String Theory Geometry - The Cosmic Investor

Tuesday, November 27, 2018

String Theory Geometry - The Cosmic Investor

String Theory Geometry Part 2

Previously, we claimed that there was no distinction between a garden hose universe with a massive radius and one with a small one. It's easy to say this but how can we make sense of this claim?

The Investor

Before we attempt cosmic investment, we have to look at normal investment.

Imagine you were an investor and you were looking at Nike and Adidas. Someone then tells you that the value of stocks for Nike and Adidas are inversely proportional. Considering they both start the day at $1, how should you invest your money?

The answer is - you should invest in both companies equally.
Worst case scenario they stay the same and you make no profit but if one doubles you'll make $2 + $0.5
If one triples you'll male $3 + $0.33
etc...

No matter how they change, you will not make a loss. It also does not matter which one increases or decreases in value. The profit you make is unaffected. If someone told you you would have $10.10 at the end of the day, you would not be able to say whether Nike went up to $10 and Adidas dropped to $0.10 or vice versa

This analogy can give you a sense of what's happening in string theory. String theory does not care whether it is the winding energy or the vibration energy that increases, all that we care about is the physical characteristics which are determined by total energy.

To improve the analogy we can say that you do not divide your money evenly and instead use $4000 to buy 3000 shares in Nike and 1000 in Adidas. Now your profit does depend on which company increases in price for example if Nike closes at $10 and Adidas closes at $0.1, you will end up with $31,100 but if the inverse happens, you end up with $10,300.

If you have a friend who gets 1000 shares in Nike and 3000 in Adidas, and Nike closes at $10 and Adidas closes at $0.1, the friend will end up with $10,300. If the inverse happens, your friend ends up with $30,100.

In terms of total stock value, swapping over which one closes high is exactly compensated by swapping over the number of shares you own of each company.

The Cosmic Investor

Imagine a garden hose universe with radius 10. A string can wrap around the dimension 1, 2 ,3 times etc... and this is called the winding number. The energy from winding is determined by the length of the string.
Length of string ∝ Radius x Winding number
Energy from winding ∝ Radius x Winding number

Strings can also undergo vibrational motion. These uniform vibrations have energies which are inversely proportional to the radius (they are proportional to 1/R).
Energy from uniform vibrations ∝ 1/Radius
And remember from the ultraviolet catastrophe that energy comes in discrete chunks? That means that the energy from uniform vibrations are whole number multiples of 1/R.
We will call these whole numbers the vibration number.

This is similar to what we had with our normal investment. R and 1/R are analogous to the the closing price per share of the companies .

You can calculate the total value of you investment from the number of shares held in each company and the closing prices.

Let's calculate the total energy carried by each string from its winding number, vibration number and radius.
Below is a table showing the total energy carried by string in a universe with radius 10 and radius 1/10. In reality the table is infinite with winding numbers and vibration numbers which can increase forever however so I can finish this post before the universe ends, I will only include winding numbers and vibration numbers up to 4.

The tables are different however the total energy columns have identical values. Take any entry and swap over the winding and vibration numbers. This gives you the same total energy in the other universe (with radius 1/R).

There is no physical distinction between a universe with radius 10 and 0.1.
If the total energy in the radius 10 universe is 40.1 (a universe with winding number 4 and vibration number 1) then it is equally valid to claim the universe has a radius 0.1 with strings of vibration number 1 and winding number 4.

The total energies are of the form
v/R + wR
where v is the vibration number
w is the winding number

And this equation is invariant when you simultaneously interchange w and v as well as R and 1/R.

The table above is also incomplete because we ignored ordinary oscillations of the string. These give additional energy to strings and determine the force charges it carries. These contributions however do not depend on the size of the universe and so even if we did include ordinary vibrations, the tables would still correspond since the additional energy effects both tables identically.

This means the mass and charges of particles in garden hose universe with radius R are identical to those in a garden-hose universe with radius 1/R.

Mass and charge govern fundamental physics and since there is no way to physically distinguish between these to geometrically distinct universe, any experiment done in one universe has a corresponding experiment that can be done in the other leading to the exact same result...

We can now see how this claim is valid but the question of how can this claim be valid still stands. How can a 6ft man fit in this tiny universe? How can a universe smaller than the Planck length exist when string theory seemingly discarded these lengths?

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