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

Debunking Extra Space Dimensions and Minimum Distance

ABSTRACT:

By means of two thought experiments and some mathematics this paper shows that extra space dimensions are untenable. This paper also shows that the minimum distance is many orders of magnitude shorter than the Planck length.

Imagine a 2D universe on an x-y plane (see diagram below). Imagine a normal vector intersecting this plane at point p. 2D-guy inhabits this universe. He can't see the vector that intersects point p. He can only detect point p, so he has no reason to believe the normal vector exists. Now, to avoid point p, he goes around it (see red arrows).

He knows it's possible to draw an imaginary line through point p that can serve as an axis. He also notices when he goes around point p he's not encircling the x-axis or the y-axis--the two dimensions of his space. Thus, he infers that the imaginary axis he's going around does not belong to his 2D universe. He realizes he has discovered a new dimension!

Now, what happens if we apply 2D-guy's process to 3D space? Will we discover a fourth dimension? Let's try it. First we must scale everything up one dimension: The universe becomes 3D; the normal vector becomes a normal plane; Point p becomes line L. Let's assume there's a fourth dimension w, and let's define the normal plane as wx. Plane wx intersects our universe at line L which runs along the x-axis. We should not be able to detect the w-axis nor the bulk of the wx plane. We illustrate this with broken lines at the diagram below:

To avoid line L, we circle around it (see red circular path). We know we can draw an imaginary plane through line L. We know that x is one dimension of the plane. We know the axis we are circling (to avoid line L) is the plane's other dimension. We note we are not going around the x-axis nor the z-axis. That leaves the w-axis, but notice that the w-axis is indistinguishable from the y-axis. Therefore, our assumption that w is a new dimension and is undectable beyond line L is false. Unlike 2D-guy, we have not discovered a new dimension. However, we learned from 2D-guy that if a new dimension exists, it should be possible to do a rotation around an axis that does not exist in our universe. Until someone demonstrates such a rotation, we can conclude, for now, that the highest dimension of space is 3D.

But what if there are extra dimensions that are very small and curled up? If that's the case we should be able to enter alternate universes and those from alternate universes should be able to enter ours. Let me demonstrate what I mean. Imagine a line and pretend it is 3D space. Extending from it is a small extra curled-up dimension:

Let's introduce an arbitrary red object that is way too big to enter the tiny curled-up dimension:

Because the red object is too big to fit, it is assumed there is no way for the big red object to enter or detect the existence of the curled-up dimension. But didn't Euclid say something about a line existing between any two points? (In this case the line would be 3D.)

There's no reason why the big red object can't follow the path of this new line (3D space)and wind up in an alternate universe adjacent to ours:

As you can see, the big red object still can't enter the small, curled-up dimension, but the curled dimension facilitates access to alternate universes. The fact that big objects don't disappear from our universe and don't seemingly emerge from nowhere is strong evidence that microscopic curled-up dimensions don't exist. But wait! Quantum particles pop into existence and vanish all the time. It is hypothetically believed they enter a curled-up dimension (vanish), then leave that dimension and re-enter our universe. However, there's an alternate hypothesis: particles are really particle-waves. Waves experience constructive and destructive interference. When there's an excitation of a field, a particle pops into existence. That excitation could be or is equivalent to constructive interference. When there's destructive interference, energy vanishes--leaving the impression that the particle has disappeared.

The foregoing arguments seem to kill any notion that there are more than three space dimensions, but what about 4D spacetime? Or, what about the 6D object that can be found in Las Vegas? Let's address the 6D object first.

The 6D object I'm referring to is the die. The die has six orthogonal sides. Each side is statistically independent. We can change the value of a side without impacting the value of the other sides. If we change, say, the one to a seven, the other sides will still be two, three, four, five, and six. The most important point we can take away from the die is it is possible to have more than three orthogonal dimensions within 3D space! The die is a 6D object but it is also a 3D cube.

