Newton-Cartan

Confirmed & Polished Summary of Type-I Newton–Cartan (NC)¶

Common to both NC and GR

• A spacetime is described by a metric structure (a single non-degenerate metric in GR; a pair of degenerate metrics in NC) together with a connection.

• In GR the metric uniquely determines the (Levi-Civita) connection.

• In NC the degenerate metrics + compatibility conditions leave residual freedom: the only free connection components are
  

  $$\Gamma^i_{00}\qquad\text{and}\qquad\Gamma^i_{0j}=-\Gamma^j_{0i}.$$

  (1)

  To fully specify the geometry one must therefore give both the (flat) metrics and the free part of the connection.

Visualization of NC spacetime
Absolute time slices are perfectly flat Euclidean 3-spaces (flat sheets of paper stacked along absolute time).
The metrics $\tau_\mu$ and $h^{\mu\nu}$ tell you how to measure time between sheets and distances/angles on each sheet.
They are not enough to decide how free particles move from one sheet to the next — that is the job of the independent connection.
When gravity is present the connection bends the free-fall trajectories, so the spacetime is curved (non-zero Riemann tensor of the connection) even though every individual spatial sheet remains flat and time is absolute.

The free connection components

• $\Gamma^i_{00}$ encodes Newtonian gravity. It is determined by the field equation
  

  $$R_{\mu\nu}=4\pi G\rho\,\tau_\mu\tau_\nu.$$

  (2)

• $\Gamma^i_{0j}$ encodes a possible rotational/Coriolis field.
  Classical Newtonian physics assumes an absolute non-rotating frame exists, so we impose the Trautman condition
  

  $$R^{[ab]}{}_{cd}=0$$

  (3)

  (coordinate-invariant). This guarantees the existence of coordinates in which $\Gamma^i_{0j}=0$ everywhere.

Rotating Earth example
In Earth-fixed coordinates both NC and GR have non-zero $\Gamma^i_{0j}$.
Most of this is a pure coordinate effect of using a rotating frame.

• In NC the Trautman condition guarantees we can always transform to a global non-rotating frame where $\Gamma^i_{0j}=0$.

• In GR the Earth (or any rotating mass) also produces a small intrinsic frame-dragging (Lense-Thirring / gravitomagnetic) field. Even after going to the best non-rotating coordinates, a residual $\Gamma^i_{0j}$ (or $g_{0i}$) remains that cannot be gauged away.
  We deliberately exclude this effect in NC by imposing Trautman, so that the theory stays 100 % equivalent to classical Newtonian gravity.

Axioms of classical Type-I NC

• A1 Absolute structures: global absolute time + Euclidean geometry on each time slice
  $\;\Rightarrow\;$ pair of degenerate flat metrics $(\tau_\mu,h^{\mu\nu})$.

• A2 Existence of a global non-rotating frame
  $\;\Rightarrow\;$ Trautman condition $R^{[ab]}{}_{cd}=0$.

• A3 Field equation
  

  $$R_{\mu\nu}=4\pi G\rho\,\tau_\mu\tau_\nu.$$

  (4)

In the non-rotating frame given by A2 the only free connection components left are $\Gamma^i_{00}$, which are completely determined by A3.

Crucial point (difference from GR)
The field equation A3 determines only the connection.
It does not determine or curve the metrics.
The metrics coming from A1 remain absolutely flat for all time; they are fixed background structures.
Gravity curves only the independent connection that is compatible with those flat metrics.

This is precisely how classical Newtonian gravity is geometrised in a fully covariant, coordinate-independent way while remaining 100 % equivalent to the original theory.

Newton-Cartan using a regulator metric¶

The regulator metric can be constructed entirely within Type-I Newton–Cartan theory. It is an auxiliary computational device, not a physical metric and not a solution of Einstein’s equation. The logical order is:

1. specify the absolute Newton–Cartan structures;

2. use the Newton–Cartan field equation to solve for the connection;

3. package that already-known connection into a non-degenerate metric whose Levi-Civita connection is easy to compute.

