Chapter 3: The ADM Decomposition

Breaking Spacetime Into Space + Time

Einstein's General Relativity naturally describes 4-dimensional spacetime as a unified whole. But for practical calculations and physical intuition, we often want to separate space and time. The ADM decomposition does exactly this.

The Fundamental Idea

Imagine slicing 4D spacetime like a loaf of bread. Each slice is a 3D spatial surface at a particular time. The ADM decomposition tells us:

• How the geometry of each slice is described (the 3-metric γ_ij)
• How much proper time elapses between slices (the lapse function N)
• How the spatial coordinates shift between slices (the shift vector N^i)

The General ADM Line Element

Any 4D spacetime metric can be written in ADM form as:

This looks complicated, but let's break it down piece by piece.

The Three Components

1. The Lapse Function N

Physical meaning: Controls how much proper time passes between spatial slices

Metaphor: Think of N as the speed setting on a cosmic clock:

• N = 1: Normal speed (far from any mass)
• N > 1: Fast-forward (negative gravity - rarely occurs naturally)
• N < 1: Slow motion (near massive objects)
• N → 0: Time nearly stops (near black hole horizon)

Examples

• GPS satellite: N ≈ 1.0000000002 (time runs slightly faster than Earth surface)
• Earth surface: N ≈ 0.9999999993 (relative to infinity)
• Neutron star surface: N ≈ 0.86 (time runs 14% slower)

2. The Shift Vector N^i

Physical meaning: How spatial coordinates "slip" or "drag" between time slices

Metaphor: Like a conveyor belt in space. Even if you're standing still in coordinates, spacetime itself can be dragging you along.

Key Properties

• Zero shift (N^i = 0): Spatial coordinates don't drag between time slices
• Non-zero shift: Space itself is "flowing" or rotating
• Near rotating objects: Space gets dragged around (frame-dragging effect)

Real-World Examples

• Around Earth: Tiny frame-dragging measured by Gravity Probe B
• Around rotating black hole: Space whirls like water going down a drain
• In our formulation: We set N^i = 0 for simplicity (diagonal gauge)

3. The Spatial 3-Metric γ_ij

Physical meaning: Describes the geometry of each spatial slice

The 3-metric tells us distances within each spatial slice:

In our spherical case:

Components Explained

• γ_rr = e^{-2Φ}: Radial distances are modified by the temporal potential
• γ_θθ = r^2: Standard spherical geometry in θ direction
• γ_φφ = r^2sin^2θ: Standard spherical geometry in φ direction

Our Specific Choice: Diagonal Gauge

In the lapse-first formulation, we make specific choices to simplify the mathematics:

1. Zero Shift Vector

This means spatial coordinates don't drag between time slices. We're using a "diagonal" gauge where space and time are orthogonal.

2. Areal Radius Condition

This ensures that spheres of radius r have area 4πr², making r the "areal radius."

3. The Key Constraint

This links the radial metric component directly to the lapse function. This is the heart of the lapse-first approach!

The Resulting Metric

With these choices, our spacetime metric becomes:

This is elegantly simple: just one function Φ(t,r) controls both time and radial space!

Physical Interpretation of Each Component

Time Component: -e^{2Φ} dt²

• Controls gravitational time dilation
• When Φ < 0: Time runs slower (gravitational redshift)
• When Φ = 0: Normal time flow

Radial Component: e^{-2Φ} dr²

• Controls radial distances
• When Φ < 0: Radial distances are stretched
• This "compensates" for the time dilation to maintain consistency

Angular Components: r² dΩ²

• Standard spherical geometry
• Ensures spheres have area 4πr²
• Unchanged by gravity in spherical symmetry

Why This Decomposition Matters

1. Computational Advantages

• 10 metric components → 1 function (in spherical symmetry)
• Equations become much simpler
• Numerical evolution is more stable

2. Physical Clarity

• Separates "what happens to time" from "what happens to space"
• Makes the temporal nature of gravity manifest
• Connects directly to what we measure (clock rates)

3. Connection to Newtonian Intuition

In the weak field limit:

This looks exactly like the Newtonian gravitational potential (up to factors of c²)!

Constraint vs. Dynamical Variables

An important distinction in the ADM formalism:

Constraint Variables (Non-Dynamical)

• Lapse N = e^Φ: Lagrange multiplier, not evolved
• Shift N^i: Lagrange multiplier, not evolved
• These are determined by constraints at each moment

Dynamical Variables

• Spatial metric γ_ij: Evolves according to Einstein's equations
• In our case: only γ_rr = e^{-2Φ} is dynamical
• Evolution driven by matter and energy

Frame-Dragging and Rotation

An important note about what's missing in our diagonal gauge:

Where Rotation Lives

In the weak-field limit, the gravitational field splits like electromagnetism:

• Gravitoelectric field ∼ ∇Φ: Like electric field from charges
• Gravitomagnetic field ∼ ∇ × N^i: Like magnetic field from currents

Frame dragging is the "curl" of the shift, just like magnetism is the curl of the vector potential.

Connection to Other Formulations

Our diagonal gauge is related to other common forms by coordinate transformations:

Eddington-Finkelstein Form

In advanced/retarded coordinates, the metric has g_tr ≠ 0 terms. Our gauge removes these cross-terms through a time redefinition.

Painlevé-Gullstrand Form

Uses "flowing space" coordinates where space itself falls into black holes. Related to our form by changing the slicing.

Isotropic Coordinates

Uses different radial coordinate to make spatial part conformally flat. All describe the same physics with different coordinate choices.

Summary

The ADM decomposition provides the framework for lapse-first GR:

• Lapse N = e^Φ: Controls time flow (our primary variable)
• Shift N^i: Controls spatial dragging (set to zero in our gauge)
• 3-metric γ_ij: Controls spatial geometry (linked to Φ)

In spherical symmetry with our gauge choice, everything reduces to understanding one function Φ(t,r). This dramatic simplification makes gravity much more tractable while preserving all the physics.

Next, we'll see how energy flow drives the evolution of this temporal potential through the famous flux law.