An Introduction to Splitting the g

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An Introduction to Splitting the g: Unpacking Spacetime Dynamics in General Relativity

I. Introduction: The Fabric of Spacetime and its Evolution

Albert Einstein’s theory of General Relativity (GR), formulated in 1915, revolutionized our understanding of gravity. It replaced Newton’s concept of a force acting instantaneously across a static Euclidean space with a dynamic picture: gravity is not a force, but a manifestation of the curvature of spacetime itself. Matter and energy tell spacetime how to curve, and the curvature of spacetime tells matter and energy how to move. This intricate dance is encoded in the Einstein Field Equations (EFE):

G_μν = R_μν - (1/2) R g_μν = (8πG / c^4) T_μν

Here, G_μν is the Einstein tensor representing spacetime curvature, R_μν is the Ricci curvature tensor, R is the Ricci scalar, g_μν is the metric tensor defining the geometry of spacetime, T_μν is the stress-energy tensor representing the distribution of matter and energy, G is Newton’s gravitational constant, and c is the speed of light. (For simplicity, we often use geometric units where G=c=1, which we will adopt for much of the discussion below).

The metric tensor g_μν is the central object. It contains all the information about the spacetime geometry, including distances, time intervals, and causal structure. In GR, spacetime is a four-dimensional manifold (three space dimensions + one time dimension), and g_μν provides the rule for measuring the infinitesimal squared interval ds^2 between nearby events:

ds^2 = g_μν dx^μ dx^ν

(Here, Greek indices μ, ν run from 0 to 3, representing time and space coordinates, and we use the Einstein summation convention where repeated upper and lower indices are summed over).

The EFEs are a complex system of ten coupled, non-linear, second-order partial differential equations for the components of the metric tensor g_μν. While they elegantly describe the relationship between geometry and matter, they present significant challenges when we want to understand the dynamics of spacetime itself. How does the geometry evolve over time? Can we predict the future state of spacetime given its state at an initial moment? This is the essence of the initial value problem in GR.

Furthermore, attempts to reconcile GR with quantum mechanics – the quest for a theory of quantum gravity – often require casting GR into a Hamiltonian framework, similar to other fundamental theories like electromagnetism and Yang-Mills theory. A Hamiltonian formulation describes the time evolution of a system in terms of its state variables (positions and momenta) defined on a “slice” of time.

However, the structure of GR poses immediate difficulties for both the initial value problem and Hamiltonian formulation. The theory possesses diffeomorphism invariance (also known as general covariance), meaning the physical laws are independent of the choice of coordinate system used to describe spacetime. This profound symmetry implies that there is no preferred way to separate space and time. Four of the ten Einstein equations do not involve second time derivatives of the metric; they act as constraints on the initial data rather than predicting evolution. This gauge freedom complicates the identification of true dynamical degrees of freedom and the formulation of a well-posed initial value problem.

To address these challenges, a powerful technique was developed, often referred to as “splitting the g” or the “3+1 decomposition”. This approach, most famously formalized by Richard Arnowitt, Stanley Deser, and Charles Misner in the late 1950s and early 1960s (hence often called the ADM formalism), provides a systematic way to foliate, or slice, the four-dimensional spacetime into a sequence of three-dimensional spatial hypersurfaces. By decomposing the 4D metric g_μν relative to this foliation, GR can be recast into a language that explicitly separates spatial geometry from time evolution, making the dynamics more transparent and amenable to analysis.

This “splitting” involves projecting the spacetime geometry and the Einstein equations onto directions normal and tangent to these spatial slices. The result is a description of gravity in terms of the intrinsic geometry of the spatial slices (described by a 3D spatial metric γ_ij), how these slices are embedded in the 4D spacetime (described by the extrinsic curvature K_ij), and how the coordinate system evolves from one slice to the next (described by the lapse function N and the shift vector N^i).

This article provides a detailed introduction to the concept of “splitting the g”. We will explore the conceptual foundations of spacetime foliation, delve into the geometric decomposition of the metric into lapse, shift, and spatial metric, examine how the Einstein equations transform under this split into constraint and evolution equations involving extrinsic curvature, and discuss the profound implications and applications of this formalism, particularly for the initial value problem, numerical relativity, and canonical quantum gravity.

II. Conceptual Foundations: Foliating Spacetime

The core idea behind the 3+1 decomposition is to view the four-dimensional spacetime (M, g_μν) not as an indivisible block, but as a continuous sequence of three-dimensional spatial “snapshots” evolving in time. This requires us to foliate the spacetime manifold M.

1. Spacelike Hypersurfaces:
A foliation involves choosing a family of non-intersecting, three-dimensional, spacelike hypersurfaces, denoted by Σ_t, that fill the spacetime (or at least a region of interest). “Spacelike” means that within each hypersurface, any two points are separated by a spacelike interval (ds^2 > 0 in the +--- metric signature, or ds^2 > 0 for spatial separations in the -+++ signature we will implicitly use more often for the final 3+1 form), ensuring that no causal signal can connect two points within the same slice. Each hypersurface Σ_t can be thought of as “space at time t”. The parameter t acts as a global time function, labeling the slices. Mathematically, t is a scalar field on M such that its gradient ∇_μ t is everywhere timelike and past-directed (or future-directed, by convention). The hypersurfaces Σ_t are then defined as the level surfaces of this time function: Σ_t = { p ∈ M | t(p) = constant }.

