Lecture Notes on ADM Formalism and Hamiltonian in General Relativity

Introduction

• Discussion on ADM formalism in general relativity.
• Transition from specific solutions in Einstein's equations to the ADM formalism.

ADM Formalism

• ADM stands for Arnowitt-Deser-Misner, referring to the Hamiltonian formalism in general relativity.
• Focuses on rewriting the Lagrangian of gravity to extract Hamiltonian, conjugate momentum, and configuration variables.

Classical Mechanics Recap

• Reminder of classical mechanics: Lagrangian, conjugate momentum, configuration variables (P and Q).
• Hamiltonian extracted from rewriting the Lagrangian with these variables.

Action in General Relativity

• Uses the Hilbert action, involving a volume integral over a four-dimensional manifold.
• Add boundary term (Gibbons-Hawking-York boundary term) for a well-defined variational principle.
• This action leads to the derivation of Einstein's equations.

Hamiltonian Formalism in General Relativity

• Aim: Reformulate Einstein's equations in Hamiltonian form (first-order differential equations).
• Converts second-order differential equations to first-order, introducing momentum variables.

Initial Value Formulation

• Facilitates defining initial values/conditions in phase space.
• Initial conditions are points in phase space, allowing trajectory determination via Hamilton's equations.

Importance of Hamiltonian Formulation

1. Initial Value Formulation:

   • Useful in numerical relativity and simulation of gravitational waves.
   • Computational convenience by using Hamilton's equations.

2. Quantization:

   • Hamiltonian needed for quantization in quantum mechanics (e.g., Schrödinger equation).
   • Necessary step before quantizing gravity.

3+1 Decomposition of Spacetime

• To define Hamiltonian formalism, a specific time 'T' must be specified, leading to 3+1 decomposition.
• 3+1 Decomposition:
  • Background manifold M = (R x Sigma), where R is time and Sigma is space with dimension 3.
  • Decomposition results in spatial slices parameterized by time.

Foliation of Spacetime

• Spacetime viewed as composed of spatial slices (foliation).
• Time evolution from past to future is depicted by a tangent vector 'TA'.

Vector Fields and Decomposition

• TA decomposed using bases and normal vectors of the spatial slice.
• Key Terms:
  • Lapse Function (N): Scalar function in the decomposition of TA.
  • Shift Vector (Na): Projection onto spatial slices, tangent to them.
  • Normal Vector (Na): Normalized; has properties like NaNa = -1.

Coordinate Systems

• Coordinate system: (T, X1, X2, X3), where 'T' is the time coordinate.
• Spatial coordinates are on Sigma.