The Quantum Differences Between Gravity and Electromagnetism

Welcome to part five of the gravity series. To read the other parts, click here and here. Are gravity and electromagnetism (EM) the same? There are some who think they are the same force. There may have been a time in the early universe when all the forces were one force, and, as time passed, the one force evolved into the ones we are familiar with. If gravity and EM are the same, they have some distinct differences we will examine at the quantum and cosmic levels.

In previous posts we discovered that gravity is a field with ever shortening spacetime wavelengths, or, increasing energy as a falling body moves closer to where there is greater mass or energy density. In addition, particle-waves' wave numbers increase. These wave numbers have the same units as curved spacetime. Gravitational acceleration is due to the difference in energy or wave number between an upper surface layer of spacetime and a lower one.

Imagine a particle falling in a gravitational field. We can model this using Schrodinger's equation with a twist: we take the difference between the Hamiltonian in the upper layer and the Hamiltonian in the lower layer:

We want to find the change in wave number (k):

We perform a couple of more steps to get the particle's change in kinetic energy:

If we take the wave number (k) and multiply it by Planck's constant (h-bar) and divide it by the particle's mass, we get the particle's change in velocity (v).

At equation 12) notice that the first term has momentum (p). Here are two kinds of momentum: mv (mass X velocity) and fh/c (frequency X Planck's constant/light speed). Velocity (v) could be a function of one or both of these momenta--or we could have two types of velocity: equations 13 and 14 below are derived from the first term of equation 12).

Equation 13) isn't fully simplified. We want to emphasize that the particle's mass cancels itself. The velocity is the same whether the mass is big or small. We can label this velocity the change in velocity due to gravity (Vg). Take note that momentum is not conserved: a big falling mass has more momentum than a smaller falling mass.

Equation 14) tells a different tale. A change in mass does change the velocity (Ve). Momentum is conserved. This equation fits the EM force.

Equations 13) and 14) reveal that when mass (m) is very large, gravity's influence stays the same; whereas, EM's impact diminishes. When mass (m) is small, EM becomes the dominant force--gravity becomes less significant.

Another key difference between gravity and EM is EM is a function of charge; whereas, gravity is a function of mass or energy density. Gravity attracts but EM obeys Column's law (like charges repel, opposite charges attract).

The crude diagrams below demonstrate EM interactions. When the field arrows are pointing towards each other, particles A and B push apart. When the arrows point away from each other, A and B separate. Particles A and B attract each other when the field lines (arrows) flow in the same direction, as if B is flowing towards A.

The following diagrams show how A and B share gravitational field lines. This sharing causes the energy field between A and B to become more intense than the fields at the far right and left of A and B. A and B want to move away from each other and move closer together. The shared energy between them makes the latter more probable. To see how this works in more detail, click here. As the distance between A and B decreases, the shared energy between them becomes stronger and the gravitational acceleration increases (the inverse square law).

One thing EM and gravity have in common are mass-less bosons. Since they are mass-less, these bosons have unlimited range and they travel at light speed. This raises a troublesome paradox, for we know EM is much much stronger than gravity. How can atoms and molecules ever get together via gravity when their outer-shell electrons have a repulsive force far greater than gravity's attractive force?

Consider two hydrogen atoms that are close together. We fully expect the electrons to repel each other, same goes for the protons. But could the proton in one atom be attracted to the electron in the other? If so, that attraction could, to some extent, cancel the repulsive force. We can use the following equations to see if gravity is stronger or weaker than the net EM repulsive force.

When we crunch the numbers we find that as distance (d) between the atoms increases, the EM force (Fe) drops more quickly than gravity (Fg). In the diagram below, where the atoms are close together, the distance between the electrons is very short compared to the distances between opposite charges, so the repulsive force is strong. This is good news! It means the atoms will remain distinct and separate. It means the ground will be solid beneath your feet.

By contrast, when the atoms are far apart, the different distances between opposites and same-charge particles are less dramatic. The charges cancel each other and gravity dominates.

Why There is Something Rather Than Nothing

Why is there something rather than nothing? Mathematicians use zero to represent nothing. The concept of zero goes way back to ancient Egypt. The hieroglyph known as 'nfr' indicated "emptiness." Aryabhatta, an Indian mathematician, introduced zero in the 5th century AD.

I was first introduced to zero in the first grade. Our teacher, Mrs. White, taught us that 2 - 2 = 0. Later in life I was introduced to the Lagrange equation, where potential energy is subtracted from kinetic energy. Of course if kinetic energy equals potential energy, you get zero on the right side of the equation.

It's safe to say that all my life I've been brainwashed to believe you can have something, take it away, and end up with nothing: zero. And I'm not alone; philosophers have asked, "Why is there something rather than nothing?" as if nothing were a viable possibility.

