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 Relativistic Nature of the Expanding Universe

Our expanding universe is not thought to be relativistic. Technically, the galaxies receding away from us are at rest. The space between them is expanding and creating the impression that the galaxies are in motion. However, each galaxy can be treated as a clock, and, it is not likely that any of these clocks tell time at the same rate.

Using the diagram below as a reference, consider Alice and Bob living in a static universe. Alice lives in galaxy A and Bob lives in galaxy B. Each has a photon gun and shoots one photon per second at the other. The vertical red arrows represent the photons fired from one galaxy to the other. When the first photon from Alice (after traveling light years) finally arrives at galaxy B, Bob intercepts it and records it as a unit of time. Since Alice fired photons at regular intervals, Bob only has to wait one second for the next photon and another second for the next one after that and so forth. He receives one photon per second from Alice, and, for similar reasons, Alice receives one photon per second from Bob. This is how Alice and Bob keep track of each other's time rate.

You will note there are some horizontal red arrows at the top of the above diagram. Also located at galaxy A is Gertrude. She also has a photon gun. She fires one photon per second at Alice. Alice therefore intercepts and records one photon per second from Gertrude. Bob also receives photons in the same manner from Norbert who is also lacated at galaxy B (see horizontal red arrows at the diagram's bottom). This is how Alice and Bob keep track of their own time rates.

When the universe is static, both Alice and Bob receive fired photons from all directions at a rate of one photon per second. All clocks seem to agree and equations 1 and 2 provide the relevant math. But suppose the galaxies move away from each other at velocity v due to expanding space. The next diagram represents this scenario:

The vertical red arrows show that the photons take longer to reach their respective destinations and fewer photons are received per second compared to photons represented by the horizontal arrows. Since each photon is counted as a unit of time, Alice and Bob have the impression that the other's time rate is slower. Since Gertrude is located in the same galaxy as Alice, she isn't receding from Alice the way Bob is, so Alice sees no change in her time rate, since she receives the same number of photons (time units) from Gertrude. Ditto for Bob and Norbert at galaxy B.

If we do a little algebra we can derive the Lorentz equation from equation 3 above. The final result is equation 10:

Equation 10 shows each galaxy's proper time (t') shrinking as they accelerate further apart. It does not matter if the galaxies are technically at rest, since the space inbetween and the photons are not at rest. If the space is at rest, here's the result after firing a photon gun for five seconds:

Notice the steady stream of photons. This is where t' = t. The next diagram shows what happens if space expands at a steady rate of v. The photons still have a steady rate but the interval between them has increased, so they are not counted as frequently. Time t' is less than t.

Finally, the next diagram is the most realistic, since the expansion of space is accelerating. Here the the interval between photons continues to grow and they are counted less and less frequently. Time t' is shrinking.

When Alice and Bob are far enough apart, neither will receive any photons from the other. Time t' will be zero. Thus the expanding universe is relativistic if one keeps track of the various time rates at different distances.

Why Time is More Than Real

"Reality is merely an illusion, albeit a very persistent one."--Albert Einstein

It is apparent from the above quote that reality distinguishes itself from ordinary illusions by its persistent nature. Reality is true even if you choose not to believe in it. Thus, if we are trying to settle the question whether time is real, we should examine time to see, if like reality, it too is a persistent illusion.

The time variable is very persistent and ubiquitous in so many physics equations. On that basis we can claim it's real, but just how real is it compared to things like matter, energy, mass, distance, force, your neighbor's barking dog? It's not like we can grab time out of the air, hold in hand and look at it like a hunk of clay. However, like clay, time can be stretched or compressed depending on how close to the speed of light you are traveling. How is that possible if time is just a product of human imagination? Surely any relative differences in time would also be limited to the human imagination and not an empirical reality.

When examining time, one has to make the distinction between how we measure time and time itself. One popular argument claims that if all particles in the universe stopped changing their states and came to rest, time would stop and cease to exist. This seems reasonable. If nothing happens, how would we experience the "flow of time"?

But what if the "flow of time" is just our experience when we measure time? If your watch stops, you don't assume that time has stopped. You only assume your ability to measure and experience "the flow of time" has stopped. So it seems reasonable to assume that time continues even if every particle comes to a grinding halt. Think of a stalled universe as one big watch that stopped.

So what exactly is time if not a flowing, evolving, ever-changing environment of entropy? The following equation, for me, is a real eye-opener and has forced me to rethink time:

E is energy and psi is the wave function, tp is the Planck time, G, c, and h-bar are the gravitational constant, light speed and Planck's constant, respectively. The above equation shows that it doesn't matter how much or little energy there is, or whether states change frequently or not at all, whether they go forward or backward. No matter what values you plug in for E and psi, you get forward time, specifically, the Planck time. Imagine having zero energy, zero change and still having a Planck time. How is that possible? Thought experiment time:

Imagine a universe with no energy, no distance or space, no charges, no masses, no momentum, no oscillators--just a single zero-dimensional point, a singularity. According to the above equation, time still exists. Why? Because the singularity is persistent--it is real. What exactly is this singularity? It's literally nothing ... except time at a single reference frame, at a single point. No clocks, no observers, just pure time.

