Raychaudhuri Equation and Singularity Theorems

Introduction: What is a Singularity?

Before we begin, we must rigorously define what we mean by a "singularity." Intuitively, we think of it as a point of infinite density or curvature. However, such a point cannot exist within the spacetime manifold itself, as the manifold is required to be smooth.

Instead, the modern and precise definition of a singularity is causal geodesic incompleteness.

As the paper explains, a spacetime is considered singular if it contains an inextendible causal geodesic of finite parameter length. Physically, this corresponds to an observer (timelike geodesic) or a light ray (null geodesic) whose existence abruptly ends—or begins—for no apparent reason. Our goal today is to show that GR predicts this very behavior.

1. The Raychaudhuri Equation: The Focusing of Geodesics

The core physical question is this: what happens to a small cloud of freely-falling test particles? Do they drift apart, or are they focused together by gravity? The Raychaudhuri equation provides the answer by tracking the evolution of a congruence of geodesics.

A congruence is a family of non-intersecting curves that fills a region of spacetime. We can describe it by the tangent vector field, let's say $U^{\mu }$ for a timelike congruence of observers. The rate of change of this vector field from one geodesic to its neighbor, encoded in the tensor ${\nabla }_{\nu }{U}_{\mu }$, tells us everything about the congruence's local behavior: its expansion, shear (distortion), and rotation.

The Raychaudhuri equation is the evolution equation for the expansion, $\theta ={\nabla }_{\mu }{U}^{\mu }$, which measures the fractional rate of change of the volume of our cloud of particles.

1.1 Timelike Congruences

A full derivation of the Raychaudhuri equation involves decomposing the velocity gradient tensor ${\nabla }_{\nu }{U}_{\mu }$ and applying the Ricci identity for commuting covariant derivatives. The resource below provides a beautifully clear and self-contained derivation.

As derived in the notes, the Raychaudhuri equation for a congruence of timelike geodesics with tangent vector $U^{\mu }$ (where $U^0=1,U^i=0$ in the chosen coordinates) is:
$\frac{d\theta }{d\tau }+\frac{1}{3}{\theta }^2=-R_{\mu \nu }{U}^{\mu }{U}^{\nu }-{\sigma }_{ij}{\sigma }^{ij}$
(Here we've assumed the congruence is hypersurface-orthogonal, so the vorticity/twist term is zero, and used 4D spacetime where $D-1=3$).

Let's analyze the terms:

• $\frac{d\theta }{d\tau }$: The rate of change of expansion along the flow.
• $\frac{1}{3}{\theta }^2$: This term is always positive. It shows that an already expanding congruence ($\theta >0$) tends to slow its expansion, while an already converging one ($\theta <0$) accelerates its convergence.
• ${\sigma }_{ij}{\sigma }^{ij}\equiv {\sigma }^2$: The shear squared. Since it's a sum of squares, ${\sigma }^2\ge 0$. Shear always promotes focusing.
• $R_{\mu \nu }{U}^{\mu }{U}^{\nu }$: This is the crucial term connecting focusing to the matter content via the Einstein Field Equations.

This leads us to a fundamental conclusion known as the Focusing Theorem. If we assume the Strong Energy Condition (SEC), which states that for any timelike vector $T^{\mu }$, $R_{\mu \nu }{T}^{\mu }{T}^{\nu }\ge 0$, then gravity is universally attractive. The SEC is physically equivalent to $(T_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T)T^{\mu }{T}^{\nu }\ge 0$, a condition satisfied by most forms of classical matter.

Under the SEC, the Raychaudhuri equation becomes an inequality:
$\frac{d\theta }{d\tau }\le -\frac{1}{3}{\theta }^2$
If we start with a converging congruence ($\theta (\tau =0)={\theta }_0<0$), this inequality can be integrated to show that $\theta (\tau )$ must diverge to $-\infty$ at or before a finite proper time $\tau =3/|{\theta }_0|$. This divergence signals the formation of a caustic or focal point, where the geodesics of the congruence cross. Gravity, if initially attractive, will inevitably focus worldlines together.

1.2 Null Congruences

A similar analysis can be done for a congruence of null geodesics with tangent vector field $k^{\mu }$. This is crucial for studying light rays and the formation of black holes.

The null Raychaudhuri equation (for a vorticity-free congruence) is:
$\frac{d\theta }{d\lambda }+\frac{1}{2}{\theta }^2=-R_{\mu \nu }{k}^{\mu }{k}^{\nu }-{\sigma }_{ab}{\sigma }^{ab}$
Here, $\lambda$ is the affine parameter along the null geodesics, and the expansion and shear are defined on the 2D "screen."

The relevant energy condition is the Null Energy Condition (NEC), which states that for any null vector $k^{\mu }$, $R_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$. This is equivalent to $T_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$ and is the weakest of the common energy conditions, satisfied by nearly all plausible classical matter.

Assuming the NEC, we again get a focusing inequality:
$\frac{d\theta }{d\lambda }\le -\frac{1}{2}{\theta }^2$
If a congruence of light rays starts converging (${\theta }_0<0$), it is guaranteed to form a caustic in a finite affine parameter $\lambda \le 2/|{\theta }_0|$.