The closest thing to nothing measured by the Wilkinson Microwave probe is about 6E-10 joules per cubic meter. The energy density of the vacuum of space is close to zero, but not quite. So the question arises, what is the probability of ever having nothing? In the case of the Lagrange equation, what is the probability that a universe could have the same amount of kinetic energy as potential energy?

Suppose we label kinetic energy "positive," and potential energy "negative." Let's assume all energy is made up of discrete quanta. We take all that quanta (an infinite amount) and put it inside a big cosmic hat. We reach in and pull some out. What is the probability we pulled out the same amount of positive quanta as negative quanta? Or what is the probability we pulled out zero quanta?

The probability we pulled out zero quanta is easy to figure out. Since there is an infinite number of quanta, the probability is 1/infinity--which is virtually zero. So we are virtually guaranteed to have more than zero quanta outside the hat. To have zero net energy, we need an even number of quanta, and equal amounts of negative and positive quanta.

If the quanta number is odd, the probability of netting zero is zero. For example, if we pulled out five quanta from our cosmic hat, the closest combination to zero would be three positives and two negatives--or vice versa. If we pulled out four quanta, there's a possibility we could have two positives and two negatives. The equations below enable us to calculate the probability of netting zero:

Equation 2) takes into account that there's a 0.5 probability that the number of quanta (n) could be even, so the probability that is calculated in equation 1) is cut in half. Thus, in equation 2's denominator, there is 2^(n+1) instead of 2^n.

The above equations show that the larger the number of quanta (n), the less likely we will have zero net energy. But shouldn't the net energy get closer to zero as we add more quanta to the mix? If we flip a coin, we get heads or tails. If we assign a value of minus one to heads and plus one to tails, we get plus one or minus one--never zero. But suppose we toss a trillion coins? We should get an equal amount of heads and tails (or pretty close).

We know if we increase the number of coins, the variance or deviation from zero is reduced.

Equations 3) and 4) clearly show, that as the number of coins (or quanta) increases, the smaller the variance becomes. The sum of coins or quanta converge to zero. This seems to contradict our earlier finding via equations 1) and 2). Let's crunch some numbers and put the data in a couple of tables to see what we get.

The first table above contains data for odd numbers of quanta. We see that the variance (v) decreases as expected. It gets closer to zero as more quanta are added. (This is also true in the second table for even numbers of quanta.) We see in the last two columns the probability of having exactly zero net energy is zero due to an odd number of quanta.

The second table shows an interesting paradox: As the variance decreases, so does the probability of having exactly zero net energy. This paradox is illustrated in the graphs below:

So, the closer we are to zero, the less likely we will have precisely zero--this is why there is something rather than nothing.

Deriving Maxwell's Equations From Heisenberg and Einstein

What is the node that connects Heisenberg's uncertainty principle, Einstein's field equations, and Maxwell's equations? To find out, let's start with Heisenberg's uncertainty principle and see what we can derive:

We derive a distance (r). We can do the same if we start with Einstein's field equations:

We can now equate the right side of equation 6) with equation 11). If we do this, we can discover the above-mentioned node.

Equation 15) reveals the "node" to be energy (E). Equation 16) shows that energy not only connects the uncertainty principle and field equations, it shows the electric field is linked as well. From 16) we can derive the integral form of Gauss's law for electric fields:

Equation 20) basically says electric charge produces an electric field. The field flux passing through a closed surface is proportionate to the charge contained within the surface.

If we take equation 20's integral and divide it by a volume (V), we can derive the differential form of Gauss's law for electric fields:

Equations 22) and 23) are just different ways of saying the electric field produced by electric charge diverges from a positive charge, or it converges upon a negative charge.

Going back to energy (E), we can begin again and derive Gauss's law for magnetic fields:

Equation 35) is the integral form. It says the magnetic flux passing through a closed surface is zero. According to equation 36), the magnetic field's divergence at any point is zero.

Given what we have derived so far, we can also derive the integral form of Faraday's law:

Equation 44) describes the electric generator. As the magnetic flux through a surface changes, an electric field is induced.

The differential form of Faraday's law is derived as follows:

According to equation 51), a magnetic field that changes with time produces a circulating electric field.

Next, we shall derive the integral form of the Ampere-Maxwell law:

Equation 63) says an electric current (I) produces a circulating magnetic field. The differential form is derived as follows:

According to equation 67), an electric field that changes with time produces a circulating magnetic field.

To put a cherry on top of the work we've done so far, we use Heisenberg's uncertainty principal and Einstein's field equations to derive the electromagnetic tensor (of equation 82) below):

Notice in equation 82) the Schur product was used to multiply the matrices to get the final result: the electromagnetic tensor.

