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Introduction to the ADM formalism

This paper introduces the ADM formalism by explicitly deriving the necessary geometrical quantities to construct the Hamiltonian formulation of Einstein's general relativity and providing examples of its applications.

Original authors: Alejandro Corichi, Dario Núñez

Published 2026-10-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Alejandro Corichi, Dario Núñez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity, as Albert Einstein described it, is not a force that pulls objects together, but a curvature of the fabric of space and time itself. Imagine a heavy ball placed on a trampoline; it creates a dip, and smaller balls rolling nearby will curve toward it, not because of an invisible tug, but because the surface they travel on has been bent. In Einstein's theory, matter tells space how to curve, and that curved space tells matter how to move. This description is beautiful and mathematically precise, but it is also notoriously difficult to use when trying to predict how the universe changes over time. The equations that govern this curvature are written in a way that treats space and time as a single, inseparable block, making it nearly impossible to isolate the "now" from the "then" or to see which parts of the equation describe the current state of the universe and which parts describe how that state will evolve.

For decades, physicists struggled with this problem. They knew the laws were correct, but they lacked a practical way to simulate the future of the cosmos or to understand the quantum mechanics of gravity. To solve this, they needed a new perspective, one that could slice the four-dimensional block of spacetime into a series of three-dimensional snapshots, much like a film reel is a series of still images. This is the challenge that a team of researchers, Alejandro Corichi and Darío Núñez, addressed in their work on the ADM formalism. Originally written in 1991 and significantly expanded in a 2026 revision, this paper serves as a comprehensive guide to a mathematical framework that turns Einstein's static equations into a dynamic story of change. It provides the tools to take the complex, four-dimensional geometry of the universe and break it down into manageable pieces that can be studied, calculated, and even simulated on a computer.

The core idea behind this work is to view the universe not as a single, unchanging object, but as a sequence of three-dimensional spaces evolving over time. The researchers show how to cut spacetime into layers, where each layer represents the entire universe at a specific moment. To do this, they introduce two new concepts that act as the controls for this evolution. The first is a "lapse" function, which determines how much time passes between one layer and the next. The second is a "shift" vector, which describes how the coordinates on one layer slide or drift relative to the layer below it. By separating the geometry of space from the flow of time in this way, the authors transform the problem. Instead of dealing with a single, impenetrable set of equations, they derive a new set of rules that clearly distinguish between the variables that change as time moves forward and the variables that act as constraints, or fixed rules, that the universe must always obey.

This distinction is crucial. The paper demonstrates that out of the ten components of the gravitational field, only six are truly dynamic; these are the parts that evolve and carry the energy of the system. The other four components are not independent variables but are instead determined by the constraints. The authors explicitly derive the mathematical expressions for these constraints, known as the Hamiltonian constraint and the momentum constraint. These are not just abstract rules; they are the conditions that any valid snapshot of the universe must satisfy. If a snapshot of space does not meet these conditions, it cannot exist in a universe governed by Einstein's laws. The paper carefully constructs the energy of the system, showing how it is entirely determined by these constraints and the boundary conditions at the edge of the system. In fact, the total energy of an isolated system, such as a star or a galaxy, is found to be a specific value calculated from how the space curves at its very edge, a concept known as the ADM energy.

The utility of this approach extends far beyond theoretical elegance. The paper explains how this formalism became the foundation for numerical relativity, the field that uses supercomputers to simulate the most violent events in the cosmos. Before this framework was fully developed and understood, it was nearly impossible to model the collision of two black holes. The ADM formalism provided the necessary structure to set up the initial conditions for such a simulation and to evolve them forward in time step by step. The authors detail how this method allows scientists to solve the constraints on an initial slice of space and then use the evolution equations to predict what happens next. This capability has turned the simulation of black hole mergers from a theoretical dream into a routine calculation, providing the waveforms that were essential for the first direct detection of gravitational waves.

