one point, we can change it to the coordinates of another point simply by multiplying a number in to both the values of abscissa and ordinate. The algorithm can be further optimized based on this important property. G y^2=x^3+ax+b α,β ∈ F_p φ(x,y)=(αx,βy) G λ φ(P)=λP P G F_p When . The reverse value of one point can be obtained only by taking the opposite number from the ordinate. Based on this feature, in the plain fast exponentiation algorithm, the scalar can be written into the non-adjacent form (NAF), namely and any two adjacent cannot be non-zero at the same time. Compared to the -bit scalars, the group operation number in the fast exponentiation algorithm can be reduced from the original average of times to times. This technique can also be used to optimize the Windowing Technique. After writing to λ = −1, α = 1, β = −1 e_{i}.2^{i}, e_{i} ∈ {−1,0,1} ∑ e_i b 3/2⋅b 4/3⋅b a_i a_i=∑_{j=0}^{b / c-1} a_{i, j} 2^{j c}, 0 <= a_{i, j}<2^{c} conduct the following operation: The result of and - can be obtained. Since the complexity of addition and subtraction in group operation of an elliptic curve is the same, we can put these terms into a bucket according to the absolute value of scalars. Therefore, the number of groups can be reduced from the original 2c to , and the overall number of group operations required is reduced from (1) to: a_i=∑_{j=0}^{b / c-1} a’_{i, j}^2^{j c} 2^{c-1} <= a’_{i, j}^ <= 2^{c-1} 2^c If the parameters of the elliptic curve are special, for example, the BLS curve can be written as , and , take third-order element α from F_p^x, then there is a corresponding that holds y^2=x^3+b p ≡ 1( mod3 ) λ λ( )=( ) x,y αx,y Algorithm 1: Signed Digits In addition, . Then the multiplication operation can be optimized as follows: λ3 ≡ 1(mod |G| ) From [1], we can find a small enough scalar | |, | |<√ to make the above equation true, which gives us a way to reduce the multiplication of bb bits into the multiplication of two bits. Use it into the Windowing Technique, and the number of group operations is reduced to m1 m2 G b/2 In summary, both the above two optimization methods can reduce the number of group operations by when . 5.5% n=10^5 References [1] Francesco Sica, Mathieu Ciet, and Jean-Jacques Quisquater. Analysis of the gallant-lambert-vanstone method based on eﬀicient endomorphisms: Elliptic and hyperelliptic curves. In International Workshop on Selected Areas in Cryptography, pages 21–36. Springer, 2002.

Optimization of Multi-Scalar Multiplication Algorithm: Sin7Y Tech Review (21)

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 4: COMPARING RESULTS IN ROTATING AND NON-ROTATING COORDINATE SYSTEMS We calculate the fractional difference between a clock on the Moon’s surface and a clock on the Earth’s surface in three different coordinate systems. These are (1) the center-of-mass locally inertial system; (2) a rotating system in which the x−axis is along the Earth-Moon line; and (3) a translated, rotating system in which the Earth is at the origin of coordinates and the Earth-Moon line is in the x′ direction. We show that in all three coordinate systems, the fractional rate difference is the same. A Keplerian orbit is assumed for the Earth-Moon system. To simplify the calculations we assume that the clocks are on the surfaces of the respective bodies. This is an approximation that can be refined when the actual positions of the clocks are specified. 4.1 Center-Of-Mass inertial coordinate system The scalar invariant in the locally inertial frame whose origin is at the center of mass of the Earth-Moon system, neglecting tidal terms, is The difference in the last term represents a difference of squares of velocities. 4.2 Rotating center-of-mass coordinates Introduce a rotating system with an Earth-Moon line along the new x-axis: X = x cos(f) − y sin(f); Y = x sin(f) + y cos(f); T = t. Then The scalar invariant becomes For a clock on the Moon, For a clock on the Moon, there is no contribution from the Sagnac term. The proper time interval is Note that there is a significant contribution from the centrifugal potential. For a clock on Earth, The proper time interval is then The fractional proper time interval difference reduces exactly to the expression in Eq. (120). 