2. The Normal Vector:
At each point p on a given hypersurface Σ_t, we can define a unique future-pointing timelike unit vector n^μ that is orthogonal to the hypersurface. Orthogonality means that n^μ is orthogonal to any vector v^μ that is tangent to Σ_t at p:
g_μν n^μ v^ν = n_μ v^μ = 0
Being a unit vector means it has a constant norm, which by convention for a timelike vector is -1:
g_μν n^μ n^ν = n_μ n^μ = -1
This unit normal vector n^μ plays a crucial role. It defines the direction “perpendicular” to the spatial slice at each point. An observer whose four-velocity is aligned with n^μ is called an “Eulerian observer” – they are momentarily at rest with respect to the spatial hypersurface Σ_t and move orthogonally from one slice to the next. The components of n^μ are related to the gradient of the time function t:
n_μ = - N ∇_μ t
where N is a positive scalar function called the lapse function, which we will define shortly. The normalization n_μ n^μ = -1 implies N = (-g^{μν} ∇_μ t ∇_ν t)^{-1/2}.

3. The Time Flow Vector:
While the normal vector n^μ describes the direction orthogonal to the slices, the way we parameterize the foliation with t introduces another important vector field. Consider a coordinate system (t, x^i) adapted to the foliation, where t is the time coordinate labeling the slices and x^i (with Latin indices i, j, k running from 1 to 3) are spatial coordinates within each slice. The vector field t^μ = (∂/∂t)^μ represents the displacement vector connecting points with the same spatial coordinates x^i on infinitesimally nearby slices Σ_t and Σ_{t+dt}. In the adapted coordinates, t^μ has components (1, 0, 0, 0).
Crucially, the time flow vector t^μ is generally not equal to the normal vector n^μ. While n^μ is geometrically determined by the hypersurface Σ_t, t^μ depends on the choice of spatial coordinates x^i and how they are propagated from one slice to the next.

4. Relating Time Flow and the Normal:
The relationship between the time flow vector t^μ and the normal vector n^μ is the key to the 3+1 decomposition. The vector t^μ can be decomposed into components normal and tangential to the hypersurface Σ_t.
The component normal to Σ_t must be proportional to n^μ. We define the proportionality factor as the lapse function N:
Component Normal to Σ_t = N n^μ
The component tangential to Σ_t is called the shift vector N^μ (often denoted N^i when referring to its spatial components in the x^i coordinates).
Therefore, the decomposition is:
t^μ = N n^μ + N^μ

Let’s understand the geometrical meaning:
* Lapse Function (N): The lapse function N relates the coordinate time interval dt to the proper time interval dτ measured by the Eulerian observers (those moving along n^μ). If you move from a point p on Σ_t to a point p' on Σ_{t+dt} along the normal n^μ, the proper time elapsed is dτ = N dt. N essentially measures how much proper time elapses normally between adjacent slices per unit coordinate time t. If N=1, coordinate time equals proper time along the normal. If N>1, proper time “lapses” faster than coordinate time; if N<1, it lapses slower. N must be positive for t to increase towards the future. We can find N by projecting t^μ onto n_μ:
t^μ n_μ = (N n^μ + N^μ) n_μ = N (n^μ n_μ) + (N^μ n_μ)
Since N^μ is tangent to Σ_t, N^μ n_μ = 0, and n^μ n_μ = -1. Thus:
N = -t^μ n_μ

  • Shift Vector (N^μ or N^i): The shift vector N^μ lies entirely within the hypersurface Σ_t (i.e., N^μ n_μ = 0). It describes how the spatial coordinates x^i “shift” or are “dragged” as we move from slice Σ_t to slice Σ_{t+dt} along the time flow vector t^μ. If N^μ = 0, points with the same spatial coordinates x^i on adjacent slices are connected by the normal vector n^μ. In this case, the spatial coordinates are “non-shifting” or “Gaussian normal coordinates” locally. If N^μ ≠ 0, the spatial coordinate lines are tilted relative to the normal n^μ. The vector N^μ represents the velocity of the spatial coordinate system relative to the Eulerian observers, measured within the slice Σ_t. We can find N^μ by projecting t^μ onto the spatial hypersurface. This projection is achieved using the spatial projector tensor γ_μν, which we define next.

This decomposition t^μ = N n^μ + N^μ is fundamental. It separates the progression of time perpendicular to the slices (N) from the tangential motion of the coordinate system within the slices (N^μ).

III. The Geometric Decomposition: Lapse, Shift, and Spatial Metric

Having established the conceptual framework of foliation and the roles of the lapse and shift, we can now perform the explicit “splitting” of the 4D spacetime metric g_μν.