Beyond the practical application to black holes, the paper explores the deeper implications for the nature of reality. It discusses the concept of "superspace," a vast mathematical landscape where every possible shape of a three-dimensional universe is a single point. In this view, the history of our universe is not a line moving through time, but a path traced through this landscape of geometries. The authors show how the equations of general relativity can be reinterpreted as the motion of a particle through this superspace, subject to specific forces and constraints. This perspective has opened new doors for quantum gravity, the attempt to merge the laws of the very large with the laws of the very small. By isolating the true degrees of freedom of the gravitational field, the formalism provides a clearer path for physicists to apply quantum mechanics to the geometry of space itself.

The revised edition of this work, updated in 2026, does not merely repeat the original findings but expands upon them to include the constraint algebra, the boundary terms that define energy, and the developments that have occurred since the early 1990s. It clarifies the mathematical structure that underpins the entire field, ensuring that the connection between the abstract theory and the concrete simulations is transparent. The authors meticulously correct earlier errors and present the material in a modern notation that aligns with current standards, making it a definitive resource for anyone wishing to understand how the universe evolves. They show that while the equations of gravity are complex, they are not chaotic; they possess a rigid structure that, once understood, allows us to predict the behavior of the cosmos with remarkable precision.

Ultimately, this paper is about making the invisible visible. It takes the abstract, four-dimensional curvature of Einstein's theory and translates it into a language of change, evolution, and calculation. By providing a clear, step-by-step construction of the Hamiltonian formulation, the authors have given the scientific community a powerful tool to explore the dynamics of the universe. Whether it is calculating the energy of a distant galaxy, simulating the merger of two black holes, or probing the quantum nature of space, the ADM formalism stands as the bridge between the static beauty of Einstein's equations and the dynamic reality of our evolving universe. It is a testament to the power of mathematical clarity, showing that even the most complex laws of nature can be broken down into understandable parts, allowing us to see the machinery of the cosmos in motion.

Technical Summary: Introduction to the ADM Formalism (Revised and Expanded Edition)

Problem Statement
The covariant formulation of Einstein's general relativity, while elegant and manifestly independent of the observer's state of motion, obscures the dynamical content of the theory. In the standard formulation, the ten components of the Einstein field equations do not clearly separate into evolution equations and constraints. This lack of separation hinders the identification of true dynamical degrees of freedom and complicates the construction of a Hamiltonian framework, which is a prerequisite for canonical quantization and numerical simulation. The central problem addressed is the reformulation of general relativity to explicitly isolate the dynamical variables, define their conjugate momenta, and derive a Hamiltonian structure that respects the theory's constraints.

Methodology
The paper employs the Arnowitt-Deser-Misner (ADM) formalism, which treats general relativity as a Cauchy problem. The methodology proceeds through the following steps:

3+1 Decomposition: Spacetime is foliated into a one-parameter family of non-intersecting spacelike hypersurfaces ( $\Sigma_t$ $Σ_{t}$ ). The spacetime metric $g_{ab}$ $g_{ab}$ is decomposed into variables intrinsic to the hypersurface and variables describing the embedding:
- The induced 3-metric $q_{ab}$ (6 components).
- The lapse function $N$ , governing the proper time separation between slices.
- The shift vector $N^a$ , governing the spatial displacement of coordinates between slices.
Geometric Construction: The paper explicitly derives the geometric quantities required for this decomposition using abstract index notation (following Wald's conventions). Key developments include:
- Defining the projection operator $q_{ab} = g_{ab} + n_a n_b$ onto the hypersurface.
- Introducing the induced covariant derivative $D_a$ and the extrinsic curvature $K_{ab}$ , defined as the Lie derivative of the metric along the normal, projected onto the hypersurface.
- Deriving the Gauss-Codazzi equations, which relate the spacetime Riemann tensor to the intrinsic curvature ( $^3R$ ) and extrinsic curvature ( $K_{ab}$ ) of the hypersurface.
Variational Principle: The Einstein-Hilbert action is rewritten in terms of the ADM variables. The scalar curvature $^4R$ is expressed as $^3R + K_{ab}K^{ab} - K^2$ plus a total divergence. The action is reformulated as an integral over time of a Lagrangian defined on the spatial hypersurface.
Hamiltonian Formulation:
- Canonical momenta $\pi^{ab}$ are defined as derivatives of the Lagrangian with respect to the time derivative of the metric $\dot{q}_{ab}$ .
- It is shown that the momenta conjugate to $N$ and $N^a$ vanish, identifying them as non-dynamical Lagrange multipliers.
- A Legendre transformation yields the Hamiltonian density, which takes the form of a linear combination of constraints: $H = \int d^3x (N \mathcal{H}_0 + N^a \mathcal{H}_a)$ .
Constraint Analysis: The paper analyzes the primary and secondary constraints using Dirac's theory of constrained systems. It demonstrates that the Hamiltonian and momentum constraints generate gauge transformations (spacetime diffeomorphisms) and calculates the constraint algebra (the Dirac algebra).
Boundary Terms: The revised edition explicitly addresses boundary terms arising from the variation of the action and the Hamiltonian. It derives the ADM energy and momentum as surface integrals at spatial infinity for asymptotically flat spacetimes, ensuring the differentiability of the Hamiltonian.

