The Initial Value Problem in General Relativity

1. Slicing Spacetime: Foliation and Hypersurfaces

The first step is to "slice" the 4D spacetime manifold $(M,g_{\mu \nu })$ into a continuous family of 3D spatial surfaces, much like stacking sheets of paper to form a book. This slicing is called a foliation.

Each slice, denoted ${\Sigma }_t$, must be a spacelike hypersurface. This means that any two points on the slice are separated by a spacelike interval. This makes ${\Sigma }_t$ a valid candidate for "space at a constant time $t$".

On each slice, we can define two fundamental geometric quantities:

1. The Induced Metric (${\gamma }_{ij}$): This is the metric on the hypersurface ${\Sigma }_t$, which measures distances within the slice. It is inherited from the full spacetime metric $g_{\mu \nu }$. Mathematically, if we think of the spatial coordinates on the slice as $x^i$, then ${\gamma }_{ij}=g_{ij}$. More formally, we can define a projection tensor ${\gamma }_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$ that projects 4-vectors onto the hypersurface. The induced metric is simply the action of this projector on the 4-metric.

2. The Extrinsic Curvature ($K_{ij}$): This tensor describes how the hypersurface ${\Sigma }_t$ is curved and embedded within the larger 4D spacetime. It measures the failure of the normal vectors to remain parallel as we move across the slice.
   It is defined as the projection of the covariant derivative of the normal vector field onto the slice:
   $K_{\mu \nu }=-{\gamma }_{\mu }^{\alpha }{\gamma }_{\nu }^{\beta }{\nabla }_{\alpha }{n}_{\beta }$
   The extrinsic curvature essentially captures the "time derivative" of the spatial metric. It tells us how the geometry of the slice is changing as we move to the next slice. This can be seen from an equivalent definition using the Lie derivative along the normal vector $n$:
   $K_{\mu \nu }=-\frac{1}{2}{\mathcal{L}}_n{\gamma }_{\mu \nu }$

The pair of tensors $({\gamma }_{ij},K_{ij})$ on a given slice ${\Sigma }_t$ constitutes the initial data. They represent the geometry of space and its rate of change at a given "moment".

2. Connecting the Slices: Lapse and Shift

We have a stack of slices, but we haven't specified how the coordinates on one slice relate to those on the next. This relationship is encoded by two quantities that arise from decomposing the "time evolution vector" ${\partial }_t$, which points from a point on slice ${\Sigma }_t$ to the point with the same spatial coordinates on the next slice ${\Sigma }_{t+dt}$.

This vector can be decomposed into a component normal to the slice and a component tangential to it:

This decomposition introduces two crucial functions:

• The Lapse Function $N$: A scalar function that specifies the amount of proper time that elapses for an observer moving along the normal vector $n^{\mu }$ between adjacent slices. It controls the "time-step" between slices.
• The Shift Vector ${\beta }^i$: A vector field within the slice that describes how the spatial coordinates are "dragged" or shifted from one slice to the next. It accounts for the freedom to relabel spatial coordinates on each slice differently.

The quantities $(N,{\beta }^i)$ are not determined by the physics; they represent our freedom to choose a coordinate system—our gauge freedom. They tell us how we stack and label our slices.

With these definitions, the entire 4D spacetime metric $g_{\mu \nu }$ can be expressed in terms of the 3D quantities ${\gamma }_{ij}$, $N$, and ${\beta }^i$. The spacetime interval $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ takes the form:

or, in a more common form:

This expression is central: it shows that the 10 components of $g_{\mu \nu }$ are replaced by the 6 components of ${\gamma }_{ij}$, the 3 components of ${\beta }^i$, and the 1 scalar $N$.

To formalize these geometric concepts, we will refer to a very clear and rigorous paper on the topic.

3. Decomposing the Einstein Equations

Now for the main event: we rewrite the 10 Einstein Field Equations in terms of our 3+1 variables. This is done by projecting the EFE onto the hypersurface ${\Sigma }_t$ and along its normal vector $n^{\mu }$. This projection splits the EFE into two distinct sets of equations.

First, let's decompose the stress-energy tensor $T_{\mu \nu }$ as measured by an observer moving with 4-velocity $n^{\mu }$ (the "Eulerian observer"):

• Energy Density: $E=T_{\mu \nu }{n}^{\mu }{n}^{\nu }$
• Momentum Density: $p_i=-T_{\mu \nu }{\gamma }_i^{\mu }{n}^{\nu }$
• Spatial Stress Tensor: $S_{ij}=T_{\mu \nu }{\gamma }_i^{\mu }{\gamma }_j^{\nu }$

Projecting the Einstein tensor $G_{\mu \nu }$ in a similar fashion and relating it to these matter quantities yields the following equations (using $G=1$):

1. The Hamiltonian Constraint (from $G_{\mu \nu }{n}^{\mu }{n}^{\nu }=8\pi E$)
   This scalar equation relates the intrinsic geometry (${}^{(3)}R$) and extrinsic geometry ($K_{ij}$) of the slice to the energy density on that slice.

   $${}^{(3)}R+K^2-K_{ij}{K}^{ij}=16\pi E$$

   Here, ${}^{(3)}R$ is the Ricci scalar of the 3-metric ${\gamma }_{ij}$, and $K={\gamma }^{ij}{K}_{ij}$ is the trace of the extrinsic curvature. Notice this equation contains no time derivatives. It is a constraint that the initial data $({\gamma }_{ij},K_{ij})$ must satisfy at $t=0$.