1. The Spatial Projector and Spatial Metric:
To isolate the geometry within a spatial hypersurface Σ_t, we need a way to project 4D tensors onto it. This is done using the spatial projector tensor, defined as:
γ_μν = g_μν + n_μ n_ν

Let’s check its properties:
* Symmetry: γ_μν = γ_νμ (since g_μν is symmetric).
* Projection Property: If v^μ is any 4-vector, γ_μ^α v^μ = (g_μ^α + n^α n_μ) v^μ = v^α + n^α (n_μ v^μ). This resulting vector is orthogonal to n_α: n_α (γ_μ^α v^μ) = n_α v^α + (n_α n^α) (n_μ v^μ) = n_α v^α - n_μ v^μ = 0. Thus, γ_μν projects vectors into the tangent space of Σ_t.
* Annihilation of Normal: γ_μν n^ν = (g_μν + n_μ n_ν) n^ν = n_μ + n_μ (n_ν n^ν) = n_μ + n_μ (-1) = 0. It projects the normal vector to zero.
* Idempotence (as a mixed tensor): γ^α_μ γ^μ_β = (g^α_μ + n^α n_μ) (g^μ_β + n^μ n_β) = g^α_β + n^α n_β + n^α n_β + n^α (n_μ n^μ) n_β = g^α_β + 2 n^α n_β - n^α n_β = g^α_β + n^α n_β = γ^α_β.

Crucially, γ_μν acts as the induced metric on the hypersurface Σ_t. If we take two vectors u^μ and v^μ tangent to Σ_t (so n_μ u^μ = n_μ v^μ = 0), their inner product calculated with the 4D metric g_μν is:
g_μν u^μ v^ν = (γ_μν - n_μ n_ν) u^μ v^ν = γ_μν u^μ v^ν - (n_μ u^μ) (n_ν v^ν) = γ_μν u^μ v^ν - 0 * 0 = γ_μν u^μ v^ν
This shows that γ_μν, when restricted to act on vectors tangent to Σ_t, behaves exactly like a metric tensor. It measures lengths and angles within the spatial slice. We call γ_μν the spatial metric or the first fundamental form.

2. Expressing the 4D Metric:
Now we can express the full 4D metric g_μν and its inverse g^μν in terms of the 3+1 quantities: the lapse N, the shift vector N^i (components of N^μ), and the spatial metric γ_ij (components of γ_μν in spatial coordinates x^i).

Let’s work in the adapted coordinate system (t, x^i). The basis vectors are ∂_t = t^μ and ∂_i. We know t^μ = N n^μ + N^μ, where N^μ = N^i ∂_i. The normal vector n^μ must be orthogonal to the spatial basis vectors ∂_i. Also, n_μ n^μ = -1. From t^μ = (1, 0, 0, 0), we have t^μ n_μ = n_0 = -N.
The spatial metric components are γ_ij = g(∂_i, ∂_j) = g_{ij}. Since ∂_i are tangent to Σ_t, we also have γ_μν ∂_i^μ ∂_j^ν = γ_{ij}.
The shift vector components are defined such that N^μ = N^i ∂_i. In coordinates, N^μ = (0, N^1, N^2, N^3). We can also find N_i:
N_i = g_{iμ} N^μ = g_{iν} N^j ∂_j^ν = g_{ij} N^j = γ_{ij} N^j.
The relation t^μ = N n^μ + N^μ implies n^μ = (t^μ - N^μ) / N.
Let’s find the components of n^μ and n_μ:
n^μ = (1/N, -N^i/N) (This uses t^μ=(1,0,0,0) and N^μ=(0, N^i))
n_μ = g_{μν} n^ν = g_{μ0} n^0 + g_{μi} n^i = g_{μ0} (1/N) + g_{μi} (-N^i/N)
Specifically, n_0 = g_{00}/N - g_{0i} N^i/N. We already found n_0 = -N, so g_{00} - g_{0i} N^i = -N^2.
And n_j = g_{j0}/N - g_{ji} N^i/N. Since n_j must be orthogonal to ∂_j (in the sense n_μ ∂_j^μ = n_j), but this only holds if n_μ is proportional to ∇_μ t. A more robust approach uses the orthogonality n_μ v^μ = 0 for tangent vectors v^μ. For v^μ = ∂_i = (0, δ^j_i), we need n_μ ∂_i^μ = n_i = 0.
So, g_{j0}/N - g_{ji} N^i/N = 0, which implies g_{0j} = g_{ij} N^i = γ_{ij} N^i = N_j.
Substituting this back into the equation for g_{00}: g_{00} - (N_i N^i) = -N^2, so g_{00} = -N^2 + N_i N^i = -N^2 + γ_{ij} N^i N^j.