Spacetime, on the other hand, involves three dimensions of space and one dimension of time. If time is multiplied by a velocity, it has units of distance and is treated as a fourth space dimension. But is it really? Let's see what the math has to say:

Equation 1 represents a photon propagating through dimensions x, y, and z over a period of time t. It covers a distance of ct or r. For the sake of keeping the math simple, at equation 2 we rotate the path r so it is along the x-axis. Equation 3 reveals that space and time are not statistically independent, i.e., orthogonal to each other. The the value of time t depends on how far the photon propagates along x, and the value of x depends on how much time t lapses. This is the consequence of converting t into distance units by multiplying it by velocity c. So ct is not a true space dimension that is orthogonal to x. However, time t without c is a very useful statistically-independent parameter. For example, coordinates x, y, z tell you where to be for your dentist appointment and time t tells you when. A change in location does not have to change the time of the appointment, nor does a change in time have to change the location. So what can be done to make ct orthogonal to x? How about multiplying ct and x by factors of g? (See equation 6.) A change in x still causes a change in t, but g-sub-tt can be adjusted so the term stays constant. By the same token, the other term stays constant if g-sub-xx is adjusted when a change in t changes x.

So can we now credibly argue that (g-sub-tt)ct is a genuine fourth space dimension? Well, no 3D space dimension (x,y,z) has to be a function of (depend on) the others. We can, for example, eliminate y and z and still have x. But we can't eliminate a photon's path (x, y and/or z) and still have ct--the distance along a non-existent path. And, if there's no ct, then there's no (g-sub-tt)ct. Therefore, (g-sub-tt)ct is a pseudo-dimension at best.

So far, it seems we've only debunked a fourth dimension of space. What about dimensions five through infinity? Well, how we label a dimension is arbitrary. Any extra dimension can be labeled the fourth dimension. Thus, all arguments we have made against dimension four apply to any extra space dimension.

Now let's turn our attention to the concept of the shortest distance. The popular choice is the Planck length. In fact some theorists quantize space with Planck-size cubes or Planck-size tetrahedrons or Planck-size strings:

In the above diagram, the cube and tetrahedron have sides that are each one Planck length. However, the red diagonal lines reveal shorter lengths all the way down to a single point. These shorter lengths are absolutely necessary to create the shapes desired. Without a zero-length point, for example, there can be no corners for cubes and tetrahedrons. Additionally, there can be no strings in any string theory, since a string is a 1D object. A 1D object implies a zero cross-section or single point. A minimum-distance-greater-than-zero requirement would be a nightmare for M-theorists, since all D-branes would have to be 10 dimensions (including strings!). To have less than 10 dimensions requires zero distance for one or more dimensions. So it can be argued that the minimum distance is really zero, at least on paper. What about the physical world?

Equation 7 tells us that the shortest wavelength is determined by the highest energy. When the universe was a singularity, how short was the singularity's wavelength? If we only account for the energy in the known universe, that wavelength would be approximately a Planck length of a Plank length of a Planck length! Not exactly zero, but far less than a Planck length. Add energy beyond our known universe, and the distance is even shorter.

From a philosophical standpoint, the very concept of length implies a 1D object in the same manner the concept of area implies a 2D object. To measure length requires that we ignore all but one dimension, i.e., we set all but one dimension to zero. So zero distance is necessary, at least in the mind's eye. Since the mind's eye lives in this universe, we can infer that the minimum distance in this universe is zero.

In conclusion, any extra space dimension would allow rotations around an imaginary axis that is not part of 3D space. It would also allow any object access to an alternate universe. The shortest distance is many orders of magnitude shorter than the Planck length, and the Planck length may only be a lower limit of what we can successfully measure.

References:

1. Greene, Brian. 2003. The Elegant Universe. W. W. Norton

2. Irwin, Klee. 04/23/2017. The Tetrahedron. Quantum Gravity Research.

3. Sutter, Paul. 02/23/2022. Loop Quantum Gravity: Does Space-time Come in Tiny Chunks? Space.com

Is Math Reality?