4.3 Rotating coordinates with Earth at origin This implies the use of a coordinate system in which the Earth is not moving. This has to be a rotating coordinate system with its origin coinciding with the Earth’s center. Therefore translating the origin to the center of the Earth, with no change in the time variable, The scalar invariant becomes The potentials in the last term have been suppressed since they do not contribute to the order of this calculation. For a clock on the surface of the Moon, There is no contribution from the Sagnac term but there is a significant contribution from the centrifugal potential, representing the transverse velocity of the Moon. The radial velocity of the Moon comes from the spatial part of the metric. The proper time interval for such a clock is For a clock on the surface of the Earth, x′ = y′ = x′ = y′ = 0. The proper time interval is It is easily seen that the fractional proper time difference reduces to expressions that have been previously derived. Thus in all three coordinate systems, the fractional proper time difference is the same. Acknowledgments We would like to acknowledge the funding we received from the NASA grant NNH12AT81I. We are grateful to Elizabeth Donley, who carefully and critically reviewed the manuscript and provided valuable suggestions. We would also like to express our gratitude to Cheryl Gramling for initiating discussions on lunar time. We extend our sincere thanks to Roger Brown, Thomas Heavner, Judah Levine, Jeffrey Sherman, and Daniel Slichter for their review of the manuscript. This work is a contribution of NIST and is not subject to US copyright. [1] Artemis Plan: NASA’s Lunar Exploration Program Overview, https://www.nasa.gov/sites/default/files/atoms/files/ artemis plan-20200921.pdf, Sept. 2020 [2] Einstein, A. The Collected Papers of Albert Einstein, Volume 6: The Berlin Years: Writings, 1914-1917 Princeton University Press, 1996. [3] Park RS, Folkner WM, Williams JG, and Boggs DH, The JPL Planetary and lunar Ephemerides DE440 and DE441, AJ. 161 105, 2021 [4] Ashby N, Relativity in the Global Positioning System, Liv. Rev. Relativity 6, [Online Article]: cited Sept. 14, 2023, http://www.livingreviews.org/lrr-2003-1. [5] International Astronomical Union (IAU), Proceedings of the Twenty-Fourth General Assembly,” Transactions of the IAU, XXIVB,pp. 34-57,2000 [6] Ashby N and Bertotti B, Relativistic effects in local inertial frames, Phys. Rev. D 14(8), 1986 (Appendix), 2246-2259. [7] Ashby N and Allan D, Practical Implications of Relativity for a Global Time Scale, Radio Science 14, 1979, pp 649-669. [8] Bertone, S, Arnold, D, Girardin, V, Lasser, M, Meyer, U, J¨aggi, A, Assessing Reduced-Dynamic Parametrizations for GRAIL Orbit Determination and the Recovery of Independent lunar Gravity Field Solutions, Earth and Space Science, 2021, https://doi.org/10.1029/2020EA001454. [9] Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). ”Progress in the determination of the gravitational coefficient of the Earth”. Geophysical Research Letters. 19 (6): 529–531. [10] CODATA 2006, physics.nist.gov/cuu/Constants [11] Mohr PJ, Taylor BN, and Newell DB, The CODATA recommended values of the fundamental physical constants: 2006, Rev. Mod. Phys., 80, pp. 633-730, 2008 [12] Konopliv AS, et al., The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission, J. Geophys. Res. Planets, 118, 1415–1434, 2013, doi:10.1002/jgre.20097 [13] Daher, H et al., Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models, Journal of Geophysical Research: Planets, 126(12), 2021, e2021JE006875. https://doi.org/10.1029/2021JE006875 [14] Petit G and Wolf P, Relativistic theory for time comparisons: a review Metrologia, 42 (3), 2005, S138 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 4: COMPARING RESULTS IN ROTATING AND NON-ROTATING COORDINATE SYSTEMS We calculate the fractional difference between a clock on the Moon’s surface and a clock on the Earth’s surface in three different coordinate systems. These are (1) the center-of-mass locally inertial system; (2) a rotating system in which the x−axis is along the Earth-Moon line; and (3) a translated, rotating system in which the Earth is at the origin of coordinates and the Earth-Moon line is in the x′ direction. We show that in all three coordinate systems, the fractional rate difference is the same. A Keplerian orbit is assumed for the Earth-Moon system. To simplify the calculations we assume that the clocks are on the surfaces of the respective bodies. This is an approximation that can be refined when the actual positions of the clocks are specified. 