Collecting these results, the components of the 4D metric g_μν in the (t, x^i) coordinates are:
g_{00} = -N^2 + N_k N^k = -N^2 + γ_{kl} N^k N^l
g_{0i} = g_{i0} = N_i = γ_{ij} N^j
g_{ij} = γ_{ij}

We can now write the spacetime line element ds^2 = g_μν dx^μ dx^ν using these components:
ds^2 = g_{00} dt^2 + 2 g_{0i} dt dx^i + g_{ij} dx^i dx^j
ds^2 = (-N^2 + N_k N^k) dt^2 + 2 N_i dt dx^i + γ_{ij} dx^i dx^j
ds^2 = -N^2 dt^2 + (γ_{ij} N^i N^j) dt^2 + 2 (γ_{ij} N^j) dt dx^i + γ_{ij} dx^i dx^j
ds^2 = -N^2 dt^2 + γ_{ij} (N^i N^j dt^2 + 2 N^j dt dx^i + dx^i dx^j)
Recognizing the term in the parenthesis as a perfect square:
ds^2 = -N^2 dt^2 + γ_ij (dx^i + N^i dt)(dx^j + N^j dt)

This is the celebrated ADM metric. It explicitly shows how the full 4D spacetime interval is constructed from:
* N: The lapse function, determining the flow of proper time between slices.
* N^i: The shift vector components, describing the shift of spatial coordinates between slices.
* γ_ij: The spatial metric tensor, describing the intrinsic geometry of each t = constant slice.

3. The Inverse Metric:
It is also essential to find the components of the inverse metric g^μν, defined by g^{μν} g_{νλ} = δ^μ_λ. Using the block matrix inversion formula for a matrix partitioned as:
M = [[A, B], [C, D]] -> M^{-1} = [[(A-BD^{-1}C)^{-1}, -(A-BD^{-1}C)^{-1}BD^{-1}], [-D^{-1}C(A-BD^{-1}C)^{-1}, D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}]]
Here, A = g_{00} = -N^2 + N_k N^k, B = [g_{0j}] = [N_j], C = [g_{i0}] = [N_i], D = [g_{ij}] = [γ_{ij}].
D^{-1} = [γ^{ij}] (where γ^{ik} γ_{kj} = δ^i_j).
N_k N^k = N_k γ^{kl} N_l.
B D^{-1} = [N_j γ^{jk}] = [N^k].
D^{-1} C = [γ^{ik} N_k] = [N^i].
B D^{-1} C = N_j γ^{jk} N_k = N_j N^j.
A - B D^{-1} C = (-N^2 + N_j N^j) - N_j N^j = -N^2.
(A - B D^{-1} C)^{-1} = -1/N^2.

Now we compute the blocks of g^{μν}:
* g^{00} = (A - B D^{-1} C)^{-1} = -1/N^2
* g^{0i} = -(A - B D^{-1} C)^{-1} B D^{-1} = -(-1/N^2) [N^k] = N^i / N^2 (Note: index k becomes i for the block)
* g^{i0} = -D^{-1} C (A - B D^{-1} C)^{-1} = -[N^i] (-1/N^2) = N^i / N^2
* g^{ij} = D^{-1} + D^{-1} C (A - B D^{-1} C)^{-1} B D^{-1} = γ^{ij} + [N^i] (-1/N^2) [N^k] (where k becomes j)
g^{ij} = γ^{ij} - N^i N^j / N^2

So, the inverse metric components are:
g^{00} = -1/N^2
g^{0i} = g^{i0} = N^i / N^2
g^{ij} = γ^{ij} - N^i N^j / N^2

We can verify these. For example, g^{0μ} g_{μ0} = g^{00} g_{00} + g^{0i} g_{i0} = (-1/N^2)(-N^2 + N_k N^k) + (N^i/N^2)(N_i) = 1 - (N_k N^k)/N^2 + (N^i N_i)/N^2 = 1 = δ^0_0.
Also, g^{iμ} g_{μj} = g^{i0} g_{0j} + g^{ik} g_{kj} = (N^i/N^2) N_j + (γ^{ik} - N^i N^k / N^2) γ_{kj}.
Using N_j = γ_{jl} N^l:
= N^i N_j / N^2 + γ^{ik} γ_{kj} - (N^i N^k / N^2) γ_{kj}
= N^i N_j / N^2 + δ^i_j - (N^i / N^2) (N^k γ_{kj})
= N^i N_j / N^2 + δ^i_j - (N^i / N^2) N_j = δ^i_j. The relations hold.

These expressions for g_{μν} and g^{μν} in terms of N, N^i, and γ_{ij} are the explicit realization of “splitting the g”. They form the foundation for rewriting the Einstein Field Equations in the 3+1 formalism.

IV. Dynamics: Extrinsic Curvature and the Einstein Equations

With the metric decomposed, the next step is to understand the dynamics – how the spatial geometry evolves over time. This involves introducing the concept of extrinsic curvature and then projecting the Einstein equations onto the spatial slices and the normal direction.

1. Extrinsic Curvature (K_ij):
The spatial metric γ_ij describes the intrinsic geometry of the hypersurface Σ_t (its curvature as a 3D manifold, characterized by the 3D Riemann tensor ³R_{ijkl}). However, it doesn’t tell us how this slice is curved or embedded within the larger 4D spacetime. This information is captured by the extrinsic curvature tensor, K_μν.

