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.

Caveat: One could ask what is the duration of a no-time state or any time state for that matter. How long must each state last in order to count as a state? The obvious answer is more than zero arbitrary time units. So do changing states really make up time or does time make up changing states? Perhaps we've gone as far down the philosophical rabbit hole as we can go.

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.

Why Entropy Happens

In the beginning, there was order, a very hot singularity, but as time progressed the universe expanded and cooled--and became more disorderly. Scientists predict a "big freeze." It's all due to entropy. As you read this blog, entropy continues. Why? That's what we will explore below. First, let's define the variables we will use:

We begin with the partition function, which has the Boltzmann factor, an exponent with a thermodynamic beta power over the base e:

If we want to determine the probabilities of the energies in a system, we make sure the probabilities add up to 1, so we normalize the partition function by dividing it by itself (Z):

However, if we want to model the universe's evolution, we need to make a slight change to the partition function. Instead of using the thermodynamic beta, we use its reciprocal. We also change the i index to a time (t) index:

Also, we want the universe's total energy to be conserved. We know dark energy is increasing and radiation energy is decreasing, so we put together an energy-conservation equation:

At equation 5, notice how an increase in the universe's volume (V) reduces the radiation energy (Er) but increases the dark energy (pV). Multiply the two energies, add a little dark and baryonic matter, and take the square root and we get a constant energy (E).

We define temperature as follows:

As volume (V) increases, the universe's temperature decreases. Equation 7 below gives us the probability of the temperature at a given time t:

A high temperature has a low probability. A low temperature has a high probability. So there is a high probability the universe's temperature will continue to decrease, and a low probability the temperature will increase. Thus, an expanding universe has a higher probability.

Now, let's take a look at entropy. We define it as follows:

We see that entropy increases as temperature decreases--so it has the same probability as temperature:

So why does entropy happen? Greater entropy has a higher probability than lower entropy. We can also say that reverse entropy is possible but less probable. A good example is the one Tyson discussed in the above video. There are pockets of order caused by star energy, so life is possible.