The $1/c$ expansion of general relativity in a $3+1$ formulation, revisited

This paper presents a novel $1/c$ expansion method for general relativity that is compatible with both ADM and Kol-Smolkin decompositions, deriving explicit results up to $c^{-3}$ order in the ADM framework and $c^{-1}$ order in the Kol-Smolkin framework to demonstrate their duality and provide all-order insights.

The Gist

Imagine you are trying to understand a massive, complex machine—like a super-advanced car engine—but you only have a wrench that works in slow motion. You want to see how the engine behaves when it's idling or moving slowly, ignoring the crazy high-speed vibrations that only happen when it's revving at maximum power.

In physics, General Relativity (Einstein's theory of gravity) is that super-complex engine. It describes how space and time bend around massive objects like stars and black holes. However, the math is incredibly difficult because it's "non-linear" (everything affects everything else in a messy way).

To make sense of it, physicists use approximations. One popular way is to look at what happens when the speed of light ($c$) is treated as "infinite." This is the $1/c$ expansion. It's like saying, "Let's pretend light travels instantly, so we can ignore the weird relativistic effects and just look at the 'everyday' gravity we see in our solar system."

This paper, by Mahmut Elbistan, is about finding a better, more universal way to do this math. Here is the breakdown using simple analogies:

1. The Two Different Maps (ADM vs. KS)

Imagine you are trying to describe a 3D landscape.

• Method A (ADM): You describe the landscape by slicing it into horizontal layers (like layers of a cake), looking at the shape of each layer and how it changes over time.
• Method B (KS): You describe the same landscape by looking at it from a different angle, perhaps focusing on the "flow" of the terrain rather than the layers.

In the past, physicists had to do the math separately for Method A and Method B. They were like two different languages describing the same mountain. A previous study showed these two methods were "dual" to each other (like a mirror image), but they only did the math up to a certain level of detail (like looking at the mountain from a distance).

2. The "Matryoshka Doll" Strategy

The author's big innovation is a new way of doing the math that doesn't care which "map" (ADM or KS) you are using. He calls this the "Matryoshka Doll" approach.

• The Old Way: You pick a map, break the math into tiny pieces, and calculate them one by one. It's tedious and easy to make mistakes.
• The New Way (Matryoshka Dolls): Imagine a set of Russian nesting dolls.
  • The Big Doll is the whole, unexpanded theory.
  • Inside it is a Medium Doll (the first level of approximation).
  • Inside that is a Small Doll (the second level), and so on.

The author realized that instead of breaking the dolls apart one by one, you can look at the structure of the nesting itself. He developed a "universal recipe" that works for the Big Doll, the Medium Doll, and the Small Doll simultaneously.

Because the two maps (ADM and KS) are mirror images (duals), this universal recipe works for both at the same time. You don't have to choose a map until the very end.

3. Going Deeper (The $c^{-3}$ Order)

Previous studies stopped their calculations at a certain depth (like looking at the mountain from 10 miles away). This paper pushes the calculation much further (looking at the mountain from 1 mile away).

• Why does this matter? It allows physicists to model gravity in "strong field" situations (like near black holes or neutron stars) with much higher precision than before, without needing to solve the impossible full equations of Einstein.
• The author successfully calculated the math up to the third level of detail ($c^{-3}$) for the ADM method and the first level for the KS method, proving that his "universal recipe" works perfectly for both.

4. The "Mirror" Effect

The most exciting part is the duality.
Think of the ADM and KS methods as two people looking at a sculpture from opposite sides.

• In the past, people thought, "If I change my view, I have to re-calculate everything from scratch."
• This paper says, "No! Because they are mirror images, if I calculate the shape for Person A, I automatically know the shape for Person B. The math is the same, just the labels are swapped."

The author proved that this "mirror symmetry" holds true even when you zoom in to the very fine details of the expansion.