Time is so essential to reality, that no "persistent illusion" can persist without it. Time can persist without anything else we would deem real, but nothing we deem real can persist without time. The words "reality," "existence," "persistence," "presence" all imply the passage of time. At this juncture, one could argue that time is not only real, but reality's most essential component. And, when we perform the above thought experiment, we witness time in its purest form.

So if you ever encounter a skeptic who believes time isn't real, that particles exist without time, ask the following question (but don't hold your breath):

"How long do particles exist without time?"

Why Gravitational Waves Fail to Confirm Extra Dimensions

According to the holographic principle, our four-dimensional universe, consisting of three space dimensions and one time dimension, is a surface area of a five-dimensional spacetime called "the bulk." The remaining dimensions of string theory or M-theory are allegedly compacted and rendered insignificant.

Gravity, compared to the other fundamental interactions, is weak due to the graviton's unique ability to move between the surface area (our spacetime) and the bulk. Other particles remain fully in our spacetime and thus have more intensity. At least that's how the story goes. Unfortunately, the gravitational-wave test described in the above video failed to confirm the existence of "the bulk" or any extra dimensions beyond our four-dimensional spacetime. This does not surprise me, given the problems extra dimensions can cause (click here to read all about it).

So why did the gravitational-wave test fail? Do we really need "the bulk" to explain the nature of gravity? We will explore these questions. First, let's define the variables we will use:

According to general relativity, gravity is a function of energy density, so let's begin with the energy density of an atom. An atom is mostly space, so let's only consider the volume of space taken up my the average nucleus and the electrons. That approximate volume can be found in the denominator of equation 1 below:

Of course if we put that volume in the numerator, we get the energy (E):

If we put a larger volume (V) in the denominator (equation 3), we get a reduced energy (E'). Reduced energy is consistent with weak gravity, so we are on the right track.

We don't want Energy units, so at 5 and 6 we use meters and Newtons to adjust the units:

Now, coincidentally, 10^-45/N is approximately equal to G/c^4, so we make the substitution:

We use distance D and the alpha scale factor to make more substitutions at equation 9. From there we derive equation 12.

Equation 12 is Newton's equation. We were able to derive this equation because we started with the premise that the intensity of gravity is determined by the actual amount of space a particle interacts with. For baryonic matter, that actual amount of space corresponds with the gravitational constant G. Note that no extra dimensions are needed to get equation 12. Our 4D spacetime is sufficient. So why should we be surprised that the gravitational-wave test failed to confirm "the bulk"?

Caveat: the above mathematics may work just fine for ordinary matter such as atoms and molecules, but what about singularities such as black holes? Theoretically, a singularity takes up no space, so there shouldn't be any interaction between the matter and space, but there is! To resolve this conundrum, we first need to establish that light speed is truly the top speed in our universe. Consider the familiar Lorentz equation:

The main problem with this equation is time (t) is arbitrary. Let's make it precise. Let's make time (t) equal to the age of the universe. When I say universe I mean everything including the megaverse if such a thing exists. What we want is the longest time ever lapsed--so we set t accordingly and define the other variables we need:

Now we derive 21 below:

Line 21 shows that no velocity (v) can exceed light speed (c). So what does this have to do with gravity and black holes? Given the fact that light speed is the top speed, we can derive the following:

Take a look at 25 and 26 above. At 25, G stays constant as long as the change in time (delta-t) is equal to or less than the age of the universe. Note that delta-t increases as radius r decreases, so G remains constant. But delta-t has an upper limit of t. If r continues to shrink, G must also shrink. Thus it appears the intensity of gravity is determined by how much space interacts with matter. The smaller the radius r, the smaller the space the matter occupies. Equation 27 shows that the intensity of gravity never exceeds the speed of light squared no matter how much radius r shrinks.

In conclusion, "the bulk" and extra dimensions are completely unnecessary to describe gravity.

What Einstein Didn't Tell You About Time

According to Einstein, if you are sitting in your chair, you are moving more through time and less through space, so time is moving faster for you than it is for a speeding jet. Unlike you, the jet is moving more through space and less through time. If a particle is completely at rest, it moves exclusively through time. If a particle is propagating at the speed of light, it is moving exclusively through space. Below is a Minkowski diagram illustrating the point. The vertical arrow represents the particle at rest and the horizontal line represents a photon. The diagonal arrow could be anything going less than light speed but not at rest.