Geometrically, K_μν measures the failure of vectors tangent to Σ_t to remain tangent when parallel transported along the normal vector n^μ. More intuitively, it measures how the normal vectors n^μ themselves change as we move tangentially within the slice. Formally, it can be defined as the projection of the covariant derivative of the normal vector field onto the hypersurface:
K_μν = γ_μ^α γ_ν^β ∇_α n_β

Let’s analyze this definition:
* It is symmetric: K_μν = K_νμ (because ∇_α n_β - ∇_β n_α = - (∂_α n_β - ∂_β n_α) = -F_{αβ}, where F is the Faraday tensor if n were related to a potential; however, for n_μ = -N ∇_μ t, the curl doesn’t necessarily vanish, but the symmetry arises from the definition involving projection and the fact n_μ n^μ = -1 implies n^μ ∇_α n_μ = 0). A clearer proof involves relating K_{μν} to the Lie derivative.
* It is purely spatial: K_μν n^μ = γ_μ^α n^μ γ_ν^β ∇_α n_β = 0 * (...) = 0. It lies entirely within the tangent space of Σ_t.

Therefore, K_μν can be fully represented by its spatial components K_{ij} in the (t, x^i) coordinates:
K_{ij} = K(∂_i, ∂_j) = K_{μν} ∂_i^μ ∂_j^ν
Using the definition: K_{ij} = γ_i^α γ_j^β ∇_α n_β = ∇_i n_j (using spatial projection property, and that ∂_i are tangent vectors).

The extrinsic curvature has a crucial dynamical interpretation: it is related to the time rate of change of the spatial metric γ_{ij}. Consider the Lie derivative of the spatial metric γ_μν along the normal vector n^ρ:
£_n γ_μν = n^ρ ∇_ρ γ_μν + γ_ρν ∇_μ n^ρ + γ_μρ ∇_ν n^ρ
Since γ_μν = g_μν + n_μ n_ν, and ∇_ρ g_μν = 0 (metric compatibility), we have ∇_ρ γ_μν = ∇_ρ (n_μ n_ν) = (∇_ρ n_μ) n_ν + n_μ (∇_ρ n_ν).
Also, γ_ρν ∇_μ n^ρ = γ_ρ^α γ_ν^β (g^{ρλ} ∇_μ n_λ) = g^{ρα} γ_α^σ γ_ν^β ∇_μ n_σ = ... This gets complicated quickly.

A more direct route connects K_{ij} to the time derivative ∂_t γ_{ij}. Recall the definition of the Lie derivative along the time flow vector t^μ = ∂_t:
£_t γ_{ij} = ∂_t γ_{ij} (in adapted coordinates where γ_{ij} components only depend on t and x^k).
We can also express the Lie derivative using the decomposition t^μ = N n^μ + N^μ:
£_t γ_{ij} = £_{Nn + N} γ_{ij} = £_{Nn} γ_{ij} + £_N γ_{ij}
£_t γ_{ij} = N £_n γ_{ij} + £_N γ_{ij}

Now, let’s relate £_n γ_{ij} to K_{ij}. Using the definition K_{ij} = ∇_i n_j (projected):
K_{ij} = γ_i^α γ_j^β ∇_α n_β. Also, £_n γ_{μν} = n^ρ ∇_ρ γ_{μν} + γ_{ρμ} ∇_ν n^ρ + γ_{ρν} ∇_μ n^ρ.
Projecting this onto spatial indices i, j:
(£_n γ)_{ij} = n^ρ ∇_ρ γ_{ij} + γ_{ρi} ∇_j n^ρ + γ_{ρj} ∇_i n^ρ
Using n^ρ ∇_ρ γ_{ij} = n^ρ (∇_ρ (g_{ij} + n_i n_j)) = n^ρ (∇_ρ n_i n_j + n_i ∇_ρ n_j) which is zero because n_i=0.
Also, γ_{ρi} ∇_j n^ρ = γ_i^α γ_ρ^σ g_{ασ} ∇_j n^ρ = γ_i^α g_{αρ} ∇_j n^ρ.
It turns out that (£_n γ_{ij}) = -2 K_{ij}. (The derivation is non-trivial, involving careful handling of projections and covariant derivatives). Intuitively, K_{ij} measures how distances within the slice change as you move normally off the slice.

Substituting this into the expression for £_t γ_{ij}:
∂_t γ_{ij} = N (-2 K_{ij}) + £_N γ_{ij}
∂_t γ_{ij} = -2 N K_{ij} + £_N γ_{ij}

The Lie derivative with respect to the shift vector N = N^k ∂_k is given by:
£_N γ_{ij} = N^k ∂_k γ_{ij} + γ_{kj} ∂_i N^k + γ_{ik} ∂_j N^k
This can also be written using the spatial covariant derivative D_i compatible with γ_{ij} (i.e., D_k γ_{ij} = 0):
£_N γ_{ij} = D_i N_j + D_j N_i
where N_i = γ_{ik} N^k. (Proof: D_i N_j = D_i (γ_{jk} N^k) = γ_{jk} D_i N^k = γ_{jk} (∂_i N^k + ³Γ^k_{il} N^l). D_j N_i = γ_{ik} (∂_j N^k + ³Γ^k_{jl} N^l). Summing them involves γ_{jk} ³Γ^k_{il} N^l + γ_{ik} ³Γ^k_{jl} N^l = (³Γ_{j,il} + ³Γ_{i,jl}) N^l. Also N^k ∂_k γ_{ij} = N^k (³Γ^l_{ki} γ_{lj} + ³Γ^l_{kj} γ_{li}) = (³Γ_{i,kj} + ³Γ_{j,ki}) N^k. Using symmetries of Christoffel symbols, the terms combine correctly).

