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How to Build a Dark Matter Galaxy and a Baryonic Matter Galaxy

According to the above video, an ultra-diffuse galaxy (far fewer stars than ours) contains little or no dark matter. Light passes throug...

Saturday, April 14, 2018

How to Build a Dark Matter Galaxy and a Baryonic Matter Galaxy

According to the above video, an ultra-diffuse galaxy (far fewer stars than ours) contains little or no dark matter. Light passes through it in a seemingly straight line (little or no gravitational lensing). Then there's the dark matter galaxy, with far fewer stars than our galaxy. Most galaxies appear to have approximately five to ten times more dark matter than baryonic matter.

Using Einstein's field equations, we shall build these different types of galaxies and derive an equation that models each of them. However, to accomplish this task, we make the following assumptions: A dark matter particle is an excitation of a field, possibly dark energy or the vacuum. Like the vacuum of space, dark matter particles are diffuse. They have no electromagnetic force. They don't interact significantly with ordinary-matter particles or with themselves. There is, however, gravitational attraction. Finally, dark matter has something similar to the Pauli exclusion principle which prevents DM particles from gravitationally collapsing into a single point in space.

Because dark matter is diffuse, its gravity is insignificant near, say, a black hole, but becomes more significant as we increase the distance (r). Below is a crude illustration of a dark-matter galaxy. It has a black hole in the center. Inside the smaller circle, there's the big black dot (black hole) surrounded by only a few small dots (dark matter). By contrast, inside the large circle the small dots dominate (make up most of the mass).

Now that we've defined "dark matter," let's define the variables we will need:

We are ready to begin working with Einstein's field equations:

Let's change the cosmological constant term and make a substitution:

We make the substitution at equation 3. At equation 4 we put the term on the other side of the equals sign:

To make the math less clunky, let's change the terms into some simple variables:

After making more substitutions we get equation 12 which has a surprising implication!

Spacetime can be flat no matter how much mass (energy density, pressure, etc.) you have! Variables C and D are the energy-stress or mass terms. As long as they are equal, spacetime is flat--which implies no gravity! How can this be?

If there is no gravity between C and D, they will not move (or they will be pushed apart by dark energy). Imagine a red bowling ball and a blue one sitting apart from each other on a trampoline. Imagine the trampoline is expanding. It might look something like this:

Each bowling ball makes a dent in the trampoline, but they don't roll to each other. But around the region of each dent, there are some marbles that roll toward each bowling ball. The analogy demonstrates the following: Gravity is local and not global. If two or more masses have equal energy densities, there is no attraction. Where they are not equal, there is attraction.

The next diagram shows why galaxies move away from each other rather than towards each other:

The above diagram is a toy universe consisting of four multi-colored galaxies. They have equal masses and are the same distance apart. The arrows assume there is gravity between them, but the red galaxy is equally attracted to the green and blue galaxies. The other galaxies are in a similar predicament. The gravity of one galaxy is cancelled by the gravity of another galaxy. So there's no movement. This universe is static. Even though there is plenty of mass, spacetime may as well be flat and gravity may as well be zero. Add some dark energy and this universe will expand indefinitely.

So how do we get some gravity going in our toy universe? We break the symmetry. One way is to alter the distances between the galaxies (see first diagram below). Another way is to vary the masses (see second diagram below).

Now, you may be thinking, this is all fine and good, but what does this have to do with dark matter? Well, to understand dark matter, we have to have a better understanding of how gravity works and how it doesn't work. The main point you want to hang on to is equal masses that are equidistant aren't attracted to each other or that attraction is cancelled. Thus galaxies stay apart and move apart except when the symmetry is broken:

Now, what is true for massive galaxies must also be true for smaller stuff like dark matter/energy particles:

So all we have to do to make a dark matter galaxy or any galaxy containing dark matter is break the symmetry by introducing, say, a huge mass:

Let's assume the big mass is baryonic matter. Why baryonic matter? Baryonic matter is electromagnetic. It can form into molecules, planets and stars. It can come in large clumps. Whereas dark matter is non-electromagnetic and does not form clumps. It remains diffuse much like dark energy or the vacuum of space. When its particles are equally spaced, they are not gravitationally attracted to each other, so they need a catalyst, a symmetry breaker. Baryonic matter comes in different-sized clumps and is a perfect symmetry breaker.

In our toy universe we can imagine little dark matter particles falling into the spacetime dent created by the huge clump of baryonic matter. The dent, of course, gets deeper, allowing even more DM particles to enter:

So when does this process end? It looks like it could go on indefinitely! Dark matter pours into the dent, the dent gets deeper, more particles pour in, and so on. There must be an equilibrium where the mass of the dark matter is around five to ten times greater than the baryonic matter. One way to create such an equilibrium is to use the inverse-square law and Hooke's law. Below we use these laws to limit how deep the spacetime dent gets. We treat it like an elastic surface that springs back into place when mass is removed.

