Mars-Centered Post-Newtonian Framework

• The Mars-centered post-Newtonian framework is a relativistic system where Mars is the local gravitating center, incorporating barycentric PN expansions for accurate orbit modeling.
• It employs effective field theory and coordinate transformations to derive local metrics, time scales (TCA), and potentials that govern Mars and surrounding bodies.
• The approach integrates GR, Newton-Cartan formulations, and Galilean-covariant methods, offering insights for improved planetary ephemerides and spacecraft dynamics.

Searching arXiv for the cited framework papers and closely related post-Newtonian formalisms. arXiv search query: (Levi, 2018, Saxcé, 2020, Tichy et al., 2011, Li, 2012, Turyshev, 11 Jun 2026) 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 (Turyshev, 11 Jun 2026, Levi, 2018, Tichy et al., 2011).

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 $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$ (Turyshev, 11 Jun 2026).

The Mars-centered celestial reference system is then introduced with coordinates $c$0, and Areocentric Coordinate Time

$c$1

as the Mars analogue of TCG. Its local metric keeps the same 1PN structure: $c$2

$c$3

$c$4

Here $c$5 is the Mars self-gravity potential, $c$6 is the external tidal potential from the Sun, Phobos, and Deimos, and $c$7 is Mars’ vector potential (Turyshev, 11 Jun 2026).

A Mars-fixed body frame is obtained by rotation,

$c$8

where $c$9 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 \equiv \mathrm{TCB}$0, and the proper times of landed and orbiting clocks. $t \equiv \mathrm{TCB}$1 is defined from TCA by a constant-rate scaling,

$t \equiv \mathrm{TCB}$2

with

$t \equiv \mathrm{TCB}$3

and is explicitly described as a coordinate scale, not a specific site’s proper time (Turyshev, 11 Jun 2026).

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,

$t \equiv \mathrm{TCB}$4

At the first EFT stage, the metric is decomposed as

$t \equiv \mathrm{TCB}$5

and short-distance structure is integrated out into Wilson coefficients on the worldline. At the second EFT stage,

$t \equiv \mathrm{TCB}$6

with $t \equiv \mathrm{TCB}$7 the near-zone orbital modes and $t \equiv \mathrm{TCB}$8 the radiation modes. For conservative dynamics one sets $t \equiv \mathrm{TCB}$9 and integrates out $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$0, obtaining an effective action purely in terms of worldline variables, from which the PN Lagrangian, Hamiltonian, and equations of motion follow (Levi, 2018).

This EFT framework uses a nonrelativistic Kaluza-Klein decomposition,

$$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$1

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 $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$2 in units $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$3, with

$$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$4

and field scalings such as $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$5, $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$6, and $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$7 (Levi, 2018).

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 $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$8, a spatial contravariant metric $$g_{00}(t,\boldsymbol{x}) = 1 - \frac{2w(t,\boldsymbol{x})}{c^2} + \frac{2w^2(t,\boldsymbol{x})}{c^4} + O(c^{-5}),$$9, a torsion-free connection $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$0, and stress-energy $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$1, together with post-Newtonian corrections $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$2 and $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$3. The combined post-Newton-Cartan theory packages Newtonian and post-1-PN structure into the covariant equations

$$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$4

$$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$5

plus a Trautman condition, a field equation for $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$6, and stress-energy conservation $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$7. The usual coordinate-dependent equations of post-Newtonian gravity are then recovered for asymptotically flat spacetimes and appropriate coordinates (Tichy et al., 2011).

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 $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$8 and with $$g_{0i}(t,\boldsymbol{x}) = -\frac{4w^i(t,\boldsymbol{x})}{c^3} + O(c^{-5}),$$9 are admissible. A Mars-centered specialization therefore treats Mars, the Sun, Phobos, Deimos, and spacecraft as worldlines $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$0, with point-particle actions of the form

$$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$1

for nonspinning bodies, and for a spinning Mars,

$$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$2

where $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$3 may include a quadrupole operator that effectively encodes $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$4 of Mars (Levi, 2018).

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 $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$5 for conservative planetary dynamics; integrate out orbital modes $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$6 to obtain an effective PN Lagrangian $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$7; 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 (Levi, 2018).

The explicit Mars-centered spatial coordinates are introduced by

$$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$8

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 $$g_{ij}(t,\boldsymbol{x}) = -\delta_{ij}\left(1+\frac{2w(t,\boldsymbol{x})}{c^2}\right)+O(c^{-4}),$$9 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 (Levi, 2018).