1. Solve the Newton-Cartan field equation¶

The absolute structures $\tau_\mu$ and $h^{\mu\nu}$ are fixed by A1; they are not determined by the field equation. Choose adapted Cartesian coordinates for these flat structures,

and use the global non-rotating frame guaranteed by the Trautman condition. Metric compatibility and Trautman then reduce the undetermined part of the connection to

For this connection the only non-zero component of the Ricci tensor is

Consequently, the Newton–Cartan field equation

is precisely Poisson’s equation,

Time acts only as a parameter in this elliptic equation. For an isolated source, imposing $\Phi\to0$ at spatial infinity gives

More general boundary conditions determine the corresponding harmonic contribution. Thus the source and boundary data determine $\Phi$, and hence the free connection component $\Gamma^i_{00}$. They do not determine $\tau_\mu$ or $h^{\mu\nu}$; those were supplied as absolute background structures.

2. Assemble the regulator metric¶

After solving for $\Phi$, choose the covariant spatial representative $h_{\mu\nu}$ associated with the adapted frame:

Now define, for a formal parameter $c$,

Equivalently,

This formula is not guessed from GR. Its $-2\Phi\,\tau_\mu\tau_\nu$ term is chosen so that the Levi-Civita formula gives the Newton–Cartan component $\Gamma^i_{00}=\partial^i\Phi$ with the desired sign. The regulator metric is therefore assembled from:

• the fixed absolute data $\tau_\mu$ and $h^{\mu\nu}$;

• the potential $\Phi$ obtained from the Newton–Cartan field equation;

• the arbitrary formal parameter $c$.

In this qualified sense, the Newton–Cartan field equation determines the gravitational part of the regulator metric. It does not determine the complete metric by itself, because its absolute temporal and spatial parts remain prescribed background data.

3. Verify that it reproduces the Newton-Cartan connection¶

The inverse regulator metric in adapted coordinates is

Its Levi-Civita connection has

exactly, for every value of $c$. The additional components are

and therefore vanish as formal powers of $c^{-2}$. Thus

The limit only removes the extra components introduced by making the degenerate Newton–Cartan structures artificially non-degenerate. It is not a non-relativistic limit of a GR solution.

4. Precise status of the construction¶

The finite-$c$ metric has no physical status in this use of the device. One does not impose Einstein’s equation on it, interpret its light cones, or regard its finite-$c$ curvature as gravitational physics. Only the limiting connection is retained.

The regulator is also not unique. Terms of order $c^{-2}$ may be added without changing the limiting Newton–Cartan connection. Therefore the field equation plus the absolute structures determines a convenient regulator representative, not a unique physical metric.

Thus the construction remains firmly inside Type-I Newton–Cartan theory: the field equation first determines the gravitational connection, and the regulator metric is built afterwards as a convenient way to reproduce that connection using the Levi-Civita formula.

A test planet orbiting a point-mass Sun¶

We now apply the same pipeline in Newton–Cartan theory and GR:

1. specify the source, boundary data, and any independent gravitational data;

2. identify the symmetry class of the desired solution;

3. choose coordinates adapted to those symmetries, fixing only coordinate freedom;

4. solve the reduced field equations for the remaining unknown functions;

5. compute the connection and solve the planet’s geodesic equation.

In both theories the final equation of motion is

The Sun is an isolated, time-independent, non-rotating, spherically symmetric body of total mass $M$. We idealize its exterior by taking its radius to zero. The planet is a point mass $m$ in the limit $m/M\to0$, so it does not modify the Sun’s field. This is therefore the test-mass limit, not the full two-body problem. Spherical symmetry supplies a distinguished center and lets us choose the orbital plane to be $\theta=\pi/2$.

To avoid confusing two different roles, $C$ will denote the formal Newton–Cartan regulator, while $c$ denotes the physical speed of light in GR.

What is assumed, determined, and chosen?¶

A coordinate system is never additional physical data. Covariant field equations determine a geometry only up to coordinate transformations, so an explicit calculation must fix that freedom. A symmetry-adapted ansatz performs two logically distinct operations:

1. requiring the solution to possess certain symmetries restricts the physical solution class;

2. choosing coordinates adapted to those symmetries fixes gauge within that class.

