A cyclic universe
A goblet, crafted by the creator of reason,
Adorned with a hundred kisses of affection on its brow.
This potter of time, shaping such a delicate cup,
Creates it and then casts it down to the ground again. (Omar Khayyam)
The notion of a cyclical universe — one that undergoes an infinite series of births and deaths with no clear beginning or end — has roots in many Eastern cultures and religions. For example, in both Buddhism and Hinduism, the concept of “Samsara” — the endless cycle of birth, death, and rebirth — applies not only to individual souls but also to the universe itself. The universe is seen as continuously cycling through creation (Brahma), preservation (Vishnu), and destruction (Shiva), known collectively as “Kalpas”. A similar idea appears in the Persian tradition of Zurvanism, where Zurvan (time) is eternal and cyclical, encompassing creation, preservation, and destruction.
We observe cycles throughout our world on many levels — from the annual rebirth of nature to the alternation of day and night, and even in life itself. While this could be dismissed as merely a coincidence of our particular position relative to the Sun in the solar system, these cycles have much deeper, intrinsic dynamical roots. The French mathematician Henri Poincaré (1890) investigated highly general dynamical systems under minimal assumptions and discovered something very profound about them. Before we dive into his findings, let’s clarify what we mean by a dynamical system and outline the assumptions involved.
A dynamical system can be thought of as a transformation of some states into a new set of states. States are the measurements that uniquely determine a system. For example, by knowing the position (x_i) and the momentum (p_i) of particles (x_i, p_i) in a system it is uniquely determined. The space of all these variables is called the phase space X. So a dynamical system is a function from the phase space into itself, i.e.
When each state maps to a single unique state, and each state is uniquely generated by exactly one state (in other words, when f is one-to-one), the dynamical system is said to be “measure-preserving.” Here, the term “measure” has been simplified to mean the way we calculate volume in the phase space. A measure-preserving dynamical system does not shrink or expand the volume of the phase space — it keeps this volume constant. But what does that imply? We will delve into that next. You can see a video of a dynamical system that moves and change its shape below.
It’s like taking a piece of dough and kneading it to reshape it; no matter how much you alter its form, its overall volume remains unchanged.
The classical (and even quantum) dynamical systems are inherently measure preserving, this means simply that they keep the energy intact during their time evolution. In other words, the measure-preserving simply means the conservation of energy. Energy is the volume of the phase space of a system!
Another key condition for Poincaré recurrence to hold is that the phase space must be finite. In practical terms, this means the phase space should be “curved” or bounded in a way that allows it to connect back to itself, much like the surface of a sphere. While this might seem like a highly restrictive condition, in reality, most dynamical systems evolve on finite manifolds, making this requirement naturally satisfied in many cases.
The Poincaré recurrence theorem states that a system with the above conditions will “always” come back to its initial condition!
Note that the condition of the mapping function to be one-to-one also means that it has an inverse! In other words, you can take the system at any point and move forward or backward as much as you want! It is like having a videotape and fast forward (or backward) without any problem. We won’t go into the details of the proof but show how this works in an interesting example named the “Arnold’s cat map”. To understand imagine that the phase space is all the points on a torus (or donut). Now, imagine yourself on the surface of the torus, you can move in two directions and come back to your initial position as shown in the image below.
Vladimir Arnold, the great Russian mathematician proposed a map that takes a set of points (p, q) and moves them across a torus
as follows:
or equivalently:
This map takes any point and moves it forward, because torus has a finite space, the points eventually come back, but this doesn’t necessarily guarantee the recurrence. The more important point to note here is that the map preserves the measure! This can be seen by looking at the determinant of the transformation which is one (note that the determinant of an operation on a shape shows the increase/decrease in the volume of that shape). According to the Poincaré recurrence theorem, if we apply this mapping to an arbitrary shape, it will eventually return to its original shape after a finite number of iterations. This implies that a collection of particles in a system, moving together, will ultimately return to their initial configuration.
To demonstrate this, we can implement the process in Python. The results are truly remarkable: an arbitrary image undergoes what appears to be an irreversible transformation, with all its parts mixed together, yet eventually, it returns to its original state! It’s akin to tossing fruits into a blender, pressing the button, and then finding, after some finite time, the fruits reassemble into their initial form!
the results on my favorite cat, Gary:
It is important to note that while a measure-preserving system will eventually return to its initial condition, the time required for this to happen in general dynamical systems can be unimaginably long — so long that it defies human comprehension. However, this does not stop us from applying the concept to the entire universe. In cosmology, there are serious discussions about this idea. Current estimates suggest that our universe might return to its initial condition in approximately exp(10¹²⁰) billion years! By that time, not only will humanity have vanished, but even the bytes of this text will be lost in the vast expansion of the cosmos. Nevertheless, the idea remains true even if we are not around to see it!
This concept, often referred to as the “bouncing universe” theory, proposes that the universe will eventually return to its starting point and begin anew. Intriguingly, this modern hypothesis echoes ancient cosmological ideas, as if we are rediscovering ancient wisdom through the lens of advanced mathematics and physics.