---
title: 1+3 Formalism in Relativity
url: https://www.emergentmind.com/topics/1-3-formalism
type: topic
---

# 1+3 Formalism in Relativity

The 1+3 formalism (sometimes called "threading") is a covariant splitting of four-dimensional Lorentzian manifolds into temporal and spatial parts relative to a chosen smooth timelike congruence $u^a$, generalizing the more restrictive 3+1 ("slicing") approach based on spacelike foliations. The 1+3 framework enables a geometric and gauge-invariant decomposition of the fields and equations of General Relativity and its generalizations, accommodating scenarios that may lack a global spacelike foliation or involve nonzero vorticity, such as rotating cosmologies, congruences in elasticity, and magnetohydrodynamic flows. This decomposition is particularly powerful for formulating constraint and evolution equations in a form adapted to the local geometry defined by $u^a$, with applications in cosmology, gravitational theory, and covariant perturbation theory.

## 1. Kinematical Decomposition and Projectors

The essential ingredient of the 1+3 formalism is the introduction of a unit future-directed timelike vector field $u^a$ on the spacetime manifold $(M,g_{ab})$, such that $g_{ab}u^a u^b = -1$ [1810.06293][1405.6319][1205.6723]. This vector represents the local "flow" (for example, the four-velocity of a set of observers or a family of fluid elements), and enables the decomposition of all tensors into components parallel and orthogonal to $u^a$.

The \emph{spatial projection tensor} is defined as
\[
h_{ab} = g_{ab} + u_a u_b,
\]
which projects vectors and tensors onto the three-dimensional "local rest space" orthogonal to $u^a$. $h_{ab}u^b = 0$ and $h^a{}_b h^b{}_c = h^a{}_c$, giving it the structure of a spatial metric on each local frame.

The covariant derivative of the flow vector $u^a$ admits the irreducible decomposition
\[
\nabla_b u_a = -u_a a_b + \tfrac{1}{3} \theta h_{ab} + \sigma_{ab} + \omega_{ab},
\]
where:
- $a_a = u^b \nabla_b u_a$ is the acceleration (vanishing for geodesic congruences);
- $\theta = \nabla_a u^a$ is the expansion scalar;
- $\sigma_{ab} = D_{\langle a} u_{b\rangle}$ is the trace-free shear tensor (symmetric, $h_{ab}$-orthogonal);
- $\omega_{ab} = D_{[a}u_{b]}$ is the antisymmetric vorticity tensor.

$D_a$ is the spatially projected covariant derivative, and angular/square brackets represent symmetric-tracefree and antisymmetric projections.

This kinematics characterizes the local deformation of congruence worldlines: $\theta$ governs isotropic expansion, $\sigma_{ab}$ parameterizes shape distortions, and $\omega_{ab}$ encodes rotation.

## 2. Covariant Derivatives and Intrinsic/Extrinsic Geometry

Once the spacetime is split via $u^a$, two natural derivative operators are defined:
- The temporal derivative, $\dot{T}_{ab...}{}^{cd...} \equiv \mathcal{L}_u T_{ab...}{}^{cd...}$, is the Lie derivative along the congruence;
- The spatial covariant derivative,
\[
D_a T_{b...}{}^{c...} = h_a{}^d h_{b}{}^e ... h^{c}{}_{f} ... \nabla_d T_{e...}{}^{f...},
\]
projects all slots orthogonally to $u^a$.

The rest space at each event is equipped with the induced metric $h_{ab}$ and the connection compatible with $h_{ab}$. For hypersurface-orthogonal congruences ($\omega_{ab}=0$), these rest spaces form bona fide spacelike hypersurfaces.

The \emph{second fundamental form} ("extrinsic curvature") is given by
\[
K_{ab} = -h_a{}^c h_b{}^d \nabla_c u_d,
\]
which in general is not symmetric for nonzero vorticity, but recovers the conventional ADM extrinsic curvature in the hypersurface-orthogonal case.

## 3. Gauss, Codazzi, and Ricci Relations

The 1+3 formalism involves projection of the full spacetime Riemann tensor onto spatial and mixed directions, yielding relations among the intrinsic spatial curvature, extrinsic curvature, and kinematical quantities:

- **Gauss equation (all spatial projection):**
  \[
  h_a{}^e h_b{}^f h_c{}^g h_d{}^h R_{efgh} = {}^{(3)}R_{abcd} + 2\sigma_{a[c}\sigma_{d]b} - 2\omega_{a[c}\omega_{d]b}
  \]
  where ${}^{(3)}R_{abcd}$ is the Riemann tensor of $D_a$ [1810.06293][1405.6319].

- **Codazzi-Mainardi equation (two spatial, one time):**
  \[
  h_a{}^e h_b{}^f h_c{}^g u^h R_{efgh} = D_{[b}\sigma_{c]a} + D_{[b}\omega_{c]a} - 2a_{[b}\omega_{c]a}
  \]

- **Ricci evolution equation (two time, one spatial):**
  \[
  h_a{}^e u^f h_b{}^g u^h R_{efgh} = -D_{(a}a_{b)} - a_a a_b + D_{\langle a} D_{b\rangle} \theta - \dot{\sigma}_{ab} + \sigma_{ac}\sigma^{c}{}_{b} - \omega_{ac}\omega^{c}{}_{b}
  \]

These relations play a central role in expressing the Einstein field equations as a set of constraint and evolution equations projected along and orthogonal to $u^a$.