At equation 21 above, the two main variables are M (total galactic mass) and x (how deep is the dent). As dark matter is added, M and x grow proportionately (see 19), but as x adds to the dent's magnitude (equation 21's second term), it has an inverse-square effect (equation 21's first term) and the gravity is reduced until equation 21 equals zero--equilibrium is achieved (see 23 below). Using a pinch of algebra, we calculate the final mass of the galaxy (see equation 28).

We see the total mass is the dark-matter constant (we set to 10) plus 1 times the original mass (Mo) we used to break the symmetry. If we assume that baryonic matter is a catalyst for the build up of dark matter within a galaxy, then we can predict that galaxies with few stars, i.e., a small amount of baryonic matter will have little or no dark matter. And we can predict that galaxies that are nearly exclusively dark matter, have or have something equivalent to a massive black hole at their center. Finally, we predict that normal galaxies with lots of dark matter also have far more stars than galaxies lacking dark matter. These predictions lead to the final equation:

Where the radius (r) is small, equation 30 reduces to Newton's equation, but when r is large, dark matter rules!

Saturday, April 7, 2018

Why Electromagnetism and Gravity Are Not the Same

According to legend, Einstein was once asked, "How does it feel to be the smartest man alive?" Einstein replied, "I don't know, you'll have to ask Nikola Tesla." Indeed, Tesla was probably the smartest man alive. He even had his own theory of gravity which was brilliantly composed and involved the aether, "ponderable bodies," "mechanical effects," "tubes of force," motion producing the illusion of time, and electromagnetism!

Tesla was also critical of Einstein's theory of gravity: General relativity. He had this to say about the concept of curved spacetime: "If mass curves space then space would have an equal and opposite force and straighten out again!" Remarkably, as brilliant as Tesla was, his theory of gravity never caught on. In this post, we will examine why. Below we define the variables we will use:

Let's take a look at the classical wave equation for both the electric and magnetic fields:

Notice that equation 1 has units of the electric field (E) over distance squared (L^-2). Equation 2 has units of the magnetic field (B) over distance squared. (Also notice the equations equal zero, which implies flat spacetime.) Now, what happens if we totally remove the magnetic and electric fields? In other words, what happens if we remove electromagnetism? We get equation 3:

At equations 4 and 5 we derive something very similar to Einstein's field equations. Notice how equations 3 and 5 have units of L^-2--or units of curvature. When mass is greater than zero, the equation no longer equals zero, i.e., spacetime is curved. We are now in a position to derive Newton's equation of gravitational acceleration:

Check out equation 7 above. It's time to do a shout-out to James Clerk Maxwell who proved that light and electromagnetism are one and the same! Equation 7 shows how the permittivity and permeability of free space equals the reciprocal of light speed squared. Equation 8, of course, is the final solution. The right side is clearly Newton's work. I will also point out that deriving Newton's gravitational acceleration would not have been possible if we didn't eliminate the electric and magnetic fields, i.e., electromagnetism.

Another key difference between electromagnetism and gravity is mass affects the rate of EM acceleration (see equation 11), but mass has no effect on gravitational acceleration (see equation 12).

Electromagnetism is a function of charge, not mass, so a change in mass causes a change in acceleration. Gravity, by contrast, has mass (m) in both the numerator and denominator of equation 12. It cancels itself, so a change in mass does not change the rate of acceleration.

As you can now plainly see, the genius who was fluent in eight languages, who gave the world the gift of alternating current and brought light and power to the world, was simply wrong about gravity. And that other dude with the wild hair was right!

Friday, March 23, 2018

The Probability of Backward Time, Forward Time and No Time

Is backward time possible? Yes it is, but what is the likelihood? What is the probability of going back in time? Imagine you have a three-sided coin. The sides are labeled -1, 0, and +1. Suppose we define time as follows: when the coin goes back to its previous state, it has gone back in time. When it goes to a new state, it goes forward in time. If it stays the same, there is no time.

Side 0 is the coin's current state, i.e., the present. Side -1 is the previous state--the past. Side +1 is the future. Below we calculate the probabilities:

At equations 2 through 4 we see the probability of going to the past is the same as going to the future, and staying in the present is just as likely. The coin above could be analogous to a simple quantum system; perhaps a single particle. Where the number of particles and states are few, backward time and "no time" are highly probable.

Now, let's make our simple system above more complex. Let's place the coin in a lattice with four cells. We decide that a change in state includes a change in position. If the coin moves to a new cell, it has moved forward in time. To move back in time, it must go back to its previous state which includes -1 and the cell it previously occupied. To have "no time" means no changes at all. Below is the relevant math. At equation 7 we normalize the total number of possibilities so the total probability is 1.

