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Deriving the Gravitational Constant G

Today we will derive the gravitational constant G, also known as Newton's constant. Here are the variables we will be working with: ...

Monday, November 21, 2016

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.

Saturday, November 19, 2016

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:

Tuesday, November 15, 2016

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.

Tuesday, November 8, 2016

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.

Monday, November 7, 2016

How Fast is the Universe Really Expanding?

How fast is the universe really expanding? Is the rate of expansion really increasing? Is dark energy really the cause? To address these questions, let's start with a simple diagram of some dots lined up in a row. Let's pretend each dot represents a galaxy.

The blue dot represents our galaxy. The red dot represents a galaxy on the other end of the universe. It is far far away, so it probably belongs to Luke Skywalker and the Star Wars cast. Anyway, the universe expands. We look through our telescope, and the other galaxies appear to be moving away from us. The red galaxy is moving faster and further than the rest.

However, at the distant red galaxy, Obi Wan Kinobe's ghost tells Luke to forget about the force for a moment and to look through his telescope. From Luke's point of view, his red galaxy is hardly moving. The way he sees it, it is our blue galaxy that is moving the fastest and furthest.

Now that's strange. But it's a major clue. Whenever two observers can't agree, we can suspect relativity is the culprit. Can relativity tell us how fast each galaxy is really moving? Here is the equation that models what Luke and we see when we all look into the great beyond (H=Hubble's Constant; v=velocity; r=radius):

The variable r is the distance you look into space. The further out you look, the greater the velocity (v) the galaxies appear to be going. This is true no matter where you are in the universe. Rumor has it that Luke's galaxy is moving away from us faster than the speed of light (c). But Luke says, "Nada!" We are the ones moving at greater-than-light speed.

The variable R(HS) below is the distance where a galaxy appears to moving at light speed (c). However, the distance of Luke's galaxy is r(u) meters from us, so we see his galaxy going faster than light, and he sees our galaxy doing the same.

The equations below (G=Newton's Constant; E=energy; m=mass; t=observer's time), however, suggest that the universe is expanding at a steady rate; i.e., Luke's galaxy and our galaxy have the same velocity--and that velocity is none other than c, the speed of light. (To see how these equations were derived, click here.)

Take a close look at equation 5 and 7. Notice that r can expand if energy (E) and mass (m) are held constant. If time (t) is increased, then the universe expands without the aid of increasing energy (dark energy).

Equations 10 and 12 below appear to be in conflict. One says the galaxies are moving at velocity v, and the other says velocity c.

So which is it? Are the galaxies going the speed of light or do their velocities vary? If the galaxies are going at a constant rate of c, then the rate of time (t') must increase as we look further out into space. If the velocities vary, then the velocity (v) is Hubble's constant (H) times the distance (r).

I mentioned earlier that relativity could resolve this conundrum. If that's true, we should be able to derive the Lorentz equation if we plug in the information we have so far.

Well there it is! Relativity is the culprit. We can now derive some useful equations below that give us the full Monty:

Equation 24 shows that velocity (v) can increase to infinity, but notice that time (t') also increases to infinity. Equation 25 tells what we already observe: When distance (r) increases, velocity increases. It also reveals why: t(H) (which is just the reciprocal of Hubble's constant) stays fixed. Equation 26 shows that the rate of time increases along with distance (r), so the velocity is no more than c for every galaxy.

So how fast the universe is expanding depends on how far you look into space and it depends on what clock you look at. The universe's clock is running faster and faster while our clock and Luke's clock tick slower. Equation 27 tells us how fast the universe's clock is ticking at any distance (r).

It looks as though we cracked the code of the expanding universe. There is, however, another problem we need to sort out. The universe is isotropic. Wherever you look, things are more similar than not. So it stands to reason, that at every point in space, we should find the same energy density. According to Einstein's Field equations, time (t') should not vary if the energy density is the same everywhere.

Hans Solo's galaxy is no more or less dense than ours, his time (t') should be the same as ours. If that's true, maybe his galaxy really is going faster than light. We need to look at this energy-density thing a little closer.

The diagram above has a little blue cube (our galaxy) and a little red cube (Luke's galaxy). Both cubes are part of the big cube (the universe). Notice that the dots (energy) are evenly spaced (isotropic). The math above shows that the more local the measurement (small radius), the greater the density. The more we look outside our locale, the bigger the radius--the smaller the density. So time (t') does indeed vary within our isotropic universe.

Now we know that if Luke's galaxy appears to be going faster than light, it's only because we are looking at our clock and not the universe's.

Update: Here's spherical diagram showing the relative nature of energy density. The dots represent galaxies. The red dot thinks it is local and that the blue dot is far away--and vice versa:

Update: What about the Doppler effect? Doesn't that prove that galaxies far from us are moving away at a faster rate? Time for a thought experiment. Alice is sitting on a tree stump, on the side of the road, blowing her horn steadily, maintaining a steady pitch. Bob is driving an ambulance with the siren turned on. Bob drives by Alice. As he drives further away, Alice hears the pitch of the siren go down (red shift). Bob, however, hears no change in the siren's pitch. He hears Alice's horn pitch go down, but Alice swears the horn pitch hasn't changed. The Doppler effect they both experience is relative.

Update: Below is some mathematics for the expanding universe. Equation 1) can be used to model the expanding universe from beginning to end. Start with a universe with zero volume. There's infinite pressure with nowhere to go but outward, so you have rapid expansion. (See equations 2 and 3.) As the volume continues to expand, the radiation-matter-energy term (E/V) becomes less significant, the overall pressure reduces, the expansion rate slows. When the pressure reaches the ground state, expansion continues at a steady rate of c if the clock used is the universe's age (t). (See equations 8 and 9.) If Hubble's constant is used, expansion velocity increases as the universe's radius increases. (See equations 10 and 11.)