A Planck of minimum length

In the quest for a TOE [Theory of Everything] which is also sometimes referred to a GUT [Grand Unified Theory], scientists are really actively looking for one complete set of elegant equations that can describe directly or from derived forms, any physical phenomena that we can think of. Turns out the way to do that is to find a way to unify the four forces in Nature: gravity, electromagnetism, strong nuclear force and weak nuclear force.

To unify isn’t necessarily being able to express each of th four forces in *one* physical equation. Instead the main aim is to be able to state the equations in a way that they are true over all energy scales.

Gravity is a purely attractive force between any two objects with mass, and is directly proportional to their masses and inversely proportional to the square of the distance between them. The constant is called G.
keep in mind that this theory has been revised in a major way since Newton proposed it. Einstein’s relativity was added to our understanding of gravity and as a result,
to get precise answers, you need relativistic equations now, not the classical Newtonian ones.

So what was the major upgrade? When Newton proposed a formula for gravity, he didn’t know what it was and how it worked. Einstein figured it out. Long story short, if you can imagine a plane in space as the surface of a trampoline, and you put a bowling ball in it, there will be a depression. THIS is the physical effect of gravity; the way it makes itself known by warping space-time. Depending on whether you put a bowling ball or a ping pong ball, the depression is less obvious or more obvious, and depends only on the mass of the object. Also from special relativity’s formula E=mc2, since Einstein found that there is a mass energy equivalence, it followed for proponents of CP conservation that objects with more energy might have a higher “mass”. So instead of using the rest mass for every object in every equations, if you plug in m=E/c2, then you find that for two identical particles, as you increase the energy of each particle by x units, the total gravitational energy increases by a factor of x2 units. So gravity doesn’t HAVE to stay weak for subatomic particles.

Electromagnetism acts between any two charged particles, and is repulsive for like charges and attractive for unlike charges. The proportionality equation is similar to gravity’s except for the constant and the [-1] multiplier depending on whether it’s repulsive or attractive. Ok…

Strong and weak nuclear forces are also electromagnetic in their interaction BUT they only operate at very small distances, ie the typical distances between protons and neutrons in a nucleus [or even exponentially lesser for the weak force]. The strong force keeps nuclei together for example. The weak force…well I have no idea what the weak force does…

Maxwell, a brilliant scientist, recognised that the previously individual formulas used for electricity and magnetism surprisingly went well together in the new field of electromagnetism and put them together. Today in 4 elegant equations, all EM interactions can be described.

So by figuring out the electricity was related to magnetism in the first place, and then discovering the existence of the strong and weak nuclear forces, and then identifying the latter as essentially electrostatic interactions, more than half the battle is solved, isn’t it? 3 out of 4 of the natural forces have been “unified”. How hard could it be to add gravity into the set?
Unfortunately, it’s not that simple.

One unfortunate side-effect of the infinitesimal numeral associated with G [which of course is not arbitrary], is the fact that gravity is actually very very weak, compared to EM forces. Meaning to say, gravity’s equations only show significant results at macro scales.

Previously that wasn’t a problem. We didn’t need to consider the effects of gravity of sub-atomic charged particles [unless you needed to be highly precise and totally theoretical, it really didn’t matter much]. BUT with the advances in cosmology, our known equations of physics break down at singularities where objects are extremely massive but also extremely small ie black holes. The relativistic gravitational equations don’t work when space-time curvature is too high. Space-time curvature is too high when mass density of a locality is outrageously high.

Speaking of localities, the smallest distances we can count with [so to speak AND NOT physcially] is Planck length. So far the closest we get to dealing with Planck length scales are sub-sub-atomic interactions [what with quarks and the like, and I suspect even that is nowhere theoretically close to and definitely larger than the actual Planck length]. And the major forces acting at these scales are largely the three EM forces with quantum statistical leeway.

Traditional gravitational equations don’t work at Planck scales. That’s cos they are non-renormalizable when the statistics of quantum theory are integrated with it, giving uncharacteristically huge fluctuations that totally mess up calculations. And so they only work for big objects. So what happens with calculating phenomena concerning black holes among other things? Our framework of physics that we’ve built up so painstakingly over so many centuries just fall apart. The black hole is a LOT of mass in a VERY small body. Significant mass means gravity should count more than EM. But small scale means only EM counts and gravitational calculations will just confuse the whole issue. And thus was born the coolness of what is now called string theory. I hope it works out for them and they find more indirect evidence soon so they at least know if they’re on the right track.

Our generation’s Einstein? Geez if he’s really that good, and it reads like he sure is, then PLEASE let’s not end up honouring him post-humously.

Anyway, Mr. Edward Witten is one of the foremost authorities on string theory in the world. If it works out, string theory might just be the TOE we’ve been looking for all this while but it’s really too early to say. One of the problems in string theory is that it requires 10 to 26 dimensions. That’s at least 6 more from the 4 [3 length dimensions and 1 time dimension] and we wonder where are the rest of the dimensions?

We can certainly argue that since we’re 4-dimensional, even if the Universe is more than 4-D we can’t actually see it. Meaning to say, if I drew a maze on paper and drew a small tear-drop representing a mouse, the mouse would have the same difficulty solving the maze as we would in one of those real-life 3D mazes. The mouse does not have the ability to climb out of the paper and plot his path just like we can’t climb the walls of the real-life maze and look for the way out. So as 4-D creatures, we can look down on 1D and 2D objects, recognise fellow 3D objects while we’re still alive [our existence on the 4thD], but how do we look past that? 5D, 6D, let alone 10D or even 26D?

So scientists say that the other 6D [or more] are actually curled up in Planck length dimensions are we don’t see them cos they’re too small. Visualise this: a horizontal 2D surface with regular-sized poles each 1m apart standing vertically. This is a good “analogy” for a 3D space yes? Assuming the 2D space is a given, the 3rd dimension is along the pole and every pole is equivalent in that sense. Now assume the 2D surface is a 3D space. The 6 extra dimensions would take the place of the poles, except they’re curved topologically complex shapes AND their net “height” above the “floor” would be measures of Planck length. So don’t blame yourself for not seeing thoe dimensions.

Of course that is just one proposal. Why it’s possible and plausible is interesting too. Ok, imagine you had a planet sized walnut. And say you had really good hearing and that volume of space has enough fluid matter to propogate sound. Someone blindfolds you, and they say, “Draw that shape of the walnut for me, as detailed as possible.”
So you start taking planet sized chunks of matter around you and you throw it at the walnut at about the same speed, listen for how long it takes and the depth of the sound, you’ll initially get an oblong pretty smooth round shape.
Now you use smaller asteroids, and you can draw most of the major grooves in the walnut.
Then you use ever smaller pieces, and consequently your drawing gets clearer and clearer. This is called probing, and typically for massive objects, the smaller the probe the clearer the definition.
Once you get to the scale of photons and electrons, you’re literally going to the limits of the detail you can draw without actually cloning the walnut. But the smallest particle you can “throw” at the walnut, the smallest probe, is still not as small as these plausible higher dimensions in Planck length. So, if these hidden dimensions are like a fine sieve framework, even the smallest probe you can throw is more likely to go through the holes in the sieve than hit the sieve mesh and actually “bounce back”.

Of course this is just an analogy, and the higher dimensions don’t make anything bounce back per se. But if we are able to probe at that scale, and have measuring instrumentation that can measure up to that amount of precision, there’s bound to be fluctuations in the probe’s motion that we can find as evidence of curled up higher dimensions.

So the smallest plank you’ll ever find is of Planck length.


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