So, we arrive at the first evolution equation:
∂_t γ_{ij} = -2 N K_{ij} + D_i N_j + D_j N_i

This equation tells us how the spatial metric γ_ij changes in time. The change is driven by the extrinsic curvature K_{ij} (scaled by the lapse N) and the spatial derivatives of the shift vector (representing the “stretching” or “shearing” caused by the coordinate flow).

2. Splitting the Einstein Equations:
Now we apply the 3+1 decomposition to the Einstein Field Equations G_{μν} = 8πG T_{μν}. We project the equations using the normal vector n^μ and the spatial projector γ_μν. This yields three sets of equations:

  • Hamiltonian Constraint: Projecting the EFE onto n^μ n^ν:
    G_{μν} n^μ n^ν = 8πG T_{μν} n^μ n^ν
    The right side is 8πG ρ_H, where ρ_H = T_{μν} n^μ n^ν is the energy density measured by the Eulerian observer.
    The left side requires relating the 4D curvature (G_{μν}) to the 3D intrinsic curvature (³R, the Ricci scalar of γ_ij) and the extrinsic curvature (K_{ij}). This is achieved via the Gauss-Codazzi equations.
    The Gauss equation relates the 4D Riemann tensor to the 3D Riemann tensor and terms involving K_{ij}. Contracting this leads to:
    G_{μν} n^μ n^ν = (1/2) (³R + K^2 - K_{ij} K^{ij})
    where ³R is the Ricci scalar of γ_ij, K = K^i_i = γ^{ij} K_{ij} is the trace of the extrinsic curvature, and K_{ij} K^{ij} = γ^{ik} γ^{jl} K_{ij} K_{kl}.
    Thus, the Hamiltonian constraint equation is:
    ³R + K^2 - K_{ij} K^{ij} = 16πG ρ_H

  • Momentum Constraint: Projecting the EFE onto n^μ and tangentially using γ_j^ν:
    γ_j^ν G_{μν} n^μ = 8πG γ_j^ν T_{μν} n^μ
    The right side is 8πG J_j, where J_j = -γ_j^ν T_{μν} n^μ is the momentum density measured by the Eulerian observer (note the minus sign convention can vary). Often written with upper index J^i = γ^{ij} J_j.
    The left side involves the Codazzi-Mainardi equation, which relates the tangential derivative of K_{ij} to the normal derivative of the 3D curvature. This leads to:
    γ_j^ν G_{μν} n^μ = D_i (K^i_j - δ^i_j K) (where D_i is the 3D covariant derivative compatible with γ_{ij}).
    Thus, the momentum constraint equations (three of them, for j=1, 2, 3) are:
    D_i (K^{ij} - γ^{ij} K) = 8πG J^j

  • Evolution Equations: Projecting the EFE purely tangentially using γ_i^μ γ_j^ν:
    γ_i^μ γ_j^ν G_{μν} = 8πG γ_i^μ γ_j^ν T_{μν}
    The right side is 8πG S_{ij}, where S_{ij} = γ_i^μ γ_j^ν T_{μν} is the spatial stress tensor measured by the Eulerian observer.
    The left side involves more complex manipulations using the Gauss-Codazzi relations and the definition of the Einstein tensor. The result relates the time evolution of the extrinsic curvature K_{ij} to spatial derivatives of the lapse and shift, the intrinsic and extrinsic curvatures, and the matter terms. The standard form is:
    ∂_t K_{ij} = -D_i D_j N + N (³R_{ij} + K K_{ij} - 2 K_{ik} K^k_j)
    + £_N K_{ij}
    - 8πG N [S_{ij} - (1/2)γ_{ij} (S - ρ_H)]

    Here, ³R_{ij} is the Ricci tensor of the spatial metric γ_{ij}, S = S^i_i = γ^{ij} S_{ij} is the trace of the spatial stress tensor, and £_N K_{ij} is the Lie derivative of K_{ij} along the shift vector N^k:
    £_N K_{ij} = N^k D_k K_{ij} + K_{kj} D_i N^k + K_{ik} D_j N^k

Summary of the ADM Equations:

The full dynamics of General Relativity, under the 3+1 split, are captured by:

  1. Definition of Extrinsic Curvature’s relation to metric time derivative:
    ∂_t γ_{ij} = -2 N K_{ij} + D_i N_j + D_j N_i (6 equations, symmetric ij)

