A relativistic framework to establish coordinate time on the Moon and beyond

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 artemis_20 . 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 einstein96 . 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 park_2021 . Section 4 discusses the time rate offsets at Earth-Moon Lagrange points $L_{1},L_{2}$, and $L_{4}/L_{5}$. 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.

An instructive example of establishing a coordinate time on Earth is the GPS time. The constellation clocks are set to beat at the average coordinate rate corresponding to clocks at rest on the surface of the rotating Earth by applying a “factory frequency offset” to the clocks before launch, which is ashby03

$$\frac{\Delta f}{f}=\frac{3GM_{e}}{2ac^{2}}+\frac{\Phi_{0}}{c^{2}}\,,$$

(1)

where $a$ is the satellite semi-major axis, $G$ is Newton’s gravitational constant, $M_{e}$ is the mass of the Earth, $c$ is the speed of light in vacuum, and $\Phi_{0}$ is the effective gravitational potential in the rotating frame, which is the sum of the static gravitational potential of the Earth, and a centripetal contribution ashby03 . The International Astronomical Union defines a “Terrestrial Time”(TT) by adopting a fixed value for $\Phi_{0}/c^{2}$ iau :

$$-L_{G}=\frac{\Phi_{0}}{c^{2}}=-6.969290134(0)\times 10^{-10}\,.$$

(2)

If we simply substitute into Eq. (1) the length of the semi-major axis of the Moon’s orbit, $3.84748\times 10^{8}$ meters, the “offset” becomes $-6.7964\times 10^{-10}$. To convert to a rate difference in microseconds per day, multiply by $86400\times 10^{6}$ $\mu$s/day, yielding $58.721\ \mu{\rm s/day}$. This does not include the effect of the Moon’s gravitational potential. Also, this approach can be questioned because it does not make sense to treat the Moon’s potential, from the point of view of an Earth-based inertial frame, as an Earth satellite; the Moon’s potential should be treated as a tidal potential. Nevertheless, a standard clock on an Earth satellite at the distance of the Moon would beat faster than a standard clock at rest on Earth by 58.721 $\mu$s per day, not including any effect from the gravitational potential of the Moon. This is a combination of Earth’s gravitational potential and second-order Doppler shifts at the orbiting satellite.

In addition, there are well-understood periodic effects arising from orbit eccentricity, with an additional contribution to the rate on the satellite clock given by ashby03

$$\Delta=-\frac{2GM_{e}}{c^{2}}\bigg{(}\frac{1}{a}-\frac{1}{r}\bigg{)}\,,$$

(3)

For a satellite in Keplerian orbit, the periodic contribution to the rate is

$$\Delta=\frac{2GM_{e}e}{c^{2}a(1-e^{2})}(\cos(f)+e),$$

(4)

where $e$ is the eccentricity and $f$ is the true anomaly. Numerical evaluation of Eq. (4) yields a value $1.2695\times 10^{-12}(\cos(f)+e)$ or $0.1097(\cos(f)+e)\ \mu{\rm s/day}$. The time average of the combination $\cos(f)+e$ is zero, so this term does not contribute to the average rate.

This model is based on using an eccentric Keplerian orbit in the local inertial frame centered on Earth’s center of mass. The center of mass of the Earth and Moon approximately follows a Keplerian orbit. However, for the Earth-Moon system, one cannot have a Keplerian orbit in a coordinate system centered on the Earth and a Keplerian orbit in a coordinate system centered on the Earth-Moon center of mass with the same orbit parameters. There are also relativistic effects arising from changes in time and length scales, Lorentz contraction, and changes in tidal effects.

In the following sections, we shall investigate a local inertial system with an origin at the Earth-Moon center of mass. This is a freely falling inertial frame only in the Sun’s gravitational field. The reason for using such a frame is that the Earth and Moon are treated more or less equivalently; tidal potentials are due only to the Sun. By addressing relativistic effects in this simple system, it may be expected that the main relativistic corrections can be better understood. We work in a plane containing the Earth, Moon, and Sun, which is inclined with respect to the ecliptic plane. Calculations are carried out only to order $c^{-2}$. Contributions from the tidal potentials of other solar system bodies are left out. The metric signature is $(-1,1,1,1)$ with Greek indices running from 0 to 3, and a negative sign is assigned for gravitational potential rather than a positive sign used in geodesy.

