An Incredibly Simple Formula For Removing Unwanted Infinities

Reality says, "It's finite," but your math says,"It's infinite." Why? And, is there a simple way to solve the problem? Today we present an incredibly simple formula for removing unwanted infinities. First, let's define the variables we will use:

Now that we got that out of the way, let's examine why infinities occur. One way to get infinity is to add up an infinite number of numbers as the following integral-summation shows:

You may have noticed something missing in equation 1 above. The thing that's missing makes the infinity finite:

The missing ingredient is none other than dx -- a really tiny number. When the infinity is multiplied by dx it magically becomes a finite number.

So one way to rid ourselves of an unwanted infinity is to multiply it by dx or something equivalent. Equation 5 below is a general formula that always provides a finite value by canceling infinity should one arise.

The first factor is equivalent to dx. The second factor could be equivalent to an infinite sum. Let's apply equation 5 to a classical example. Suppose we have a lattice with N cells. Each lattice cell contains some fraction (epsilon) of the total energy (E):

We could just add up all the parts to get the total energy (E):

We could calculate the average energy per cell (bar-epsilon*E) and multiply that value by the number of cells (N) to get the total energy:

But let's see what happens when we apply our new formula:

At equations 13 and 14, the dx equivalent is simply 1 which is often true in a classical case. It's a bit redundant but yields the right answer as do the more familiar methods above. In this example, it's like counting bricks to get the total. Very simple but effective. It's effective because the average energy per cell is simply the total energy divided by N--the number of cells. Thus, the bar-epsilon in our formula is the reciprocal of N--and the two variables cancel, leaving the total energy E. But look what happens when bar-epsilon doesn't equal the reciprocal of N:

In the above example we have infinite or infinitely uncertain energy in each cell, but the bar-epsilons and N's cancel, the uncertainty and/or infinity is nullified and we end up with the finite energy E.

The following example shows how infinity can rear its ugly head in a seemingly classical lattice.

Here's a data table and chart for different values of n:

Notice how the density increases as n decreases. As n increases the density levels off. The same is true for a 3D lattice:

When n is increased to infinity, the density is virtually constant and behaves like bricks. Increasing and decreasing the number of bricks does not change their mass density.

When n is reduced to zero, the density blows up to infinity! No more brick-density behavior:

We can translate the lattice equation into something resembling perturbation theory:

At equation 29 above we take epsilon to infinity (which is equivalent to taking n to zero) and the answer is 1 + infinity. We can use a famous ad hoc method to rid ourselves of the infinity: simply subtract it or ignore it. By pretending it isn't there, we get a legitimate finite energy value. On the other hand, it would make more mathematical sense to reduce epsilon to zero:

Let's apply our new formula to this problem. We begin with the original lattice equation and reduce n to zero to get the infinite result. At equation 33 we multiply the infinite density by the lattice volume to get the total energy--which is also infinite.

At 33b we restate the formula for convenience. At 34 we plug in the values we get from equation 33. At 35 we see the total energy is no-longer infinite, but the correct finite value.

Now, let's consider an even more classical setup. Here's the lattice diagram:

Surely the density should always behave like bricks, since we have one dot per cell. Sixteen dots per sixteen cells has the same density as one dot per one cell:

But even bricks have a limit. If you divide a brick up into smaller and smaller pieces, eventually you end up with something that no-longer resembles a brick. We can simulate this by reducing n to 1, and k to 0 (see 39 and 40). Once again we get the dreaded infinity--but we can fix that using the new formula (see 41 through 44).

How about a real-world example? At 45 below we set the total energy of two electrons equal to their Coulomb energy at a fixed radius (ro). You will note the total energy (2Ee) is quantized at a fixed integer value (no).

If the radius were to shrink to zero, one would think the energy would explode to infinity. To solve this problem we do some algebra to derive equation 52 below:

At 51 and 52 we see the radius cancelling itself. The total energy of the electrons is determined by the quantum number n. Apparently it is possible to have zero to infinite Coulomb energy (or force) without impacting the overall energy? The crude diagrams below sheds some light on this:

The diagrams show two Coulomb energies: E1 and E2 with radii of r1 and r2, respectively. There is a unit area A. Over a large surface area there are more units of A and vice versa. There are two circles; each representing two different surface areas. Using this information we derive equation 58:

Equation 58 tells us that a large Coulomb energy over a small surface area is equal to a small Coulomb energy over a large surface area, and the total energy is the Coulomb energy over unit area A times the surface area. The total energy is, of course, the total energy of the two electrons--and that energy is finite and conserved! This result is consistent with our formula:

If we plug in the following values and do the math, we get the finite energy of the two electrons:

Thus, we now have an incredibly simple and redundant formula that shows the connection between the parts and the whole at the classical and quantum levels.

Update: Below is a proof of the formula:

Conquering the Infinite Slit Experiment

There once was a physics student named Richard Feynman who asked his professor (re: the double-slit experiment), "What happens if you increase the number of slits to infinity?" This question led to summing the infinite number of paths a particle can take from point A to point B. If each path has a finite energy and we add up all those energies, we should get infinite energy! But that can't be right, since we started with one particle with a finite energy. Energy should be conserved, so what's wrong here? That's what we shall address in this post. First, let's define the variables we will use:

To tame this infinity problem we borrow a great idea from calculus: the integral. The integral is used to find the area beneath a curve (see diagram below). Equations 1 and 2 define the integral in terms of a summation.