The first operation is not, in general, a consequence of the field equation alone. If the complete source, boundary, and initial data are invariant under a group and the corresponding solution is unique, then the solution inherits that symmetry. This argument applies directly to the Newtonian Poisson problem below. For the well-posed GR initial-value problem, complete initial data determine the spacetime geometry uniquely only up to diffeomorphism. A mass and asymptotic flatness do not constitute complete initial data and do not exclude independent gravitational radiation. One must also provide suitable initial data or restrict attention to the spherically symmetric sector. Birkhoff’s theorem then says that every spherically symmetric vacuum exterior is locally Schwarzschild, and hence is automatically static.

Thus neither calculation obtains spherical symmetry merely by writing spherical coordinates. Spherical symmetry characterizes the physical problem; the coordinates are chosen afterwards to represent that symmetry economically.

1. Newton-Cartan theory¶

Absolute structure, source, and coordinate gauge¶

Let $\delta_h$ denote the delta distribution with respect to the Euclidean spatial volume element. The point-mass density is

Newton–Cartan theory also supplies the absolute clock and flat Euclidean spatial geometry as fixed background structures. Choose time-independent spherical coordinates adapted to them:

In these coordinates,

This is only a coordinate choice on the prescribed flat spatial geometry. It does not assume that gravity curves space in a particular way. The source, the condition $\Phi\to0$ at spatial infinity, and the flat background are rotationally invariant. Uniqueness of the Poisson problem therefore implies that $\Phi$ is rotationally invariant, so $\Phi=\Phi(r)$.

Reduced field equation¶

In the global non-rotating frame guaranteed by the Trautman condition, the undetermined gravitational connection is

where $D_i$ is the covariant derivative of the flat spatial metric. The Newton–Cartan field equation becomes the coordinate-invariant Poisson equation

For a radial scalar in the spherical coordinates above,

Hence $\Phi=A+B/r$. The boundary condition at infinity sets $A=0$, while integrating the field equation over a small ball containing the source gives

Therefore

The source normalization and boundary condition have now fixed the solution completely. In these coordinates its radial gravitational component is

Metric structure and connection¶

Package the result into the auxiliary metric

on the orbital plane. Its Levi-Civita connection has the components that survive as $C\to\infty$,

The first component is gravitational; the last two are the ordinary flat-space polar-coordinate connection.

Geodesic and orbit¶

In the regulator limit, the time geodesic equation gives $\mathrm{d}^2t/\mathrm{d}\lambda^2=0$, so we may use absolute time $t$ as the affine parameter. The spatial geodesic equations are

where dots denote derivatives with respect to $t$. The angular equation integrates to the conserved specific angular momentum

Set $u(\phi)=1/r$. Eliminating $t$ from the radial equation gives the Binet equation

Its bound-orbit solution is

or

The orbit is a closed ellipse with the Sun at one focus. Its period obeys

2. General relativity¶

Source, symmetry, and coordinate gauge¶

The GR field equation is

Outside the Sun, $T_{\mu\nu}=0$. Unlike the Newton–Cartan Poisson equation, Einstein’s equation has independent gravitational degrees of freedom, so the mass alone does not select one spacetime. In Newton–Cartan theory, once $\rho$ and the spatial boundary condition are specified, Poisson’s equation uniquely determines $\Phi$; there is no source-free gravitational radiation carrying additional information. In GR, by contrast, the same matter source and total mass can coexist with different incoming or outgoing gravitational-wave configurations. These configurations solve the vacuum Einstein equation outside the source but describe different spacetimes. One must therefore specify gravitational initial data in addition to the matter source, or restrict the solution class so that this freedom is absent. We choose the latter option by requiring a spherically symmetric exterior. Spherical symmetry excludes gravitational waves, and Birkhoff’s theorem then implies that the vacuum exterior is static and locally Schwarzschild.

Use coordinates adapted to the resulting time-translation and rotation symmetries. Choose $r$ to be the areal radius, so every symmetry sphere has area $4\pi r^2$, and normalize $t$ to the proper time of stationary observers at infinity. The most general metric after these gauge choices is

The functions $A$ and $B$ have not been guessed. They represent all metric freedom left after imposing spherical symmetry, staticity, and the stated coordinate gauge.

