Assuming Einstein's general theory of relativity, some large scale features of the structure of spacetime and an energy condition, Stephen Hawking famously showed that such a universe must contain a singularity. Up until this point I've written a series of posts on the beginning of the universe, and the Penrose-Hawking singularity theorem is one of the most well-known results in cosmology, so this post was inevitable.
Most of the work was done between 1965 and 1970, Roger Penrose, Stephen Hawking and George Ellis proved a set of singularity theorems together, all of which make varied assumptions. When I refer to the "Penrose-Hawking Singularity theorem" in this post I’m mostly referring to the last theorem, which was proven in 1969 in a famous paper titled "The singularities of gravitational collapse and cosmology". This is a theorem which proves the incompleteness of timelike geodesics, I’ll leave aside Penrose’s theorem in 1965 on null geodesics for another time.
The theorem is not very complicated but we need to first introduce a few concepts, we write the causal path through spacetime in a parametric form
Where x is the local coordinate in Minkowski spacetime and s is a parameter. We need the tangent vector $\frac{dx^{\mu }\left ( s \right )}{ds}$ to be non-zero for all s, then a path is considered causal if its tangent is everywhere timelike or null.
Causal paths in a Minkowski spacetime, M from p to q lie on a "causal diamond" $D_{p}^{q}$ (the intersection between the causal past of q and the causal future of p), and we say that in a suitable sense p and q are "compact". Causality is a restriction on spacetime, for example, if I have a two dimensional Minkowski spacetime with the metric
Where for any pairing (x, t), the differential of x with respect to t is less than one, for coordinates q = (0,0) and p = (1,0), and where the maximum Euclidean length is $\sqrt{2}$ (an upper bound on Euclidean length implies compactness). This has some important consequences, the proper time for a causal path is
The proper time has an upper bound on all causal paths from p to q, were this not the case, and proper time grows without a limit, then the set of causal paths would not converge (and wouldn't be compact). The only real assumption we needed to derive the compactness of space, was the compactness of some causal diamond $D_{p}^{q}$ and therefore it follows that there is some geodesic which maximizes the proper time.
To make this argument we needed to assume a causality condition, as well. We could imagine for example on our causal diamond that q follows a closed causal curve, from q back to itself and then to p. We have to assume that if p is close enough to q, p is contained in a local Minkowski neighbourhood of q.
Penrose-Hawking had to assume some condition that ensured the general compactness of the causal diamond $D_{p}^{q}$ this is the assumption of "general hyperbolicity". It's standard in physics, that we imagine some initial hypersurface S and specify equations of motion that determines how S evolves. In general relativity, one has to assume that S is a spacelike surface, that nearby points on S are spacelike separated because for example, $t = 0$ .
For a singularity theorem, this isn't enough, we need to assume that our initial hypersurface is achronal as well, meaning there is no timelike path connecting both q and p. Taken together, this means there is no causal relationship between q and p. But we also, of course, need the idea of an inextendible causal path, simply a causal curve that cannot be extended.
A spacetime is globally hyperbolic if it contains a Cauchy hypersurface, meaning it is an achronal spacelike hypersurface S and if some point is in M but not in S then every inextendible causal path through p intersects S. If there is a past directed causal path passing through p, that cannot be extended until it reaches S then one needs to set an initial condition on that causal path. Since S cannot be used to predict the behaviour of p.
Hawking assumes that the universe is globally hyperbolic with a Cauchy hypersurface S. He showed that the local Hubble constant has a positive minimum on an initial hypersurface S, and there is no causal path in the past of S with a point that is a proper time more than one over a local $H_{min}$.
Since the universe is globally hyperbolic, every point on that spacetime S is connected by a causal path of maximum proper time. Such a causal path is a timelike geodesic without focal points (described by the Raychaudhuri equation). Second, since the local Hubble constant is below some minimum, any past-directed timelike geodesics develop a focal point if they're orthogonal to S. Or in other words, there is an upper bound on the time the universe could have existed for.
To reach this conclusion we had to make a series of assumptions, that
A more serious alternative, option four is violated by quantum fields which include a "negative pressure vacuum" these are proposed to be what drives cosmic inflation and option five is violated if general relativity is replaced with a quantum theory of gravity, that smears out the singularity with quantum effects (adding some correction to the second Friedman equation).
The theorem is not very complicated but we need to first introduce a few concepts, we write the causal path through spacetime in a parametric form
Where x is the local coordinate in Minkowski spacetime and s is a parameter. We need the tangent vector $\frac{dx^{\mu }\left ( s \right )}{ds}$ to be non-zero for all s, then a path is considered causal if its tangent is everywhere timelike or null.
Causal paths in a Minkowski spacetime, M from p to q lie on a "causal diamond" $D_{p}^{q}$ (the intersection between the causal past of q and the causal future of p), and we say that in a suitable sense p and q are "compact". Causality is a restriction on spacetime, for example, if I have a two dimensional Minkowski spacetime with the metric
Where for any pairing (x, t), the differential of x with respect to t is less than one, for coordinates q = (0,0) and p = (1,0), and where the maximum Euclidean length is $\sqrt{2}$ (an upper bound on Euclidean length implies compactness). This has some important consequences, the proper time for a causal path is
The proper time has an upper bound on all causal paths from p to q, were this not the case, and proper time grows without a limit, then the set of causal paths would not converge (and wouldn't be compact). The only real assumption we needed to derive the compactness of space, was the compactness of some causal diamond $D_{p}^{q}$ and therefore it follows that there is some geodesic which maximizes the proper time.
