The Geometry of the Universe¶
A bold claim for a magnificent book, published July 2021.
The Geometry of the Universe covers every important aspect of modern cosmology: dark matter, the Big Bang, general relativity, Mach’s Principle, black holes, quasars, galactic rotation curves and gamma-ray bursts.
It makes some very bold claims, and backs them up with very strong philosophical arguments, rigorous mathematics, and a wonderful historical account of how we got where we are.
The book was written by Colin Rourke, emeritus professor of mathematics at the University of Warwick.
Recently retired after completing 50 years of lectures.
His interests were in topology and he had a particular interest in the Poincaré Conjecture.
Poincaré Conjecture¶
Poincaré very much plays a part in the story told in the book, a little diversion is in order.
The conjecture is about 3-dimensional spaces and states:
Every simply connected, closed 3-manifold is homeomorphic to the
3-sphere.
For our purpose, the conjecture says any 3-dimensional space is the same as a 3-sphere, provided the space has certain properties.
Poincaré Conjecture is an excellent essay on the subject, written by none other than, Colin Rourke.
The theorem was eventually proven by Grigori Perelman in 2006.
Knowing that the theorem holds, we can proceed forward knowing that if part of the universe has the qualities described in the conjecture, then it is essentially a 3-sphere.
Locally, we can construct frames of reference that entail a 3-sphere of space dimensions intertwined with a dimension of time. If we fix one dimension, then 3-spheres emerge.
A not so brief history¶
The book many ways it is the antidote to the biggest selling book on theoretical physics, A brief history of time.
Time is also central to this story of geometry, but one of the book’s central arguments is that the history of time if very likely anything but brief!
Stephen Hawking went out of his way to avoid mathematical formulae in his book, believing that every equation had the potential to halve his audience.
His book delved into the physics of the event horizon’s of black holes and it’s curious similarity to the physics of the big bang.
Colin Rourke takes the approach of splitting the book into three parts. The first three chapters are intended to be accessible to a wider audience, this is followed by a more technical treatment in the following chapters, with six appendices of yet more technical detail.
Chapter 1, leads us on a journey from the ancient Greeks to Einstein.
It is only page 8 before we are introduced to some geometry, but it is perfectly possible to skip the proofs and follow the narrative of the book.
The first two chapters of the book introduce a `cast`_ of some 20 or more key characters in the story as we are taken on a journey, from the Greeks to Einstein. Some familiar names: Kepler, Newton, Einstein, but also some less well known, who played key parts in the story.
It becomes clear that there were many others very heavily involved in the evolution of theories. There was considerable debate about many aspects of modern cosmology that finally lead to the adoption the Big Bang theory.
Fred Hoyle wrote a book, The Nature of the Universe almost 70 years ago, with a second edition around 1960.
It was Hoyle, an advocate of a static universe, that coined the phrase Big Bang, mocking the idea. He proposed a continuous creation model, whereby the expansion we observe is balanced by continuous creation of new matter.
Only a single particle needs to be created per year, for each cubic kilometer of space, in order to balance the expansion.
Observations of quasars, which have high red-shift were assumed to be very distant, extraordinarily powerful, systems. We do not see any of these objects nearby and so this was taken as evidence supporting an early stage, post big bang, of the evolution of the universe.
The Geometry of the Universe argues that a large part of the quasar red shift we observe is, in fact, gravitational and that, quasars are relatively close objects, of very modest power, where the light they emit is subject to considerable gravitational red-shift, in accordance with general relativity.
One of the less known characters in the story is Willem de Sitter, who came up with the concept of de Sitter space, as a solution to Einstein’s general relativity equations.
de Sitter Space is, in-fact, a sphere, a 4-sphere, 3 of space, with a dimension of time interweaved.
The book proposes that de Sitter Space is a natural model for the universe. Indeed, it is generally accepted that the universe is entering a de Sitter phase. One of the book’s key arguments is that it has in fact likely been in such a state for a very long time indeed.