Is math reality or just a proposed description of reality which may falsely describe, or not accurately or fully describe reality? According to some, the fact that 2 + 2 = 4 is proof that math is reality and vice versa. But what about 2 + 2 = 5? One could argue that this is wrong and fits no reality and isn't real. The counter argument is 2 + 2 = 5 is real (like many string theories) in some parallel universe inside a megaverse.

Unfortunately, there is no empirical evidence of parallel universes--only mathematical evidence. Yet, it is argued that mathematical evidence is every bit as valid as empirical evidence. When these two types of evidence disagree, we simply invent a new universe, i.e., a reality where they do agree. Problem solved?

Hmmmm ..., if math is always right somewhere within a megaverse, wouldn't every crackpot idea be right somewhere within a megaverse? For example, our earth isn't flat, but surely there is some parallel universe that contains a flat earth? It looks as though the scientific method becomes a joke if we are allowed to move the goalpost (invent new unobserved universes) when the math or theory is wrong in this universe.

To claim that math is reality is to ignore infinities--that have never been observed or verified--and negative probabilities (predicted by quantum physics math) and all the instances where the math is wrong, or, where the math is approximate. When calculating the area of a circle, who knows and uses the exact value of pi? No one to my knowledge. If we don't know and can't determine the exact value of pi, then how can we say with confidence that it is real? Our math in general is riddled with approximations and error margins. From an empirical standpoint, we have yet to make a perfect measurement of the circumference of an ellipse or calculate it with a perfect value of pi; yet, some argue that math is reality and reality is math.

Do Electrons Leave Our Universe When They Move Inside the Atom?

Using a quantum microscope, it is possible to view the wave function of the hydrogen atom (see diagram above). Note the electron can only be found in the lit areas and not in the dark areas between. When the electron moves between energy levels, it seems to mysteriously vanish from our universe and then mysteriously reappears, hence the alternating pattern of dark and light circular bands.

Of course there's a string theory that explains this phenomenon: when the electron leaves an energy level it literally leaves our 3D space and enters a higher curled dimension. It loops around then re-enters our space:

At the diagram above you can imagine the electron (red dot ) moving left to right. The loop represents the extra hidden dimension.

Now notice the diagram below. It is the famous double-slit experiment. Notice the target screen has light and dark areas. We could postulate that the dark areas are due to the light disappearing into a higher dimension or an alternate universe--or perhaps it was abducted by extraterrestrials.

Of course this is all nonsense. We know why the dark sections are dark and why the light sections are light: constructive and destructive interference wave patterns. Where the red and black lines are parallel there's light (constructive interference), where they cross or move in opposite directions, there is darkness (destructive interference).

Since there is no empirical evidence of extra dimensions, it makes sense to propound an alternate hypothesis that uses established physics as its basis--the established physics being constructive and destructive interference. If you compare the two diagrams above, you will note a striking similarity: both have alternating light and dark areas on their respective target screens.

We know what causes the interference pattern of the double-slit experiment--it's the two slits. But what could possibly cause the interference pattern (discrete energy levels) of the hydrogen atom? That's what this post shall cover. First, let's define the variables needed:

At equation 1 below, energy (E) is charge (q) times voltage (V). Assuming energy E is an eigenvalue, it must have a probability (P(E)). That brings us to equation 2. Quantum mechanics tells us the probability is an amplitude squared (A^2, see equation 3). A pinch of algebra gives equations 5 and 6.

At equation 5 we see the positive charge from the hydrogen atom's proton corresponds to a positive amplitude squared. At 6 the negative charge from the electron corresponds to a negative amplitude squared. To get a negative amplitude squared we multiply the positive amplitude of a sine wave with its negative amplitude:

Notice there's no way to get a positive amplitude squared using this method. To get the positive squared amplitude, we need to do the following:

We use the absolute value of sine to create a positive squared amplitude. We now have what we need to model the hydrogen atom's constructive and destructive interference wave pattern. Check out equations 7 through 10 below:

When we plug in values for equation 7, we get the wave pattern below which maps beautifully to the light and dark areas of the hydrogen atom:

No extra dimensions needed, just old-school physics.