Reduced field equation¶

For $r>0$, the independent vacuum Einstein equations reduce to

Their general solution is

Asymptotic flatness with the chosen normalization of $t$ gives $K=1$. Matching to a spherical interior, or equivalently identifying the Arnowitt–Deser–Misner (ADM) mass measured at infinity, gives

Metric structure and connection¶

The resulting physical spacetime metric is the Schwarzschild metric. On the orbital plane,

Here this is the physical spacetime metric, not a regulator. The point-mass language means that we retain this exterior solution down to $r=0$; it is not necessary to insert a three-dimensional delta function directly into the nonlinear vacuum calculation.

The relevant Levi-Civita components are

Geodesic and orbit¶

Insert this connection into the same affine geodesic equation used above. Let $\tau_p$ be the planet’s proper time, and let dots in this subsection denote derivatives with respect to $\tau_p$. The Killing symmetries give two conserved quantities per unit planet mass:

The timelike normalization

then gives the radial first integral

Again set $u(\phi)=1/r$ and use $\dot\phi=\ell/r^2$. Differentiating the radial first integral produces the exact Schwarzschild orbit equation

The first two terms are exactly the Newton–Cartan Binet equation. The additional term,

comes from the Schwarzschild metric and prevents a generic bound orbit from closing. To first order in $GM/(c^2p)$, the perihelion advance per radial period is

Thus the relativistic orbit is approximately

3. The common pipeline¶

The machinery after the geometry is obtained is the same: form the connection and solve the geodesic equation. The theories differ both in what the field equation determines and in what data are needed for uniqueness. Newton–Cartan theory keeps its degenerate temporal and spatial metrics fixed; its elliptic field equation, source, and spatial boundary condition determine the remaining connection. GR solves for a non-degenerate spacetime metric and normally requires gravitational initial data in addition to matter data. In the spherically symmetric vacuum sector, Birkhoff’s theorem collapses that freedom, and the mass and asymptotic normalization determine the Schwarzschild geometry up to coordinates.

A test particle viewed from a rotating disk¶

We now consider a different example: a source-free test particle described entirely in coordinates attached to a rotating disk. The disk is only a reference system. It has no mass, produces no gravitational field, and does not represent a rotating material source. Consequently, the GR solution is flat spacetime written in a rotating gauge, not the Kerr metric.

This distinction is essential. Rotation of the coordinates changes the metric components and connection coefficients, producing fictitious forces in the geodesic equation, but it does not generate curvature. We will not write an equation of motion in a non-rotating frame and transform it. Instead we specify the rotating geometry directly, verify the tensor field equations in these coordinates, calculate its connection, and write its geodesics.

1. Specify the disk-fixed coordinates intrinsically¶

Let

be coordinates fixed to the disk: a point painted on the disk has constant $(x,y,z)$. The disk rotates about the $z$ axis with angular velocity

Define the rotational shift vector

This specifies the rotating observer congruence directly. In an ADM description, choose flat spatial slices, lapse $k$, and this shift. Equivalently, introduce the coframe

For a constant $\omega$, the congruence describes a uniformly rotating disk. We temporarily allow $\omega(t)$ so that the Euler term is also visible; setting $\dot\omega=0$ gives the rigid, uniformly rotating example.

For either the formal Newton–Cartan regulator speed $C$ or the physical GR speed $c$, the corresponding line element has the common form

or explicitly,

No other coordinate system is needed to define this metric. The shift is part of the coordinate gauge: it says that the coordinate worldlines are the rotating observers.

2. Calculate the connection directly¶

The Levi-Civita formula applied to the metric above gives the following non-zero coefficients:

All coefficients with an upper $t$ index vanish. In vector notation, the spatial coefficients are

These expressions are independent of $k$. Direct substitution into the Riemann tensor gives

Thus the connection is non-zero but flat. Its coefficients describe the acceleration and rotation of the coordinate grid, not a gravitational field.

3. Newton-Cartan theory¶

There is no mass source:

The fixed Newton–Cartan structures in the disk coordinates are

The connection calculated above is compatible with these structures. Because its Riemann tensor vanishes, it satisfies both the vacuum field equation and the Trautman condition directly in the rotating coordinates:

The field equation does not determine $\omega$. That function specifies the chosen rotating observer congruence and is therefore coordinate gauge, not source data.

