1/c Expansion of General Relativity

• The 1/c expansion is a systematic method that expands GR’s field equations in powers of the inverse speed of light, linking Newtonian and post-Newtonian frameworks.
• It employs a 3+1 decomposition in both ADM and KS schemes, revealing a duality that simplifies technical computations and clarifies geometric structures.
• The nested 'matryoshka' methodology organizes higher-order corrections explicitly, enabling detailed analyses of strong and weak gravitational fields and matter couplings.

The $1/c$ expansion of General Relativity is a systematic approach in which the field equations and action of GR are analytically expanded in powers of the inverse speed of light, $c^{-1}$. This procedure generates a hierarchy of nonrelativistic gravitational theories, each capturing successive relativistic corrections, and provides a unified framework to relate Newtonian gravity, post-Newtonian (PN) theory, and extended nonrelativistic gravity models. By formulating the expansion within a $3+1$ decomposition of spacetime, the approach allows simultaneous treatment of both the Arnowitt–Deser–Misner (ADM) and Kol–Smolkin (KS) schemes, exposing their duality and facilitating technical computations up to high orders. This methodology yields insights into the structure of strong and weak gravitational fields, the emergence of Newton–Cartan geometry, and the coupling to nontrivial matter sectors.

1. Dual $3+1$ Decomposition and Unified Expansion Formalism

The $1/c$ expansion is most naturally performed in a $3+1$ decomposition where spacetime is sliced into spatial hypersurfaces parameterized by a time coordinate. The two principal decompositions relevant to this expansion are:

• ADM decomposition: The metric is written as $$ds^2 = -N^2 dt^2 + h_{ij}(dx^i + N^i dt)(dx^j + N^j dt)$$, with $N$ the lapse, $N^i$ the shift, and $h_{ij}$ the spatial metric.
• KS decomposition: The "dual" form $ds^2 = -M^2(dt + M_i dx^i)^2 + h_{ij} dx^i dx^j$, with $M$ and $M_i$ as lapse and shift–like variables but with the roles of tangent and cotangent spaces interchanged.

Despite their differing assignment of variables, both decompositions yield the same parent action (Einstein–Hilbert in $3+1$ form): $$S_{EH} = \int\left(\widehat{R} + K_{ij}K^{ij} - K^2\right)n_t \sqrt{h}\, d^{d+1}x,$$ where $n_t$ is the lapse (either $N$ or $M$), $K_{ij}$ is the extrinsic curvature, and $\widehat{R}$ is the Ricci scalar of $h_{ij}$. The $1/c$ expansion developed in this framework is compatible with both the ADM and KS forms, thus revealing their underlying duality (Elbistan, 5 Aug 2025).

2. The "Matryoshka" or Nested Expansion Methodology

A key technical innovation is the "matryoshka" (nested) expansion, wherein each term in the Lagrangian is recursively organized by order in $c^{-1}$: $$[\mathcal{L}]_n = [R]_n[\sqrt{-g}]_0 + [R]_{n-1}[\sqrt{-g}]_1 + \ldots + [R]_0[\sqrt{-g}]_n.$$ The expansion proceeds by:

• Analytically expanding all fields (e.g., $h_{ij}$, lapse, shift) as $X = X_0 + c^{-1} X_1 + c^{-2} X_2 + \cdots$.
• Expanding inverse metrics and determinants using canonical determinant and inverse formulas.
• Leveraging geometric identities—especially the metric compatibility condition and the properties of the spatial connection—to express higher-order contributions in terms of geometric quantities constructed from the spatial metric and its covariant derivatives.

This staged expansion allows computations to an arbitrary order (demonstrated explicitly up to $c^{-3}$), with lower-order terms persistently embedded, ensuring that each advance in order builds upon the established structure.