  2. Hamiltonian Constraint:
    ³R + K^2 - K_{ij} K^{ij} = 16πG ρ_H (1 equation)

  3. Momentum Constraints:
    D_j (K^{ij} - γ^{ij} K) = 8πG J^i (3 equations)

  4. Evolution Equations for Extrinsic Curvature:
    ∂_t K_{ij} = -D_i D_j N + N (³R_{ij} + K K_{ij} - 2 K_{ik} K^k_j) + £_N K_{ij} - 8πG N [S_{ij} - (1/2)γ_{ij} (S - ρ_H)] (6 equations, symmetric ij)

This set of equations replaces the original ten Einstein Field Equations. Notice the structure:
* The variables describing the gravitational field on a spatial slice are the spatial metric γ_ij (6 components) and the extrinsic curvature K_{ij} (6 components). These constitute 12 variables per spatial point.
* The Hamiltonian and Momentum constraints (1+3 = 4 equations) do not involve time derivatives. They are constraints that the initial data (γ_ij, K_{ij}) must satisfy on the initial slice Σ_0. Furthermore, if they are satisfied initially, the evolution equations guarantee they remain satisfied on all subsequent slices.
* The evolution equations (6 for ∂_t γ_{ij} and 6 for ∂_t K_{ij}) determine how γ_ij and K_{ij} evolve in time, given the lapse N and shift N^i.
* The lapse N and shift N^i (1+3 = 4 variables) are not determined by these equations. They must be specified freely. This reflects the gauge freedom of General Relativity – the freedom to choose the coordinate system (how to label the time slices and how spatial coordinates behave).

V. Interpretation and Significance

The ADM formalism, or the “splitting of g”, provides profound insights into the structure and dynamics of General Relativity.

1. Hamiltonian Formulation:
The ADM framework naturally leads to a Hamiltonian formulation of GR. The spatial metric γ_ij can be viewed as the configuration variable (analogous to position q in particle mechanics). Its canonically conjugate momentum π^{ij} turns out to be related to the extrinsic curvature:
π^{ij} = -√γ (K^{ij} - γ^{ij} K)
where √γ is the determinant of the spatial metric γ_ij.

In terms of these canonical variables (γ_ij, π^{ij}), the Hamiltonian and momentum constraints can be written as H = 0 and H_i = 0, where H and H_i are functions of γ_{ij} and π^{ij} involving spatial derivatives. The total Hamiltonian governing the evolution is found to be a linear combination of these constraints integrated over the spatial slice:
H_total = ∫_Σ (N H + N^i H_i) d^3x

The lapse N and shift N^i appear as Lagrange multipliers enforcing the constraints H=0 and H_i=0. This structure is characteristic of gauge theories. The constraints generate the gauge transformations: H generates time reparameterizations (related to the choice of N), and H_i generate spatial diffeomorphisms (related to the choice of N^i). The fact that the Hamiltonian is purely a combination of constraints indicates that there is no absolute time evolution in GR; all evolution is relative to the chosen coordinate system or gauge. This Hamiltonian formulation is the starting point for canonical quantization approaches to quantum gravity, leading to the Wheeler-DeWitt equation (H Ψ = 0, where Ψ is the wave function of the universe).

2. The Initial Value Problem:
The ADM formalism provides a clear framework for the initial value problem of GR. To predict the future evolution of spacetime, one must:
1. Specify an initial 3D spacelike hypersurface Σ_0.
2. Define initial data (γ_ij, K_{ij}) on Σ_0 that satisfies the Hamiltonian and momentum constraints (H=0, H_i=0). Finding such constraint-satisfying initial data is a non-trivial mathematical problem in itself (e.g., using the conformal method developed by York and Lichnerowicz).
3. Choose a prescription for the lapse function N(t, x^i) and the shift vector N^i(t, x^i) for all future times t > 0. This is a gauge choice, determining the coordinate system used to describe the evolving spacetime. Different choices yield different coordinate representations of the same physical spacetime geometry. Common choices include:
* Geodesic Slicing: N=1, N^i=0. Observers with constant x^i follow geodesics normal to the initial slice. Slices might converge or intersect (caustics).
* Maximal Slicing: Choose N such that the trace of the extrinsic curvature K = γ^{ij} K_{ij} remains zero (or constant, usually zero) on all slices (∂_t K = 0 or K=0). This requires solving an elliptic PDE for N at each time step. It tends to avoid singularities.
* Harmonic Slicing: Choose N and N^i such that the time coordinate t satisfies the wave equation □t = 0. Leads to harmonic coordinates.
4. Solve the evolution equations for γ_ij and K_{ij} (∂_t γ_{ij} = ..., ∂_t K_{ij} = ...) forward in time, using the chosen N and N^i and the initial data.

The existence and uniqueness theorems (primarily due to Choquet-Bruhat) guarantee that for sufficiently smooth initial data satisfying the constraints, and reasonable choices of N and N^i (leading to globally hyperbolic spacetimes), a unique solution exists for at least a finite time interval.