Key Contributions and Results

Explicit Geometric Derivations: The paper provides a self-contained, step-by-step derivation of all geometric quantities (induced metric, extrinsic curvature, projection operators) and their relationships to the spacetime curvature tensors, correcting sign conventions and errors present in earlier versions of the text.
Constraint Algebra: It details the Poisson bracket structure of the constraints, showing that they form a first-class system. The algebra closes, but the structure "constants" are actually structure functions dependent on the metric $q_{ab}$ , distinguishing it from a standard Lie algebra.
Degrees of Freedom Counting: By analyzing the 12 phase-space components of $(q_{ab}, \pi^{ab})$ against the 4 first-class constraints, the paper confirms that general relativity possesses two physical degrees of freedom per spatial point, corresponding to the two polarizations of gravitational waves.
3+1 Evolution Equations: The paper presents the evolution equations for the metric and extrinsic curvature (or momenta) in the 3+1 form, clarifying the distinction between the constraint equations (which restrict initial data) and the evolution equations (which propagate the data).
ADM Energy: The paper derives the expression for the ADM energy as a surface integral at infinity, linking the Hamiltonian formulation to the physical mass of isolated systems.
Historical and Modern Context: The revised edition includes new sections (3.8–3.10 and Section 5) that update the formalism with developments since 1991, specifically:
- The application of the formalism to Numerical Relativity, including the BSSN formulation and the "moving puncture" method used in black hole merger simulations.
- Canonical Quantum Gravity, discussing the Wheeler-DeWitt equation, the problem of time, and the transition to Loop Quantum Gravity via Ashtekar variables.
- Cosmology, detailing the application of the formalism to homogeneous models (minisuperspace) and the study of cosmological singularities.

Significance and Claims
The authors state that the primary goal of this manuscript is to fill a gap in the literature by providing a clear, complete, and accessible introduction to the ADM formalism and its applications. The paper claims that by reformulating general relativity in the Hamiltonian framework, the dynamical content of the theory is made explicit, separating constraints from evolution equations.

The significance of the formalism is highlighted in three areas:

Quantization: It provides the necessary canonical structure (variables and constraints) for attempts to quantize gravity, whether through the Wheeler-DeWitt equation or modern approaches like Loop Quantum Gravity.
Numerical Relativity: It serves as the foundational tool for solving Einstein's equations numerically, enabling the simulation of complex astrophysical phenomena such as binary black hole mergers.
Conceptual Clarity: It offers a geometric interpretation of spacetime as the evolution of three-geometries (Geometrodynamics) and clarifies the nature of the space of solutions (Superspace).

The paper modestly notes that while the formalism is powerful, solving the constraints and reducing the system to a true Hamiltonian with a finite number of degrees of freedom has only been achieved formally or in highly symmetric cases (like cosmology). It emphasizes that the 3+1 decomposition is a tool for analysis and computation rather than a unique physical description, as the choice of foliation (lapse and shift) remains a gauge freedom.

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