In the above video, Sabine Hossenfelder discusses one of the shortcomings of quantizing gravity. At high energies or short distances things go haywire and you get crazy big numbers or infinities. In this post I present one possible solution to this problem. It is not the only solution, and, only experiments will reveal which solution is correct, or, reveal that none are correct. Here is a list of variables we will be working with:
Let's start things off by taking two arbitrary masses (m',m) and creating a reduced-mass Schwarzschild radius:
At equation 2 below, we express the maximum energy of the gravitational field between the two masses. Notice at equation 3, if the radius between the two masses is taken to the zero limit, you don't end up with an infinity. Instead the maximum gravitational energy is conserved, and, said energy never exceeds the maximum energy available--which is always finite.
The equation at lines 2 and 3 can be written in a more familiar form of work (energy) equals force times distance (see 3a below):
If the distance (carrot r) is great, the gravitational force (mg) is small, but if the distance goes to zero, the force blows up to infinity, but ... the energy is conserved, since the infinite force is only along a zero distance.
The next step is to quantize what we have so far. Let's take equation 3a and use a scale factor (alpha). At equation 4, we multiply alpha by a time derivative of h-bar. That takes care of energy (E). On the right side of 4 we have alpha times the time derivative of momentum (p) times the distance. The time derivative of momentum is, of course, the force.
From 4 we derive Heisenberg's uncertainty principle for the singularity (reduced by the alpha factor):
What does equation 7 tell us? It indicates that if we know the exact position of a singularity, we are completely uncertain about its momentum, and vice versa.
In the previous post entitled "Why Strings Don't Exist", we showed it is possible to have a length shorter than the famed Planck length. The question becomes, what is the minimum discrete distance, assuming there is such a thing? Normally, we use infinitesimal points to build geometrical objects like lines, planes, rectangles, etc. What happens if we use a line with the smallest magnitude possible that is greater than zero? Before we delve into these questions, let's define the variables:
OK, let's assume the fundamental building block is a line somewhere between zero and the Planck length. We'll call it d:
Let's try building a square with d:
So far, so good. All the distances appear to be no less than d. But what is the distance along the diagonal (or hypotenuse)?
The diagonal distance is a little bit more than d. The additional distance is marked in red. Notice this distance is not an integer multiple of d. To make this distance, we need a distance d plus a distance less than d. There can be no distance less than d, so we can't draw the above square. Here's an idea: draw a rectangle with 3X4 d-units. Thanks to Pythagoras, the diagonal will be 5 d-units:
Because all distances must be integer units of d, our geometry does not include squares, rectangles or triangles that have diagonals and sides that fail to have magnitudes that are integer multiples of d. But at least we found one rectangle that works--or maybe not:
If we draw lines from each d to every other d, we once again have distances that are not integer multiples of d. In the example above, we have a distance (c) of 3.6d. To have that distance, we need 3d plus a 0.6d. In our geometry, there's no such thing as 0.6d. Thus we can't draw the 3X4 rectangle. In fact, every rectangle and triangle we draw will have some distance that includes a fraction of d.
Perhaps we'll have better luck with circles? Check this out:
If we take distance d and shape it into a circle circumference (C), the diameter (D) will be less than d! D = d/pi (where C=d). If we try to draw any circle with d-units, we hit a brick wall. You see, you calculate the circumference with pi * D. If D is an integer (n) multiple of d, pi * n won't equal a circumference with an integer multiple of d. Pi is an irrational number.
OK, so perfect circles are out. How about imperfect circles? Perhaps we can replace pi with something more rational. Even if we do, circles have the same problem we encountered earlier:
There's always a line between two d's, that is not an integer multiple of d. This is also true with any shape imaginable:
So far, things look pretty hopeless for our geometry based on d. But unfortunately, there's more pain. Let's go back to the beginning and reexamine d:
Distance d is a line along the x-axis. But what is it along the y and z axis, i.e., what is its cross section?
Line d has a cross section of zero magnitude! That zero magnitude is just a single point in space with zero distance! Zero distance is not allowed, so a distance-d line is not allowed. Perhaps we can convert the line into a cylinder, so the cross-section will have a d-magnitude. Let's look at our new cross section:
Oops! The cross section is shaped like a circle. We can easily draw a red line from one d to the other that isn't an integer multiple of d. Thus it is now abundantly clear that a geometry based on distance d (instead of a point) is a dismal failure.
Update: Here is a formal proof that shows that, at any two non-parallel adjacent unit lengths (in this case the Planck length), it is possible to draw lines (lines a, b) shorter than the unit length. The following diagram is an example of an arbitrary shape:
Note that the shape is made up of connected unit lengths. Here are two examples of adjacent-unit lengths blown up to a convenient size for inspection:
Now let's take one arbitrary pair of connected unit lengths and label the lines and angles:
Now we are ready to do the proof. Here it is:
Update: The following proof demonstrates why a curved Planck-length string creates distances shorter than the so-called shortest distance. Below are some examples of curved strings:
The examples above clearly make the point, but is the point generally true? Here's the proof:
Once again we have an x-y plane or system zooming along the z-axis with momentum p. The frequency of the ground-state oscillator is n/2. We want to add up all the values of n and get infinity minus one. At step one below we multiply the sum by the exponent e to the minus epsilon power--which is equivalent to multiplying by one. Epsilon is a very tiny number that is practically zero.
The sum is equivalent to minus one-half the derivative of the sum without multiplying by n.
Since the sum adds to infinity we can replace it with another expression that amounts to infinity: an exponent e divided by one minus the exponent e.
We then convert the expression's numerator and denominator into a Taylor expansion of exponent e, and then simplify.
Next, pull out one-over-epsilon from the fraction.
The variable s, like epsilon, is a tiny number. We can use the rule below to convert the denominator.
We can now multiply the numerator by the converted denominator. Note that the simplification leaves out the terms that cancel and have high powers of epsilon. These high-power terms drop to zero when epsilon goes to its zero limit.
We simplify further by multiplying the parenthetic terms by one-over-episilon, taking the derivative with respect to epsilon and multiplying by minus one-half.
We end up with infinity minus 1/8. We need a minus one, so we need to multiply -1/8 by eight. That gives us eight dimensions. Add to that the z-axis plus time--and we get a total of ten dimensions.
So then why does M-theory have eleven dimensions if there are only ten? Back in the 1990's there were five string theories. (Now there are around E500 string theories!) How could any of those theories be the unifying theory when there were five of them? M-theory to the rescue! According to M-theory, those five theories (or E500 theories) are just different forms of the same theory.
Imagine that each of the string theories describes a unique 10D universe. Imagine all those 10D universes existing in a higher dimension. Yes, the eleventh dimension. Think of the eleventh dimension as a line (or string if you prefer), and each 10D universe is a point along that line.
With a little math similar to what we were doing above, we can derive the eleventh dimension below. We take our infinity minus 1/8 (which includes the z-axis and time) and add up an infinite number of them.
We get infinity minus one. The minus one times one is one more dimension to add to our collection. That brings us up to 11D.
But as you probably know, we can also derive 26 dimensions. We can also derive four dimensions:
So how many dimensions does our universe have? Which mathematics is telling it like it is? Well, this is where our powers of observation come in handy. The most useful mathematical models are the ones that are consistent with the reality we observe. For now, it is prudent to go with 4D, since that is what we have observed.
