Gauss–Bonnet Newton–Cartan Gravity

• Gauss–Bonnet Newton–Cartan gravity is a non-relativistic theory derived from the Newton–Cartan limit of Einstein–Gauss–Bonnet gravity, yielding a corrected Poisson equation.
• The formulation employs auxiliary Maxwell and scalar fields to enforce a central U(1) symmetry and zero-torsion constraint for isolating the Poisson sector.
• A systematic 1/c expansion cancels divergent terms and produces a closed set of equations under Galilean boosts with definitive Newton–Cartan curvature corrections.

Searching arXiv for the cited paper and closely related Newton-Cartan / higher-order gravity work. arxiv_search(query="Higher-Order Newton-Cartan Gravity Cardona Romano", max_results=10) arXiv search: "Higher-Order Newton-Cartan Gravity" Gauss–Bonnet Newton–Cartan gravity is the zero-torsion non-relativistic theory obtained from the Newton–Cartan limit of Einstein–Gauss–Bonnet gravity in $D$ spacetime dimensions. In the construction summarized by Cardona and Romano, the relativistic theory is supplemented by a Maxwell 1-form $A_\mu$ to produce the central $U(1)$ needed for Bargmann and, for isolating the Poisson equation at the level of the equations of motion, optionally by a scalar $\Phi$. The resulting non-relativistic system is characterized by a finite magnetic $1/c$ limit, a Gauss–Bonnet-corrected Poisson equation written in Newton–Cartan curvature variables, and a full set of non-relativistic equations forming a closed multiplet under Galilean boosts (Cardona et al., 7 Jul 2025).

1. Position within higher-order Newton–Cartan gravity

The construction of Gauss–Bonnet Newton–Cartan gravity appears within a broader study of the non-relativistic Newton–Cartan limit of higher-order gravity theories in arbitrary dimensions. That study treats the limit both at the level of the action and at the level of the equations of motion, and extends the action-level procedure to theories whose Lagrangian is a function of the Ricci scalar, quadratic Ricci tensor, and quadratic Riemann tensor. Two explicit equation-of-motion models are then analyzed: Einstein–Gauss–Bonnet gravity and quadratic Ricci scalar theory (Cardona et al., 7 Jul 2025).

Within that framework, the Gauss–Bonnet sector is distinguished by the statement that it is possible to obtain the Poisson equation by introducing a scalar field and imposing an appropriate constraint. The corresponding non-relativistic theory is identified as zero-torsion Gauss–Bonnet Newton–Cartan gravity. The qualifier “zero-torsion” is tied to the condition

$2\partial_{[\mu}\tau_{\nu]}=0,$

which is the non-relativistic limit of a Maxwell-field constraint and is imposed as part of the derivation (Cardona et al., 7 Jul 2025).

A common source of confusion is to treat the higher-curvature correction as a perturbation requiring an independent non-relativistic scaling. In this construction, that is not the case: the Gauss–Bonnet coupling $\alpha$ enters at the same non-trivial order $c^0$ in the expansion and does not require a separate rescaling of $\alpha$ (Cardona et al., 7 Jul 2025).

2. Relativistic starting point and auxiliary fields

The relativistic starting point is the Einstein–Gauss–Bonnet action in $D$ dimensions, supplemented by a Maxwell 1-form $A_\mu$0 and a scalar $A_\mu$1: $A_\mu$2 Here $A_\mu$3 is the Riemann tensor, $A_\mu$4, $A_\mu$5, and

$A_\mu$6

The Maxwell field is introduced to produce the central $A_\mu$7 needed for Bargmann. The last term, proportional to $A_\mu$8 and $A_\mu$9, is used in the Poisson-equation trick; alternatively, one may set $U(1)$0 if one prefers imposing a second on-shell constraint instead (Cardona et al., 7 Jul 2025).

This structure makes clear that the non-relativistic limit is not taken from a purely metric higher-curvature theory alone. Rather, the limiting procedure is organized so that the central extension associated with Bargmann symmetry is already present in the relativistic parent theory through $U(1)$1, while $U(1)$2 can be used to cancel subleading incompatibilities in the Poisson-sector combination of equations of motion. This suggests that the Newton–Cartan limit of higher-order gravity is controlled not only by the scaling of the metric sector but also by the scaling and constraint structure of auxiliary gauge and scalar sectors.

3. $U(1)$3 expansion and the finite magnetic limit

The relativistic vielbein $U(1)$4, with $U(1)$5 and $U(1)$6, is decomposed as

$U(1)$7

with inverse

$U(1)$8

and

$U(1)$9

The metric and inverse metric expand as

$\Phi$0

and

$\Phi$1

For the Maxwell field, the magnetic ansatz is

$\Phi$2

so that

$\Phi$3

Upon inserting these expansions into the Gauss–Bonnet–Maxwell action, all divergences $\Phi$4 and $\Phi$5 cancel, provided one also adds the requisite higher-derivative Maxwell terms in precise combinations, leaving a finite order-$\Phi$6 action that defines the non-relativistic Gauss–Bonnet Newton–Cartan action. The summary of the construction further states that one cancels all divergences in the action by adding higher-derivative Maxwell couplings fixed uniquely by the requirement of finiteness (Cardona et al., 7 Jul 2025).

