According to string theory, d-branes come in one or more dimensions. They provide an anchor for strings. Some string theories have only even-numbered d-branes; others have odd numbered d-branes. As we shall see later in this post, the odd d-branes, or systems with an odd number of space dimensions, have a better shot at being real. One real example is space in our universe. We perceive it as 3D, and it can be thought of as a giant 3-brane.
But what about 4-branes, 5-branes and beyond? Today we are going to put various multi-dimensional branes to a rigorous test. The test is designed to show whether extra dimensions truly exist. It is the cross-product test. The diagram below shows how the cross product works in three dimensions:
The results are in red. When we calculate the cross-product of two dimensions, we get a third dimension that is perpendicular to the others. To test for extra dimensions, we need to define what we mean when we say "dimension." Dimensions are lines in space that are perpendicular to each other.
Because they are perpendicular to each other, their cross-products must yield a kind of symmetry, i.e., there must be the same number of each dimension. The indexes can also be added or subtracted to get the cross-product. For example, in 3D space, D3D1 = D2 (3-1=2) and D1D2 = D3 (1+2=3) However, there must not be any index sharing. For example, D4D8 = D4 is invalid, since the D4 on the left side of the equation is clearly the same dimension as the D4 on the right. We could change the index(s) to cover our tracks, but then we lose the index symmetry--and such a loss reveals the problem in a different way.
We also need at least three dimensions to get started. Applying the test to 1-branes and 2-branes yields a bunch of zeros. But at least there is symmetry, so the dimensions in the 2-brane might be perpendicular. The fact there is no index sharing is also a good sign.
As you can see, I put the results in tables. Each row element and each column element yield a corresponding result in the tables (let your finger be your guide). Now let's look at a 3D system or 3-brane:
You'll notice cross-products that share the same index yield zero as they should. Three dimensions obey the index rule and have perfect symmetry. The bottom table above shows there are two results for each dimension. A 3-brane is perfect; it meets all the requirements, and we'll use it as a yard stick to judge systems and objects with extra dimensions.
Let's check out 4D:
The 4-brane isn't looking good. The results in red violate the index-sharing rule. There is also a lack of symmetry--there is not the same number of each dimension (see right-hand table). The cross-products are also fake. I'll explain what that means later when I'm done laying the groundwork. For now, let's move up to five dimensions:
Ah! Perfect symmetry! Four of each dimension. Could we be living in a 5D universe? A 5-brane sure looks feasible--but then there are those nasty index violations in red. Oh well.
The 5-brane does show, however, that odd numbers of dimensions have symmetry. What is true for five dimensions is also true for seven dimensions and the nine dimensions of E500 string theories:
Once again we have perfect symmetry and index violations in red. What about the 10 space dimensions of M-theory?
Holy d-brane, Batman! What a mess! Ten dimensions lack symmetry and have bleeding-red index violations. It's unlikely these dimensions are perpendicular to each other. But what about the 9D system? That one looks pretty good, not perfect like 3D, but close. Thus it is time to explain why its cross-products are fake as a presidential campaign promise.
If there are really nine dimensions of space then any cross-product between any two dimensions should yield the remaining seven--since the remaining seven are allegedly perpendicular to the two. The above tables only show single results for each cross-product. Such a strategy is useful in that it exposes which cross-products aren't perpendicular (see index violations marked in red). That being said, D1D2 should yield D9 and the rest. D3D6 should yield D8 and the rest. If these extra dimensions were real, there would be more uncertainty due to multiple results. We would not get a nice, single result from any cross-product. If we could choose the ideal d-brane or universe, 3D would be the best choice. The marvelous thing about 3D is you have perfect symmetry and the same number of dot-products as cross-products. With higher dimensions this is never the case.
The diagram above shows how D1D2 equals seven possible results in a 9D system. Such a system has 81 dot-products and 567 cross-products! Not exactly a balanced system. Then again, what if all those extra dimensions are tiny and curled up? That would sweep the multiple results under the rug. However, if dimensions are curled, their angles are no longer perpendicular. The diagram below shows the x-axis being curled to the y-axis. Note how the angle is no longer ninety degrees.
If dimensions don't have to be right angles to each other, then we can have an infinite number of them within right-angled 2D or 3D space.
The above diagram shows how a 2D triangle can be expanded into a 4D triangle that occupies a flat 2D space. But why stop at four dimensions when we can have eight or more?
As you can see, it is easy to fool ourselves into believing there are extra dimensions. What we call extra dimensions could really just be a bunch of posers occupying a flat 2D or 3D space. If there are really four or more dimensions that are right angles to each other, cross-product results would be uncertain. A lack of symmetry implies extra dimensions do not exist.
For more on this topic, check out theses links: "Debunking Bosonic String Theory's 26 Dimensions" and "Are String Theory's Extra Dimensions Real?" and How to Derive M-Theory's Eleven Dimensions and Reduce Them to Four Dimensions.
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