## 4. Projected Einstein Equations and Constraint Structure

Projecting Einstein's equations $G_{ab} = 8\pi T_{ab}$ along and orthogonal to $u^a$ yields a manifestly covariant, gauge-invariant initial-value system [1810.06293][1405.6319][1205.6723]:

- **Hamiltonian (energy) constraint:**
  \[
  \tfrac12({}^{(3)}R - \sigma_{cd}\sigma^{cd} + \omega_{cd}\omega^{cd} + \tfrac23\theta^2) = 8\pi \rho
  \]
  where $\rho = T_{ab}u^a u^b$.

- **Momentum constraint:**
  \[
  D^b(\sigma_{ab} + \omega_{ab}) - \tfrac23 D_a \theta + 2\omega_{ab} a^b = 8\pi q_a
  \]
  where $q_a = -h_a{}^b T_{bc}u^c$.

- **Raychaudhuri (expansion) evolution:**
  \[
  \dot{\theta} = -\tfrac13\theta^2 - \sigma_{ab}\sigma^{ab} + \omega_{ab}\omega^{ab} + D_a a^a + a_a a^a - 4\pi(\rho + 3p)
  \]

- **Shear and vorticity evolution:**
  Evolution equations for $\sigma_{ab}$ and $\omega_{ab}$ are similarly projected, with precise expressions for the effect of anisotropic pressure, acceleration, and spatial derivatives.

- **Shear-balance (momentum transport) and Bianchi constraints** appear naturally in this language and are essential for the closure and consistency of the system [1205.6723].

These equations, together with conservation laws for the energy-momentum tensor and an equation of state, form a closed system suitable for cosmological and astrophysical modelling.

## 5. Generalizations and Applications

The 1+3 formalism is particularly advantageous in situations where spacelike hypersurface foliations may be ill-defined or inconvenient:
- Rotating spacetimes or congruences with nonzero vorticity, where spatial slices are ill-defined but threading structure remains [1405.6319].
- Covariant cosmological perturbation theory: all kinematical and dynamical fields admit gauge-invariant, observer-related interpretation.
- Reconstruction of modified gravity models: for example, $f(R)$-gravity, with combined electromagnetic fields and anisotropic stresses, can be efficiently handled using 1+3 projections of the modified field equations and Raychaudhuri equation [2001.03613].
- Magnetohydrodynamics, elasticity, and electromagnetic field evolution naturally take their canonical form in this language [1205.6723][2001.03613].

Table: Key fields in 1+3 formalism

| Symbol            | Geometric/Dynamical Role         | Orthogonality Condition          |
|-------------------|----------------------------------|----------------------------------|
| $u^a$             | Flow/congruence vector           | $g_{ab}u^a u^b = -1$             |
| $h_{ab}$          | Spatial metric (projector)       | $h_{ab}u^b = 0$                  |
| $a_a$             | Acceleration                     | $a_a u^a = 0$                    |
| $\theta$          | Expansion scalar                 | Scalar                           |
| $\sigma_{ab}$     | Shear tensor                     | $u^a\sigma_{ab} = 0$, traceless  |
| $\omega_{ab}$     | Vorticity tensor                 | $u^a\omega_{ab} = 0$, antisymm.  |

A plausible implication is that gauge-invariant and observer-dependent effects (such as anisotropic stress, vorticity, or expansion history) can be separated unambiguously in strongly gravitating or nontrivially topological spacetimes.

## 6. Relation to 3+1 Formalism and Reduction

The 1+3 and 3+1 approaches are closely related but distinct in generality and applicability. In the limit where the congruence $u^a$ is hypersurface-orthogonal ($\omega_{ab} = 0$), one recovers the 3+1 "slicing" approach, with $u^a$ as the normal to spacelike hypersurfaces. In this limit:
- $K_{ab}$ becomes symmetric;
- The spatial metric $h_{ab}$ can be interpreted as the induced metric on the slicing;
- The ADM (Arnowitt-Deser-Misner) formalism's coordinate-based evolution/constraint equations are recovered in covariant notation [1810.06293][1405.6319].

However, for rotating fluids, electromagnetic fields with nontrivial topology, and nontrivial matter congruences, 1+3 retains applicability where 3+1 may not be available.

Illustrative comparison:

| Feature          | 3+1 ("Slicing")      | 1+3 ("Threading")        |
|------------------|---------------------|--------------------------|
| Split direction  | Normal to $\Sigma_t$ | Timelike congruence $u^a$|
| Vorticity        | Typically zero       | Arbitrary                |
| Slicing required | Yes                  | No                       |
| Extrinsic curv.  | Symmetric            | Can be asymmetric        |

This suggests that formal differences—antisymmetric extrinsic curvature, torsion terms in Codazzi, arbitrary acceleration—are essential from the perspective of physical observer congruences and non-foliable geometries.

## 7. Practical Computation and Extensions

The 1+3 formalism has found extensive use in both analytical and numerical relativity:
- Derivation of ODE systems for spatially homogeneous cosmologies, anisotropic elastic media, and rotating fluids reduces higher-dimensional PDEs to tractable equations [1205.6723].
- Covariant perturbation theory in $f(R)$ and anisotropic cosmologies is enabled by clean separation of spatial/temporal projections [2001.03613].
- Maxwell and Einstein–Maxwell equations admit systematic split, with electric and magnetic fields decomposed directly relative to $u^a$ [2001.03613][1205.6723].
- The presence and algebraic structure of the commutator (Jacobi) and Bianchi identities is manifest in the frame approach, ensuring completeness of the field equation hierarchy.

The 1+3 formalism transparently enforces physical viability conditions—such as positivity of energy, gauge invariance, and proper decomposition of electromagnetic and gravitational stresses—across general spacetimes and matter models.

In summary, 1+3 formalism constitutes a foundational geometric tool for the study of dynamical and constraint structures in relativistic gravity, offering generality, gauge-invariance, and adaptability in a variety of physical and mathematical settings.

Source: https://www.emergentmind.com/topics/1-3-formalism