One misconception would be to expect the Poisson equation to emerge already at leading order. In the equation-of-motion analysis, this does not occur: the expanded system contains leading $\Phi$7, subleading $\Phi$8, and only then the Poisson equation at $\Phi$9. The finite action and the emergence of the Poisson sector are therefore related but distinct features of the limit.

4. On-shell constraints and extraction of the Poisson sector

To isolate the geometric Poisson equation from the expanded equations of motion, three ingredients are imposed.

First, there is the Maxwell-field constraint

$1/c$0

which becomes the zero-torsion condition

$1/c$1

in the non-relativistic limit.

Second, there is the scalar-field constraint

$1/c$2

which allows the subleading $1/c$3 incompatibilities in the Poisson-combination to be cancelled by the variation $1/c$4.

Third, a specific linear combination of equations is constructed,

$1/c$5

with constants tuned so as to kill the $1/c$6 and $1/c$7 pieces. After these steps, the leading surviving equation is the $1/c$8 term $1/c$9, which becomes the Gauss–Bonnet-corrected Poisson equation (Cardona et al., 7 Jul 2025).

In this formulation, zero torsion is not an independent postulate added after the fact. It is enforced through the condition $2\partial_{[\mu}\tau_{\nu]}=0,$0, and the scalar $2\partial_{[\mu}\tau_{\nu]}=0,$1 is not introduced as propagating Newtonian matter but as part of the mechanism that isolates the desired Poisson-sector equation. A plausible implication is that the non-relativistic limit is controlled by a constrained multiplet of fields rather than by the metric variables alone.

5. Newton–Cartan curvature and the Gauss–Bonnet-corrected Poisson equation

The Newton–Cartan connection used in the construction is

$2\partial_{[\mu}\tau_{\nu]}=0,$2

with associated Newton–Cartan Riemann tensor

$2\partial_{[\mu}\tau_{\nu]}=0,$3

and Ricci tensor

$2\partial_{[\mu}\tau_{\nu]}=0,$4

The final Poisson equation is

$2\partial_{[\mu}\tau_{\nu]}=0,$5

Here

$2\partial_{[\mu}\tau_{\nu]}=0,$6

is the index-$2\partial_{[\mu}\tau_{\nu]}=0,$7 component of the Newton–Cartan Ricci tensor, identified in the summary as the usual Laplacian of the Newton potential, while the remaining terms are Gauss–Bonnet corrections built from purely spatial Newton–Cartan curvature. They are given by

$2\partial_{[\mu}\tau_{\nu]}=0,$8

and

$2\partial_{[\mu}\tau_{\nu]}=0,$9

A schematic form is also given: $\alpha$0 where $\alpha$1 is the usual spatial Laplacian and $\alpha$2 etc.; however, the compact form in terms of Newton–Cartan Ricci components is identified as the preferred expression (Cardona et al., 7 Jul 2025).

The corrected Poisson equation is the central output of the Gauss–Bonnet Newton–Cartan construction. It shows that the Newtonian sector is modified not by an arbitrary higher-derivative scalar operator but by specific contractions of spatial Newton–Cartan curvature inherited from the Einstein–Gauss–Bonnet parent theory. This suggests a geometrically rigid relation between the relativistic higher-curvature term and the non-relativistic Poisson sector.

6. Full field equations and Galilean boost structure

The full zero-torsion Gauss–Bonnet Newton–Cartan system includes the constraints

$\alpha$3

together with a set of non-relativistic equations of motion that split according to their order in the $\alpha$4 expansion.

The Poisson equation appears at order $\alpha$5: $\alpha$6

A second scalar equation appears at order $\alpha$7: $\alpha$8

There is a mixed tensor equation at order $\alpha$9: $c^0$0 and a traceless symmetric spatial tensor equation at order $c^0$1: $c^0$2 where $c^0$3 denotes the traceless projection. The explicit index formulas are stated to be given in section 4.3 of the paper (Cardona et al., 7 Jul 2025).

Under an infinitesimal Galilean boost with parameter $c^0$4, the system transforms as a closed, indecomposable multiplet: $c^0$5

$c^0$6

$c^0$7

Hence the set

$c^0$8

is invariant and closes under the Bargmann algebra (Cardona et al., 7 Jul 2025).

This boost structure is significant because it shows that the Gauss–Bonnet-corrected Poisson equation is not an isolated scalar relation. It is one component of a larger non-relativistic multiplet whose covariance is organized by Bargmann symmetry. In that sense, zero-torsion Gauss–Bonnet Newton–Cartan gravity is defined not only by the corrected Poisson equation

$c^0$9

but by the full closed set of scalar, mixed, and traceless spatial equations obtained from the $\alpha$0 expansion of the relativistic Einstein–Gauss–Bonnet system.