It is the simplest possible model for the universe, essentially being just a sphere.
It is also a vacuum solution to Einstein’s general relativity, which creates a fascinating twist in the story.
A key observation is that modern cosmology is in fact ignoring precisely half[1] of de Sitter space.
[1] Or rather, mapping half the universe onto a subspace of `dss`_, a space which had a common origin a mere 13.7 billion years ago.
The Geometry of the Universe tells a compelling tale explaining how we arrived at that situation, whilst shining a bright light to the way ahead.
Imagine an observer such as ourselves, here on earth, and a source of light, a distant galaxy arriving at the edge of our visible universe.
It is shown that any such galaxy is seen red-shifted for all but a small, finite time, creating an extreme case of observer selection bias: we only see a source blue shifted for a small, finite time of the infinite time it is visible to us.
Hence our observations are dominated by red-shifted sources and we mis-interpret the small sample of blue-shifted signals, which we see as gamma-ray bursts.
Growing evidence that the big bang is a mistake is presented. Fully formed spiral galaxies are visible in the Hubble deep field, just a few hundred million years after the supposed big bang.
Each time we are able to gaze further out into space we keep seeing more of the same: deep space looks pretty much like our local neighbourhood of space. Galaxies as far as the lenses can see.
It is going to be interesting to see what the James Webb Space Telescope reveals when it is launched later this year.
As a sensitive infra-red telescope it should pick up nearby baby galaxies with a significant red-shift and light from the edge of our universe too.
What to do?¶
One way forward is to ask what an essentially infinite space filled with galaxies would actually look like?
This is essentially what is done in the book. Imagine a universe with galaxies way beyond the Hubble distance. At the centre of each galaxy a giant, rotating black hole. What would it all look like? What effect would these distant, giant rotating mass have on our frame of reference?
This subject is introduced with a discussion of Mach’s Principle. Ernst Mach, another important member of the cast, criticised Newton’s hypothesis of a universal inertial frame as having no basis in reality.
He proposed that all assumptions about space and time should have their origin in observable quantities.
As the book notes, there are lots of ways to interpret Mach’s principle.
Rourke provides a mechanism for Mach’s principle by proposing that rotating masses drag the nearby inertial frames coherently with the rotation.
More specificly, that the effect is proportional to the mass of the body and drops of with the reciprocal of the distance.
With this assumption, he is also able to reproduce galactic rotation curves without need for dark matter.
With that not inconsiderable problem cleared up, it is time to get back to Einstein.
At the end of Chapter 1, it is recommended that readers take a look at Appendix A. It is indeed worth a visit, as special and general relativity are central to the story.
Not only does the appendix cover a complex subject in just a few pages, it does so in a way that highlights how the equations of general relativity change when the rotational frame dragging is added to the picture.
Amongst the formula in the appendix there are some wonderful nuggets, including the statement that that Einstein’s biggest blunder was not in fact his introduction of the cosmological constant, but the reintroduction of a universal time in his models for the universe in the large.
This in turn leads to the Big Bang theory.
And a book published over 30 years ago, Stephen Hawking’s, A Brief History of Time.
It is interesting to note how much theory had shifted in the intervening time.
The Big Bang theory was now firmly established. New observations in the intervening time had been deemed to be an excellent fit with the theory, and hence confirmation, of the big bang theory.
Of particular note is the cosmic micro-wave background (CMB).
Thermal radiation, with curious spherical harmonics.
The large scale harmonics in the cosmic microwave background are much less of a puzzle if the universe has had rather more time than 13.7 billion years for the harmonics to emerge.
The radiation we see as the `cmb`_? It is the thermalised radiation from the missing parts of de Sitter Space.
References¶
Geometry of the Universe World Scientific Publishing.
Poincaré Conjecture Essay by Colin Rourke on the Poincaré Conjecture.