3. Gauge Freedom and Degrees of Freedom:
The ADM formalism makes the gauge freedom of GR explicit through the arbitrary choice of N and N^i. These four functions correspond to the four coordinate freedoms in 4D spacetime. The remaining degrees of freedom describe the physical, propagating gravitational field.
We started with 10 components of g_μν. The 3+1 split uses γ_ij (6 components), K_{ij} (6 components), N (1 component), and N^i (3 components) – total 16 variables initially.
However, γ_ij and K_{ij} are related by the ∂_t γ_{ij} equation, so they are not entirely independent dynamically (they are like q and p=m dq/dt). The fundamental phase space variables are (γ_ij, π^{ij}), which have 12 components per point.
The 4 constraint equations (H=0, H_i=0) reduce the number of independent degrees of freedom.
The 4 gauge choices (N, N^i) also reflect non-physical freedom.
The net result is 12 - 4 (constraints) - 4 (gauge) leaves 4 phase space degrees of freedom per point. Since phase space has pairs of canonical variables (like position and momentum), this corresponds to 2 physical propagating degrees of freedom per point. These represent the two polarization states of gravitational waves. The ADM formalism helps isolate these true dynamical modes from the gauge artifacts.

VI. Applications

The “splitting of g” or ADM formalism is not just a theoretical curiosity; it is a cornerstone of several major areas within gravitational physics.

1. Numerical Relativity:
Perhaps the most significant application is in numerical relativity, the field dedicated to solving the Einstein Field Equations numerically on computers. Simulating highly dynamic and strong-field scenarios – such as the merger of black holes and neutron stars, supernova core collapse, or the evolution of the early universe – requires solving the initial value problem. The ADM equations provide a framework for this, casting GR as a Cauchy problem.
However, the original ADM equations, while correct, suffer from numerical instabilities when used directly for long-term simulations. This led to the development of alternative formulations, most notably the BSSN (Baumgarte-Shapiro-Shibata-Nakamura) formalism. BSSN is derived from the ADM split but introduces new variables (like the conformal factor of the metric, the trace of K, and conformal connection functions) and reformulates the equations to have better stability properties, making it the workhorse of modern numerical relativity simulations, especially for binary mergers that produce detectable gravitational waves. Nonetheless, BSSN is fundamentally rooted in the 3+1 decomposition pioneered by ADM.

2. Canonical Quantum Gravity:
As mentioned earlier, the Hamiltonian formulation derived from the ADM split is the starting point for canonical quantization attempts. The Wheeler-DeWitt equation, H Ψ = 0, arises directly from quantizing the Hamiltonian constraint. While this approach faces significant technical and conceptual challenges (the “problem of time”, operator ordering ambiguities, defining an inner product), it remains a major research direction in quantum gravity. Loop Quantum Gravity (LQG), another prominent approach, also heavily utilizes a connection-dynamics formulation which is related to the ADM phase space through a canonical transformation.

3. Theoretical Cosmology and Perturbation Theory:
The 3+1 formalism is useful for studying cosmological models beyond the highly symmetric Friedmann-Lemaître-Robertson-Walker (FLRW) solutions. It allows for the analysis of inhomogeneities and anisotropies in the universe and their evolution. Cosmological perturbation theory, which studies the growth of structure on top of a background FLRW spacetime, can also be effectively formulated using the 3+1 split, particularly when analyzing scalar, vector, and tensor modes (gravitational waves) in a gauge-invariant way.

4. Mathematical Relativity:
The ADM formalism plays a crucial role in proving fundamental theorems about GR, such as the positive energy theorem (proven by Schoen and Yau, and later by Witten, using initial data on spacelike hypersurfaces) and theorems regarding singularity formation (Penrose-Hawking singularity theorems often rely on concepts related to foliations and trapped surfaces).

VII. Conclusion

“Splitting the g” – the 3+1 decomposition of spacetime formalized by Arnowitt, Deser, and Misner – is a pivotal technique in General Relativity. By conceptually slicing the 4D spacetime into a sequence of 3D spatial hypersurfaces, it allows us to disentangle the intricate geometry of Einstein’s theory into more manageable components: the intrinsic geometry of space (γ_ij), its embedding in spacetime (K_{ij}), and the coordinate system’s evolution (N, N^i).

This decomposition transforms the elegant but opaque Einstein Field Equations into a set of constraint equations, governing the allowed initial states, and evolution equations, dictating how the spatial geometry changes over time. This structure illuminates the nature of GR as a constrained dynamical system, makes the gauge freedom associated with coordinate choices explicit, and provides a well-defined framework for the initial value problem.

Its significance cannot be overstated. It laid the foundation for the Hamiltonian formulation of GR, enabling decades of research into canonical quantum gravity. It is the bedrock upon which numerical relativity is built, allowing us to simulate the most extreme gravitational phenomena in the universe and predict the gravitational wave signals observed by detectors like LIGO and Virgo. Furthermore, it provides essential tools for theoretical studies in cosmology and mathematical relativity.

While seemingly a mathematical manipulation, the 3+1 split offers deep physical insights into the nature of time, space, and gravity in Einstein’s theory. It reveals how the seemingly static block universe picture of spacetime can be understood dynamically, as an evolving “now” represented by the spatial hypersurfaces, whose geometry warps and curves under the influence of matter and energy, propagating the force we call gravity. It remains an indispensable tool for anyone seeking to understand or work with the dynamics of spacetime itself.

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