The tale of how our intuitions are not scale invariant
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Humans appear to live in a small bubble of intelligibility. We seem to have a good, intuitive grip of the world, as long as things are relatively close to our size and move relatively close to our speed. Otherwise, all of our common sense breaks down. I want to explore how reality manifests so differently at different scales. Moving from small to big, slow to fast, and from near to far — the impossible becomes possible.
Big and Small
After a long walk, on a quiet fall afternoon, you take a moment to rest on a bench. No birds chirping, no wind, no people — only silence. It would be easy to say: “Ah, the world rests in stillness.” Would you just magnify things slightly, you’d be witness to an astonishing battle. Insects fighting to the death — the cells in your body constantly waging war against intruders. The equilibrium you experience on the surface is a result of a dynamic interplay of opposites just below the surface. The world is teeming with activity, yet from our limited perspective, the forest is quiet — and so are we. If we zoom in even further, to the atomic level, the chaos would be made even more salient. Particles constantly being created and annihilated out of the quantum vacuum. Every molecule in your body oscillating and bumping into its neighbors randomly. In the realm of the tiny, things only get weirder. On your park bench, you spot an ant crawling along. You localize it, and can roughly say how fast it is moving. You can say with confidence that you know both its position, and its momentum (represented canonically by the letter p, momentum is the mass of an object multiplied by its velocity). In the quantum realm, the rules are different, as given by the famous Heisenberg Uncertainty Principle.
This relation involves two quantities, the uncertainty in a position measurement, and the uncertainty in a momentum measurement. With any measurement, even our measurement of the ants position and momentum, there is some error or uncertainty involved, since no measuring device (even our eyes) is perfect. This equation says that the product of these two quantities must be larger than a given number — in this case, the reduced Planck’s constant over two. Strangely, the smaller one of these quantities is, the larger the other must be in order for the relation to hold. Simply put: the more certain we are about the position of a particle(which corresponds to a small uncertainty), the less we know about its momentum. If we completely localize a particle, and can confidently say: “Here it is!”, we now know nothing about its momentum (and vice versa).
A second strange consequence of zooming into the world is the disappearance of continuity. As the name suggests, the quantum world is quantized. Things come in discrete packets — including units of time and distance. The smallest possible distance is called the Planck length, roughly 1.61×10−35m. The smallest unit of time is the Planck time, roughly 5.39 × 10 −44 s. This means that the apparent smoothness of our macroscopic world is an illusion — and time actually jumps with Planck time intervals. As the ant appears to glide along the forest trail, he’s actually making discrete Planck length jumps. The fluid nature of our reality emerges out of ignorance.
Slow and Fast
When it comes to objects moving slowly in comparison to the speed of light, our intuition has us covered. If your car is moving down the street, I can predict with a high degree of precision where it will be at subsequent moments in time. However, as found by Einstein, our classical laws of motion do not hold close to the speed of light (represented by the letter c=300,000,000m/s).
As discovered in his theories of relativity, spacetime itself is warped when travelling at great velocities. Depending on your relative perspective, while moving close to c, time can be dilated and lengths can be contracted. The factor by which this happens is given above. As can be seen, for small values of velocity, the gamma has a negligible effect. However, the closer you get to c, the more pronounced the effect (test this out by inserting large values for v in the above equations). In the astonishing tale of the Muon, this effect plays an extraordinary role. The muon is a subatomic particle created when cosmic rays interact with our atmosphere. This particle is highly unstable and decays very rapidly. Due to its brief lifespan, about halfway down from the atmosphere to the surface of the Earth, it should decay. Thus, we detect no muons on the surface of the Earth, since they disappear before they have time to reach us. But tens of thousands of muons are detected every second. This is where our Lorentz Factor enters the picture. These particles are moving very close to the speed of light, so relativistic effects begin to play a role. From the muon’s frame of reference — it still lives for the same amount of time. However, the distance between the atmosphere and surface of the Earth is literally contracted. It has less distance to cover, giving it time to reach us before it decays. However, from our perspective, the Earth is not being contracted. We see the muons as having a longer existence (half-life), giving them time to reach the surface.
Near and Far
Our fascination with the stars dates back to the time of Aristotle. After Kepler formulated the laws of planetary motion, many of our celestial questions where answered. With the introduction of Einstein’s General Relativity, questions pertaining to the movement of bodies in the cosmos seemed to have been solved. Yet we are faced with a new, astonishing problem.
“There is little evidence that Newtonian gravity is accurate on scales much larger than 0.1pc” -Scott Tremaine
Despite what Star Wars would have you think, pc, or parsec, is a unit of length. 1 parsec is equal to approximately 3.26 light years, or 30 trillion kilometers. Once again, as we move far out from the human bubble, our intuition begins to break down. One of our formulations of gravity — Newton’s inverse square relation, does not hold at far out distances. The postulated existence of Dark Matter is a direct consequence of this. Suppose we take a typical galactic cluster, and calculate the masses of all the visible objects. From here, we can derive the gravitational forces involved. Surprisingly, given the velocity of these galaxies, the force is not enough to bind them and the cluster should scatter. Yet, the clusters remain tightly bound, and so there must be something else contributing to the gravitational attraction. Similar failures of classical gravity occur when observing distant galaxies, clusters and nebulas. At the aforementioned cosmic scales, our notions of gravity breakdown, and new theoretical entities are required to make this anomalous behaviour intelligible. Another alternative: rewrite the equations that govern acceleration and gravity at these galactic distances, and decide upon a critical point when these new laws become valid.
Understanding the true scope of our scientific and cognitive frameworks is an important step in humility. As has been shown, the more we scale away from our relative size, the more reality slips through our fingers.
“Say not, ‘I have found the truth,’ but rather, ‘I have found a truth’. Say not, ‘I have found the path of the soul.’ Say rather, ‘I have met the soul walking upon my path.’
For the soul walks upon all paths.
The soul walks not upon a line, neither does it grow like a reed.
The soul unfolds itself, like a lotus of countless petals.”