---
title: Mars-Centered Post-Newtonian Framework
url: https://www.emergentmind.com/topics/mars-centered-post-newtonian-framework
type: topic
---

# Mars-Centered Post-Newtonian Framework

Searching arXiv for the cited framework papers and closely related post-Newtonian formalisms.
arXiv search query: 1811.12401 2004.03513 1101.0588 1206.2637 2606.13726 Mars-centered post-Newtonian framework EFTofPNG Galilean covariance Newton-Cartan
A Mars-centered post-Newtonian framework is a relativistic reference-system and dynamical framework in which Mars plays the role of the local gravitating center, while the global background is described in a barycentric post-Newtonian expansion. In its standard General Relativity realization, the construction follows the International Astronomical Union BCRS/TCB formalism, defines a Mars-centered celestial reference system with Areocentric Coordinate Time \(T \equiv \mathrm{TCA}\), and relates barycentric coordinate time, Mars-centered coordinate time, a conventional Mars surface time scale, and the proper times of landed and orbiting clocks. In a broader methodological sense, the same topic is connected to effective field theory formulations of post-Newtonian gravity, coordinate-independent post-1-Newtonian Newton-Cartan constructions, and Galilean-covariant \(c\)-expansions that make local body-centered frames explicit [2606.13726][1811.12401][1101.0588].

## 1. Reference-system architecture in General Relativity

The formal starting point is the IAU Barycentric Celestial Reference System with coordinate time \(t \equiv \mathrm{TCB}\) and metric
\[
g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),
\]
\[
g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),
\]
\[
g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),
\]
with Mars’ barycentric position, velocity, and acceleration denoted by \(\boldsymbol{x}_{Ma}(t)\), \(\boldsymbol{v}_{Ma}\), and \(\boldsymbol{a}_{Ma}\), and local BCRS position relative to Mars given by \(\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)\) [2606.13726].

The Mars-centered celestial reference system is then introduced with coordinates \((cT,\boldsymbol{X})\), and **Areocentric Coordinate Time**
\[
T \equiv \mathrm{TCA}
\]
as the Mars analogue of TCG. Its local metric keeps the same 1PN structure:
\[
G_{00}(T,\boldsymbol{X}) = 1 - \frac{2}{c^2}\Big(U_{Ma}(T,\boldsymbol{X}) + U_{\rm tid}(T,\boldsymbol{X})\Big) + \frac{2U_{Ma}^2}{c^4} + O(c^{-5}),
\]
\[
G_{0i}(T,\boldsymbol{X}) = -\frac{4}{c^3} W^i_{Ma}(T,\boldsymbol{X}) + O(c^{-5}),
\]
\[
G_{ij}(T,\boldsymbol{X}) = -\delta_{ij}\Big\{1+\frac{2}{c^2}\big(U_{Ma}+U_{\rm tid}\big)\Big\} + O(c^{-4}).
\]
Here \(U_{Ma}\) is the Mars self-gravity potential, \(U_{\rm tid}\) is the external tidal potential from the Sun, Phobos, and Deimos, and \(W^i_{Ma}\) is Mars’ vector potential [2606.13726].

A Mars-fixed body frame is obtained by rotation,
\[
\boldsymbol{X} = \mathcal{R}_{Ma}(T)\,\boldsymbol{X}_\text{bf},
\]
where \(\mathcal{R}_{Ma}(T)\) is built from the IAU/WGCCRE Mars orientation model. Within this hierarchy, the paper distinguishes **TCB**, **TDB**, **TCA**, a candidate Mars surface coordinate time \(T_M\), and the proper times of landed and orbiting clocks. \(T_M\) is defined from TCA by a constant-rate scaling,
\[
T_M = TCA - L_{\rm surf}^{\rm def}\,(TCA-T_{M0}),
\]
with
\[
L_{\rm surf}^{\rm def} = 1.406355\times10^{-10} = 12.15091~\mu\mathrm{s\,d^{-1}},
\]
and is explicitly described as a coordinate scale, not a specific site’s proper time [2606.13726].

## 2. Post-Newtonian foundations and equivalent formalisms

One formal underpinning is the effective field theory treatment of post-Newtonian gravity. The starting action is General Relativity with the Einstein-Hilbert term, a gauge-fixing term, and a sum of worldline actions,
\[
S_{\rm tot} = S_{\rm EH}[g_{\mu\nu}] + S_{\rm GF}[g_{\mu\nu}] + \sum_a S_{\rm pp}^{(a)}[g_{\mu\nu},y_a^\mu,\dots].
\]
At the first EFT stage, the metric is decomposed as
\[
g_{\mu\nu}=g^s_{\mu\nu}+\bar g_{\mu\nu},
\]
and short-distance structure is integrated out into Wilson coefficients on the worldline. At the second EFT stage,
\[
\bar g_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}+\tilde h_{\mu\nu},
\]
with \(H_{\mu\nu}\) the near-zone orbital modes and \(\tilde h_{\mu\nu}\) the radiation modes. For conservative dynamics one sets \(\tilde h_{\mu\nu}=0\) and integrates out \(H_{\mu\nu}\), obtaining an effective action purely in terms of worldline variables, from which the PN Lagrangian, Hamiltonian, and equations of motion follow [1811.12401].