4.1 Center-Of-Mass inertial coordinate system The scalar invariant in the locally inertial frame whose origin is at the center of mass of the Earth-Moon system, neglecting tidal terms, is The difference in the last term represents a difference of squares of velocities. 4.2 Rotating center-of-mass coordinates Introduce a rotating system with an Earth-Moon line along the new x-axis: X = x cos(f) − y sin(f); Y = x sin(f) + y cos(f); T = t. Then The scalar invariant becomes For a clock on the Moon, For a clock on the Moon, there is no contribution from the Sagnac term. The proper time interval is Note that there is a significant contribution from the centrifugal potential. For a clock on Earth, The proper time interval is then The fractional proper time interval difference reduces exactly to the expression in Eq. (120). 4.3 Rotating coordinates with Earth at origin This implies the use of a coordinate system in which the Earth is not moving. This has to be a rotating coordinate system with its origin coinciding with the Earth’s center. Therefore translating the origin to the center of the Earth, with no change in the time variable, The scalar invariant becomes The potentials in the last term have been suppressed since they do not contribute to the order of this calculation. For a clock on the surface of the Moon, There is no contribution from the Sagnac term but there is a significant contribution from the centrifugal potential, representing the transverse velocity of the Moon. The radial velocity of the Moon comes from the spatial part of the metric. The proper time interval for such a clock is For a clock on the surface of the Earth, x′ = y′ = x′ = y′ = 0. The proper time interval is It is easily seen that the fractional proper time difference reduces to expressions that have been previously derived. Thus in all three coordinate systems, the fractional proper time difference is the same. Acknowledgments We would like to acknowledge the funding we received from the NASA grant NNH12AT81I. We are grateful to Elizabeth Donley, who carefully and critically reviewed the manuscript and provided valuable suggestions. We would also like to express our gratitude to Cheryl Gramling for initiating discussions on lunar time. We extend our sincere thanks to Roger Brown, Thomas Heavner, Judah Levine, Jeffrey Sherman, and Daniel Slichter for their review of the manuscript. This work is a contribution of NIST and is not subject to US copyright. [1] Artemis Plan: NASA’s Lunar Exploration Program Overview, https://www.nasa.gov/sites/default/files/atoms/files/ artemis plan-20200921.pdf, Sept. 2020 [2] Einstein, A. The Collected Papers of Albert Einstein, Volume 6: The Berlin Years: Writings, 1914-1917 Princeton University Press, 1996. [3] Park RS, Folkner WM, Williams JG, and Boggs DH, The JPL Planetary and lunar Ephemerides DE440 and DE441, AJ. 161 105, 2021 [4] Ashby N, Relativity in the Global Positioning System, Liv. Rev. Relativity 6, [Online Article]: cited Sept. 14, 2023, http://www.livingreviews.org/lrr-2003-1. [5] International Astronomical Union (IAU), Proceedings of the Twenty-Fourth General Assembly,” Transactions of the IAU, XXIVB,pp. 34-57,2000 [6] Ashby N and Bertotti B, Relativistic effects in local inertial frames, Phys. Rev. D 14(8), 1986 (Appendix), 2246-2259. [7] Ashby N and Allan D, Practical Implications of Relativity for a Global Time Scale, Radio Science 14, 1979, pp 649-669. [8] Bertone, S, Arnold, D, Girardin, V, Lasser, M, Meyer, U, J¨aggi, A, Assessing Reduced-Dynamic Parametrizations for GRAIL Orbit Determination and the Recovery of Independent lunar Gravity Field Solutions, Earth and Space Science, 2021, https://doi.org/10.1029/2020EA001454. [9] Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). ”Progress in the determination of the gravitational coefficient of the Earth”. Geophysical Research Letters. 19 (6): 529–531. [10] CODATA 2006, physics.nist.gov/cuu/Constants [11] Mohr PJ, Taylor BN, and Newell DB, The CODATA recommended values of the fundamental physical constants: 2006, Rev. Mod. Phys., 80, pp. 633-730, 2008 [12] Konopliv AS, et al., The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission, J. Geophys. Res. Planets, 118, 1415–1434, 2013, doi:10.1002/jgre.20097 [13] Daher, H et al., Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models, Journal of Geophysical Research: Planets, 126(12), 2021, e2021JE006875. https://doi.org/10.1029/2021JE006875 [14] Petit G and Wolf P, Relativistic theory for time comparisons: a review Metrologia, 42 (3), 2005, S138 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