When the coordinate form of Poisson’s equation fails¶

Newton–Cartan theory has a physical absolute time, represented by the one-form $\tau$. Nevertheless, the coordinate called “time” need not coincide with it. A useful visualization is a continuous version of time zones. Let $T$ denote absolute time and assign clocks the displayed coordinate time

Clocks at different positions have different offsets but run at the same rate. Two events with the same $T$ are physically simultaneous, yet their displayed values differ by

Conversely, a collection of events with the same displayed time $q^0$ contains different absolute times,

This resembles describing Earth “at 12:00 local time everywhere”: the description combines events occurring at different UTC times. Geometrically, the hypersurfaces $q^0=\text{constant}$ are tilted relative to the absolute simultaneity hypersurfaces $T=\text{constant}$.

Ordinary Newtonian calculations almost always use adapted coordinates, in which the coordinate-time slices are the absolute-time slices. Allowing non-adapted coordinates does not remove or relativize absolute time: $\tau$ still identifies it. The purpose is instead passive coordinate covariance. There are four reasons to require the Newton–Cartan equations to work in such coordinates:

1. Physical laws must not depend on clock labels. Changing position-dependent clock offsets or other bookkeeping conventions must not change the gravitational field.

2. The physical time structure must be separated from notation. Absolute time is the geometric one-form $\tau$, not whichever coordinate happens to be called $x^0$.

3. Coordinate artifacts must not become apparent sources. Mixed time-space derivatives introduced by a synchronization convention must not be mistaken for additional mass or curvature.

4. Arbitrary charts must be usable directly. Overlapping chart patches or computational grids need not first be converted into one global adapted coordinate system.

The four-dimensional field equation accomplishes this by containing the absolute clock as a tensor field:

In adapted coordinates,

so the only sourced component is $R_{00}=4\pi G\rho$. After a general coordinate transformation, $\tau_\mu$ may have several non-zero components. The right-hand side then transforms with it and identifies which combination of coordinate directions is temporal. Its kernel,

identifies the physical spatial directions independently of the coordinate labels. The Ricci tensor on the left transforms in exactly the same way, so the equation continues to match curvature to mass in every chart.

This is the precise sense in which the tensor equation carries the absolute clock explicitly: one never has to infer physical time from the name or position of a coordinate. Non-adapted coordinates add no new physics and are usually less convenient, but they test whether the formulation describes Newton–Cartan geometry rather than one privileged coordinate system.

The rotating coordinates above remain adapted to absolute time because $\tau=\mathrm dt$. Consequently, the physical values

still satisfy $D_iD^i\Phi=4\pi G\rho$. To see a genuine limitation of that three-dimensional notation, consider coordinates that are not adapted to the absolute-time foliation.

Begin in adapted Cartesian coordinates with

On every absolute-time slice,

Now introduce coordinates that mix time and space:

where $\kappa$ has units of inverse velocity. Since $\Phi$ and $\rho$ are scalars,

If one incorrectly treats $t'$ as absolute time and applies the ordinary spatial Laplacian while holding $t'$ fixed, the result is

which is generally not equal to $4\pi G\rho'=6a$. Here $a$, $a'$, and $a''$ are evaluated at $t'-\kappa x'$.

The failure occurs because the hypersurfaces $t'=\text{constant}$ are not the absolute-time slices:

Thus the three coordinate directions labelled by $i$ are not tangent to $\ker\tau$, and $\delta^{ij}\partial'_i\partial'_j$ at fixed $t'$ is not the intrinsic spatial operator $D_iD^i$. The fundamental equation

continues to hold without modification. Therefore the formula $D_iD^i\Phi=4\pi G\rho$ is coordinate independent only when $D_i$ genuinely denotes differentiation within the absolute spatial slices; it is not obtained in arbitrary spacetime coordinates by simply calling the three non-time coordinate indices “spatial.”

