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Geometry and Experience Remembered

Section 6 from The Pressure of Light by Malcolm

Also available on YouTube

“A specific inertia-generating envelope is not assumed; rather, all inertia-generating matter will consist of stars, as those in the portion of our universe accessible to our telescopes. This is compatible with the facts only when we imagine that the portion of the universe visible to us must be considered extremely small (with regard to mass) against the universe as a whole. This view played an important role for me psychologically, since it gave me the courage to continue to work at the problem when I absolutely could not find a way of obtaining covariant field equations.”

– Letter from Einstein to William de Sitter, 1916, in which Einstein remembers-remembering failing to obtain covariant field equations, then remembers visualizing the universe as it’s seen from telescopes, then remembers seeing the enormity of a universe where what’s seen from telescopes is minutely small, then remembers an awareness of a feeling of courage, which, through subsequent analysis, he remembered as connected with the immensity of the universe.

https://einsteinpapers.press.princeton.edu/vol8-trans/290

“In practice I must, and in theory I can make do with this, and I am not at all unhappy when you reject all questions that delve further. On the other hand, you must not scold me for being curious enough still to ask: Can I imagine a universe or the universe in such a way that inertia stems entirely from the masses and not at all from the boundary conditions? As long as I am clearly aware that this whim does not touch the core of the theory, it is innocent; by no means do I expect you to share this curiosity!”

– Same letter from Albert Einstein to William de Sitter, 1916, in which he reveals, shyly, weakly, his in fact firm belief that the universe is both bounded, and infinite.

https://einsteinpapers.press.princeton.edu/vol8-trans/290

Also available on YouTube

I want you to remember visualizing a glass sphere hovering above a large white table that is otherwise surrounded by the darkness of an empty universe. Don’t try to put the image in your head, that doesn’t make sense anymore, you can only remember it. Now, it’s been the case that the human mind should add a touch perception to three-dimensional visualizations, ever since artists of ancient times discovered that to create the concept of a three-dimensional object in the mind, like a sphere, it works best to show a sphere held in the hands, so that the mind can feel the curves, or balancing on a rim, so that the mind can see the weight of the sphere teetering from one side to the other, than to simply show the sight of a sphere, which the mind can only imagine, without a concept of touch, in a very low quality way. So, you may want to remember touching the glass sphere, or remember having pushed it, so that it wobbled and bounced in its hover a little, revealing its mass. 

Next, remember that there was a lamp over the sphere, directly over top. Then, remember that there was a knitted cup-coaster in your hand, and that you placed it on the side of the glass sphere, and then left it, because just as the sphere magically hovered, the coasters magically stuck. Where was the coaster’s shadow? How big was it? Can you remember? The light from the lamp did in fact reach out to infinity, that’s what light does, albeit in the faintest of brightness as it extends to the ends of the universe. This coaster, however, placed on the side of the sphere, cast a shadow much closer than infinity, to a spot on the white table just a little ways to the side of the sphere.

Remember bringing that shadow closer to the sphere without having detached the coaster from the surface of the sphere. How did you do it? You moved the coaster down the sphere towards the point that was closest to the table. As you continued to move the coaster closer and closer to the bottom, you saw the shadow get smaller and smaller. Once the coaster reached the point closest to the table, the shadow shrunk to the exact size of the coaster, and sat directly beneath it. Remember continuing the experiment by moving the coaster up to the top of the sphere. As the coaster approached the equator of the sphere, the shadow as a whole moved away from the sphere. After the coaster passed the equator, the closer edge of the shadow moved closer to the sphere, but the outer edge continued its journey towards infinity. Do you remember what happened when the coaster reached the top of the sphere, directly beneath the lamp so that it had blocked all it’s light? The shadow’s outer edge extended to infinity, creating darkness across the entirety of the white-table universe.

Einstein’s visualization is describing an infinite universe, because the coaster can move around the sphere for as long as it wants, for an infinite amount of time, even creating infinitely new path-shapes to trace. As it does, the shadow of the coaster also moves around in an infinite universe. The three-dimensional glass-sphere and it’s perfect re-representation on a two-dimensional table shows how a universe can be perfectly the same under dramatically different conceptual circumstances. 

Remember that there was a stack of coasters in your pocket, and that they were six-sided-polygons, so that you could fit them together around the sphere, and so you did. You put one on the bottom point and started stacking them around it in a connected-circle. Eventually you covered the whole sphere, and therefore all of infinity on the two dimensional table. But how was that possible? First, the coaster roamed forever around an infinite universe, exploring the two-dimensional white-table in a way that seemed very similar to space-exploration in this universe, which often sorts its solar-systems and galaxies out in flat planes like the table. When you started stacking coasters, however, the sphere quite quickly filled up in it’s very non-infinite capacity, and the two-dimensional plane became enveloped in the shadow of a fully-covered sphere. This was, without a doubt, a universe both bounded and infinite.

With this visualization, Einstein was describing exactly what he felt general-relativity said about this universe. The two-dimensional table is like three-dimensional space, the place that human minds see themselves moving around in, while time ticks by in the background without any direct influence on any object. A universe where the position of things is described as much by time as by space, is the sphere, and this sphere is the sphere on whose surface Einstein believed we lived, bounded, but infinite. I love this visualization, and before I move onto my projection of the visualization, I just want to make Einstein’s point a little clearer with a slightly different tweak.