Timekeeping in Space: Comparing Rotating and Inertial Frames

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

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

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 3. CLOCK RATE DIFFERENCES BETWEEN EARTH AND MOON The Earth and the Moon orbit around their mutual center of mass in different Keplerian orbits. Meanwhile, the center of mass of the Earth-Moon system orbits around the Sun in an approximately Keplerian orbit. To calculate the rate differences between clocks on Earth and on the Moon, a fictitious locally freely-falling inertial frame is introduced at the Earth-Moon center of mass. This makes it convenient to calculate the proper times elapsed on moving clocks in terms of Keplerian motions of the Earth and the Moon. The Sun’s contribution is only tidal effects. If we omit the tidal potential of the Sun, the scalar invariant takes a simple form (Appendix 2) Consider a clock fixed on the surface of the rotating geoid of Earth. Since the geoid is a surface of approximate hydrostatic equilibrium, if such clocks are viewed from the local inertial frame they beat at the same rate, which can be evaluated at the equator. The proper time on the Earth-based clock becomes The first two contributions can be identified with the quantity Φ0. The contribution from the Moon’s potential can be well approximated by setting where D is the Earth-Moon distance. The dot product term between velocities will depend on the specific position of the clock and will vary with a daily period; this variation is similar to the corrections to the gravitational potential contribution from the Moon arising from the fact that the clock is not at the center of the Earth. Omitting such contributions give A similar argument applied to a clock fixed on the rotating Moon’s surface of hydrostatic equilibrium gives the proper time The following combination of quantities occurs frequently and can be reduced to a simpler expression The difference of squares of velocities is then Using Eq.(27) in Eq.(18), we obtain This term will have a period of approximately 27 days. Cross terms between velocities of the center of mass and rotational velocities have been omitted. One such term is This contribution will also vary with the rotation period of the Earth. Another such term is Since the Moon is tidally locked to the Earth, its center of mass velocity and the velocity of a clock on the selenoid will be highly correlated. Therefore, this term might give rise to a constant long-term average. Omitting the position-dependent terms, we have used the constants listed in Table I to evaluate the constant contribution and the amplitude of the periodic term. We find: None of the above estimates include tidal effects. This omission is because as a tidal force pushes back and forth on a satellite, two other side effects have to be accounted for. These are a change in the position of the satellite that entails a change in the gravitational potential of the body about which the satellite is orbiting and a change in the velocity of the satellite clock that changes its second-order Doppler shift. The residuals of the gravitational potential and second-order Doppler shift for the Earth-Moon system obtained by subtracting the Keplerian model from that obtained from DE440 are graphed in Fig. 1. Previous work on such problems has shown that these changes are of similar orders of magnitude. Summarizing, there are three contributions to the frequency shift of a clock in a satellite that are of similar orders of magnitude: (1) the perturbing tidal potential itself; (2) the perturbed position that changes the contribution from the main potential; (3) the perturbed velocity that changes the time dilation contribution. Although the perturbing tidal potential can easily be estimated, calculating the other two contributions is more complicated. When the Keplerian model is compared with DE440 ephemerides, the effects of solar tides are plotted in Fig. 2. It might appear that the second-order Doppler contribution to the rate difference depends on the coordinate system used. In the center-of-mass coordinates, a difference of squares of velocities appears, but in a system in which the Earth is at rest, the square of the relative velocity appears. Appendix 4 shows that contributions from the centrifugal potential, which occurs in a rotating coordinate system resolve the apparent discrepancy. 