Now, think about how we humans measure time. We see repeating patterns like day and night, the four seasons, the moon's phases. We then assign time units to these patterns. But imagine a universe where there are no patterns or events, where everything is at rest. How could time be measured? Who would do the measuring?

According to Einstein, everything in that universe is moving through time only, but how can we verify that? There are no clocks to track the alleged passing time. Consider his thought experiment where he has a clock consisting of a light beam oscillating in a boxcar. In the diagram below we mimic that thought experiment with an imaginary photon oscillating vertically inside a box:

The photon is moving at light speed, so special relativity says it experiences no time, but we do. Why? Because we see something happening. We see something that is not at rest. We decided to make that moving photon our clock. Ironically, in order to have time or know time exists, we need something to move through space.

Here's another irony: If all the particles in our imaginary universe were to change to a new overall state, we would say time has moved forward. If all the particles returned to a previous state, we'd say time has gone backwards. If you grow older, time is progressing; if you grow younger, time is regressing. Now, suppose you didn't age or grow younger? Suppose all particles stopped changing states? We would no doubt say time has stopped.

But how can that be? Didn't Einstein say when particles are at rest they move through time?

What gives a sense of time is ongoing change and/or repeated patterns. Time feels real when stuff is happening. When nothing happens, it is as if time has stopped. When things happen faster, it's as if time has sped up, when things slow down, time seems slower. However, if the photon box moves through space, the photon does not appear to complete the cycle as fast, so from our point of view, time has slowed:

But here's the thing: do all clocks slow down when they move faster through space? We will demonstrate the answer to this question is definitely NOT a "yes." First, let's define the variables we will be using:

Here's a new thought experiment: We take a photon clock like the one illustrated above. We have it oscillate in a larger box at velocity u. The larger box is moving at velocity v. Let's assume both velocities are well below the speed of light. We use the Pythagorean theorem to calculate the combined velocity. Below is a diagram for visual reference:

Of course the photon in the smaller box is still moving at velocity c, but this is no longer the clock we want to track time with. We decide to use the larger clock with the small box ticking off the time at a steady rate of velocity u. The question is, what happens to this new rate of time when velocity v is increased? To figure this out requires a little math. We start with equation 1 below which gives the relative mass of the entire system. From there we derive equation 9:

Equation 9 is just Einstein's full energy equation. Let's assume energy is conserved and the combined velocity of v and u is held constant. If we increase velocity v, u must decrease accordingly. So far, our new clock is consistent with relativity's prediction that time slows when velocity v is increased.

Now suppose we add energy to the system. Since both v ad u are well below light speed, we can use the additional energy to increase v without decreasing u! The system mass will increase, but time stays the same!

And what of the photon clock? It slows down as expected. What we now have are three time rates: t', t, and u. We can relate them as follows:

If we only change velocity v, velocity u can remain steady while time t' changes. However, if the combined velocity of u and v is light speed, when v increases, u must decrease.

So to keep our new clock ticking at the same rate, we must keep our combined velocity below light speed. That shouldn't be too hard.

Below we derive equations 16 and 17. These equations express u in terms of time rather than velocity:

Thus when when something moves through space it doesn't necessarily move less through time, and vice versa.

How Entropy, Temperature and Chemical Reactions Impact Time Dilation

How do we measure time? Normally we take some event that happens over and over again, and, we assign it a time unit. For example, the complete rotation of the earth we call a day. We can take that time unit (or any fraction thereof) and assign it to other events--that may or may not be periodic--such as a chemical reaction, for instance. We say the chemical reaction happened in time t.

What if that chemical reaction were to slow down? Can we say its rate of time has slowed? If we define time as the rate of change, then the change in the rate of chemical reactions, entropy, or the earth's rotation would indeed impact the rate of time.

Consider the famous twin paradox, where one twin boards a rocket that hurls him into outer space close to light speed. According to Einstein, his time slows, he ages more slowly than his twin back on earth. His body's biochemical reactions are slower, entropy is reduced--at least that is the implication.

When he arrives back on earth, he will be younger than his twin brother. But his twin has a plan: while he's out in space, his twin cryogenically freezes himself. His twin will be thawed out by the time he gets back to earth. If his twin's plan works, they will both be the same age. Now, did the twin on earth reduce his time rate? Can the rate of time be reduced by means other than high velocities and mass density? This post shall address these questions, but first, let's define the variables we will use:

Let's start with entropy. If we somehow reduce the rate of entropy, will the time rate also be reduced?