Remember that you also had a pocket full of rectangular-knitted-pieces that you could place around the sphere, stacking up rows like a mason. Remember that you did this, leaving the bottom half of the sphere already filled, and the top half still clear, so that a circular shadow cast out to a significant distance from the sphere. Remember that you gave yourself the challenge of moving the shadow towards infinity on the two dimensional white-table one step, equal to the distance of the last step, at a time. It’s possible of course, all you had to do was remove some yarn from each new row of knitted pieces, so that the smaller pieces cast a shadow of equal size to the previous row. How did it all play out? Yes, of course you were able to continue to add layers of rectangles, expanding the shadow-sphere by shadow-rows of equal size, but the actual knitted pieces had to shrink in width with each new layer, until they were so small, and progress so slow, that not only did the knitted pieces never reach the top, they never really got that close. I believe this is wonderful way to understand the limits, and also freedoms, of space travel. It is bounded, in our small capacity to cover pathetically small distances in three-dimensional space during any one particular incremental step. But it’s also infinite in this limited-region of the universe, at least if its remembered that time is a dimension that the mind moves around in too. This was Einstein’s point about the bounded, but infinite universe, described by general relativity. 

Now, I would like you to remember how this sphere could have said just as much about infinite heterogeneity-in-time. If you only had remembered one coaster on the sphere moving on its own, like an asteroid, magically exempt from the decay of entropy, moving with inertia through the emptiness of space, it would be hard to have imagined it moving infinitely in infinitely different types of paths, since any change in its motion, without anything to change it, would violate Newton’s First Law, the law of Inertia. With Newton in mind, it seems unlikely two coasters would create much heterogeneity either, since the most intuitive dynamics for two objects in a universe is that they fall into orbit with each other, and in this magically entropy-less universe, that orbit would achieve a perfectly time-reversible state. Now with orbits in mind, you can remember a most famous example of infinite heterogeneity, the problem of three bodies in orbit. Three coasters alone in a universe would not only trace infinitely variable paths all around the sphere, but any other object added to this universe would also have to conform to a life of infinite heterogeneity, because of the inevitable intersection with the complex of orbits. 

Now, remember a single cup-coaster again, and remember it travelled a cyclical path that went from the bottom, where it cast its smallest shadow, to the top, where it cast a shadow to infinity. Remember what its shadow looked like when you moved the coaster around the sphere in a homogenous cycle that indicated no direction in time. The shadow moved like a wave, oscillating between infinite reach and smallest possible size. The coaster did not experience heterogeneity in time, because the coaster did not experience anything in its worldline that would indicate forward motion in time, towards infinite heterogeneity or any other directional concept for time. If you had placed a rectangular-knitted-piece in the path of the cup-coaster, the cup-coaster would have experienced heterogeneity-in-time, because the rectangular-knitted-piece got in its way and caused an erratic chain of events that veered the coaster away from its homogenous experience of time. What if the knitted-rectangle wasn’t in the path of the coaster, would it still have created heterogeneity? Yes, because while the knitted-cup-coaster would not have run into the knitted-rectangle, its shadow would’ve, if not simply when the shadow had increased to infinite coverage as the coaster passed beneath the lamp. This is the same thing as saying that two objects, while not intersecting in three-dimensions, nevertheless intersected in two-dimensions, or, in the dimensions of the real universe, that the two objects didn’t connect uniformly in space, but did, at some point, in time.

This continuation of Einstein’s Geometry and Experience provides some fortification for the principle of infinite-heterogeneity-in-time, because it shows that the principle has application to the philosophical foundations of quantum mechanics. Rather than predicting where the universe might decide to place an electron at a given time, or what might make a wave decide to collapse into a particle in certain experiments, a confusingly non-deterministic perspective that has prevailed since 1927, the principle of infinite-heterogeneity-in-time says the predictions are instead regarding the conditions for which a particle can appear, rather than existing in a form guaranteed to experience heterogeneity-in-time, such as a wave with infinite reach. For example, a single electron emitted so that it may travel a free path to a point of annihilation has no opportunity to experience heterogeneity-in-time, so it must exist as a wave reaching into the universe’s infinitely-heterogeneous dimensions-of-space. However, anything that is set up along that path, that in anyway creates a heterogeneous experience of time, no matter how subtly, would allow the wave to collapse into a particle, since it now has the capacity to experience heterogeneity-in-time as a particle.

This is a wonderful way to imagine quantum mechanics, because it immediately leads to the question about what other kinds of particles might waves collapse into given they have the opportunity to experience our world heterogeneously. It’s a wonderful principle, derived from a wonderful argument, built into an amazing essay, if I don’t say so myself. Of course, it could all be wrong. My expectation of my audience is that they not only read and digest this material, but respond with support, or a challenge, or a successful falsification of a statement made, or an indication that some other theory is better, or similar, but already disproven. I like to think that The Pressure of Light presents proposals that are risky, that could easily be torn down if they are wrong, because just imagine if it turns out all attempts fail and I’m right. Just imagine if all the chaos the human mind is capable of, all the infiltration the human mind is vulnerable too, the violence it can feel, the darkness, the loneliness, what if all that stress dissipated, leaving nothing but the feeling, of the pressure of light.

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