This EFT framework uses a nonrelativistic Kaluza-Klein decomposition,
\[
ds^2 = -e^{2\phi}(dt - A_i dx^i)^2 + e^{-2\phi}\gamma_{ij}dx^i dx^j,
\]
together with Schwinger’s time gauge for the tetrad field and the canonical gauge for worldline rotational variables. The post-Newtonian bookkeeping is organized by the small parameter \(v\ll1\) in units \(c=1\), with
\[
n\,\text{PN}\equiv \mathcal{O}(v^{2n})\quad\text{beyond Newtonian},
\]
and field scalings such as \(\phi\sim v^2\), \(A_i\sim v^3\), and \(\gamma_{ij}-\delta_{ij}\sim v^2\) [1811.12401].

A complementary geometric formulation is the coordinate-independent post-1-Newtonian generalization of Newton-Cartan theory. In that formulation the basic objects are a temporal 1-form \(t_a\), a spatial contravariant metric \(h^{ab}\), a torsion-free connection \(D_a\), and stress-energy \(T^{ab}\), together with post-Newtonian corrections \(k^{ab}\) and \(p_{ab}\). The combined post-Newton-Cartan theory packages Newtonian and post-1-PN structure into the covariant equations
\[
h^{ab} t_b = 0,\qquad h^{ab} p_{bc} - k^{ab} t_b t_c = 0,
\]
\[
D_a(h^{bc}+k^{bc})=0,\qquad D_a(-t_b t_c+p_{bc})=0,
\]
plus a Trautman condition, a field equation for \(R_{ab}[D]\), and stress-energy conservation \(D_aT^{ab}=0\). The usual coordinate-dependent equations of post-Newtonian gravity are then recovered for asymptotically flat spacetimes and appropriate coordinates [1101.0588].

## 3. Construction of a Mars-centered dynamical system

The EFT/PN formalism is explicitly stated to be general: bodies need not be black holes; any gravitating bodies moving with small velocities relative to \(c\) and with \(G M/(rc^2)\ll 1\) are admissible. A Mars-centered specialization therefore treats Mars, the Sun, Phobos, Deimos, and spacecraft as worldlines \(y_a^\mu(\sigma)\), with point-particle actions of the form
\[
S_{\rm pp}^{(a)}=-m_a\int d\tau_a
\]
for nonspinning bodies, and for a spinning Mars,
\[
S_{\rm pp}^{(M)}=\int d\sigma \left[-M_M\sqrt{u_M^2}
-\frac{1}{2}S_{M,\mu\nu}\Omega_M^{\mu\nu}+L_{\rm NMC}^{(M)}\right],
\]
where \(L_{\rm NMC}^{(M)}\) may include a quadrupole operator that effectively encodes \(J_2\) of Mars [1811.12401].

The practical derivation proceeds in a barycentric PN frame and only afterward passes to local coordinates. The sequence given for a Mars-centered application is: integrate out strong-field modes around each body to obtain worldline Wilson coefficients; decompose the metric into orbital and radiation modes and set \(\tilde h_{\mu\nu}=0\) for conservative planetary dynamics; integrate out orbital modes \(H_{\mu\nu}\) to obtain an effective PN Lagrangian \(L_{\rm eff}(x_a^i,v_a^i,S_a^{ij},C_i^{(a)};t)\); then construct Mars-centered coordinates from Mars’ barycentric trajectory. Because orbital velocities in the Mars-Sun system are small, the data state that one needs only a low PN order: for Solar-System ephemerides, the standard is roughly 1PN GR, plus certain Solar multipole and frame-dragging effects, while 2PN-4PN are far below current observational requirements [1811.12401].

The explicit Mars-centered spatial coordinates are introduced by
\[
X^i = x^i - x_M^i(t),
\]
and at PN order this translation must be supplemented by velocity-dependent and gravitational potential-dependent corrections. The same data describe the Mars frame metric \(g_{\alpha\beta}^{(M)}(T,X^i)\) as obtained by a PN expansion of the global barycentric metric around the Mars worldline, using matched asymptotic expansions or standard PN coordinate transformations. This suggests that a Mars-centered framework is not merely a shifted Newtonian chart, but a local chart derived from global multi-body PN dynamics [1811.12401].