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)

$\displaystyle-ds^{2}=-\left(1+\frac{2\Phi_{e}}{c^{2}}+\frac{2\Phi_{m}}{c^{2}}\right)(dx^{0})^{2}+\left(1-\frac{2\Phi_{e}}{c^{2}}-\frac{2\Phi_{m}}{c^{2}}\right)(dx^{2}+dy^{2}+dz^{2})\,.$

(12)

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

$\displaystyle-c^{2}d\tau_{e}^{2}=-\left(1+\frac{2\Phi_{e}}{c^{2}}\big{|}_{R_{Eeq}}+\frac{2\Phi_{m}}{c^{2}}\big{|}_{R_{Eeq}}\right)(dx^{0})^{2}+\frac{\big{(}{\bf V}_{e}+{\bf v}_{e}\big{)}^{2}}{c^{2}}(dx^{0})^{2}\,,$

(13)

where the equatorial radius of the Earth is denoted by ${\bf R}_{Eeq}$, ${\bf V}_{e}$ is the velocity of the Earth’s center of mass in the Earth-Moon coordinate system, and where ${\bf v}_{e}$ represents the velocity of the clock on the equator due to Earth rotation. Expanding the velocity term, taking square roots of both sides and rearranging,

$\displaystyle cd\tau_{e}=\left(1+\frac{\Phi_{e}}{c^{2}}\big{|}_{R_{Eeq}}-\frac{{\bf v}_{e}^{2}}{2c^{2}}+\frac{\Phi_{m}}{c^{2}}\big{|}_{R_{Eeq}}-\frac{V_{e}^{2}}{2c^{2}}-\frac{({\bf V}_{e}\cdot{\bf v}_{e})^{2}}{c^{2}}\right)(dx^{0})\,.$

(14)

The first two contributions can be identified with the quantity $\Phi_{0}$. The contribution from the Moon’s potential can be well approximated by setting

$$\frac{\Phi_{m}}{c^{2}}\big{|}_{R_{Eeq}}=-\frac{GM_{m}}{c^{2}D}\,,$$

(15)

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 gives

$$cd\tau_{e}=\left(1+\frac{\Phi_{0}}{c^{2}}-\frac{GM_{m}}{c^{2}D}-\frac{V_{e}^{2}}{2c^{2}}\right)dx^{0}\,.$$

(16)

A similar argument applied to a clock fixed on the rotating Moon’s surface of hydrostatic equilibrium gives the proper time

$$cd\tau_{m}=\left(1+\frac{\Phi_{0m}}{c^{2}}-\frac{GM_{e}}{c^{2}D}-\frac{V_{m}^{2}}{2c^{2}}\right)dx^{0}\,,$$

(17)

where ${\bf V}_{m}$ is the velocity of the Moon’s center of mass, and $\Phi_{m}$, discussed above, is the combination of the Moon’s gravitational potential on the selenoid, and second-order Doppler shift due to the rotation of a clock on the Moon’s equator. Therefore, the fractional frequency shift of a clock on the Moon’s equator relative to a clock on Earth’s equator is

$\displaystyle\frac{d\tau_{m}-d\tau_{e}}{d\tau_{e}}=\frac{(GM_{m}-GM_{e})}{c^{2}D}+\frac{\Phi_{0m}-\Phi_{0}}{c^{2}}-\frac{1}{2c^{2}}(V_{m}^{2}-V_{e}^{2})\,,$

(18)

where $GM_{e}$ and $GM_{m}$ are the standard gravitational parameters for the Earth and Moon. The distance to the Moon from the Earth, for a Keplerian orbit, is given by (see Appendix 3)

$$D=\frac{a(1-e^{2})}{1+e\cos(f)}\,,$$

(19)

where $f$ is the true anomaly plus possibly a constant, $a$ is the length of the semi-major axis, $M_{T}=M_{e}+M_{m}$, and $e$ is the eccentricity of the Moon’s orbit. Then

$$\dot{D}=\frac{(1-e^{2})ae\sin(f)}{(1+e\cos(f))^{2}}\frac{df}{dt}\,,$$

(20)

$$\dot{f}=\frac{df}{dt}=\frac{n(1+e\cos(f))^{2}}{(1-e^{2})^{3/2}}\,,$$

(21)

$$n^{2}=\frac{GM_{T}}{a^{3}}\,.$$

(22)

The following combination of quantities occurs frequently and can be reduced to a simpler expression

$$D^{2}{\dot{f}}^{2}+(\dot{D})^{2}=\frac{GM_{T}}{a}\left(\frac{1+2e\cos(f)+e^{2}}{1-e^{2}}\right)\,.$$