If we take the Boltzmann constant (which has entropy units) and multiply it by the unit-less Lorentz factor's squared reciprocal, we derive equation 2 below:

Equation 2 shows that entropy is reduced when velocity (v) is increased. The time rate is also reduced. So far it appears we have a correlation between entropy and time. If equation 2 is valid, we should be able to use it to derive a more standard entropy equation:

Equation 6 above confirms the validity of equation 2. So we can say the twin in outer space, traveling near light speed, has reduced his entropy. Now the twin on earth wants to freeze himself, i.e., lower his body's temperature. Will this reduce his entropy? Equation 8 below confirms that it will. From 8 we derive equation 12 which shows lowering the temperature (T) reduces the time rate.

According to equation 12, the twin on earth will age more slowly. Does this imply the rate of his body's biochemical reactions will slow down?

Equation 13 is the Arrhenius equation, where k is the rate of a chemical reaction. When temperature T is reduced, so is the rate of the chemical reaction. From 13 we derive a new entropy equation at 19:

Equation 19 tells us that when the rate of a chemical reaction is reduced, so is entropy. And the reduced chemical-reaction rate reduces the rate of time:

So the age difference between the twins could be nil when the space twin arrives back on earth. If the earth twin cryogenically freezes himself, he may be younger than the space twin. Both twins have found a way to reduce their respective time rates, they have found ways to time travel into the future. Here are four ways to reduce the time rate:

Why Einstein's Time Theory Works So Well

Experiments involving mu-mesons confirm Einstein's special relativity theory and the value of the Lorentz factor. When a particle is moving fast along a straight line, its time slows down. But is this the only circumstance where time slows? This post will address that question and more. First we define the variables we will use:

Einstein's classic thought experiment involved a speeding train passing by an observer who sees the inside of a boxcar as the train whisks by. The rectangular diagram below represents the inside of the moving boxcar. A person inside the boxcar shines a light from the floor to the ceiling (see red line in diagram). If the train is at rest, the light beam goes straight up to the ceiling. When the train moves, the beam goes up at an angle as indicated in the diagram. Using the Pythagorean theorem, we can derive the Lorentz equation (see equation 7 below):

The train and the vertical light beam can be considered a clock. We can say that one unit of time passes when the beam goes from the floor to the ceiling and vice versa. The Lorentz factor is perfect for this clock. As the train moves faster, it takes longer for the light to reach the ceiling--thus our time unit takes longer, i.e., time slows down.

Now, suppose the light beam went from side to side instead of up and down? Would the Lorentz factor still work or does time behave differently? In the diagram below, we have the beam going from right to left. From that, we derive equation 12 below. Note that equation 8 is the velocity addition formula where the total velocity never exceeds light speed.

In the next diagram, the beam is moving left to right. From that we derive equation 18:

Let's take the average of the right sides of equations 12 and 18 at 19. Doing the math we see that the result at 20 is equal to the Lorentz equation at 21. So when the beam goes horizontally, time behaves the same way as when the beam is vertical. But what if the beam goes at an angle? It would have a horizontal and vertical component. At 22 we use sine and cosine for the vertical and horizontal components.

Equation 23 shows it doesn't matter whether the light beam is horizontal, vertical or some arbitrary angle. In any case, we get the Lorentz-equation value of time (t'). It appears the Lorentz factor works splendidly in flat spacetime and where particles, trains, light all move in straight lines. What about curves?

At 24 and 25 below we start with flat spacetime. At 26 we include the metric tensor. From there we derive equation 31:

Equation 31 confirms the Lorentz factor works in curved spacetime. Now what about crazy, oddball geometries (see diagrams below)? We can still derive the Lorentz equation (see equation 41).

Equation 41 will accommodate horizontal light beams, and beams going at any arbitrary angle or curve. At 42 below we calculate the horizontal component which is equal to the Lorentz equation at 41:

To cap things off, we use basic calculus for curves and squiggly lines to derive the Lorentz equation yet again (see 50 below):

We know that the velocity along a curve varies from point to point, so why is velocity (v) in the above equations unvaried? Because velocity v is the average velocity along the whole curve distance. We take the velocity at each point, add them all together, but that would give us infinity, so we need to divide by n, the number of velocities:

OK, so we've established the Lorentz factor works in all situations where a particle, train or whatever is moving at velocity v along any line. We know that mass can slow time as well as velocity. We also know that protons have their mass due to quark oscillations. So we can think of mass as oscillating velocity. The equations below show the relationships between time, mass, angular velocity and the Lorentz factor:

Update: Below are some diagrams (and calculus) that help one visualize how a light beam (or photon) moves along a curved horizontal component. The first diagram below shows the photon (red dot) moving from left to right along epsilon-x. Epsilon-x is also moving from left to right along x. The total distance covered is x + epsilon-x. The second diagram shows the same except the photon moves right to left. The total distance is x - epsilon-x.

In the next two diagrams, everything is the same except x and epsilon-x are now warped, distorted, curved, etc.

Below is a mathematical proof showing that the magnitudes of x and epsilon-x are invariant no matter how they are distorted: