Hamilton formulation of General Relativity
The Hamiltonian formulation of general relativity is a reformulation of Einstein’s theory of gravity in terms of the language of Hamiltonian mechanics. This approach is important for understanding the dynamics of spacetime and is widely used in areas such as numerical relativity and quantum gravity.
Overview:
In general relativity, spacetime is described by Einstein’s field equations, which relate the curvature of spacetime to the energy and momentum of matter. These equations are covariant, meaning they treat space and time on an equal footing. However, the Hamiltonian formulation breaks the problem down into a way that explicitly separates space and time, allowing for a time evolution approach.
This reformulation is essential for understanding how spacetime evolves over time, which is particularly useful when solving dynamic problems like black hole mergers or gravitational wave propagation. It also plays a critical role in attempts to quantize gravity, such as in canonical quantum gravity.
Key Concepts:
- Lagrangian vs. Hamiltonian Formulation:
- In classical mechanics, systems can be described either by a Lagrangian or a Hamiltonian approach.
- The Lagrangian formulation focuses on a quantity called the Lagrangian, which is a function of the configuration of a system and its velocity. The equations of motion are derived from minimizing the action.
- The Hamiltonian formulation, on the other hand, is built around a function called the Hamiltonian, which represents the total energy of the system. It focuses on the positions and momenta of a system and uses Hamilton’s equations to describe the system’s time evolution.
- In classical mechanics, systems can be described either by a Lagrangian or a Hamiltonian approach.
- 3+1 Decomposition:
- The Hamiltonian formulation of general relativity is based on a 3+1 decomposition of spacetime, where spacetime is split into three spatial dimensions and one time dimension. This allows us to think of general relativity as the evolution of space over time, instead of dealing with the full four-dimensional spacetime all at once.
- In this approach, spacetime is foliated into a series of 3D spatial hypersurfaces that evolve with time. The main idea is to track how these 3D slices change from one moment to the next.
- The Hamiltonian formulation of general relativity is based on a 3+1 decomposition of spacetime, where spacetime is split into three spatial dimensions and one time dimension. This allows us to think of general relativity as the evolution of space over time, instead of dealing with the full four-dimensional spacetime all at once.
- ADM Formalism:
- The Hamiltonian formulation of general relativity is often referred to as the ADM formalism, named after physicists Richard Arnowitt, Stanley Deser, and Charles Misner, who developed this approach in the late 1950s and early 1960s.
- In the ADM formalism, spacetime is described by a metric on the 3D spatial hypersurfaces and a set of variables that describe how these hypersurfaces evolve over time.
- The Hamiltonian formulation of general relativity is often referred to as the ADM formalism, named after physicists Richard Arnowitt, Stanley Deser, and Charles Misner, who developed this approach in the late 1950s and early 1960s.
Components in the ADM Formalism:
- Lapse Function (N): This function describes how much proper time elapses between two neighboring spatial hypersurfaces.
- Shift Vector (Nᵢ): This describes how coordinates on the spatial slice shift from one hypersurface to the next.
- 3-Metric (γᵢⱼ): This is the metric that describes the geometry of the 3D spatial slice at a given moment of time. It contains information about distances and curvature within that slice.
- Conjugate Momenta (πᵢⱼ): These are the momenta conjugate to the 3-metric. They describe how the geometry of the spatial slice evolves over time.
- Constraints:
- The Hamiltonian formulation of general relativity includes a set of constraints that must be satisfied. These constraints arise because general relativity is a gauge theory, and not all degrees of freedom are physical.
- There are two main constraints:
- Hamiltonian constraint: Ensures that the energy density of the gravitational field matches the matter-energy content.
- Momentum constraint: Ensures that the momentum density is consistent with the flow of matter and energy.
- Hamiltonian constraint: Ensures that the energy density of the gravitational field matches the matter-energy content.
- These constraints are akin to conservation laws and must hold for the system at every moment in time.
- The Hamiltonian formulation of general relativity includes a set of constraints that must be satisfied. These constraints arise because general relativity is a gauge theory, and not all degrees of freedom are physical.