At equation 2, notice how we can take N (the number of y's) to infinity and end up with a finite area beneath the curve. A finite number is definitely what we are after. Another way to get the finite area under the curve is to determine the average y (see equation 3), then map our curve to a simple rectangle (see diagram below) that has the same area as the curve shape. Equation 4 shows that the integral is equal to 'a' (the x value) times the average y value.

Now, let's apply what we know to the slit experiment. At the bottom of the next diagram, a gun fires an electron with energy Ee. Let's assume that all lights and detectors are off and the electron goes through all the slits simultaneously. The energy at each slit is some multiple or fraction of Ee. These energies can vary due to constructive and destructive interference. As with our y's above, we determine the average epsilon or coefficient for each Ee.

We don't know the average energy per slit. We do know it must be greater than zero according to the Heisenberg uncertainty principle and the Compton wavelength formula (see equations 5 and 5b). To have zero energy requires infinite time or an infinite wavelength. Obviously the space we use for the experiment is less than infinity, and, the longest time on record is the age of the universe--also less than infinity, so yes, the energy per slit must be greater than zero.

If N equals infinity, we might naively multiply N by the average energy at each slit to get infinite energy:

But here's a better approach. First let's reset our dimension variables:

Let's express x and y in terms of the electron's initial energy:

Next, let's express x and y in terms of the average energy per slit:

To get dx we divide by the number of slits (N).

At 12 we sum all the y's. At 13 we convert that sum to its energy equivalent:

We solve the integral at 14. At 16 the infinities and epsilons cancel. That brings us to 17, the area under the curve, but expressed in terms of energy.

Multiply both sides by E^2/xy, find the square root which gives a finite energy for the electron.

Thus energy is conserved. Our final energy is the same as our initial energy instead of being absurdly infinite.

Can the Jet Take Off from a Conveyor Belt? Yes and No

The above physics problem has caused a lot of controversy and heated debates. You might wonder why. After all, didn't Myth Busters settle the question? Here's their video clip:

The above video certainly nearly settles the following question: "Can a plane take off from a conveyor belt if the plane and conveyor belt are going equal and opposite speeds?" Well, not if both the plane and belt speeds equal zero. LOL! Obviously the answer is also no if the plane taxis below the minimum take-off speed. In any case, the Myth Buster question is not the one before the court. The question is, "Can the 747 take off if the belt is designed to exactly match the speed of the wheels?" This question is subtly different from the Myth Buster question. It focuses on the wheel speed rather than the plane speed. As we shall see below, focusing on the wheel speed can change your answer to "Will the Jet Fly?"

To solve the problem we should first examine why the plane flies in the Myth Buster video. Below is a free body diagram:

The plane flies because the thrust force (or propeller force) is greater than the friction forces. But what happens if the thrust force exactly equals the friction forces? The plane stays on the conveyor belt. Suppose the friction forces are greater than the thrust force? The plane slides backwards. The video below provides an excellent demonstration:

Now, with all the above in mind, let's examine the question again: "Can the 747 take off if the belt is designed to exactly match the speed of the wheels?" To answer the question we should think of a scenario where the wheel speed matches the conveyor belt speed. How about when the thrust force equals the friction forces? That would keep the jet in the same spot on the conveyor belt. We can imagine the belt driving the wheels (like gears) and their respective velocities matching. Below is the relevant math:

At equation 3 we have the gear formula that shows if the conveyor belt turns at a velocity v, the wheels also turn at velocity v. At equation 4 we see that these equal and opposite velocities net zero. This corresponds and is consistent with equation 5 where the net force is zero. So we can conclude that when the thrust force is equal to the friction forces, The wheels of the jet have the same speed or velocity as the conveyor belt. The jet obviously fails to take off under this condition. Note the jet is staying in place. It's displacement velocity is zero or less than the belt velocity. The jet can only take off when the wheel speed exceeds the belt speed; i.e., when the thrust force overcomes the friction forces--when its displacement velocity is greater than zero.

So, can the 747 take off if the belt is designed to exactly match the speed of the wheels? No.

Update: If Myth Busters had done their experiment with a car, the car's speedometer would have shown the car going twice as fast as the conveyor belt. Car speed is measured from the rotation of the wheels. The plane, by contrast, measures its speed with air speed. So even though the plane's wheel speed is twice that of the belt's, the pilot thinks the plane's speed exactly matches the belt's . Given the fact the wheel speed is twice as fast, it is no surprise the plane takes off.

Update: How about a little Einstein's equivalence principle? Imagine the jet on the conveyor belt with its engines shut off. The conveyor belt accelerates until it overcomes any friction. The jet stays where is, its wheels spinning at the same rate as the conveyor belt. It kind of reminds me of the old tablecloth trick, where you yank the table cloth from under the dishes.