To make this argument we needed to assume a causality condition, as well. We could imagine for example on our causal diamond that q follows a closed causal curve, from q back to itself and then to p. We have to assume that if p is close enough to q, p is contained in a local Minkowski neighbourhood of q.
Penrose-Hawking had to assume some condition that ensured the general compactness of the causal diamond $D_{p}^{q}$ this is the assumption of "general hyperbolicity". It's standard in physics, that we imagine some initial hypersurface S and specify equations of motion that determines how S evolves. In general relativity, one has to assume that S is a spacelike surface, that nearby points on S are spacelike separated because for example, $t = 0$ .
For a singularity theorem, this isn't enough, we need to assume that our initial hypersurface is achronal as well, meaning there is no timelike path connecting both q and p. Taken together, this means there is no causal relationship between q and p. But we also, of course, need the idea of an inextendible causal path, simply a causal curve that cannot be extended.
A spacetime is globally hyperbolic if it contains a Cauchy hypersurface, meaning it is an achronal spacelike hypersurface S and if some point is in M but not in S then every inextendible causal path through p intersects S. If there is a past directed causal path passing through p, that cannot be extended until it reaches S then one needs to set an initial condition on that causal path. Since S cannot be used to predict the behaviour of p.
Hawking assumes that the universe is globally hyperbolic with a Cauchy hypersurface S. He showed that the local Hubble constant has a positive minimum on an initial hypersurface S, and there is no causal path in the past of S with a point that is a proper time more than one over a local $H_{min}$.
Since the universe is globally hyperbolic, every point on that spacetime S is connected by a causal path of maximum proper time. Such a causal path is a timelike geodesic without focal points (described by the Raychaudhuri equation). Second, since the local Hubble constant is below some minimum, any past-directed timelike geodesics develop a focal point if they're orthogonal to S. Or in other words, there is an upper bound on the time the universe could have existed for.
To reach this conclusion we had to make a series of assumptions, that
- There are no closed time-like curves in our past. ***
- The Universe contains enough matter to create a closed trapped surface.
- Generic energy condition (geodesics encounter curvature in our past, that isn't specially aligned with it).
- Strong energy condition (gravity is always an attractive force).
- Our spacetime satisfies the equations of Einstein's General Theory of Relativity.
It's perfectly possible to violate any of these conditions, a closed timelike curve would violate our causality condition and a model exploiting this option was proposed by Gott and Li. **** Option two could be violated in some exotic spacetimes if the universe perfectly counterbalances the effects of gravity, with some de-focusing characteristics. Option three could also be violated in some physically unreasonable models, that are very specially constructed.
A more serious alternative, option four is violated by quantum fields which include a "negative pressure vacuum" these are proposed to be what drives cosmic inflation and option five is violated if general relativity is replaced with a quantum theory of gravity, that smears out the singularity with quantum effects (adding some correction to the second Friedman equation).
Although Einstein's theory of general relativity is supported overwhelmingly by evidence
- Gravitational lensing, during an eclipse a beam of light from a distant star is deflected by the gravitational field of the sun.
- Precise calculation of the precision in Mercury's orbit.
- Hulse-Taylor binary pulsar experiment, the loss of energy found in two neutron stars orbiting each other, is in exact agreement with the predictions of general relativity.
- Detection of gravitational waves.
It's still only an approximation to a proper theory of quantum gravity. Each of these assumptions equates to either assuming that geodesics start out focused, that they evolve in a way that continues focusing or that there are no focal points. A spacetime satisfying these conditions will be geodesically incomplete into the past.
End Notes
* In 1965 Roger Penrose showed in a paper titled "Gravitational collapse and spacetime singularities" that once a star collapsed passed the point where r = 2m it would not come out again. This result was already suggested by the Schwarzschild solution to GR field equations, but this involved integrating over a perfectly symmetrical sphere. Even small amounts of angular momentum in Newtonian mechanics allowed that solution to be violated, and the star could re-expand. However, Penrose's proof introduced the concept of a closed trapped surface, a region where light and matter cannot escape due to the intensity of gravitational forces and was thereby able to drop the assumption of exact symmetry.
** Hawking and Ellis were inspired by Penrose's 1965 paper and wrote in a letter to the editor a new singularity theorem generalizing the work of another cosmologist, Shepley from 1964 who worked on homogeneous models.
*** Hawking was able to prove a singularity theorem three years earlier, in 1966 which made no assumption about closed time like curves but this theorem only applied to closed, 'everywhere expanding' models of the universe, that possessed a certain geometry. This was then replaced with the 'closed trapped surface' assumption.
**** Hawking showed in 1992, that in classic theory a CTC would destroy itself, forming a singularity. In other words CTCs violate what became known as the "Chronology Protection Conjecture". Though this assumption is no longer necessary, more modern singularity theorems do not have a 'no-CTC' assumption.
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