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):
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.
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.
To derive Dirac's Lagrangian, we begin with the Dirac field's adjoint spinor (bar-psi) and spinor (psi). Note that each spinor contains right-handed (psi-R) and left-handed (psi-L) fields. (Right-handed means the spin and momentum of the field particles are in the same direction. Left-handed means spin and momentum are opposite.)
Let's put the fields into an algebraic form:
Next, we multiply them together:
On the right side of the equal sign the first two terms are each zero and add to zero. Here's why:
The mixed terms don't equal zero. Here's why:
The first two terms that equal zero won't equal zero if we multiply them both by Dirac's matrix.
Now we can set those first two terms equal to the mixed terms:
On the left side, we take the derivative of the fields and multiply by -i, Planck's constant, and the speed of light (c). That gives us kinetic energy. On the right side we multiply by mass (m) and c^2. That gives us potential energy.
The Lagrangian is kinetic energy minus potential energy, so we subtract the potential energy from both sides to get the Lagrange (L):
There's a new theory that explains every mystery in physics. I call it the magic-dust theory. Like other modern theories it is mathematically consistent. But like other modern theories it is currently not testable. To observe the magic dust requires a particle accelerator the size of our galaxy, maybe bigger.
Here's an example of the mathematics of this promising new theory:
Thanks to this new innovative theory, we now know what caused the Big Bang. There is no empirical evidence, but, as I said before, this theory is mathematically consistent. Here is an example:
All kidding aside, the "magic-dust" theory demonstrates what is wrong with modern theoretical physics. If you stop and think about it, there isn't much difference between magic dust, and strings.
Like magic dust, strings have virtually unlimited power: they can vibrate and make the different particles; they can stretch up to infinity in multiple dimensions to give us d-branes. They can do whatever is needed to explain any mystery. As long as the math is consistent, we can call it science. However, Issac Newton would disagree. Below he discusses gravity and his philosophy:
"I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction."
Note how Newton uses the word "hypothesis" rather than "theory." Technically, string theories and other modern theories aren't theories at all. At best, they are hypotheses, since the entities they invoke (e.g. strings, extra dimensions, etc.) have never been observed. Below, William Whewell has this to add:
"What is requisite is, that the hypotheses should be close to the facts, and not connected with them by other arbitrary and untried facts; and that the philosopher should be ready to resign it as soon as the facts refuse to confirm it."
I have to agree. The challenge is to explain the mysteries without magical thinking; i.e., adding extra entities that have never been observed like extra dimensions, strings, branes, magic dust, etc. The whole idea of science, in my opinion, is to reduce mystery, not add to it. If you have to create a new mysterious object to explain an old mystery, then that is the analogue of digging one hole to fill another.
Using new, unobserved, untested entities to explain a mystery has the disadvantage of increasing your burden of proof. This is why the best hypotheses use known facts as much as possible to explain the unknown.
On the flip side it's true that Democritus dreamed up the atom centuries before it was physically observed. And maybe sometime in the distant future, humans will discover strings, extra dimensions, gods, fairies, pixies, and dragons in their basements ... or ... maybe not. The thing I love about science is we are free to change our minds as soon as the evidence becomes available.
Within quantum field theory we find two equations based on Einstein's equation for energy (E = energy; p = momentum; c = light speed; m = mass):
The first is Dirac's equation. The second, the Klein Gordon equation (i = imaginary number; h-bar = Planck's constant; bold y with arrow = Dirac-Pauli matrices; a-sub-u = first derivative; psi = quantum field; t = time; up-side-down triangle = Laplacian).
The goal is to take elements and ideas from the above equations and combine them with Einstein's field equation below (Guv = Einstein's tensor; Gn = Newton's constant; Tuv = the energy-stress tensor; and then there's pi):
Let's first write Guv in quantum terms (psi-sub-r = relativity-field-wave function; Aj = a coefficient element in series j). Note there's a double derivative of psi with respect to an x-vector, and tensor indices (uv) after the closing parenthesis. There is also psi's complex conjugate, which, when multiplied my psi and the rest, gives the expectation value; i.e., a result we expect on the classical scale.
Next, we write Tuv in quantum terms (V = volume). Again we get an expectation value, and the entire term is placed in parenthesis and made a tensor with indices uv.
Now that we have Einstein's equation in quantum terms, let's define each element in greater detail. Below is the definition of psi:
Next we define k. The k represents wave number(s) for one or more particles summed (j=1 to n) by the summation sign. The arrow over k indicates it is a 4-vector. The "a" with a dagger is the creation operator. The zero in the ket represents the field in the ground state.
The variable x is also a 4-vector:
The bold b with an arrow can be any one of three types of spin matrices: Dirac-Pauli matrices (y) for half-spin particles; S-matrices for integer-spin particles; and the number one for zero-spin particles.
Below is a full demo of the Dirac-Pauli matrices:
Here is an example of one Dirac matrix acting on the k 4-vector:
The following are 4X4 matrices I designed for spin-1 particles. I always admired Dirac's equation and matrices, but they are limited to half-spin particles. Now it's possible to include bosons in the field.
We now have a quantized version of Einstein's field equations where we can work with both matter and anti-matter. We can work with zero-spin particles (Higgs field), spin-1 bosons, fermions, atoms, molecules and beyond ... and see how they curve spacetime in terms of the wave number(s) k.
The interesting thing about k^2 is it has the same units as Einstein's field equations: 1/L^2. Thus if we take the double derivative of psi with respect to x, we get k^2. Multiply that by the unit-less coefficient A; multiply psi by its complex conjugate--and we get the equivalent of an Einstein tensor element. Put 16 elements together and we get the complete Einstein tensor.
