Of Surreal Geometry 4

 

Say you have a function f(t) from time t0, to infinity, that describes a spiral from an infinite diameter at point A = (ω, ½ω) to zero, revolving counterclockwise around and converging in limit at the point P = (½ω, ½ω). This function f(t) maps a duration of time to some point (x, y) with components of nonnegative value, in such a way that, for every revolution of this spiral, the point f(t) is half a Planck length closer to point P. Note that the duration of each successive revolution lasts less than its predecessor's.

Say you have a clock that can measure any amount of time. When you invoke this clock, it gives you a timestamp of any event whatsoever in terms of whichever definition of time you want to use to represent it. For example, in this world you can have a process like writing a book. You start with zero pages, just a blank canvas and, in your head, in your mind, there must be at least a few ideas cooking. And you start churning away. Maybe, you design an outline. Or you may just start typing away, if you're so bold and audacious.

Point of the matter is, from the moment you start, to the moment you say your manuscript is finished, our clock can tell us how much time has passed between these two moments. It can say: three months, for example. And it can also say that of those three months, you spent only about 210 hours actively interacting with your text editor, typing and deleting text sometimes, and whatnot. Those hours are not exactly evenly distributed in time. Some days, you felt under the weather and decided to binge watch the rest of that season you've been watching through your media streaming platform of preference. Some other days, you were extra motivated. So, you put in the extra work to compensate for that time you spent the whole night crying, because something happened in one of the episodes that you felt strongly identified with, or whatever.

Now, it could very well be that, if we ask this clock what is the difference between the moment at which you would end up spending half of the time you did interacting with the text editor, and half the linear time, from start to finish, our clock would report a non zero duration of linear time. Hence, that we have two different measurements that describe the "time" at which you were halfway through writing your book. And they could be many other measurements, but two suffice for this treatise, as an explanation of the problem with Einstein's relativistic curvature of spacetime. When, in reality, there is no such thing the bending of space, but of the fields that carry the particles whose force interactions we are presently aware of. So, I say, that if mass is what causes the "curvature of spacetime," what is actually curving is only the Higgs Field, that we know of. For the Higgs Field is what gives the particles of our Standard Model mass. Of this fact, Einstein was inevitably unnotified. And I'm sure he'd probably at least modify General relativity to account for this. That is, if I'm right about the existence of absolute space. Or, in general, that mathematical objects exist regardless of the human mind, and are fundamental for the understanding of the building blocks, or the elements, of what we know exists.

Let's say that, if we have a counter that takes exactly 1 unit of duration, say, of one Planck time, to count one number after the other, from zero, to infinity. Then, once it has counted all the whole numbers, we say it took ω Planck times to count them.

Let us have an infinite tape, like in a Turing Machine, and let our counter print "1" and move to the next frame in the tape every time it counts a number, every frame measuring exactly one plank length. Let us unroll our tape over the number line in the x-axis, with the first frame lying between zero and one Planck length, stretching to infinity. Let us project the number line to a circle of a Planck Length in diameter, whose bottom lies tangent to our number line at x=0, projecting through the point at the top of the circle. Then the distance of one plank length on the x-axis lies at its intersection with the projecting line when it has an angle of 45° from the y-axis. That is our scale.

If you halve the diameter of the circle, then you can say that the distance that represented one Planck length before, now represents two Planck lengths. Every time we halve the circle, our scale represents twice the length it used to have. This is a geometric progression. Then, if we halve our circle n times, our scale length will represent 2^n times the Planck length. Meaning that, if we halve it ω times, our scale will represent 2^ω times the Planck length and that the diameter of our circle would be 1/(2^ω) times the Planck length, an infinitesimal. We know then that we need to halve the circle at most k times, equal to the base-2 logarithm of ω, in order for our scale to represent the length of ω times the Planck length. We also note that for our scale length to represent ½ω, k-1 must be equal to the base-2 logarithm of (½ω), all plus one, which is consistent enough to carry this investigation further. So, I'm going to have to read a little more about arithmetic on Surreal Numbers to do so.