3+1 ADM Decomposition of Spacetime

1. Foliating Spacetime

The first step in separating space and time is to slice, or foliate, the 4D spacetime manifold $(M,g_{\mu \nu })$ into a continuous family of 3D spacelike hypersurfaces, which we'll label ${\Sigma }_t$.

As the reading explains, a Hamiltonian formulation requires canonical variables (like position and momentum) defined at an instant in time. To achieve this in GR, we must choose a notion of "an instant in time." We do this by assuming our spacetime is globally hyperbolic, allowing it to be foliated by a family of spacelike surfaces ${\Sigma }_t$, where each slice represents "space" at a coordinate time t.

2. Geometry on the Slices: The Induced Metric

With our spacetime sliced into hypersurfaces ${\Sigma }_t$, we need to describe the geometry on these slices.

Each ${\Sigma }_t$ is a 3D Riemannian manifold whose metric is inherited from the 4D spacetime metric $g_{\mu \nu }$. To define this formally, we first introduce the timelike unit normal vector field $n^{\mu }$ to the hypersurfaces. By definition, $n^{\mu }$ is orthogonal to any vector lying within the hypersurface and is normalized such that $n^{\mu }{n}_{\mu }=-1$.

The induced metric (or spatial metric) ${\gamma }_{\mu \nu }$ is then defined as the projection of the 4D metric onto the hypersurface. It acts as a projection operator that takes any 4-vector and gives you its component tangent to ${\Sigma }_t$.

As detailed in the resource, the induced metric ${\gamma }_{\mu \nu }$ is given by:

You can verify that this tensor acts as a spatial projector. If you contract it with the normal vector $n^{\mu }$, you get zero:

And if you contract it with a vector $V^{\mu }$ that is already spatial (i.e., tangent to ${\Sigma }_t$, so $V^{\mu }{n}_{\mu }=0$), it leaves the vector unchanged:

The components of ${\gamma }_{\mu \nu }$, often written as ${\gamma }_{ij}$ using spatial indices $\{i,j,k\}=\{1,2,3\}$, give us the metric for measuring distances within a given slice ${\Sigma }_t$.

3. Geometry between the Slices: Lapse and Shift

We now have the geometry on each slice. But how do these slices stack together? We need to relate the coordinate systems on adjacent slices, ${\Sigma }_t$ and ${\Sigma }_{t+dt}$. This is accomplished by the lapse function and the shift vector.

Consider the time-evolution vector field, $t^{\mu }=(\partial /\partial t{)}^{\mu }$. This vector connects points that have the same spatial coordinates $x^i$ on successive slices. In general, this vector is not orthogonal to the slices. We can decompose it into a normal and a tangential component with respect to ${\Sigma }_t$. This decomposition defines the lapse and shift.

Following the logic in the resource, we decompose the time evolution vector $t^{\mu }$ as:

Here:

• $N$ is the lapse function. It is the magnitude of the component of $t^{\mu }$ normal to the slice. It relates coordinate time $dt$ to the proper time $d\tau$ for a "normal observer" (one whose worldline is always orthogonal to the slices): $d\tau =Ndt$. It tells you how much proper time elapses between slices.
• $N^{\mu }$ is the shift vector. It is the component of $t^{\mu }$ tangent to the slice. It describes how the spatial coordinates are dragged or shifted from one slice to the next. If you move from a point $p\in {\Sigma }_t$ to the point on ${\Sigma }_{t+dt}$ along the normal vector, your spatial coordinates will have changed by $-N^{i}dt$.

With these definitions, we have all the pieces—${\gamma }_{ij}$, $N$, and $N^i$—to reconstruct the full 4D metric. The 4D line element $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ can be written as:

where $N_i={\gamma }_{ij}{N}^j$. A more intuitive and standard way to write this is:

This expression beautifully captures the geometric meaning of the decomposition. The total spacetime interval is separated into the proper time elapsed between slices ($-N^{2}d{t}^2$) and the spatial distance on the slice ${\gamma }_{ij}d{x}^{i}d{x}^j$, corrected for the fact that the coordinates are being dragged by the shift vector.

The 10 independent components of the symmetric 4-metric $g_{\mu \nu }$ are now replaced by:

• 6 components of the symmetric spatial metric ${\gamma }_{ij}$
• 3 components of the shift vector $N^i$
• 1 component of the lapse function $N$

This gives a total of 10 components, as expected.

The final result from the reading gives the components of the metric and its inverse explicitly:

Here, ${\gamma }^{ij}$ is the matrix inverse of ${\gamma }_{ij}$.

4. A Glimpse of Dynamics: Extrinsic Curvature

We have successfully decomposed the spacetime metric. However, $g_{\mu \nu }$ only describes the geometry at one "moment". To make this a dynamical theory, we need a "velocity" variable that tells us how the geometry evolves from one slice to the next.

This role is played by the extrinsic curvature tensor, $K_{ij}$. It measures how the spatial slice ${\Sigma }_t$ is bent or embedded within the 4D spacetime. This is distinct from the intrinsic curvature of the slice (given by its 3D Ricci scalar $(3)R$), which can be calculated purely from ${\gamma }_{ij}$.

Formally, the extrinsic curvature measures how the normal vector field changes as we move across the hypersurface:

The crucial insight for dynamics, which you can see derived as K_μν = -1/2 * L_n(γ_μν) in the SciPost notes (page 12), is that $K_{ij}$ is proportional to the Lie derivative of the spatial metric along the normal vector. A more practical form, derived in the resources, relates it to the time derivative of the spatial metric:

where ${\dot{\gamma }}_{ij}={\partial }_t{\gamma }_{ij}$ and $D_i$ is the covariant derivative compatible with ${\gamma }_{ij}$.

This equation shows that $K_{ij}$ is essentially the "velocity" of the spatial metric ${\gamma }_{ij}$. In the Hamiltonian formulation, $({\gamma }_{ij},K_{ij})$ form a pair of canonical variables, analogous to $(q,p)$ in classical mechanics. The lapse $N$ and shift $N^i$ are not dynamical variables but act as Lagrange multipliers that enforce the constraint equations of GR.

Conclusion

In this lesson, we have taken the 4D block universe of General Relativity and sliced it into a dynamical sequence of 3D spaces evolving in time. This is the heart of the 3+1 decomposition.

Key Takeaways:

• The ADM formalism foliates spacetime into a series of spacelike hypersurfaces ${\Sigma }_t$.
• The spacetime geometry, encoded by $g_{\mu \nu }$, is decomposed into three fundamental ingredients:
  1. The spatial metric ${\gamma }_{ij}$, which measures distances within each slice.
  2. The lapse function $N$, which measures the flow of proper time between slices.
  3. The shift vector $N^i$, which describes how spatial coordinates are dragged between slices.
• The 4D line element takes the intuitive form $d{s}^2=-N^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt)$.
• The extrinsic curvature $K_{ij}$ measures the embedding of the spatial slices in spacetime and serves as the conjugate momentum to the spatial metric ${\gamma }_{ij}$, providing the "velocity" information needed for a dynamical description.

Preview of the Next Lesson:

We have introduced the extrinsic curvature $K_{ij}$ as the second key piece of the ADM puzzle. In our next lesson, we will delve into its properties more deeply. We will formally derive the extrinsic curvature of the spatial slice and explore its geometric and physical meaning in greater detail, setting the stage for deriving the fundamental constraint equations of General Relativity.