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 3. CLOCK RATE DIFFERENCES BETWEEN EARTH AND MOON The Earth and the Moon orbit around their mutual center of mass in different Keplerian orbits. Meanwhile, the center of mass of the Earth-Moon system orbits around the Sun in an approximately Keplerian orbit. To calculate the rate differences between clocks on Earth and on the Moon, a fictitious locally freely-falling inertial frame is introduced at the Earth-Moon center of mass. This makes it convenient to calculate the proper times elapsed on moving clocks in terms of Keplerian motions of the Earth and the Moon. The Sun’s contribution is only tidal effects. If we omit the tidal potential of the Sun, the scalar invariant takes a simple form (Appendix 2) Consider a clock fixed on the surface of the rotating geoid of Earth. Since the geoid is a surface of approximate hydrostatic equilibrium, if such clocks are viewed from the local inertial frame they beat at the same rate, which can be evaluated at the equator. The proper time on the Earth-based clock becomes The first two contributions can be identified with the quantity Φ0. The contribution from the Moon’s potential can be well approximated by setting where D is the Earth-Moon distance. The dot product term between velocities will depend on the specific position of the clock and will vary with a daily period; this variation is similar to the corrections to the gravitational potential contribution from the Moon arising from the fact that the clock is not at the center of the Earth. Omitting such contributions give A similar argument applied to a clock fixed on the rotating Moon’s surface of hydrostatic equilibrium gives the proper time The following combination of quantities occurs frequently and can be reduced to a simpler expression The difference of squares of velocities is then Using Eq.(27) in Eq.(18), we obtain This term will have a period of approximately 27 days. Cross terms between velocities of the center of mass and rotational velocities have been omitted. One such term is This contribution will also vary with the rotation period of the Earth. Another such term is Since the Moon is tidally locked to the Earth, its center of mass velocity and the velocity of a clock on the selenoid will be highly correlated. Therefore, this term might give rise to a constant long-term average. Omitting the position-dependent terms, we have used the constants listed in Table I to evaluate the constant contribution and the amplitude of the periodic term. We find: None of the above estimates include tidal effects. This omission is because as a tidal force pushes back and forth on a satellite, two other side effects have to be accounted for. These are a change in the position of the satellite that entails a change in the gravitational potential of the body about which the satellite is orbiting and a change in the velocity of the satellite clock that changes its second-order Doppler shift. The residuals of the gravitational potential and second-order Doppler shift for the Earth-Moon system obtained by subtracting the Keplerian model from that obtained from DE440 are graphed in Fig. 1. Previous work on such problems has shown that these changes are of similar orders of magnitude. Summarizing, there are three contributions to the frequency shift of a clock in a satellite that are of similar orders of magnitude: (1) the perturbing tidal potential itself; (2) the perturbed position that changes the contribution from the main potential; (3) the perturbed velocity that changes the time dilation contribution. Although the perturbing tidal potential can easily be estimated, calculating the other two contributions is more complicated. When the Keplerian model is compared with DE440 ephemerides, the effects of solar tides are plotted in Fig. 2. It might appear that the second-order Doppler contribution to the rate difference depends on the coordinate system used. In the center-of-mass coordinates, a difference of squares of velocities appears, but in a system in which the Earth is at rest, the square of the relative velocity appears. Appendix 4 shows that contributions from the centrifugal potential, which occurs in a rotating coordinate system resolve the apparent discrepancy. 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