Here are the probabilities for our more complex system:

At equation 10, notice how the probability of forward time has increased to 10/12. At equations 8 and 9 we can see that backward time and "no time" have lost some ground--they are now less probable. Their likelihood decreases as we add more and more particles, states, and cells, and, the likelihood of forward time increases. Below are some general equations that determine the probability of past, present, and future.

But take note of equation 14. There's a question mark. Our model of time is incomplete. So far, we have only included what happens when the coin is tossed, i.e., when there is an interaction between, say, you and the coin. Assuming you toss the coin at a steady rate, there are three basic states: you toss the coin and get -1, you toss the coin and get +1, you toss it and get 0. But what happens if you don't toss it, i.e., if there isn't any interaction? Nothing changes and time stands still.

Thus there are two ways time can be zero: no interactions or an interaction where you get back the same state. To get the true probabilities of time, we need to take relativity into account. (The variables we will be using are defined below.) We know that time can slow down at high velocities and where there are large masses. The slowing of time implies that there are more instances where time is "no time" and fewer instances where time is moving forward or backward.

Equations 15 through 19 give us the relationships between mass, energy, velocity and time:

Equations 15 and 16 show how increased linear velocity (v) reduces the time rate (t') and increases the mass (m'). It follows that there is a correlation between reduced time rate and increased mass. It is understood that increased mass reduces the time rate, but how? Especially if it is at rest. Oscillators are the key. Equations 17 through 19, which involve Hooke's law and Einstein's mass-energy equivalence, show how oscillators increase mass.

If all variables, except angular velocity, are held constant at equation 19, It becomes apparent that increased angular velocity increases mass, or, mass is a function of angular velocity. Looking carefully at equations 15 through 19, it follows that increased linear velocity (v) causes increased angular velocity. In the case of both mass and linear velocity, there is increased angular velocity or oscillations. Could increased oscillations cause slower time? If so, that would explain why both linear velocity and mass slow time. Let's see if we can prove it:

Equation 34 clinches it! Increased oscillations cause a reduced rate of time. Anytime we add energy to a system, the oscillators oscillate more. Why does this reduce the time rate? Take a look at the left side of equation 34. It has the variable "u"--the 'relative' emission and absorption rate of gauge bosons within a system of harmonic oscillators. Bosons move no faster than light speed. They can't speed up when fermions speed up. When fermions are at rest, gauge bosons are relatively faster and carry force faster than when fermions are in motion (oscillating), so time is faster when fermions are at rest and slower when fermions are in motion.

Using our coin-toss analogy, if you can't increase your speed and have to chase the coin and catch it before you can change its state, you can change its state more often if the coin is at rest than if it is moving at high speed. So time, as we defined it earlier, has more instances of "no time" if the coin is hard to catch (i.e. time is slower). Also, at 34, notice the plus and minus sign in front of the radical. The square root can be negative as well as positive. This allows for backward time. The only question that remains is, "What are the odds?"

If slower time increases the instances of "no time" due to no interaction, we must reduce the probabilities of the other three possibilities: forward time, backward time, "no time" with interactions. To do this we use the Lorentz factor from equation 34.

Equations 36 through 38 show that forward time still dominates within complex systems due to more degrees of freedom. To get the probability of "no time" due to no interactions, we subtract the above probabilities from the total of 1:

This probability is zero when matter is at rest, and it grows when matter is in motion. We now have a complete probability distribution for backward time, forward time, and no time.

Update: What about the oscillations? Couldn't they count as changes of state? Sure, why not? But the net value of time would still be "no time." Take for example a pendulum. We decide if it swings right, time is going forward, but when it swings left, time goes backward and washes out the forward time, so time makes no progress until something more happens than mere oscillations.

Saturday, March 17, 2018

Taming Infinities--Introducing n-space

Each line has an infinite number of points. We tame this infinity by creating an arbitrary finite unit. For example, take the set of real numbers. Between 0 and 1 there are an infinite number of numbers:

Normally, we count using integers: 1, 2, 3, etc. But we don't have to do it that way. We could count like this: 1-infinity, 2-infinity, 3-infinity--all the way up to infinity-infinity. If 3 is greater than 2, than 3-infinity is greater than 2-infinity. So what we have are different magnitudes of infinity that make up our finite numbers. With this in mind, it seems reasonable to assume we could add up an infinite set of numbers and get a finite number. For example, we could take the entire line of positive real numbers ...

...and shape it into a circle:

Now infinity is equal to zero and 2pi radians, i.e., finite numbers.

Let's imagine we are extremely naive, we don't know the first integer greater than zero. So we decide to add up all the real numbers from zero to the next integer point. That gives us an infinity:

The vertical lines represent the infinite quantity of real numbers between zero and the question mark. They make a nice 2D drawing of a triangle. The average real number is at the half-way point. If we take this number (.5) and multiply it by 2, we get the right answer: not infinity, but 1. This is the basic logic behind n-space. We take an infinite number of points in space of any number of dimensions and map them to a 2D space. The average value (expectation value) becomes our vertical axis. We multiply this value by the horizontal axis to get the total area which is a finite value.