Example with two overlapping non-adapted charts¶

The same issue becomes unavoidable when a solution is supplied on an atlas whose transition functions mix temporal and spatial coordinate labels. Cover a region by two charts $U$ and $V$. On $U$, use

On $V$, use

On the overlap $U\cap V$, the transition map includes

so the coordinate called “time” in one chart depends on both temporal and spatial coordinates in the other. The absolute clock has the consistent representations

Return to the local solution used above,

In chart $U$, let

If the three indices $1,2,3$ are incorrectly treated as spatial directions at fixed $q_U^0$, their ordinary Laplacian gives

In chart $V$, the same incorrect prescription gives

On the overlap these expressions generally differ:

and neither is generally equal to

Thus the naïve three-coordinate Poisson equation does not even patch consistently between $U$ and $V$. It is not sufficient to describe the solution in this atlas.

The tensor equation does patch consistently. In chart $U$ its source side contains

and hence, for example,

In chart $V$,

so the corresponding non-zero mixed spatial components occur in the 02 and 22 positions instead. The transition map transforms these two sets of components into one another exactly because both are representations of

Therefore, if one chooses to work directly in the supplied charts without reconstructing an adapted time coordinate, the four-dimensional tensor equation is necessary. It carries both the absolute clock and the curvature information required to glue the local descriptions together.

Regulator metric and equations of motion¶

To obtain the connection using the regulator construction, take

Its relevant Christoffel symbols are independent of $C$, while all unwanted regulator components vanish identically. The formal limit therefore gives exactly the flat Newton–Cartan connection above:

Since $\Gamma^t_{\mu\nu}=0$, the time component of the affine geodesic equation is

We may therefore use absolute time $t$ as the affine parameter. The spatial equations are

The three terms are, respectively, the Coriolis, centrifugal, and Euler accelerations. In components,

For a uniformly rotating disk, $\dot\omega=0$, so only the Coriolis and centrifugal terms remain. There is no translational fictitious force because the disk’s origin was chosen not to accelerate.

4. General relativity¶

There is likewise no stress-energy source:

We also specify that there is no source-free gravitational radiation: the initial geometry is flat. The exact physical metric in the disk-fixed coordinates is

Because its full Riemann tensor vanishes, it solves the vacuum Einstein equation directly:

The rotating observers, whose spatial coordinates are constant, are timelike only where

The surface $x^2+y^2=c^2/\omega^2$ is the light cylinder of the rotating congruence. This restriction has no Newton–Cartan analogue.

The finite-$c$ metric has exactly the same connection coefficients listed above. Since $\Gamma^t_{\mu\nu}=0$, $t$ is again an affine parameter along every geodesic, up to a constant rescaling. Consequently, the exact GR coordinate equations for a freely falling test particle are

This equality with the Newton–Cartan coordinate equations is exact, not a low-velocity approximation. The relativistic difference lies in the metric’s causal and clock structure. Along a timelike trajectory,

so a physical test particle must satisfy

5. What the field equations did and did not determine¶

The field equations determine or constrain the physical geometry; they do not choose the observer. Here the physical solution is flat in both theories, while the disk-fixed lapse and shift specify a rotating coordinate gauge. The fictitious forces arise when the flat connection is expressed in that gauge. They are derived intrinsically from the connection in the disk coordinates, without transforming an equation of motion from any other frame.

A test particle around a rotating spherical Earth¶

We now combine the previous two examples. The source is an isolated spherical body of radius $R$, mass $M$, and angular momentum

where $\boldsymbol{\Omega}$ is constant and defines the rotation axis. We keep $I$ general; for a uniform solid sphere,

We consider only the exterior region $r>R$. In Newton–Cartan theory the result is exact. In GR a rotating material body is not exactly spherical, so there is no exact solution characterized only by $M$ and $\mathbf J$. We use the systematic Earth approximation:

• weak gravitational field, $GM/(c^2r)\ll1$;

• slow source rotation, retaining terms linear in $\mathbf J$;

• first post-Newtonian accuracy in the mass field;

• no incoming gravitational radiation;

• no rotational quadrupole or higher multipole moments.

The neglected deformation begins at order $\Omega^2$, whereas frame dragging is linear in $\mathbf J$. This approximation therefore isolates the leading intrinsically relativistic effect of rotation.