(23)

The velocities have radial as well as transverse components. Consider first the quantity ${\bf V}_{e}^{2}$. The radial and transverse components of this velocity are

$$V_{r}=-\frac{M_{m}}{M_{T}}\dot{D}\,,\quad V_{t}=\frac{M_{m}}{M_{T}}D\dot{f}.$$

(24)

Therefore

$\displaystyle\frac{V_{e}^{2}}{2c^{2}}=\frac{1}{c^{2}}\left(\frac{M_{m}^{2}}{M_{T}^{2}}\right)(\dot{D}^{2}+D^{2}\dot{f}^{2})=\mu^{2}\frac{GM_{T}}{2ac^{2}}\left(\frac{1+2e\cos(f)+e^{2}}{1-e^{2}}\right),$

(25)

where $\mu=M_{m}/M_{T}=0.012150$. A similar calculation for $V_{m}$ yields

$\displaystyle\frac{V_{m}^{2}}{2c^{2}}=(1-\mu)^{2}\frac{GM_{T}}{2ac^{2}}\left(\frac{1+2e\cos(f)+e^{2}}{1-e^{2}}\right)\,.$

(26)

The difference of squares of velocities is then

$\displaystyle\frac{V_{e}^{2}-V_{m}^{2}}{2c^{2}}=(2\mu-1)\left(\frac{GM_{T}}{2ac^{2}}\right)\left(\frac{1+2e\cos(f)+e^{2}}{1-e^{2}}\right)\,.$

(27)

Using Eq.(27) in Eq.(18), we obtain

$$\frac{d\tau_{m}-d\tau_{e}}{d\tau_{e}}=\frac{(GM_{m}-GM_{e})}{c^{2}D}+\frac{\Phi_{0m}-\Phi_{0}}{c^{2}}-(1-2\mu)\left(\frac{GM_{T}}{2ac^{2}}\right)\left(\frac{1+2e\cos(f)+e^{2}}{1-e^{2}}\right).$$

(28)

Now we’ll discuss the small position-dependent terms that have been omitted. ${\bf D}$ represents the vector from the center of the Earth to the center of the Moon. The actual distance from the center of the Moon to a clock on Earth’s geoid is $|-{\bf D}+{\bf r}_{e}|$ , where ${\bf r}_{e}$ is the vector from the Earth’s center to the clock on the equator. Then

$\displaystyle-\frac{GM_{m}}{c^{2}|-{\bf D}+{\bf r}_{e}|}\approx-\frac{GM_{m}}{c^{2}D}-\frac{GM_{m}}{c^{2}D}\frac{{\bf D}\cdot{\bf r_{e}}}{D^{2}}\,.$

(29)

The second term will vary with the rotation of the Earth. Its magnitude is approximately

$$\frac{GM_{m}a_{e}}{c^{2}D^{2}}\approx.0002\ \mu{\rm s/day}.$$

(30)

A similar term has been omitted from the calculation of the gravitational potential of the Earth on a Moon-fixed clock:

$$\frac{GM_{e}a_{m}}{c^{2}D^{2}}\approx.0045\ \mu{\rm s/day}.$$

(31)

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

$$\frac{{\bf V}_{e}\cdot{\bf v}_{e}}{c^{2}}\approx\frac{V_{e}\omega_{e}a_{e}}{c^{2}}\approx 0.0055\ \mu{\rm s/day}.$$

(32)

This contribution will also vary with the rotation period of the Earth. Another such term is

$$\frac{{\bf V}_{m}\cdot{\bf v}_{m}}{c^{2}}\approx\frac{V_{m}\omega_{m}a_{m}}{c^{2}}\approx 0.0045\ \mu{\rm s/day}.$$

(33)

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 1 to evaluate the constant contribution and the amplitude of the periodic term. We find:

$$\frac{d\tau_{m}-d\tau_{e}}{d\tau_{e}}=6.48378(15)\times 10^{-10}-1.25502518(89)\times 10^{-12}\cos(f)\,.$$

(34)

Multiplying by $10^{6}\times 86400$ $\mu$s to obtain a time difference per day gives

$$56.0199(12)-0.10843417(89)\cos(f)\ \mu{\rm s/day}\,.$$

(35)

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.

