Singularity Theorems of Penrose and Hawking – Revolutions in General Relativity

Vishrut Kinikar

General relativity, since its advent in 1915, has predicted a wide range of phenomena, from a geometrical interpretation of space-time to mass-energy equivalence. Its insights on gravitational lensing and singularities serve as the bedrock for British physicist Roger Penrose’s revolutionary singularity theorem. However, a major development that perhaps contributed to Penrose’s discovery came in 1915 itself shortly after Einstein’s publication of general relativity through Karl Schwarzschild’s solution to Einstein’s equations. This solution predicted a singularity (a point where gravity is so intense that space-time itself breaks down) that is spherical, non-rotating, and has an event horizon – i.e. a black hole. Robert Oppenheimer and Hartland Snyder expanded on this discovery in 1939, predicting the formation of singularities with pressure-less and spherical dust clouds under idealized conditions.

Penrose’s Singularity Theorem

Roger Penrose then expanded this formation of singularities to conditions where arrangements deviate from a perfect sphere. In his 1965 theorem, he made use of a few foundational principles to reach this heavily generalized result. The theorem demonstrated that singularities were in fact, much more common than physicists previously thought. Penrose’s singularity theorem states that for a surface which is trapped, has null geodesic incompleteness, obeys the Weak Energy Condition, is causally well-behaved, a singularity must exist. What these conditions mean must be unpacked. A surface which is trapped is a surface where light rays traveling outward and inward with respect to the surface at right angles cannot escape the surface’s gravitational field. Think of a trapped surface as a cage for light rays. The null geodesic incompleteness requirement is essentially saying the same thing. A geodesic is the shortest path from one given point to another in space-time. A null geodesic is a type of geodesic known as a pseudo-Riemannian geodesic. There are three types of pseudo-Riemannian geodesics. These are timelike geodesics, null geodesics, and spacelike geodesics. These three geodesics are governed by the following equation:

ds² = −c²dt² + dx² + dy² + dz²

“ds²” refers to the space-time interval and it is this value that sets the three different pseudo-Riemannian geodesics apart. Timelike geodesics are geodesics where ds² < 0 and are related to the paths of massive bodies. Null geodesics, which are central to Penrose’s theorem, are geodesics where ds² = 0, and refer to paths of massless particles (such as light). Spacelike geodesics, on a final note, are geodesics that describe faster-than-light travel (which is not physically possible), and arise when ds² > 0. In the context of Penrose’s theorem, the null geodesics (light rays) need to have “incompleteness”, which means they cannot extend indefinitely in both directions in space-time; “completeness” refers to the ability of a geodesic to extend indefinitely in both directions of space-time. The next requirement, the fulfillment of the Weak Energy Condition, posits that the energy density recorded by any timelike observer (observer slower than the speed of light) must be nonnegative. This is represented mathematically by the following inequality:

T_μvv^μv^v ≥ 0

where the tensor product of the 4-dimensional velocities v^μv^v contracted with the energy-momentum tensor T_μv must be greater than or equal to zero. In the context of Penrose’s singularity theorem, the given surface’s energy density must be nonnegative according to the measurements of any timelike observer. The final requirement is that the surface must be causally well-behaved, meaning that time at the surface must maintain a causal order, i.e. always follow a cause-and-effect relationship. In the simplest words, time must flow from past to future how we normally experience it. Therefore, Penrose’s singularity theorem states that a surface where light (null geodesics) coming to and from it at right angles cannot extend indefinitely, which has a nonnegative energy density, and has a cause-and-effect structure of time, guarantees the existence of a singularity. The diagram below illustrates this perfectly:

Hawking’s Singularity Theorem

Shortly after Penrose’s breakthroughs, renowned British physicist Stephen Hawking applied Penrose’s work to cosmology, showing that the universe must have begun at a singularity (Big Bang). He did so by using the same assumptions of Penrose’s theorem, coupled with the Strong Energy Condition and the Raychaudhuri equation. Hawking started his work on this theorem with a simple question: If future-directed geodesics in a trapped surface are incomplete like Penrose showed, could the same be the case for past-directed geodesics?

With this question in mind, Hawking turned the Raychaudhuri equation into a simple-to-use inequality. The Raychaudhuri equation is a groundbreaking result derived by the Indian physicist Amal Kumar Raychaudhuri which elucidates the behavior of geodesics, i.e. their convergence and divergence. The Raychaudhuri equation for timelike geodesics is as follows:

dθ/dτ​ = -1/3 * θ² − σ_μvσ^μv + ω_μvω^μv − R_μvv^μv^v

“θ” is the expansion scalar – the rate of change of the volume of nearby geodesics as they travel through space-time and R_μvv^μv^v is part of the Strong Energy Condition – which states that gravity is always attractive.

The easy-to-use inequality that Hawking simplified the equation to is as follows:

dθ/dτ​ ≤ -1/3 * θ²

Hawking realized that as the proper time τ approaches a critical proper time τ_s in the past (which has an upper bound of 3/|θ₀|), θ blows up infinitely in the negative direction. Mathematically, as τ → τ_s, θ → -∞. This result posits that there exists a time in our past in which the geodesics in our universe were condensed infinitely, i.e. a singularity. Therefore, this led Hawking to conclude that space-time must have begun with a singularity.

Conclusion

The joint work of Hawking and Penrose reinforced Schwarzschild’s result with a newfound vigor, corroborating the once tentative idea of singularities. The theorems of Hawking and Penrose were perhaps the greatest successes of general relativity when applied to cosmology. These theorems altered the traditional definition of a singularities; a singularity – erstwhile thought of exclusively as a place in space-time of infinite curvature, was now conceived of as a place where geodesics are not able to extend indefinitely. This new definition was revolutionary and much more convenient for physicists. These theorems stand as towering achievements, inspiring physicists with the hope that the fields of physics and cosmology continue to undergo revolutionary breakthroughs such as these.

Image Sources:

Einstein Online