Questions: ADM Formalism (Introduction)

Question 1

In the ADM formalism, the lapse function N and shift vector N^i are:

• A. Dynamical degrees of freedom that propagate as gravitational waves
• B. Gauge variables that specify how the coordinate system evolves from one spatial slice to the next, with no dynamical content
• C. Components of the stress-energy tensor
• D. Determined by the spatial metric γ_ij through the constraint equations

Answer: (B)

The lapse N specifies how much proper time elapses between adjacent spatial hypersurfaces at each point, and the shift N^i specifies how spatial coordinates shift laterally between slices. They encode the four coordinate degrees of freedom (gauge freedom) of GR and can be freely chosen — different choices correspond to different coordinate systems (slicing conditions). They have no dynamical content: they do not appear with time derivatives in the action, and their equations of motion are the constraint equations (which constrain the initial data, not the evolution). The physical degrees of freedom reside entirely in the spatial metric γ_ij and its conjugate momentum π^{ij}.

Question 2

The ADM constraint equations must be satisfied only on the initial time slice — they are automatically preserved by the evolution equations.

• T. True
• F. False

Answer: True

The Hamiltonian constraint H = 0 and momentum constraints H_i = 0 are conditions on the initial data (γ_ij, π^{ij}) on a single spatial slice. The evolution equations (derived from the remaining Einstein equations) propagate the data forward in time, and the Bianchi identities guarantee that if the constraints are satisfied initially, they remain satisfied at all later times. This structure is analogous to electromagnetism, where Gauss's law (∇·E = ρ/ε₀) is a constraint on the initial data that is preserved by Maxwell's evolution equations.

Question 3

Explain why the ADM formalism is essential for numerical relativity.

Model answer: Numerical relativity requires solving the Einstein equations on a computer, which means evolving initial data forward in time step by step. The covariant form of the Einstein equations (G_μν = 8πG T_μν /c⁴) is not directly suitable for time evolution because it mixes space and time without a clear notion of 'initial data' and 'time stepping.' The ADM formalism provides the 3+1 decomposition needed: it identifies the initial data (γ_ij, K_ij on a spatial slice), the constraints the data must satisfy, and the evolution equations that advance the data in time. The lapse and shift provide the gauge freedom needed to choose computationally stable coordinate conditions. Modern numerical relativity codes (e.g., those used to predict LIGO waveforms for binary black hole mergers) are all based on the ADM decomposition or its refinements (BSSN formulation, generalized harmonic coordinates).

The first successful numerical simulation of binary black hole mergers (Pretorius, 2005; Campanelli et al., 2006; Baker et al., 2006) was a breakthrough that required decades of work on the ADM framework, stable gauge conditions, and computational infrastructure. These simulations now provide the gravitational wave templates used by LIGO/Virgo for signal detection and parameter estimation.

Question 4

How many physical degrees of freedom does the gravitational field have per spatial point, and how does the ADM counting arrive at this number?

Model answer: The spatial metric γ_ij has 6 independent components (symmetric 3×3 matrix). Each has a conjugate momentum π^{ij}, giving 12 phase-space variables per point. The 4 constraint equations (1 Hamiltonian + 3 momentum) remove 4 phase-space degrees of freedom. The 4 gauge degrees of freedom (lapse + 3 shift components) remove another 4 through gauge-fixing. This leaves 12 - 4 - 4 = 4 phase-space variables, corresponding to 2 physical configuration-space degrees of freedom per point. These are the two polarizations of gravitational waves — the only true propagating degrees of freedom of the gravitational field.

The ADM counting confirms from the Hamiltonian perspective what the linearized theory shows from the wave perspective: gravity has exactly two propagating degrees of freedom. This is the same number as electromagnetism (two photon polarizations), despite the gravitational field having a much larger number of metric components.