In the inertial frame centered on Earth’s center, the Earth’s velocity is zero, and the time is defined by a standard clock at the origin—Geocentric Coordinate Time (TCG) iau . A further scale change $t_{TCG}\rightarrow(1+L_{G})t$, see for example Eq. (7), defines a coordinate time whose rate is the same as that of standard clocks at rest on Earth’s geoid petit_2005 . Similarly, in a freely falling inertial frame centered on the Moon’s center, the Moon’s velocity is zero, and a local time is defined by a standard clock at the Moon’s origin; this could be called TCL. A further time change such as $t_{TCL}\rightarrow(1+L_{m})t$ would define a coordinate time whose rate equals that of standard clocks at rest on the Moon’s selenoid. Then, using Eq. (18),

	$\displaystyle dt_{TCL}\approx(1+L_{Gm})dt_{TCG},\quad{\rm where}$
	$\displaystyle L_{Gm}=\frac{(GM_{m}-GM_{e})}{c^{2}D}-\frac{1}{2c^{2}}(V_{m}^{2}-V_{e}^{2}).$		(36)

$L_{Gm}$ in Eq (36) is a rate with periodic contribution arising from Earth-Moon orbital eccentricity, which varies as the true anomaly   $-1.49373-0.10967\cos(f)~{}\mu$s/day. The computed offset of $L_{Gm}$ compared with data from DE440 is given in Fig. 3.

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.

The Lagrange points offer a cost-effective and low-noise environment for stationing spacecraft with clocks because there is no net acceleration. As a result, the orbits of clocks at these points with respect to the Earth-Moon barycenter are well-determined and the corresponding frequency offsets are precisely determined. Therefore, the proper times of clocks at Lagrange points can be related to the proper times of clocks on the Earth and Moon with high precision. This information is crucial for synchronizing remote clocks in cislunar space. Both $L_{4}$ and $L_{5}$ are stable points, but $L_{1}$ and $L_{2}$ are only metastable locations.

We presented a model based on Keplerian orbits for establishing coordinate time on the Moon and rates of clocks at Lagrange points in cislunar space. We have used values for Keplerian orbit parameters that can be looked up; the only parameters that fit were the times of periapsis passage. The main numerical results obtained using our approach are given in Table 2. We assumed a fixed eccentricity and fixed value for the semi-major axis for the Moon’s orbit around the Earth, as the present-day values for these parameters are very slowly varying moon_fact .

The planetary ephemeris DE440 was used to calculate the potentials and velocities of Eq. (28); the difference between the DE440 calculation and the Keplerian model calculations is only of the order of a few ns per day. Such differences are due to tidal potentials arising from solar system bodies. Tidal effects can be readily modeled using available orbit data and added as corrections to the Keplerian model for synchronizing remote clocks on the Moon to within a few hundreds of ps or better. Changes in time coordinate entail changes in length scale, which should be of higher order than the $c^{-2}$ effects we have considered here (see, for example, the length scale change in Eq. (69)).

This approach is also useful in calculating time comparisons between Earth and clocks in the neighborhood of other solar system bodies such as Mars. The available spherical harmonic gravity potential for Mars allows an estimate of the quantity $L_{M}$ for Mars that includes the average equatorial potential and rotational effects, analogous to $L_{G}$ for Earth. In the case of Mars, the only available coordinate systems for the description of the problem are barycentric coordinates. The Earth-Mars rate difference is dominated by the difference in the Sun’s gravitational potential at the two locations. Keplerian models, as well as computations using DE440, can be usefully compared; this will be the subject of a future paper. Spatial transformations accompanying time transformations also remain to be examined as part of future work.

The Moon’s center of mass is in free fall, and therefore its path is a geodesic. It is useful to construct Fermi coordinates with origin at this point, since then the only forces on an object in the neighborhood of the Moon due to external bodies are tidal forces. In this coordinate system, the Christoffel symbols due to external bodies are all zero at the origin, while contributions to Christoffel symbols from the Moon itself must be “effaced”, or discarded since they are infinite and such terms cannot cause acceleration of the Moon itself. The following calculation is taken only to order $c^{-2}$. The geodesic in question is complicated because the Earth-Moon system orbits the Sun in an approximately Keplerian orbit, while the Moon and Earth revolve around each other in a different, approximately Keplerian orbit; this latter orbit is perturbed by the Sun’s tidal potential so is not known analytically. We can still construct Fermi coordinates since many unknown quantities cancel out. Here, we show how the metric given in Eq. (12) arises.