2. The Momentum Constraint (from $G_{\mu \nu }{\gamma }_i^{\mu }{n}^{\nu }=8\pi p_i$)
   This set of 3 vector equations relates the spatial derivatives of the extrinsic curvature to the momentum density.

   $$D_j{K}_i^j-D_{i}K=8\pi p_i$$

   Here, $D_j$ is the covariant derivative associated with the 3-metric ${\gamma }_{ij}$. Like the Hamiltonian constraint, these equations contain no time derivatives and must be satisfied by the initial data.

3. The Evolution Equations (from $G_{\mu \nu }{\gamma }_i^{\mu }{\gamma }_j^{\nu }=8\pi S_{ij}$)
   This set of 6 tensor equations contains time derivatives and governs the evolution of the geometry from one slice to the next. They can be written as evolution equations for ${\gamma }_{ij}$ and $K_{ij}$:

   $$\frac{\partial }{\partial t}{\gamma }_{ij}=-2N{K}_{ij}+D_i{\beta }_j+D_j{\beta }_i$$ $$\frac{\partial }{\partial t}{K}_{ij}=-D_i{D}_{j}N+N\left({}^{(3)}{R}_{ij}+K{K}_{ij}-2K_{ik}{K}_j^k\right)+{\mathcal{L}}_{\beta }{K}_{ij}-8\pi N\left(S_{ij}-\frac{1}{2}{\gamma }_{ij}(S-E)\right)$$

   (where ${\mathcal{L}}_{\beta }$ is the Lie derivative along the shift vector).

These equations are the 3+1 decomposition of the EFE. The initial value problem is now clear:

1. On a 3-manifold $\Sigma$, choose initial data $({\gamma }_{ij},K_{ij})$ that solves the 4 constraint equations.
2. Choose a gauge, i.e., specify the lapse $N$ and shift ${\beta }^i$ for all time.
3. Use the 6 evolution equations to evolve $({\gamma }_{ij},K_{ij})$ forward in time, generating the full 4D spacetime.

This entire procedure is described in detail in the following resource.

4. The Hamiltonian Viewpoint: ADM Formalism

Given your background in Hamiltonian mechanics, you will appreciate that this entire structure can be derived from an action principle, leading to a Hamiltonian formulation of General Relativity. This is the ADM formalism.

In this view:

• The configuration variables are the components of the 3-metric, ${\gamma }_{ij}(x)$.
• Their conjugate momenta, ${\pi }^{ij}(x)$, are constructed from the extrinsic curvature: ${\pi }^{ij}=\sqrt{\gamma }(K^{ij}-{\gamma }^{ij}K)$.
• The Hamiltonian is not a single function but a linear combination of the constraints:
  $H_{ADM}={\int }_{\Sigma }{d}^{3}x(N\mathcal{H}+{\beta }_i{\mathcal{H}}^i)$
  where $\mathcal{H}=0$ and ${\mathcal{H}}^i=0$ are precisely the Hamiltonian and momentum constraint equations.

From this perspective, the lapse $N$ and shift ${\beta }^i$ are not dynamical variables. They are Lagrange multipliers that enforce the constraints. Hamilton's equations for $({\gamma }_{ij},{\pi }^{ij})$ generate the time evolution, while varying the action with respect to $N$ and ${\beta }^i$ gives back the constraint equations.

This Hamiltonian structure is not just elegant; it's essential for attempts to quantize gravity and it reveals the deep geometric meaning of the constraints:

• The momentum constraint ${\mathcal{H}}^i$ generates spatial diffeomorphisms (coordinate changes within the slice).
• The Hamiltonian constraint $\mathcal{H}$ generates "time" evolution normal to the slice.

The following resource provides a beautiful and complete treatment of this Hamiltonian approach.

Conclusion

Today, we have fundamentally reframed the Einstein Field Equations from a static description of a 4D block universe into a dynamic, evolutionary system. This is a crucial shift in perspective that makes the theory computationally tractable.

Key Takeaways:

• The Initial Value Problem in GR involves specifying initial data on a 3D spacelike hypersurface and evolving it to construct a full 4D spacetime.
• The initial data consists of the induced 3-metric ${\gamma }_{ij}$ (the geometry of the slice) and the extrinsic curvature $K_{ij}$ (how the slice is embedded in spacetime).
• The lapse function $N$ and shift vector ${\beta }^i$ represent the gauge freedom in how we foliate spacetime.
• The 10 Einstein equations split into 4 constraint equations (1 Hamiltonian, 3 Momentum) that are algebraic in the initial data, and 6 evolution equations that are differential in time.
• In the ADM Hamiltonian formalism, the constraints are generated by the Hamiltonian, which is a linear combination of the constraint equations themselves, with the lapse and shift acting as Lagrange multipliers.

Preview of Next Lesson:

Now that we understand how the EFE function as a dynamic system, we will turn to one of the most important concepts in analyzing their solutions: symmetries. Spacetime symmetries, such as time-independence or rotational invariance, lead to conserved quantities, just as they do in classical mechanics. We will formalize this by defining Killing vector fields and exploring their connection to conservation laws via Noether's theorem. This will be an essential tool for finding and interpreting exact solutions to the EFE, starting with the Schwarzschild black hole.