3. Expansion of Geometric Quantities and Lagrangian Structure

A salient step is the expansion of primary geometric quantities—spatial metric inverse, determinant, Christoffel symbols, and Ricci tensor/scalar:

• The inverse metric expansion uses relations such as $A^{ij} = -h^{ik} a_{kl} h^{lj}$ etc., for each correction $a_{ij}, b_{ij}, c_{ij},\ldots$.
• The expansion of the spatial metric determinant is given recursively, e.g.,

$$[\sqrt{-g}]_1 = \sqrt{h}\, n_{1t},\quad [\sqrt{-g}]_2 = \sqrt{h}\left(n_{2t} - \frac{1}{4}n_t a^{ij}a_{ij}\right)$$

• The Ricci tensor and scalar expansion exploits identities from the spatial metric's covariant constancy and properties of the Wheeler–DeWitt metric ($h^{ij}h^{kl} - h^{ik}h^{jl}$), leading to expressions of the form

$$[\widehat{R}]_n = h^{ij}[\widehat{R}_{ij}]_n = h_n^{ij}[\widehat{R}_{ij}]_0 + h_{n-1}^{ij}[\widehat{R}_{ij}]_1 + \ldots + h^{ij}[\widehat{R}_{ij}]_n$$

with all dependence on the nth-order fields being universal and, due to contraction with the DeWitt metric, reducing to well-controlled structures.

4. Preservation of Duality and Comparison of ADM and KS Schemes

By working with the action in its universal $3+1$ form and carrying out the expansion prior to specifying decompositions, the duality between ADM and KS is manifest in all computations. Explicit application shows that:

• The structure of the expanded Lagrangians in both ADM and KS decompositions is identical (modulo variable definitions), with both sharing the same sequence of corrections at each order.
• The technical implementation in the ADM scheme is notably simpler, since higher-order corrections to the frame fields vanish ($e_i^{(n)} = 0$ for $n \geq 1$), extrinsic curvature is symmetric, and antisymmetric complications appearing in KS are avoided.
• For the KS case, the nested expansion—applied up to $c^{-1}$—reproduces all even/odd ordering structure and confirms the interplay of the dual variables.

5. All-Order Observations and Hierarchical Structure

The expansion reveals recurrent structures:

• The nth-order Ricci scalar contraction with the DeWitt metric can be written as

$$h^{ij}[\widehat{R}_{ij}]_n = (h^{ij}h^{kl} - h^{ik}h^{jl})\widehat{\mathcal{D}}_k(\ldots),$$

with $\widehat{\mathcal{D}}_k$ the spatial covariant derivative.

• On variation with respect to the lapse, new contributions at each order enter as second derivatives of nth-order metric corrections or as combinations mixing lower-order corrections to reach order $n$.
• This property enforces a recursive structure akin to the cascading form of the equations in the KS "1/c expansion" (Elbistan et al., 2022), with each new order depending linearly on the higher-order fields and their spatial derivatives, and with source terms constructed from known lower-order quantities.

6. Practical Implications and Extensions

The methodology enables:

• Systematic computation of nonrelativistic gravitational dynamics beyond traditional post-Newtonian theory, incorporating strong-field corrections and nontrivial spatial curvature effects.
• Straightforward inclusion of matter couplings and extension to theories beyond GR (e.g., higher derivative gravity, curvature-matter couplings) by applying the nested expansion scheme to the relevant action.
• All-order control over the expansion, suggesting possible resummation schemes or efficient computation in strong-gravity astrophysical regimes such as neutron stars or binary mergers.

The preservation of duality at every order ensures that both ADM- and KS-structured calculations for gravitational systems (both in classical and quantum contexts) remain in correspondence, facilitating cross-comparisons and checks.

7. Summary Table: Expansion Structure and Duality

Decomposition	Metric Form	Technical Simplicity	Expansion Order Demonstrated
ADM	$$ds^2 = -N^2 dt^2 + h_{ij}(dx^i + N^i dt)(dx^j + N^j dt)$$	Maximal (frame corrections vanish)	$c^{-3}$
KS	$ds^2 = -M^2(dt + M_i dx^i)^2 + h_{ij} dx^i dx^j$	Additional complications (antisymmetry)	$c^{-1}$

This demonstrates that the expansion structure and duality are preserved in both schemes, with explicit calculations possible to arbitrarily high order, depending on technical tractability.

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The $1/c$ expansion in the $3+1$ formalism, as detailed in (Elbistan, 5 Aug 2025), provides a robust, covariant, and technically systematic framework to derive and analyze nonrelativistic gravity theories as natural limits of general relativity, maintaining full compatibility with both ADM and KS decompositions, and facilitating extensions to higher orders and more general field content.