We work in the plane of the Sun, Earth, and Moon and denote the total gravitational potential by

$$\Phi=\Phi_{e}+\Phi_{s}+\Phi_{m}\,.$$

(66)

the subscripts e, s, and m represent the potentials of the Earth, Sun, and Moon, respectively. Beginning with the metric in the solar system barycentric coordinates,

$$-ds^{2}=-\left(1+\frac{2\Phi}{c^{2}}\right)(dX^{0})^{2}+\left(1-\frac{2\Phi}{c^{2}}\right)(dX^{2}+dY^{2}+dZ^{2})\,,$$

(67)

where $dX^{0}$ is the time and $dX,dY$ and $dZ$ are the space coordinate displacements in the barycentric coordinate system. Lowercase letters will be reserved for corresponding quantities in the local Fermi normal coordinate system.

We give the transformation equations between barycentric coordinates and Fermi normal coordinates with the center at the Moon as follows:ashbybertotti86

	$\displaystyle X^{0}=\int_{0}^{x^{0}}\left(1-\frac{\Phi_{e}(m)+\Phi_{s}(m)}{c^{2}}+\frac{V(m)^{2}}{2c^{2}}\right)dx^{0}+\frac{{\bf V}(m)\cdot{\bf r}}{c}\,;\hbox to101.17755pt{}$		(68)
	$\displaystyle X^{k}=X^{k}(m)+x^{k}\left(1+\frac{\Phi_{e}(m)+\Phi_{s}(m)}{c^{2}}-\frac{{\bf A}(m)\cdot{\bf r}}{c^{2}}\right)+\frac{r^{2}A(m)^{k}}{2c^{2}}+\frac{V(m)^{k}{\bf V}(m)\cdot{\bf r}}{2c^{2}}\,.$		(69)

Here, the notation $(m)$ as in ${\bf V}(m)$ 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:

	$\displaystyle\frac{\partial X^{0}}{\partial x^{0}}=1-\frac{\Phi_{s}(m)+\Phi_{e}(m)}{c^{2}}+\frac{V(m)^{2}}{2c^{2}}+\frac{{\bf A}(m)\cdot{\bf r}}{c^{2}}\,,\quad\frac{\partial X^{0}}{\partial x^{k}}=\frac{V(m)^{k}}{c}\,,\quad\frac{\partial X^{k}}{\partial x^{0}}=\frac{V(m)^{k}}{c}\,,$		(70)
	$\displaystyle\frac{\partial X^{k}}{\partial x^{j}}=\delta_{j}^{k}\left(1+\frac{\Phi_{s}(m)+\Phi_{e}(m)}{c^{2}}-\frac{{\bf A}(m)\cdot{\bf r}}{c^{2}}\right)+\frac{V(m)^{k}V(m)^{j}}{2c^{2}}-\frac{x^{k}A(m)^{j}-x^{j}A(m)^{k}}{c^{2}}\,.$		(71)

Transformation of the metric tensor is accomplished with the usual formula:

$$g_{\alpha\beta}=\frac{\partial X^{\mu}}{\partial x^{\alpha}}\frac{\partial X^{\nu}}{\partial x^{\beta}}G_{\mu\nu}\,,$$

(72)

where the summation convention for repeated indices applies. Thus, for the time-time component of the metric tensor in the freely falling frame,

	$\displaystyle g_{00}=\left(\frac{\partial X^{0}}{\partial x^{0}}\right)^{2}G_{00}+\sum_{j=1}^{j=3}\left(\frac{\partial X^{j}}{\partial x^{0}}\right)^{2}G_{jj}\hbox to180.67499pt{}$
	$\displaystyle=\left(1-\frac{\Phi_{s}(m)+\Phi_{e}(m)}{c^{2}}+\frac{V(m)^{2}}{2c^{2}}+\frac{{\bf A}(m)\cdot{\bf r}}{c^{2}}\right)^{2}(-1)\left(1+\frac{2(\Phi_{e}+\Phi_{m}+\Phi_{s})}{c^{2}}\right)+$
	$\displaystyle\frac{V(m)^{2}}{c^{2}}\left(1-\frac{2(\Phi_{e}+\Phi_{m}+\Phi_{s})}{c^{2}}\right)\,.\hbox to72.26999pt{}$		(73)

Expanding and keeping terms of order $c^{-2}$,

$\displaystyle g_{00}=-\left(1+\frac{2\Phi_{m}}{c^{2}}+\frac{2(\Phi_{e}+\Phi_{s})}{c^{2}}-\frac{2(\Phi_{e}(m)+\Phi_{s}(m))}{c^{2}}+\frac{2{\bf A}(m)\cdot{\bf r}}{c^{2}}\right)\,.$