A related but distinct reformulation is the Galilean-covariant PN expansion. There the relevant local fields are a scalar potential \(\phi(x,t)\) and a vector potential \(A(x,t)\), with
\[
g = -\nabla\phi - 2\,\dot A,\qquad \Omega = \tfrac12 \operatorname{curl} A,
\]
and body-centered charts are written as galileomorphisms
\[
x' = R(t)(x - x_0(t)),\qquad t' = t+t_0.
\]
For a body-centered frame one takes \(x_0(t)\) as the position of Mars’s center and \(R(t)\) as a time-dependent rotation aligning axes with Mars-fixed axes. The framework is explicitly designed so that the field equations remain form-invariant under these transformations [2004.03513].

## 4. Equations of motion, potentials, and orbital effects

For conservative motion, integrating out orbital modes yields an effective Lagrangian
\[
S_{\rm eff}=\int dt\,L_{\rm eff}(x_a^i(t),v_a^i(t),S_a^{ij},\dots),
\]
from which Euler-Lagrange equations and, by Legendre transform, PN Hamiltonians are obtained. The public EFTofPNG package organizes this pipeline through the modules **FeynRul**, **FeynGen**, **NLoop**, and **Gauge Invariant Observables**, and its public version handles the point-mass sector to 4PN and all spin sectors up to 4PN in the conservative sector [1811.12401].

For a spacecraft in the PN metric generated by Mars and external bodies, the data give the schematic Mars-generated metric terms
\[
g_{00} = -1 + 2U_M - 2U_M^2 + \cdots,\qquad
g_{0i} = -4 V_i^{(M)} + \cdots,\qquad
g_{ij} = \delta_{ij}(1+2U_M)+\cdots,
\]
with \(U_M=GM_M/r_M\). The resulting acceleration is written as
\[
\mathbf{a}_s = -\nabla U_M + \mathcal{O}(v_s^2/c^2,U_M^2/c^2,v_Mv_s/c^2,\dots),
\]
and in Mars-centered coordinates as
\[
\ddot{\mathbf{X}}_s=\mathbf{a}_N^{(M)}(\mathbf{X}_s)+\frac{1}{c^2}\mathbf{a}_{1\rm PN}^{(M)}(\mathbf{X}_s,\dot{\mathbf{X}}_s,t)+\cdots,
\]
where \(\mathbf{a}_N^{(M)}=-GM_M\mathbf{X}_s/|\mathbf{X}_s|^3\). The \(1/c^2\) terms include \(v_s^2\nabla U_M\), \(\nabla(U_M^2)\), and frame-dragging corrections if spin is included [1811.12401].

In the two-body PPN description of orbital elements, the perturbing acceleration is decomposed into radial, transverse, and normal components \((R,S,W)\), with \(W=0\) in the model described. The Gaussian planetary equations then imply
\[
\Delta a=\Delta e=\Delta i=\Delta\Omega=0,
\]
while the longitude of periastron and mean longitude at epoch acquire both secular and periodic shifts. In the GR specialization,
\[
\Delta\tilde{\omega}_{\text{GR}}=\frac{6\pi Gm}{c^2 a(1-e^2)}\quad\text{rad/cycle},
\]
and
\[
\dot{\tilde{\omega}}_{\text{GR}}=\frac{3nGm}{c^2 a(1-e^2)}.
\]
For a Mars-centered problem this applies directly to a Mars-satellite relative orbit with \(m=m_{\text{Mars}}+m_{\text{sat}}\), or to the Sun-Mars orbit with \(m=m_\odot+m_{\text{Mars}}\) [1206.2637].

## 5. Time transformations and clock modeling around Mars

The BCRS-to-MCRS time transformation specialized to Mars is
\[
T = t - \frac{1}{c^2}\big[A(t) + v_{Ma}^i r_{Ma}^i\big] + \frac{1}{c^4}\big[B(t) + B^i(t) r_{Ma}^i + B^{ij}(t)r_{Ma}^i r_{Ma}^j + C(t,\boldsymbol{x})\big] + O(c^{-5}),
\]
with coefficients determined by Mars’ barycentric kinetic energy and the external BCRS potentials. The large position-dependent term \((\boldsymbol{v}_{Ma}\cdot\boldsymbol{r}_{Ma})/c^2\) is stated to reach \(0.91~\mu\mathrm{s}\) at a surface site, \(0.99~\mu\mathrm{s}\) at \(300\) km, \(5.48~\mu\mathrm{s}\) at areostationary radius, and \(6.30~\mu\mathrm{s}\) at Deimos distance, and is described as mandatory in any TCB↔TCA transformation [2606.13726].