2. The Singularity Theorems

The Raychaudhuri equation provides the engine for the singularity theorems. The general structure of these theorems, as identified by José Senovilla, is:

(E) Energy Condition + (C) Causality Condition + (I) Initial/Boundary Condition $⟹$ Geodesic Incompleteness

The energy condition (E) ensures gravity is attractive, driving the focusing via the Raychaudhuri equation. The initial condition (I) guarantees that a congruence starts converging somewhere. The causality condition (C) then creates a contradiction between this inevitable focusing and the global structure of spacetime, which can only be resolved if a geodesic fails to exist long enough to reach its focal point.

2.1 The Penrose Singularity Theorem (1965)

Penrose's theorem applies to the gravitational collapse of an isolated object, like a star forming a black hole. Its key innovation was the concept of a trapped surface.

Let's summarize the theorem and its proof:

Theorem (Penrose, 1965): Let $(M,g)$ be a spacetime that:

1. (E) Satisfies the Null Energy Condition ($R_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$).
2. (C) Is globally hyperbolic with a non-compact Cauchy surface (modeling an asymptotically flat, open universe).
3. (I) Contains a trapped surface (a compact, spacelike 2-surface where both families of orthogonal null geodesics are converging).

Then, $M$ is null geodesically incomplete.

• Proof Sketch:
  1. The existence of a trapped surface $S$ means, by definition, that null geodesics emerging from it are already converging ($\theta <0$).
  2. The NEC and the null Raychaudhuri equation imply these geodesics must form caustics in a finite affine parameter.
  3. Consider the boundary of the future of $S$, denoted $\partial J^{+}(S)$. This boundary is generated by null geodesics from $S$. A key result from causality theory is that these generators cannot have focal points. (If they did, one could find a timelike shortcut past the focal point, meaning it wasn't on the boundary after all).
  4. Therefore, if the spacetime were geodesically complete, the boundary $\partial J^{+}(S)$ would be compact.
  5. However, another theorem of causality states that in a globally hyperbolic spacetime with a non-compact Cauchy surface, any such achronal boundary must also be non-compact.
  6. We have a contradiction: $\partial J^{+}(S)$ must be both compact and non-compact. The only way to resolve this is to abandon the initial assumption of null geodesic completeness. The geodesics generating $\partial J^{+}(S)$ must terminate before they have a chance to form the full compact boundary.

2.2 The Hawking Singularity Theorem (1966)

Hawking's theorem applies not to a single collapsing object, but to the entire universe. It provides the mathematical foundation for the Big Bang.

Theorem (Hawking, 1966): Let $(M,g)$ be a spacetime that:

1. (E) Satisfies the Strong Energy Condition ($R_{\mu \nu }{T}^{\mu }{T}^{\nu }\ge 0$ for any timelike $T^{\mu }$).
2. (C) Is globally hyperbolic.
3. (I) Contains a compact Cauchy surface $\Sigma$ on which the expansion of the normal timelike geodesics is everywhere negative (i.e., the universe is everywhere re-collapsing).

Then, $M$ is timelike geodesically incomplete. (The time-reversed version, with everywhere positive expansion, implies past incompleteness—a Big Bang).

• Proof Sketch (for a Big Bang):
  1. Assume our present-day Cauchy surface $\Sigma$ is expanding everywhere. This means the congruence of past-directed timelike geodesics orthogonal to $\Sigma$ starts with $\theta <0$.
  2. The SEC and the timelike Raychaudhuri equation imply that every one of these past-directed geodesics must form a focal point within some maximum finite proper time, say $t_{\text{max}}$.
  3. Now, invoke global hyperbolicity. For any point $p$ in the past of $\Sigma$, there must exist a longest timelike geodesic from $p$ to $\Sigma$. Such a maximizing geodesic must intersect $\Sigma$ orthogonally.
  4. But here is the contradiction: a geodesic ceases to be a maximizing curve beyond its first focal point.
  5. Therefore, no past-directed timelike geodesic from $\Sigma$ can be extended beyond $t_{\text{max}}$ into the past, because it would have to contain a focal point and would thus fail to be the longest curve to any point beyond that. This contradicts the existence of such points in a globally hyperbolic spacetime.
  6. The conclusion is that the spacetime is past-timelike geodesically incomplete. There was a beginning.

Conclusion

Today we have journeyed to the very edge of classical General Relativity. We have shown that singularities, defined as the abrupt termination of worldlines, are not mere mathematical curiosities but are robust predictions of the theory.

Key Takeaways:

• A singularity in GR is precisely defined as causal geodesic incompleteness.
• The Raychaudhuri equation quantifies how gravity focuses congruences of geodesics, relating the change in expansion $\theta$ to curvature, shear, and vorticity.
• Energy conditions (like the NEC and SEC), which posit that gravity is attractive, are the key physical input that drives focusing.
• The Penrose Singularity Theorem shows that the formation of a trapped surface during gravitational collapse inevitably leads to a null geodesic incompleteness (a singularity).
• The Hawking Singularity Theorem shows that a re-collapsing (or, time-reversed, an expanding) universe with a compact spatial topology must be timelike geodesically incomplete (implying a Big Crunch or Big Bang).

These theorems marked a paradigm shift, demonstrating the inherent limitations of General Relativity and pointing toward the necessity of a more fundamental theory—quantum gravity—to describe the physics of singularities.

Preview of Next Module:

Having explored the extreme, strong-field limit of GR that leads to singularities, we will now change gears completely. In the next module, we will step back into the weak-field limit to study one of the most celebrated predictions of the theory: gravitational waves. Our first lesson will involve deriving the linearized Einstein equations, which describe how small perturbations of the metric propagate through spacetime as ripples, a far cry from the cataclysmic breakdown of spacetime we studied today.