Summary

• The Problem: Einstein's gravity equations are too hard to solve exactly. We need approximations for slow-moving, strong-gravity systems.
• The Old Solution: Do the math separately for two different ways of slicing space-time (ADM and KS). It was slow and limited.
• The New Solution: Use a "Matryoshka Doll" framework. Create a single, universal set of rules that works for both ways of slicing space-time simultaneously.
• The Result: The author calculated the gravity equations with much higher precision than ever before, proving that the two different methods are perfectly symmetrical even at the deepest levels of calculation.

In a nutshell: The author built a master key that opens two different locks at once, allowing us to see the universe's gravity in much sharper detail than before.

Technical Summary

1. Problem Statement

General Relativity (GR) is notoriously difficult to solve exactly due to the non-linearity of its field equations. Approximation schemes are essential for studying physical systems. The Post-Newtonian (PN) expansion is a standard weak-field, slow-velocity approximation. However, the $1/c$ expansion (large $c$ expansion) is a distinct non-relativistic (NR) approximation that allows for strong-field interactions while maintaining slow velocities, making it suitable for systems like dense neutron stars and black hole mergers.

Previous work (specifically Ref. [20]) established a $1/c$ expansion within the Kol-Smolkin (KS) decomposition of the 3+1 formalism up to order $c^{-2}$. The ADM (Arnowitt-Deser-Misner) and KS decompositions are known to be dual to each other (roles of the metric and its inverse are interchanged), and they share a common form of the Einstein-Hilbert action. However, existing expansion methods required choosing a specific decomposition (ADM or KS) at the outset, obscuring the duality at the level of the expansion and limiting the order of computation.

The paper addresses the following questions:

1. Does the duality between ADM and KS decompositions hold at the level of the $1/c$ expansion?
2. Can a unified expansion scheme be developed that avoids choosing a specific slicing until the final step?
3. Can the expansion be pushed beyond the previously achieved $c^{-2}$ order (specifically to $c^{-3}$)?

2. Methodology

The author proposes a novel, unified framework based on the "parent action" valid for both decompositions, rather than expanding specific decompositions separately.

A. The Parent Action and Duality

The starting point is the Einstein-Hilbert action in a 3+1 form:
$$S_{EH} = \int \left( \hat{R} + K_{ij}K^{ij} - K^2 \right) n_t \sqrt{h} \, d^{d+1}x$$
where:

• $n_t$ is the lapse function ($N$ in ADM, $M$ in KS).
• $h_{ij}$ is the spatial metric.
• $\hat{R}$ is the spatial Ricci scalar defined via a specific connection $\hat{\Gamma}$.
• $K_{ij}$ is the extrinsic curvature tensor.
• The definitions of $K_{ij}$ and the frame vectors differ between ADM and KS, but the form of the action remains identical.

B. The "Matryoshka Dolls" Expansion Scheme

Instead of the traditional method (choose decomposition $\to$ expand fields $\to$ collect orders), the author introduces a step-by-step expansion of the parent Lagrangian $[L] = [R\sqrt{-g}]$.

• Concept: The expansion is treated as a nested structure (like Matryoshka dolls).
• Step Definition: The $n$-th step, $[L]_n$, contains terms of order $c^{-n}, c^{-(n+1)}, c^{-(n+2)}$.
• Recursive Structure:
  $$[L]_n = [R]_n [\sqrt{-g}]_0 + [R]_{n-1} [\sqrt{-g}]_1 + \dots + [R]_0 [\sqrt{-g}]_n$$
  This allows the calculation of higher-order terms by only computing the new "incremental" terms $[R]_n$ and $[\sqrt{-g}]_n$, while reusing results from previous steps.

