In this post we once again derive the Heisenberg uncertainty principle, but this time we make use of the Schwartz inequality and the position-momentum commutator. We begin our proof by defining the variables:
Below we have the Schwartz inequality:
Is it true? Let's prove it. At lines 2 and 3 we map inner products with multiple (n) dimensions to simpler 2D Pythagorean expressions. At 4 through 7 we define the bras and kets and their inner products in terms of a, b, c, d.
Next, we take the products of the inner products and derive 11 below:
At 12 and 13 we convert the variables ad and bc to x and x+h. You may recognize h from calculus texts. In this case, it is just any arbitrary number. At 14 and 15 we do a little algebra to get 16:
At 16 it is obvious the absolute value of h^2 is greater than or equal to zero. Thus, the Schwartz inequality is true.
Before we put it to work, we need to define energy (E) and time (t). E is the lowest possible energy (ground-state) and t is the reciprocal of frequency (f). We sandwich these between the normalized bras and kets at 20. That brings us to the energy-time uncertainty at 21.
At 22 and 23 we do a quick-and-dirty derivation of the momentum-position uncertainty:
We can also find the momentum-position uncertainty by making use of its commutator and a wave function (psi). At 24 we define the commutator; at 25 we define momentum (p). After making a substitution for p at 26, we do some more algebra until we get the desired outcome at 30.
Equation 30 is looking good, but there is a slight problem: it's an equation! We want an inequality, so we make use of the Schwartz inequality one more time:
This post is a sequel to "Dark Energy In - Dark Energy Out = Gravity." Today we are going to find the relationship between Hubble's observations (i.e. Hubble's constant), dark energy and gravity--and we are going to derive it from Heisenberg's uncertainty principle. Let's kick things off with defining the variables:
Equation(and inequality) 1 below is the energy-time version of Heisenberg's uncertainty principle:
The idea here is to build an expanding universe by taking a bottom-up approach. We build the very large by starting with something very small. We derive a simple energy equation (see equation 4 below).
Note the change in energy or energy difference variable on the left side of equation 4. We can substitute some arbitrary energy (E) minus the ground state (epsilon * E):
Let's bring all the terms to the left side and derive equation 10 below:
We now have an energy squared minus another energy squared minus the ground state squared equals the final energy (Ef) squared. We get equations 11 and 12 below by using Planck's reduced constant (h-bar), the wave number (k), and the light-speed constant (c)--and making substitutions.
Checking the units, we find equation 12 to be eerily similar to Einstein's field equations. Not a bad thing, by the way. It allows us to rewrite equation 12 to get 13:
Multiply both sides of 13 by the volume (D^3) to get 14 and 15:
Multiply both sides by Hubble's constant (H):
Multiply both sides by c^2/D:
From here we can derive 21 below:
Equation 21 is a power equation that has two components: the force of gravity, and the velocity the universe is expanding at distance D. However, this is only part of the story. Equation 21 does not take into account the mass density of spacetime or vacuum. A more complete equation is 22:
Note that as distance D increases, Volume V increases. The vacuum-mass-density gravity grows(dark matter effect) while classical Newton's gravity shrinks. If we utilize the cosmological constant, we can see a more precise relation between gravity and dark energy. First, we need to go back a few steps and work the cosmological constant into the math. Let's start with equation 13 and work forward:
A note re: equation 23. We want epsilon/D^2 to represent a ground state and a ground state does not increase or decrease, so epsilon must be proportionate to D^2. Thus we can set the term equal to the cosmological constant.
Equation 29 reveals something interesting: the instant velocity of expansion (HD) appears to be unaffected by gravity. The gravity in the numerator seems to be proportionate to the gravity in the denominator. This suggests the big crunch ain't gonna happen. But wait! It gets better. Suppose the universe expands to a point where Newtonian gravity (GM/r^2) is insignificant? We can drop it and get equations 30 and 31:
Look at equation 31. The only variable that isn't a constant is distance D. Now here's the awesome part: When D increases, so does gravity and so does the rate of expansion. The expansion rate (HD) is a function of gravity ... or is it dark energy? They both appear to be two sides of the same coin. And why not? They are both components of vacuum power (P).
Update: We can take equation 31 and derive the value of the cosmological constant:
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.
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.
What is the shortest distance between two points? In free, flat space it would have to be a perfectly straight line. Any other line is going to be greater or equal to that perfectly straight line. Does Heisenberg's Uncertainty Principle work the same way?
Suppose we have a perfect line between points A and B. The perfectly straight line between them is xo, and that distance is equal to Planck's constant (h-bar) divided by momentum (p). If we can draw a straight line (x) between A and B, with no margin of error, then that line will be equal to h-bar/p and xo. If there is a margin of error, then x will be greater than or equal to h-bar/p--with an emphasis on "greater than."
Assuming error margins exist, it seems reasonable to assume the Heisenberg Uncertainty Principle is bullet proof. Since our line x is greater than or equal to xo, we can use this fact to show, in terms of momentum (where m=mass, v=velocity), that any arbitrary velocity is less than or equal to the speed of light (c).
From equation 2, we can derive a new uncertainty principle between mass (m) and position (x). Check out equations 6 and 7 below.
It has been argued that mass-less particles, i.e., photons are unaffected by the Uncertainty Principle. At equation 8 we substitute a photon's mass equivalent (fh/c^2). You can see that it is possible to know the photon's position and frequency (f) with great precision (h-bar/c), but there is still some uncertainty, some margin for error.
At equation 9 it appears the Uncertainty Principle is debunked--at least in part. We multiply both sides of equation 8 by velocity (v) and we get equation 9. If velocity is zero, it is possible to know the exact position and momentum of a particle. If velocity is light speed (c), then h-bar is the best we can do (with the exception of a mass-less particle). The median velocity is 1/2c. With that we can derive the original Heisenberg Uncertainty Principle.