At Mars’ origin, the mean TCA-TCB rate is organized through the Mars barycentric energy function
\[
\mathcal{E}_{Ma}(t)=L_A+\dot P_A(t),\qquad \langle \dot P_A\rangle=0.
\]
A simple Sun-only Keplerian estimate gives
\[
L_A^{(\odot)} \simeq \frac{3GM_\odot}{2a_{Ma}c^2} \approx 9.72\times10^{-9}=0.8396~\mathrm{ms\,d^{-1}},
\]
while the leading eccentricity modulation yields \(\sim1.21\times10^{-9}\) fractional rate, \(104.6~\mu\mathrm{s\,d^{-1}}\), and \(\sim11\,\mathrm{ms}\) amplitude over a Martian year [2606.13726].

Proper time in the MCRS satisfies
\[
\frac{d\tau}{dT} = 1 - \frac{1}{c^2}\Big\{\tfrac{1}{2}V^2 + U_{Ma}(T,\boldsymbol{X}) + U_{\rm tid}(T,\boldsymbol{X})\Big\} + O(c^{-4}),
\]
and the paper rewrites \(\tau-T_M\) as a secular rate plus zero-mean periodic terms. The retention policy is explicit:
\[
\epsilon_f = 5\times 10^{-18},\qquad \epsilon_t = 0.1~\mathrm{ps}.
\]
Within that policy, the numerical realization uses the GMM-3 Mars gravity field through degree and order 120, exact point-mass tides from the Sun, Phobos, and Deimos with origin and dipole terms removed, and bounds on omitted local \(c^{-4}\) and external-perturber terms [2606.13726].

Representative regimes show that, relative to the adopted Mars surface scale, a \(300\) km near-polar clock is slower by \(4.56\) microseconds per day, while areostationary and Deimos-distance clocks are faster by \(9.13\) and \(9.52\) microseconds per day. The leading Mars-\(J_2\) timing line is approximated by
\[
P_{J_2}(T) \simeq -\frac{3}{8}\frac{GM_{Ma} R_0^2 J_{2Ma}}{c^2 r^3 n}\sin(2nT+\varphi_{J2}),
\]
with amplitude about \(87~\mathrm{ps}\) at \(300\) km altitude and several ps near areostationary radius and Deimos distance [2606.13726].

## 6. Scope, limitations, and related interpretations

The standard Mars-centered construction is explicitly presented as a **reference-system and model-retention framework, not a final operational Mars Time Ephemeris**. A realized sub-ps system is stated to require a selected planetary ephemeris, Mars orientation and seasonal-gravity model, spacecraft orbit determination, calibrated link delays, and covariance analysis. The leading unclosed surface-realization term is time-variable low-degree gravity from seasonal \(\mathrm{CO}_2\) exchange, which must be modeled, monitored, or empirically bounded before sub-ps Mars surface-scale claims are made [2606.13726].

On the dynamics side, a further limitation is that EFTofPNG is optimized and tested for two-body problems. Mars-centered applications require Mars, the Sun, Earth, Phobos, Deimos, and spacecraft, so the formalism either superposes pairwise PN interactions at 1PN where true three-body corrections are extremely small, or extends the diagrammatic machinery to \(3+\) bodies with increased combinatorics and code-performance cost. Non-gravitational forces such as radiation pressure and drag are outside the GR EFT sector and must be added separately [1811.12401].

A common misconception is that a Mars-centered PN framework is exhausted by a Mars-fixed Newtonian potential. The constructions summarized here require barycentric-to-local transformations, velocity-dependent and gravitational-potential-dependent corrections in those transformations, and a distinction between coordinate times and proper times. Another misconception is that the subject belongs only to compact-binary theory: the EFT/PN formalism is stated to be applicable to any gravitating bodies with small velocities and \(G M/(rc^2)\ll1\), which includes planetary motions around the Sun and spacecraft dynamics around Mars [1811.12401].

Related literature also shows that the phrase can be interpreted more broadly. One line of work merges PPN and Newton-Cartan Theory into a Galilean-covariant \(1/c^2\) expansion in which each coefficient is a Galilean scalar, vector, or tensor and the local fields are \(\phi\) and \(A\) [2004.03513]. Another line derives 1PN equations of motion of mass centers in a scalar theory of gravity with a preferred frame, using a global preferred inertial frame and then constructing body-centered coordinates by subtracting the chosen body’s trajectory [2604.15397]. These formulations are not identical to the IAU-based GR Mars-centered framework, but they clarify that Mars-centered post-Newtonian modeling can be approached either as a local chart within standard GR or as part of a more general program of covariant nonrelativistic expansions and body-centered frame constructions.

Source: https://www.emergentmind.com/topics/mars-centered-post-newtonian-framework