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Physics · Document 7

ADM Formalism

The 3+1 decomposition of spacetime for canonical gravity

1. The 3+1 Decomposition

The ADM formalism (Arnowitt-Deser-Misner) reformulates general relativity by splitting 4D spacetime into 3D space evolving in time. This is essential for canonical quantization and numerical relativity.

🎮 Interactive: Spacetime Foliation
t (time)Sigma_tn^muSpacetime foliated into spatial slices
3+1 decomposition: Spacetime is sliced into a family of 3D spatial hypersurfaces Sigma_t, each labeled by time t. The normal vector n^mu points in the time direction.
The 3+1 decomposition rewrites the 4D metric as a 3D spatial metric plus lapse and shift functions that describe how time flows and coordinates move.

2. ADM Variables

The 4D metric g_uv is decomposed into:

VariableSymbolMeaning
Spatial metricgamma_ij3D geometry of each slice
LapseNProper time between slices
ShiftN^iCoordinate drift between slices
🎮 Interactive: Lapse and Shift
tt + dtPN*dt
Lapse N:
How fast proper time advances between slices. N = 1 means coordinate time equals proper time.
Shift N^i:
How spatial coordinates slide between slices. Non-zero shift means coordinates move in space.

The ADM Metric

ds^2 = -N^2 dt^2 + gamma_ij (dx^i + N^i dt)(dx^j + N^j dt)
4D line element

This decomposes the 10 components of g_uv into:

  • 6 components in gamma_ij (symmetric 3x3 matrix)
  • 1 component in N (lapse function)
  • 3 components in N^i (shift vector)

3. Extrinsic Curvature

The extrinsic curvature K_ij measures how each spatial slice is embedded in 4D spacetime - how much it curves in the time direction.

🎮 Interactive: Extrinsic Curvature
Normal vectors change direction = extrinsic curvature
Surface curvature
K ~ 0.15
Flat embedding (K=0)
No
Extrinsic curvature K_ij: Measures how the spatial slice bends within the higher-dimensional spacetime. Unlike intrinsic curvature (within the slice), this captures the embedding.
K_ij = -(1/2N)(d_t gamma_ij - D_i N_j - D_j N_i)
Definition
Intrinsic curvature (R):
Curvature within the 3D slice
Extrinsic curvature (K):
How the slice bends in 4D spacetime
The pair (gamma_ij, K_ij) forms the canonical variables for gravity, analogous to position and momentum in mechanics. This is the starting point for quantum gravity!

4. Constraints

Not all (gamma_ij, K_ij) configurations are valid initial data for Einstein's equations. They must satisfy constraint equations:

Hamiltonian Constraint

H = R + K^2 - K_ij K^ij = 16 pi G rho

Relates the spatial curvature and extrinsic curvature to energy density. This constraint generates time evolution.

Momentum Constraints

H_i = D_j(K^j_i - delta^j_i K) = 8 pi G j_i

Relates the gradient of extrinsic curvature to momentum density. These generate spatial diffeomorphisms.

Physical Meaning

  • - H = 0: Energy is conserved (no creation from nothing)
  • - H_i = 0: Momentum is conserved (gauge invariance)
  • - 4 constraints reduce 12 variables to 8 physical degrees of freedom
  • - 2 polarizations of gravitational waves (4 more gauge freedoms)

5. Evolution Equations

Given valid initial data, the ADM formalism provides evolution equations:

d_t gamma_ij = -2N K_ij + D_i N_j + D_j N_i
Evolution of spatial metric
d_t K_ij = -D_i D_j N + N(R_ij + K K_ij - 2 K_ik K^k_j) + ...
Evolution of extrinsic curvature
Numerical relativity: These equations are solved on supercomputers to simulate black hole mergers and neutron star collisions. LIGO detections are compared to these simulations!

6. Connection to Z2 Framework

Z²
(gamma_ij, K_ij)
Canonical variables for gravity enable quantization and cosmological calculations
Z²
H = R + K^2 - K_ij K^ij
The Hamiltonian constraint must be solved on the T^3/Z_2 compactification
Z²
3+1 split
Separates compact spatial dimensions (T^3/Z_2) from cosmic time evolution

Why ADM Matters for Z2

  • - Canonical quantization: Path to quantum gravity
  • - Constraint algebra: Tests consistency of compactification
  • - Moduli dynamics: Extra dimension sizes as dynamical variables
  • - Cosmology: FLRW metric in ADM form gives Friedmann equations

7. BSSN Formulation

For numerical stability, the ADM equations are often recast in the BSSN(Baumgarte-Shapiro-Shibata-Nakamura) form:

ADM:
Original formulation, elegant but numerically unstable
BSSN:
Conformal decomposition, stable for simulations
gamma_ij = e^(4 phi) * gamma_tilde_ij (det gamma_tilde = 1)
Conformal decomposition

Exercises

  1. Write the Minkowski metric ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 in ADM form. What are N and N^i?
  2. For the Schwarzschild solution in Schwarzschild coordinates, find the lapse N(r).
  3. Show that a flat spatial slice (K_ij = 0) in flat spacetime satisfies both constraints.
  4. How many independent components does K_ij have? (Hint: it is symmetric)
  5. Why are the constraints called "initial value" constraints?