Hamilton’s Equations in General Relativity:
- In the Hamiltonian formulation, the equations of motion are derived from Hamilton’s equations, which describe how the 3-metric and the conjugate momenta evolve with time. The Hamiltonian function (or Hamiltonian constraint) governs this evolution.
Hamilton’s Equations:
\[ \dot{\gamma_{ij}} = \frac{\delta \mathcal{H}}{\delta \pi^{ij}} \]
\[ \dot{\pi^{ij}} = -\frac{\delta \mathcal{H}}{\delta \gamma_{ij}} \]
Where:
- \( \dot{\gamma_{ij}} \) represents the time derivative of the 3-metric.
- \( \pi^{ij} \) is the conjugate momentum.
- \( \mathcal{H} \) is the Hamiltonian.
These equations describe how the geometry of space (the 3-metric) and its conjugate momenta evolve over time.
- In the Hamiltonian formulation, the equations of motion are derived from Hamilton’s equations, which describe how the 3-metric and the conjugate momenta evolve with time. The Hamiltonian function (or Hamiltonian constraint) governs this evolution.
Why Use the Hamiltonian Formulation?
- Time Evolution:
- The Hamiltonian formulation explicitly separates space and time, making it easier to study how space evolves over time. This is particularly important in numerical relativity, where simulations of dynamic spacetimes (such as black hole mergers or gravitational wave generation) are carried out.
- The Hamiltonian formulation explicitly separates space and time, making it easier to study how space evolves over time. This is particularly important in numerical relativity, where simulations of dynamic spacetimes (such as black hole mergers or gravitational wave generation) are carried out.
- Canonical Quantization:
- The Hamiltonian formulation is the basis for the canonical quantization of general relativity, which is one of the approaches to developing a theory of quantum gravity. In canonical quantum gravity, the idea is to treat the 3-metric and its conjugate momenta as quantum operators and develop a quantum version of general relativity.
- The Wheeler-DeWitt equation, which is central to canonical quantum gravity, emerges from the Hamiltonian constraint in this formalism.
- The Hamiltonian formulation is the basis for the canonical quantization of general relativity, which is one of the approaches to developing a theory of quantum gravity. In canonical quantum gravity, the idea is to treat the 3-metric and its conjugate momenta as quantum operators and develop a quantum version of general relativity.
- Understanding Initial Value Problems:
- The Hamiltonian formulation provides a natural way to frame the initial value problem in general relativity. Given the geometry and momenta of a spatial slice (i.e., the 3-metric and conjugate momenta), the Hamiltonian formulation allows one to evolve these quantities forward in time, which is crucial for solving Einstein’s equations in dynamic scenarios.
- The Hamiltonian formulation provides a natural way to frame the initial value problem in general relativity. Given the geometry and momenta of a spatial slice (i.e., the 3-metric and conjugate momenta), the Hamiltonian formulation allows one to evolve these quantities forward in time, which is crucial for solving Einstein’s equations in dynamic scenarios.
Summary of Steps:
- Foliation: Spacetime is foliated into 3D spatial slices that evolve with time.
- Metric and Momenta: The geometry of space at each moment is described by the 3-metric \( \gamma_{ij} \), and its time evolution is controlled by conjugate momenta \( \pi^{ij} \).
- Hamiltonian and Constraints: The system’s evolution is governed by the Hamiltonian function, subject to constraints (Hamiltonian and momentum).
- Time Evolution: Hamilton’s equations describe how the 3-metric and momenta change over time.
Historical Context:
- Paul Dirac initiated the idea of a Hamiltonian formulation for general relativity in the 1950s.
- Arnowitt, Deser, and Misner (ADM) built on this foundation and formalized the approach, which remains central to modern treatments of gravitational dynamics and quantum gravity.
In essence, the Hamiltonian formulation of general relativity rewrites Einstein’s field equations in a way that highlights the time evolution of spacetime, providing powerful tools for both theoretical and numerical work in gravitational physics.