C. Generic Identities

To handle the expansion without fixing the decomposition, the author derives generic identities valid for both ADM and KS:

1. Inverse Metric Expansion: Derives recursive formulas for the expansion of $h^{ij}$ based on $h_{ik}h^{kj} = \delta^j_i$.
2. Covariant Derivative Identities: Uses the identity $\hat{D}_i h_{jk} = 0$ to generate expressions for the expansion of Christoffel symbols $[\hat{\Gamma}]_n$ and the Ricci tensor $[\hat{R}_{ij}]_n$ in terms of elementary fields and their derivatives.
3. Wheeler-DeWitt Metric: Utilizes the symmetry of the tensor $(h_{ij}h_{kl} - h_{ik}h_{jl})$ to simplify contractions, eliminating problematic antisymmetric terms during the calculation of the Ricci scalar.

3. Key Contributions

1. Unified Expansion Framework: The paper establishes a method to perform the $1/c$ expansion directly on the parent action without committing to ADM or KS initially. This preserves the duality of the decompositions throughout the calculation.
2. Extension to $c^{-3}$ Order: The author successfully pushes the expansion of the ADM decomposition up to $c^{-3}$ order. Previous results were limited to $c^{-2}$ (1PN order) for the KS decomposition.
3. Verification of Duality: By applying the generic formulas to both decompositions, the paper demonstrates that the duality between ADM and KS holds at the level of the expanded Lagrangian. The differences arise solely from the specific form of the extrinsic curvature tensor $K_{ij}$ and the frame vectors.
4. Systematic Algorithm: The "Matryoshka" approach provides a systematic algorithm to generate compact formulas for any order $n$, reducing the computational complexity compared to term-by-term expansion.

4. Key Results

• General Formulas: The paper provides explicit, generic expressions for the expansion of the Ricci scalar $[\hat{R}]_n$ and the determinant $[\sqrt{-g}]_n$ up to $n=3$ (Section 6).
• ADM Expansion ($c^{-3}$):
  • Zeroth Order ($c^0$): Recovers the stationary limit (no time derivatives).
  • First Order ($c^{-1}$): Includes single time derivatives and expanded fields.
  • Second Order ($c^{-2}$): Includes terms with two time derivatives and quadratic expansions.
  • Third Order ($c^{-3}$): A novel result containing complex interactions of expanded fields and their derivatives. The author notes that the ADM decomposition is technically superior for this expansion because the frame vector components $(n)^e_i$ vanish for $n \ge 1$, simplifying the derivatives compared to the KS case.
• KS Expansion ($c^{-1}$): The author applies the framework to the KS decomposition up to $c^{-1}$ to explicitly demonstrate the duality. The results match previous literature but are derived via the new unified method.
• All-Order Observation: The paper identifies a universal structure in the $n$-th order Lagrangian involving a variation with respect to the $n$-th order lapse field acting as a Lagrange multiplier, leading to second-order derivatives of the $n$-th order fields.
• Newtonian Limit: In Appendix C, the author shows how a conformal redefinition of the fields ($n_t \to e^{\psi/2}$) recovers the Poisson equation in the Newtonian limit, confirming the physical consistency of the expansion.

5. Significance and Future Applications

• Strong Field Physics: The $c^{-3}$ expansion is crucial for modeling strong gravitational fields where standard PN expansions (which assume weak fields) may fail or require higher-order corrections. This is particularly relevant for neutron stars and black hole mergers.
• Theoretical Unification: By proving that the duality between ADM and KS survives the expansion process, the paper unifies two major approaches to non-relativistic gravity.
• Connection to Newton-Cartan and Carroll Gravity: The framework bridges the gap between standard GR, Newton-Cartan theory, and Carrollian gravity (small $c$ expansion), offering a potential pathway to study the relation between Galilean and Carrollian limits.
• Condensed Matter Applications: The author suggests potential applications in condensed matter physics, specifically relating the ADM/KS decompositions to Zermelo and Randers metrics found in graphene and black hole analogues.

In conclusion, this paper provides a robust, systematic, and duality-preserving tool for high-order non-relativistic expansions of General Relativity, significantly extending the known results for the ADM formalism and offering a new perspective on the structure of non-relativistic gravity.