(74)

Except for the moon’s potential, terms in the last line add up to the solar tidal potential, for expanding $\Phi_{e}+\Phi_{s}$ about the origin and using

$$A(m)^{k}=-\frac{\partial(\Phi_{e}+\Phi_{s})}{\partial X^{k}}\,,$$

(75)

we find

	$\displaystyle\frac{\Phi_{e}}{c^{2}}+\frac{\Phi_{s}}{c^{2}}-\frac{\Phi_{e}(m)+\Phi_{s}(m)}{c^{2}}+\frac{{\bf A}_{cm}\cdot{\bf r}}{c^{2}}=\frac{\Phi_{t}}{c^{2}}$
	$\displaystyle=-\frac{GM_{e}}{c^{2}r_{e}^{5}}\frac{3({\bf r_{e}}\cdot{\bf r})^{2}-r_{e}^{2}r^{2}}{2}-\frac{GM_{s}}{c^{2}R^{5}}\frac{3({\bf R}\cdot{\bf r})^{2}-R^{2}r^{2}}{2},$		(76)

where ${\bf R}$ is the vector from the Sun to the center of the Moon, ${\bf r_{e}}$ is the vector from the Earth to the center of the Moon, and ${\bf r}$ is the vector from the center of the Moon to the observation point, in local Fermi normal coordinates. Eq. (VI) gives the total tidal potential $\Phi_{t}/c^{2}$ in the vicinity of the Moon due to the Earth and the Sun. Thus

$$g_{00}=-\left(1+\frac{2\Phi_{m}}{c^{2}}+\frac{2\Phi_{t}}{c^{2}}\right)\,.$$

(77)

For the spatial component $g_{11}$ we have

	$\displaystyle g_{11}=\left(\frac{\partial X^{0}}{\partial x^{1}}\right)^{2}G_{00}+\sum_{j=1}^{j=3}\left(\frac{\partial X^{j}}{\partial x^{1}}\right)^{2}G_{jj}$
	$\displaystyle=\left(\frac{V(m)^{1}}{c}\right)^{2}(-1)+\left(\frac{\partial X^{1}}{\partial x^{1}}\right)^{2}G_{11}$
	$\displaystyle=1-\frac{2\Phi_{m}}{c^{2}}-\frac{2\Phi_{t}}{c^{2}}\,,\hbox to36.135pt{}$		(78)

where we have again expanded and kept only terms of order $c^{-2}$. Similarly,

$$g_{22}=g_{33}=g_{11}\,.\hbox to21.68121pt{}$$

(79)

The metric component $g_{12}$ is given by

$$g_{12}=\frac{\partial X^{0}}{\partial x^{1}}\frac{\partial X^{0}}{\partial x^{2}}G_{00}+\sum_{j}\frac{\partial X^{j}}{\partial x^{1}}\frac{\partial X^{j}}{\partial x^{2}}G_{jj}\,.$$

(80)

Keeping only terms of order $c^{-2}$, this becomes

$\displaystyle g_{12}=-\frac{V(m)^{1}V(m)^{2}}{c^{2}}+\bigg{(}\frac{V(m)^{1}V(m)^{2}}{2c^{2}}+\frac{V(m)^{2}V(m)^{1}}{2c^{2}}\bigg{)}=0\,.$

(81)

Similarly,

$$g_{13}=g_{23}=0\,.$$

(82)

Summarizing, the scalar invariant with origin at the Moon’s center is

$\displaystyle-ds^{2}=-\left(1+\frac{2\Phi_{m}}{c^{2}}+\frac{2\Phi_{t}}{c^{2}}\right)(dx^{0})^{2}+\left(1-\frac{2\Phi_{m}}{c^{2}}-\frac{2\Phi_{t}}{c^{2}}\right)(dx^{2}+dy^{2}+dz^{2})\,.$

(83)

The speed of light $c$ is a defined quantity, which does not change when transforming coordinates. However, because the time scale changes, the length scale will also change. A quantity such as $\Phi_{m}/c^{2}$ has been carried forward from barycentric coordinates and one might question whether it should change due to time and length scale changes. However, such quantities are already of order $c^{-2}$ and any such changes would be of higher order and are therefore negligible. In these coordinates, contributions to Christoffel symbols of the second kind due to external bodies are zero since tidal potentials have been neglected.