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.

Why a Discrete Minimum Distance Fails

In the previous post entitled "Why Strings Don't Exist", we showed it is possible to have a length shorter than the famed Planck length. The question becomes, what is the minimum discrete distance, assuming there is such a thing? Normally, we use infinitesimal points to build geometrical objects like lines, planes, rectangles, etc. What happens if we use a line with the smallest magnitude possible that is greater than zero? Before we delve into these questions, let's define the variables:

OK, let's assume the fundamental building block is a line somewhere between zero and the Planck length. We'll call it d:

Let's try building a square with d:

So far, so good. All the distances appear to be no less than d. But what is the distance along the diagonal (or hypotenuse)?

The diagonal distance is a little bit more than d. The additional distance is marked in red. Notice this distance is not an integer multiple of d. To make this distance, we need a distance d plus a distance less than d. There can be no distance less than d, so we can't draw the above square. Here's an idea: draw a rectangle with 3X4 d-units. Thanks to Pythagoras, the diagonal will be 5 d-units:

Because all distances must be integer units of d, our geometry does not include squares, rectangles or triangles that have diagonals and sides that fail to have magnitudes that are integer multiples of d. But at least we found one rectangle that works--or maybe not:

If we draw lines from each d to every other d, we once again have distances that are not integer multiples of d. In the example above, we have a distance (c) of 3.6d. To have that distance, we need 3d plus a 0.6d. In our geometry, there's no such thing as 0.6d. Thus we can't draw the 3X4 rectangle. In fact, every rectangle and triangle we draw will have some distance that includes a fraction of d.

Perhaps we'll have better luck with circles? Check this out:

If we take distance d and shape it into a circle circumference (C), the diameter (D) will be less than d! D = d/pi (where C=d). If we try to draw any circle with d-units, we hit a brick wall. You see, you calculate the circumference with pi * D. If D is an integer (n) multiple of d, pi * n won't equal a circumference with an integer multiple of d. Pi is an irrational number.

OK, so perfect circles are out. How about imperfect circles? Perhaps we can replace pi with something more rational. Even if we do, circles have the same problem we encountered earlier:

There's always a line between two d's, that is not an integer multiple of d. This is also true with any shape imaginable:

So far, things look pretty hopeless for our geometry based on d. But unfortunately, there's more pain. Let's go back to the beginning and reexamine d:

Distance d is a line along the x-axis. But what is it along the y and z axis, i.e., what is its cross section?

Line d has a cross section of zero magnitude! That zero magnitude is just a single point in space with zero distance! Zero distance is not allowed, so a distance-d line is not allowed. Perhaps we can convert the line into a cylinder, so the cross-section will have a d-magnitude. Let's look at our new cross section:

Oops! The cross section is shaped like a circle. We can easily draw a red line from one d to the other that isn't an integer multiple of d. Thus it is now abundantly clear that a geometry based on distance d (instead of a point) is a dismal failure.

Update: Here is a formal proof that shows that, at any two non-parallel adjacent unit lengths (in this case the Planck length), it is possible to draw lines (lines a, b) shorter than the unit length. The following diagram is an example of an arbitrary shape:

Note that the shape is made up of connected unit lengths. Here are two examples of adjacent-unit lengths blown up to a convenient size for inspection:

Now let's take one arbitrary pair of connected unit lengths and label the lines and angles:

Now we are ready to do the proof. Here it is:

Update: The following proof demonstrates why a curved Planck-length string creates distances shorter than the so-called shortest distance. Below are some examples of curved strings:

The examples above clearly make the point, but is the point generally true? Here's the proof: