How to Have Unlimited Orthogonal Space and Time Dimensions within 4D Spacetime

ABSTRACT:

By means of a thought experiment and a mathematical proof, it can be shown that unlimited space and time dimensions are possible, and, that only three space dimensions are mutually perpendicular.

Imagine you are throwing a party. You have invited n number of guests. You want to know the following: the starting location and time of each guest and the time each guest arrives at your party. You know the location of the party and you know what time the guests are supposed to arrive. All of this information consists of 2n+1 bits of time and n+1 bits of location. Assuming each location has coordinates x, y and z, the total bits of space information is 3n + 3.

Each bit of information is statistically independent, i.e., orthogonal to all the rest. We can think of any two bits as having a 90 degree separation. In the case of coordinates x, y, and z, such separation can be easily drawn on graph paper. In all other cases, such separation may be purely abstract and unimaginable. In any case, we can argue that 3n+3 space dimensions and 2n+1 time dimensions are necessary. To have all the information you want, you need 5n+4 dimensions. If n = 100 guests, you only need 504 spacetime dimensions!

It's fairly obvious that the above thought experiment only involves three space dimensions that are mutually perpendicular. But is this universally true? Is there, say, a mathematical proof? The following equation suggests there's no upper limit to how many space dimensions you can have:

It seems like the only constraint re: the number of space dimensions is an empirical one. Let's see if we can find a mathematical one. Let's begin with the following premises:

1. w, x, y and z are unit vectors.

2. w is an arbitrary extra space dimension. What is true for w is true for any extra space dimension.

3. A unit vector is consistent with 1D space and points in only one direction.

4. if two unit vectors (x, y) are perpendicular, they define a plane that is consistent with 2D space. Thus plane xy is not perpendicular to plane xy.

Following these premises we have:

At steps 2 through 4 we assume that all four unit vectors are mutually perpendicular. At 5 we assume w is perpendicular to plane xy. At 6 we assume z is also perpendicular to plane xy. At 7 we conclude that either w is parallel to z or if w is perpendicular to z, then plane xy must be perpendicular to plane xy--which violates premise 4. So w is not an extra dimension. According to premise 2, what is true for w is true for any alleged extra space dimension. Thus, we can further conclude there are only three space dimensions that are mutually perpendicular. If such a conclusion is valid, we should be able to falsify the following 7D cross product table:

From this table we can gather that the unit vector e1 is the solution to three cross-products involving 6 dimensions (see equation 8 below). Premise 3 stipulates that a unit vector only points in one direction. It is also obvious that e2e3, e4e5, e6e7 make three planes persuant to premise 4. Vector e1 can't point in just one direction if it is normal to all three planes, unless all three planes are subsets of the same plane. The inevitable conclusion is not all of these dimensions are mutually perpendicular.

Now, let's take a look at the so-called extra dimensions 4 through 7. At each of the equations 9 through 12 below, the unit vectors circled in red contribute to planes with normal vectors pointing in different directions. Thus they can't all have the same normal vector or cross-product solution.

Therefore, it is safe to say that dimensions 4 through 7 do not behave like mutually perpendicular dimensions.

Circling back to our hypercube at equation 1, we can conclude that there is no upper limit to how many dimensions the cube can have, but only three are mutually perpendicular. The rest may or may not be orthogonal in the sense that they are statistically independent.

References:

1. Seven-dimensional Cross Product. Wikipedia

2. Octonian. Wikipedia

The Origin of Time, Space, Etc.

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.

Debunking Bosonic String Theory's 26 Dimensions

About one hour into the above video, Leonard Susskind (one of my favorite professors) shows how string theorists mathematically derived 26 dimensions. Are you thinking what I'm thinking? If there are really 26 dimensions, then why do most string theories have fewer dimensions? This question made me watch the video with a CSI-investigator disposition.

The original goal was to get a minus-one ground state, so when a creation operator is applied, the photon will have zero mass. Imagine an x-y plane or system going down the z axis with momentum p. The frequency of the ground-state oscillator is n/2. Here is the desired equation and result:

If the result is infinity plus -1, that's OK, according to Susskind, it's just absorbed into the momentum along z and can be ignored. After some fancy math, including a Taylor expansion, the final result is ...

According to Susskind, if we drop the infinity (that lop-sided 8) we end up with -1/24. We need -1, so multiply by 24--that gives us 24 dimensions. Add the z axis and time, and that brings us to 26 dimensions. Voila!

But here's the rub: What if the result was the desired -1? How many dimensions would there be then? Let's see ... multiply by 1. That gives us one dimension. Add the z-axis and time--that brings us to two space dimensions and one time dimension!

Obviously the math and/or logic is seriously flawed.

Since it is OK to do away with infinities, we could take another approach. If you stop and think about it, we ignore infinities whenever we measure anything. How long is a meter? It's an infinite number of points. How big is a circle? It contains and infinite number of points. Everything we measure would be infinite if we didn't invent arbitrary units of measurement. We can do that here:

So instead of having 1/2 of an infinity, why not have 1/2 of a unit of ground state? And, while we're making changes, why do the frequencies of the ground state oscillators have to be a positive number when the ground state is negative? Why not be consistent? You'll note I made both sides of the above equation negative. The result is -1/2. Multiply by 2, add z and time--and voila! We live in a universe with three space and one time dimension after all!

Update: Here is another way to arrive at four dimensions instead of 26. It's based on the premise that the desired result is infinity plus -1/2. This ensures that within the infinity, there is a negative ground state.

Why A Quantum Gravity Theory May Be Impractical

Finding that elusive graviton would be a boon to the Standard Model--but is a quantum theory of gravity practical?

According to the Wilkinson Microwave Anisotropy Probe, the ground-state energy density of the vacuum of space is E-10 joules per cubic meter.  This is essentially the energy of nothing.  Energy less than this is less than nothing.  For all intents and purposes, energies less than the vacuum don't exist.  For the graviton to exist, it should have an energy density greater than E-10.

Suppose we take two protons and place them a Bohr radius apart.  How much is the gravitational energy?  What is the gravitational energy density?

The formula used to measure gravitational energy is E = Gmm/r.  Where E is energy; G is Newton's constant (E-11); m is the proton mass (E-27) and r is the bohr radius (E-11).

E = (E-11)(E-27)(E-27)/E-11 = E-54 joules.

To determine the energy density, divide by the the volume, the cube of the Bohr radius:

E-54/E-33 = E-21 joules per cubic meter!

This is approximately 100 billion times less than the vacuum energy density!  No wonder the graviton has not been found.

Our gravitational energy formula suggests that we could reduce the radius to an infinitesimal size.  Surely we would get an astronomical amount of gravitational energy and gravitons would be as common as sand on on the beach.

There are a couple of problems with this strategy:  1.  Gmm/r can't be greater than (m^2c^4 + p^2c^2)^.5--the total energy of the protons.  2.  Most of that energy is due to the strong force and the quarks that make up the protons.  For quarks and protons to exist, they need the lion share of the available energy.

To get an energy density greater than the vacuum's there needs to be a lot more particles than just two protons.  According to the gravitational-energy formula, when you double the mass, you increase the gravitational energy four times.  The chances of finding a graviton increase exponentially as you add more mass.  Or does it?

When you add more mass, you add a haystack of particles for the graviton to hide in.  You might say that the graviton has its own uncertainty principle: the closer you get (quantum scale), the harder it is to find.  If you step back and look at the big picture (the mass of a double-star system) you can detect a very faint gravitational wave.  Such a wave has little or nothing to do with quantum physics, since you need huge masses just to get started.

These are some of the reasons why a quantum gravity theory may be impractical.

Update:  An elementary particle such as an electron at rest needs E-14 joules of energy to exist.  The gravitational energy of a proton, using the proton's classical radius, is (E-11)(E-54)/E-15 = E-50 joules.  This is not nearly enough energy to create an elementary particle such as the electron--so it appears the graviton does not exist at the quantum level, unless its energy requirement is a thousand billion trillion trillion times lower than the electron's.