We derive the motion in two coordinate systems:

1. asymptotically non-rotating coordinates, whose axes are fixed relative to distant stars;

2. Earth-fixed coordinates, in which points painted on the rotating surface have constant spatial coordinates.

The word “global” below refers to the first choice. In GR it means asymptotically non-rotating, not a preferred coordinate system built into the theory.

Define

1. Newton-Cartan field in global coordinates¶

Let the spherical density be $\rho(r)$, with

Rigid rotation changes the mass current but not the spherical mass density. The classical Newton–Cartan field equation is sourced only by $\rho$:

In asymptotically non-rotating spherical coordinates it reduces to

For $r>R$, the source vanishes. Asymptotic flatness and the total mass fix

In global Cartesian coordinates the non-zero gravitational connection is

Equivalently, it is obtained from the regulator metric

by taking $C\to\infty$. Since $\Gamma^t_{\mu\nu}=0$, absolute time is an affine parameter. The geodesic equation gives

There is no term involving $\mathbf J$. In classical Newtonian gravity, a spherical body’s rotation does not alter its exterior gravitational field.

2. Newton-Cartan field in Earth-fixed coordinates¶

Now use coordinates fixed to the Earth and specify the rotational shift directly:

Because $\Phi$ is spherical and time independent, it has the same function of the Earth-fixed coordinates. The appropriate regulator metric is

This metric is specified directly in the Earth-fixed chart. Its $C\to\infty$ connection has

The rotation-dependent part of this connection is flat and source independent. The curvature still satisfies

the fictitious terms do not act as additional mass density. Writing

for the Earth-fixed coordinate velocity, the geodesic equation gives

The last two terms are the Coriolis and centrifugal accelerations. No Euler term appears because $\boldsymbol{\Omega}$ is constant, and no translational term appears because the Earth’s center is the coordinate origin.

3. GR field in asymptotically non-rotating coordinates¶

The GR source contains both energy density and mass current. To leading order,

In Lorenz gauge, the linearized Einstein equation is

The $T_{tt}$ part produces the monopole potential $U$, while the mass-current part $T_{ti}$ produces the gravitomagnetic field determined by $\mathbf J$. Solving with asymptotic flatness and no incoming radiation, then including the nonlinear monopole term required at first post-Newtonian order, gives the exterior metric in asymptotically non-rotating harmonic Cartesian coordinates:

Terms beyond first post-Newtonian order in the monopole, terms beyond first order in $\mathbf J$, mixed higher-order terms proportional to $GMJ/c^4$, and source multipoles beyond $M$ and $\mathbf J$ are omitted.

Let

be the velocity in these coordinates. Substituting the metric directly into the geodesic equation and consistently expanding gives

where the first post-Newtonian Schwarzschild correction is

and the leading Lense–Thirring acceleration is

Equivalently,

Unlike the Newtonian exterior field, the global GR connection therefore depends on the source’s angular momentum. This is intrinsic frame dragging: it remains present in asymptotically non-rotating coordinates and cannot be removed globally by choosing axes fixed to distant stars.

4. GR field in Earth-fixed coordinates¶

Choose the $z$ axis along both $\boldsymbol{\Omega}$ and $\mathbf J$, and define the Earth-fixed shift

The alignment $\mathbf J\parallel\boldsymbol{\Omega}$ makes the spin components time independent in the Earth-fixed chart. Without this alignment, the gravitomagnetic coefficients would depend explicitly on time.

It is convenient to introduce the disk-fixed spatial coframe

The same physical exterior geometry, written directly in the Earth-fixed chart to the stated approximation, is

This is not obtained by transforming an already-derived equation of motion. It is the weak-field solution expressed using the lapse, spatial metric, and shift of the Earth-fixed coordinate congruence. When $M=\mathbf J=0$, it reduces exactly to the source-free rotating-disk metric. When $\boldsymbol{\Omega}=0$, it reduces to the asymptotically non-rotating metric above.

For a test particle define

Here $\mathbf v$ is the Earth-fixed coordinate velocity, while $\mathbf u$ is the velocity entering the local asymptotically non-rotating gravitational terms. A direct calculation of the Christoffel symbols of the Earth-fixed metric and substitution into the non-affinely parametrized geodesic equation with parameter $t$ gives