A related but distinct reformulation is the Galilean-covariant PN expansion. There the relevant local fields are a scalar potential $\boldsymbol{x}_{Ma}(t)$0 and a vector potential $\boldsymbol{x}_{Ma}(t)$1, with

$\boldsymbol{x}_{Ma}(t)$2

and body-centered charts are written as galileomorphisms

$\boldsymbol{x}_{Ma}(t)$3

For a body-centered frame one takes $\boldsymbol{x}_{Ma}(t)$4 as the position of Mars’s center and $\boldsymbol{x}_{Ma}(t)$5 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 (Saxcé, 2020).

4. Equations of motion, potentials, and orbital effects

For conservative motion, integrating out orbital modes yields an effective Lagrangian

$\boldsymbol{x}_{Ma}(t)$6

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 (Levi, 2018).

For a spacecraft in the PN metric generated by Mars and external bodies, the data give the schematic Mars-generated metric terms

$\boldsymbol{x}_{Ma}(t)$7

with $\boldsymbol{x}_{Ma}(t)$8. The resulting acceleration is written as

$\boldsymbol{x}_{Ma}(t)$9

and in Mars-centered coordinates as

$\boldsymbol{v}_{Ma}$0

where $\boldsymbol{v}_{Ma}$1. The $\boldsymbol{v}_{Ma}$2 terms include $\boldsymbol{v}_{Ma}$3, $\boldsymbol{v}_{Ma}$4, and frame-dragging corrections if spin is included (Levi, 2018).

In the two-body PPN description of orbital elements, the perturbing acceleration is decomposed into radial, transverse, and normal components $\boldsymbol{v}_{Ma}$5, with $\boldsymbol{v}_{Ma}$6 in the model described. The Gaussian planetary equations then imply

$\boldsymbol{v}_{Ma}$7

while the longitude of periastron and mean longitude at epoch acquire both secular and periodic shifts. In the GR specialization,

$\boldsymbol{v}_{Ma}$8

and

$\boldsymbol{v}_{Ma}$9

For a Mars-centered problem this applies directly to a Mars-satellite relative orbit with $\boldsymbol{a}_{Ma}$0, or to the Sun-Mars orbit with $\boldsymbol{a}_{Ma}$1 (Li, 2012).

5. Time transformations and clock modeling around Mars

The BCRS-to-MCRS time transformation specialized to Mars is

$\boldsymbol{a}_{Ma}$2

with coefficients determined by Mars’ barycentric kinetic energy and the external BCRS potentials. The large position-dependent term $\boldsymbol{a}_{Ma}$3 is stated to reach $\boldsymbol{a}_{Ma}$4 at a surface site, $\boldsymbol{a}_{Ma}$5 at $\boldsymbol{a}_{Ma}$6 km, $\boldsymbol{a}_{Ma}$7 at areostationary radius, and $\boldsymbol{a}_{Ma}$8 at Deimos distance, and is described as mandatory in any TCB↔TCA transformation (Turyshev, 11 Jun 2026).

At Mars’ origin, the mean TCA-TCB rate is organized through the Mars barycentric energy function

$\boldsymbol{a}_{Ma}$9

A simple Sun-only Keplerian estimate gives

$$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$0

while the leading eccentricity modulation yields $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$1 fractional rate, $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$2, and $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$3 amplitude over a Martian year (Turyshev, 11 Jun 2026).

Proper time in the MCRS satisfies

$$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$4

and the paper rewrites $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$5 as a secular rate plus zero-mean periodic terms. The retention policy is explicit: $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$6 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 $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$7 and external-perturber terms (Turyshev, 11 Jun 2026).

Representative regimes show that, relative to the adopted Mars surface scale, a $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$8 km near-polar clock is slower by $$\boldsymbol{r}_{Ma}=\boldsymbol{x}-\boldsymbol{x}_{Ma}(t)$$9 microseconds per day, while areostationary and Deimos-distance clocks are faster by $c$00 and $c$01 microseconds per day. The leading Mars-$c$02 timing line is approximated by

$c$03

with amplitude about $c$04 at $c$05 km altitude and several ps near areostationary radius and Deimos distance (Turyshev, 11 Jun 2026).

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 $c$06 exchange, which must be modeled, monitored, or empirically bounded before sub-ps Mars surface-scale claims are made (Turyshev, 11 Jun 2026).

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 $c$07 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 (Levi, 2018).

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 $c$08, which includes planetary motions around the Sun and spacecraft dynamics around Mars (Levi, 2018).

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 $c$09 expansion in which each coefficient is a Galilean scalar, vector, or tensor and the local fields are $c$10 and $c$11 (Saxcé, 2020). 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 (Arminjon, 16 Apr 2026). 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.