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
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