To learn more about Maxwell's equations, there is an excellent book entitled "A Student's Guide to Maxwell's Equations" by Daniel Fleisch who explains every detail, symbol, and nuance. There's also this video which is also excellent:

Which Way Will the Pencil Fall?--Playing with Perturbation Theory

On a smooth, flat table you have a pencil balancing on its tip. Which way will it fall? Or will it just stay balanced on its tip? Variable L is the displacement. If the pencil stays balanced, L equals zero. If the cat jumps on the table ... L is greater than zero. In fact, L is the length of the pencil laying on its side, thanks to Felix the feline pencil flopper.

We can mathematically represent this feline faux pas with the following equation straight out of perturbation theory:

Whether the displacement (L) is zero or greater depends on the value of epsilon. If epsilon is zero, then L just equals Lo which is zero. In that instance, the pencil is perfectly balanced on that smooth, shiny tabletop. Epsilon represents a small disturbance (or a big one). When Felix jumps on the table, epsilon is greater than zero, and that causes L1 and L2 to come into play, which causes L to be greater than zero--so the pencil falls on its side.

The above equation, however, does not tell us which way the pencil fell (or the color of Felix's fur). It does not tell us much about the forces that make up epsilon. We want to know which way the pencil fell and the magnitude and direction of the forces involved. We could represent the direction and angle of displacement (L) if we use a complex number: a + bi. We can also use angle phi:

The maximum displacement (L) (i.e., the length of the pencil laying on its side) is equal to the square root of the complex number times its complex conjugate:

Epsilon can be divided into three dimensions of force: epsilon(a) is the force(s) that causes the pencil to fall in the "a" direction or "x" direction. Epsilon(b) is the force(s) pertaining to the "bi" direction or "y" direction. Since the pencil does not rise vertically along the "z" axis, or drill into the table, we only need to consider two dimensions.

The equation below tells us what epsilon is along directions "a" and "b." You will note that the complex number is divided by itself. This yields a one or a zero. The value of N equals one if the pencil falls, and zero if it does not.

We can find the values of a and b as follows:

And let's not forget the values of epsilon(a) and epsilon(b):

Last, but not least, we can find angle phi:

Click here for a totally excellent tutorial on perturbation theory.

How Did the Ancients Build Those Mysterious Stone Structures?

Several years ago I watched a documentary (I don't recall the title) about ancient pagan sun-worshipers who built these mysterious stone houses and shrines in Britain. I don't recall who they were exactly. They may have been Celts or Druids--or a different group. I do remember the problem the archeologists were having. They were trying to figure out how these ancient people managed to build these stone structures.

They weren't just ordinary stone structures. The stones were heat-fused together. An incredible feat by these ancients, considering their stone-age/bronze-age technology. The popular theory, of course, is that ancient people were lunkheads who couldn't do anything without the help of visiting extraterrestrials.

Being scientists, the makers of the documentary set out to prove the popular theory wrong. They tried to construct one of these stone structures, using only the materials available in ancient times. They first created a mound of dirt, then piled stones on top to get a kind of igloo shape:

Next, they surrounded the structure with timber:

They then set it ablaze:

What happened next? Well not much. They hoped the stones would fuse together after being exposed to a pyre that would make any Viking funeral a hit. Unfortunately, the stones failed to fuse together. But these archeologists were stubborn and did not give up. They tried the same procedure again and again, hoping for a different result: success. But the stones "stubbornly" refused to fuse together.

At the time I watched this documentary, I wasn't even a physics student yet, so I, like the archeologists, had no clue what went wrong. But that was then. Now it's patently obvious what went wrong. There wasn't enough heat because the second law of thermodynamics reared its ugly head!

When the fire burned, most of the heat was lost to the open air. The convective-heat-transfer equation below spells this out:

The fire's temperature is approximately 1571 degrees Fahrenheit or 855 degrees Celsius. The air temperature was far colder, so nearly 100% of the heat was lost. What these archeologists needed to do was somehow raise the air temperature to 855C, then little or no heat would have been lost.

What I suspect the ancients did was something the archeologists failed to do: The ancients used some form of insulation. If insulation is used, the following equation shows the benefits:

Thick insulation with low thermal conductivity will not only save energy, but can also cause the temperature to increase. This is good news, considering the minimum temperature needed to melt stones is higher than the fire's temperature. Here are the numbers:

So now the question is what materials were available that could be used to insulate? Here is a short list that includes each material's thermal conductivity (kt):

Using stones for insulation seems like an obvious choice, but if there is no mortar to work with, then we would need to heat-fuse these stones together so we can insulate the fire so the stones we started with will be heat-fused.

Wood has a slightly higher thermal conductivity: .5. It is easy to build a wooden insulating frame around the stone structure. The main drawback is it will burn up. That leaves soot. Soot has a very low thermal conductivity: .07. Insulation made from soot will raise the temperature approximately seven times higher than stone or wood--up to 10,500 degrees Fahrenheit or 6,000 degrees Celsius.