The Role of Gravity and Motion in Space-Time Coordination

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 2.1 Coordinate Time In establishing a coordinate time on and near the Earth, two relativistic effects are compensated by adjusting the rates of standard clocks. These are (a) the gravitational potential at the geoid and (b) the second-order Doppler shift due to the Earth’s rotation. The gravitational potential at the equator can be estimated from existing models of Earth’s gravitational potential. Viewed from an Earth-centered inertial frame, the second-order Doppler shift is where ωe is Earth’s angular rotation rate and ae is the equatorial radius. Because the Earth’s geoid is nearly a surface of effective hydrostatic equilibrium, all atomic clocks on the geoid beat at equal rates, and this rate can be calculated on the Earth’s geoid at the equator. The effective potential Φ0 in Eq. (2) represents the fractional rate difference between an atomic clock at rest at infinity if the Earth were the only celestial body and an atomic clock fixed on the geoid of the rotating Earth. With this adjustment in scale, apart from tidal effects which average to near zero, clocks at rest on the geoid beat at the rate of International Atomic Time (TAI), defined by atomic clocks at rest on the geoid. Coordinate time suitable for use in navigation and timekeeping near the Earth’s surface is then obtained by synchronizing clocks in the local inertial frame [7]. The proper time on a clock at rest on the geoid then, apart from tidal contributions, beats at the rate of coordinate time because the term Φ0 cancels the potential and second-order Doppler shifts on the geoid. 2.2 Local Frame for the Moon While the Moon appears fairly rigid, it is nearly spherical due to hydrostatic equilibrium. One can imagine a locally inertial, freely falling reference frame with its origin at the Moon’s center of mass, see Appendix 1. Near the Moon, the scalar invariant will be 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 2.1 Coordinate Time In establishing a coordinate time on and near the Earth, two relativistic effects are compensated by adjusting the rates of standard clocks. These are (a) the gravitational potential at the geoid and (b) the second-order Doppler shift due to the Earth’s rotation. The gravitational potential at the equator can be estimated from existing models of Earth’s gravitational potential. Viewed from an Earth-centered inertial frame, the second-order Doppler shift is where ωe is Earth’s angular rotation rate and ae is the equatorial radius. Because the Earth’s geoid is nearly a surface of effective hydrostatic equilibrium, all atomic clocks on the geoid beat at equal rates, and this rate can be calculated on the Earth’s geoid at the equator. The effective potential Φ0 in Eq. (2) represents the fractional rate difference between an atomic clock at rest at infinity if the Earth were the only celestial body and an atomic clock fixed on the geoid of the rotating Earth. With this adjustment in scale, apart from tidal effects which average to near zero, clocks at rest on the geoid beat at the rate of International Atomic Time (TAI), defined by atomic clocks at rest on the geoid. Coordinate time suitable for use in navigation and timekeeping near the Earth’s surface is then obtained by synchronizing clocks in the local inertial frame [7]. The proper time on a clock at rest on the geoid then, apart from tidal contributions, beats at the rate of coordinate time because the term Φ0 cancels the potential and second-order Doppler shifts on the geoid. 2.2 Local Frame for the Moon While the Moon appears fairly rigid, it is nearly spherical due to hydrostatic equilibrium. One can imagine a locally inertial, freely falling reference frame with its origin at the Moon’s center of mass, see Appendix 1. Near the Moon, the scalar invariant will be 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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