Hierarchy Problem - Solution Via Extra Dimensions

Solution Via Extra Dimensions

If we live in a 3+1 dimensional world, then we calculate the Gravitational Force via Gauss' law for gravity:

(1)

which is simply Newton's law of gravitation. Note that Newton's constant G can be rewritten in terms of the Planck mass.

If we extend this idea to extra dimensions, then we get:

(2)

where is the 3+1+ dimensional Planck mass. However, we are assuming that these extra dimensions are the same size as the normal 3+1 dimensions. Let us say that the extra dimensions are of size n <<< than normal dimensions. If we let r << n, then we get (2). However, if we let r >> n, then we get our usual Newton's law. However, when r >> n, the flux in the extra dimensions becomes a constant, because there is no extra room for gravitational flux to flow through. Thus the flux will be proportional to because this is the flux in the extra dimensions. The formula is:

which gives:

Thus the fundamental Planck mass (the extra dimensional one) could actually be small, meaning that gravity is actually strong, but this must be compensated by the number of the extra dimensions and their size. Physically, this means that gravity is weak because there is a loss of flux to the extra dimensions.

This section adapted from "Quantum Field Theory in a Nutshell" by A. Zee.

Read more about this topic:  Hierarchy Problem

Famous quotes containing the words solution, extra and/or dimensions:

    All the followers of science are fully persuaded that the processes of investigation, if only pushed far enough, will give one certain solution to each question to which they can be applied.... This great law is embodied in the conception of truth and reality. The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth, and the object represented in this opinion is the real.
    Charles Sanders Peirce (1839–1914)

    When a lady of wealth, is seen roaming about in search of cheaper articles, or trying to beat down a shopkeeper, or making a close bargain with those she employs, the impropriety is glaring to all minds. A person of wealth has no occasion to spend time in looking for extra cheap articles; her time could be more profitably employed in distributing to the wants of others. And the practice of beating down tradespeople, is vulgar and degrading, in any one.
    Catherine E. Beecher (1800–1878)

    I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.
    Henry David Thoreau (1817–1862)