The above diagram shows how each point in the original lattice space is mapped to n-space. Each point in the original space becomes a vertical line in n-space. So an infinite number of points, lines, planes or cubes (lattice cells) become an infinite number of vertical lines. The average vertical line (bar-np) is multiplied by the horizontal line (nx) to get the area--which is the correct finite answer.

Why is the n-space area the correct answer--and not infinity? Consider the following diagram:

Max Planck found that if he added up a set of finite discrete energies, he got the correct finite value. The above diagram shows we can also add up an infinite set of continuous energies and get the same finite value! Whether the energies are discrete or continuous, the area under the curve is the same. Thus, finding the area under the n-space curve is a way to find the correct answer. (Take note that, throughout this post, we take the energy term normally reserved for a single particle and use it to represent any energy. Sometimes the frequency and Planck's constant are set to one.)

The following relationships show us how to get the values in n-space we need to calculate the correct, finite answer:

Now, we want n-space to help us solve infinity problems in the quantum as well as the classical realm. This is why n-space was derived from Heisenberg's uncertainty principle. Here are the variables involved:

Here is the derivation:

Equation 12 shows the n-space area is always greater than or equal to 1/2--or the ground-state:

According to equation 13, the total energy in a system, like Planck's constant, has two components, dimensions, or factors (nx, np). The horizontal dimension (nx) is derived from position, and the vertical dimension is derived from momentum. The total energy is equal to or greater than the ground-state energy. Using equation 12 we can derive equation 14:

At 14, k is a constant, so if equation 14 represents the total energy in the system, that energy is conserved. It does not matter how big or small the average energy is at any given point in the original space or lattice. Nor does it matter if there are an infinite number of such points. That energy or quantum number (np) is offset by quantum number (nx), giving the conserved quantity.

Now that we've laid the groundwork for n-space, let's attempt to solve a classic problem: calculating the total energy in a sphere, where each point in that sphere has a given amount of energy, momentum, and/or mass. And, of course, there are an infinite number of points in the sphere.

Immediately we run into a problem: if we know the exact energy at each point, we know the exact momentum if we divide the energy by c (light speed), and, it is obvious we know the exact position as well--a clear violation of the uncertainty principle. If we zero in on a point in space, according to Heisenberg, we should be totally uncertain about the energy and momentum. According to the de Broglie wavelength formula, if we reduce a wavelength to zero, i.e., a single point, we should have infinite energy! And, we can only know that if we have no clue where that point is located!

Below is the relevant math:

Realistically, each point of energy is not a point at all, but a wavelength with a one-dimensional magnitude. If the average wavelength is greater than zero, then we have a finite energy at each wavelength.

We can think of each wavelength as a line. Assuming we know the energies and momentums, we don't know the positions, but we can make this fact unimportant if we calculate the average energy/momentum. Then we know that at any randomly chosen position the average energy is always the same. We can then map each energy/momentum to n-space.

Using equations 21 and 22 we find the average vertical factor (np).

We use the following equations to find the horizontal factor (nx):

At equation 23 we see a problem. To find nx we must first know nt--the total that we are trying to calculate! So we move on to equation 24. We know the total volume but we don't know this thing called the unit volume. We get a unit volume by taking the volume of another system, where we know all the variable values, and multiplying that volume by a factor of np/nt (see equation 25). Once we have our unit volume, we can plug that into equation 24 to get the nx value for the instant problem.

We do the final steps below:

The strategy we used works as long as the following is true:

Suppose we have a scenario where we have a volume of energy, say, a star. The energy is conserved as follows:

The star collapses into a black hole. All wavelengths allegedly shrink to a zero limit. That forces the average momentum factor np to blow up to infinity:

At equation 31, the star's radius also shrinks to a zero limit. We should end up with a singularity that has a position unknown to us, assuming we know the total energy, mass, and momentum. We can imagine the singularity being anywhere within the Schwarzschild radius. Nevertheless, we can crudely map the star to n-space as follows:

As explained earlier, the positions of the wavelengths and the position of the singularity become irrelevant when we determine the average np for each wavelength. Now, let's assume we don't know the star's total energy. We want to find it, so we need to find the value of nx. The star volume is its radius cubed times 4/3 pi. The Schwarzschild radius cubed times 4/3 pi shall serve as the unit volume. We divide the volume by the unit volume--the 4/3 pi's cancel:

When we do the math we see that the star's total energy is finite and is equal to the black hole's (assuming energy is constant and none was transferred).

So in the case of the black hole, np had an infinite limit, but nx had a zero limit--so the total energy ended up being finite and conserved.