I don't know what those ancient people did exactly, but here's how I would engineer a heat-fused stone structure. I would pile stones on a mound of dirt, leaving openings where I want doors and windows. I'd put the timber on top, and over all that I'd build a wooden insulating structure. (I use cutaway views in the diagrams below.)

Of course the wooden planks I'd use would be coated with a paste containing soot, or be charred wood. I would also use a bellows to feed the fire more oxygen. I'd also bury the whole shebang under a pile of mud that would be allowed to harden. Finally, I'd light the fire and insert the bellows and pump away.

To save some time, it might not be necessary to coat the wood with soot or use burnt planks. Fresh wood could be used. It would no doubt burn and may become soot that sticks to the mud structure that remains. This would be ideal.

When enough time has lapsed, the fire would be doused, the mud structure removed. The final step involves removing the dirt from inside the stone structure. If all goes well, it should stand firm because the stones would be heat-fused together.

The Origin of Time, Space, Etc.

Is time eternal? Or is it finite? If time is eternal, then an infinite amount of time has passed. Thus, there will be no future. If there is a future, then there is more time left. Thus, an infinite amount of time has not passed. Time is then finite; it had a beginning.

So how did time begin? For that matter, how did the universe begin? Where did energy, matter and space come from? Did something come from nothing? If we decide that nothing caused something, what does that mean? It could mean that time, space, etc. arose from the great void and black abyss of nothingness, or it could mean that these things always existed--and were, therefore, caused by nothing; i.e., had no cause.

Is your head spinning yet?

Let's assume, for starters, that time had a beginning, where time (t) equaled zero. The equation below reveals something interesting. To have zero time requires infinite energy:

Unfortunately our universe does not have infinite energy. Furthermore, it's a non-sequitur that there would be any energy if there was no time. Energy can't exist for any period of time without time.

There's also Heisenberg's Uncertainty Principle to consider.

As you can see, if time was ever zero, the Uncertainty Principle was violated. Without time, there was clearly no momentum or motion. Today we have momentum, so momentum was not always conserved. If nothing existed (if and when there was no time) then the current energy and mass were not always conserved either. Then again, why would any laws of physics exist in the "great nothing abyss"?

If time was at zero the challenge before us is to figure out how everything emerged out of nothing. If we start with nothing, the concept of "cause and effect" is useless. We're back to the nothing-caused-something paradox discussed above.

If we start with something, "cause and effect" remains intact, but if we regress far enough into the past, we find nothing again--or we have the infinite-time paradox (also discussed above).

We must also consider relativity. Photons, for example, experience zero time, so zero time is possible if there is another reference frame where time progresses. The equations below show that time (t') can be zero as long as time (t) is greater than zero. The syntax t'/t means time (t') per time (t)--e.g. zero time (t') lasted for a period of time (t) seconds.

In the beginning there was no time for a period of zero seconds. In other words, a state of no time can't exist without time. Yet there was a beginning? A big bang? What caused it? Well, nothing. If something caused it, then we are not at the beginning. We need to move back in time another step or more.

There are several theorists who have proposed various models that allegedly explain how time, space and everything else emerged. But their models consist of shapes, objects, dimensions and other devices that are all functions of time and space. Their reasoning is circular. A geometric object can't cause time or space, since the geometric object requires time and space (and the imagination of the physicist who created it) to exist.

What if time is both eternal and finite? Relativity suggests this could be the case. We know that as the universe expands, its energy density decreases and its time rate increases. If we reverse the process, go back in time, the rate of time would decrease. It would slow to a crawl as we get closer and closer to the beginning.

Imagine you're wearing a watch that gives the time (t') illustrated above. If you start at the beginning and wait approximately 13.8 billion years, you experience the entire span of time. For you, the total time is finite.

Now imagine you're wearing a watch that gives the time (t). Recall the concept of the limit you find in elementary calculus texts. Imagine taking a string and cutting it in half, then cutting one of the halves in half. Repeat this process an infinite number of times. You find you get closer and closer to a zero length, but you never reach it.

Time (t) is an eternity, since a full 6.9 billion years passes for every fraction of that time (t')--and there are an infinite number of those fractions of (t').

So here's the scoop: Whether time is finite or eternal depends on which time you are looking at. Historical time (t') gives us a finite amount of time. But if we use current time (t), the universe's beginning was an infinite number of years ago. We can say that momentum and energy have always been conserved. We can say Heisenberg's Uncertainty Principle is eternal. We can say these things because the beginning of time is a limit that can never be reached. Yet, we have a future because time (t') is finite. So go ahead and eat the cake because we can have it too.

So what about space? Why did space expand? Well, I think it had no choice:

You see, space (x) is light speed (c) times time (t). If time grows, so must space. The first equation above shows what would happen if this were not the case. If time (t) grew and space (x) did not, energy would not be conserved and light speed would be less than c.