Professor Marolf May 11, 2000
Since the earlier part of the 1980’s, people have suspected, and consequently built theories around, the fact that there may be more to the dimensionality of the universe than what we experience. A more recent theory, the theory that will be discussed in more detail in this paper, takes advantage of these ideas to try to solve a problem that is on the minds of physicists. I will discuss this shortly as a prelude to the main body of the work I have done. Through personal correspondence with Paul Steinhardt, a man who is currently working in this particular field, I have learned that there are indeed three approaches to the subject of extra dimensions and what their consequences are. These include the approach of Dimopoulos, the approach of Randall-Sundrum, and the approach of Ovrut. The latter two of these three approaches involves supersymmetry, another theory altogether. However, the view that this paper is based upon is the view of Dimopoulos, which involves working with extra dimensions on a fairly free basis.
The problem that I mentioned before is that it seems that the Planck mass (the smallest unit of mass that can be considered before quantum physics takes over), according to theory, is huge as compared to a nucleon (such as an atom). The jump in scale is quite eye catching. A few calculations are given here that illustrate just how radical this jump in scale is. For instance, if we take the numerical planck mass (Mp), and compare it to the mass contained in 1 atomic mass unit, or 1 nucleon:
By taking the ratio of Mp to amu (atomic mass units), we can find approximately how many nucleons are in a Planck mass.
Which amounts to 2.74x1018 carbon atoms (if we so choose to make our comparison using carbon atoms, which I have on the basis that the amu scale is based on 1/12 of a carbon atoms with six protons and 6 neutrons. If, for the most useful comparison, we decide to see how large a sphere of carbon we could form by using a Planck’s mass worth of carbon, we obtain a suprisingly large result, slightly larger than a speck of dust. For this calculations I have used the value for the density of carbon at 298 K. By using the desnity of carbon and the volume of a sphere, we can obtain a hypothetical radius. Using the basic definitions for density and Volume of a sphere, we can obtain the relation:
This relation, when the density of carbon at 298 K is factored in (2267 kg/m3), we obtain that the radius of such a sphere would have the value:
While this is extremely small, it is quite large on the atomic scale.
The theory that Dimopoulos and company has put out, in order in part to solve this problem, is that the universe contains small extra dimensions, which would, when factored in, make the Planck mass grow smaller as to correspond with nature. How this is derived mathematically will be shown in the following sections. In addition, the theory takes into account the fact that there is a weakness of gravity at distances much greater than 1 mm, the size of their proposed small dimensions.
The Dimopoulos paper takes into account several things, but one of the main things discusses is how gravity (or in their case gravitational potential) will behave under the new system that is being proposed. The following equations are used to describe gravitational potential in two places – the first within the bounds of the dimension, where the distance separating two masses attracting each other is much less than the size of the small dimension. In the second case, the separation distance is much larger than the dimension size:
In order to understand these equations, it is essential that week take them apart carefully and see how we can derive these equations using basic physics laws.
Two laws that will play most of roles here include Gauss’ Law and Newton’s universal law of gravitation. By observing the equations, we can see that they do bear a resemblance to the equations that would describe regular Newtonian gravitation. However, we can derive the “normal” gravitation law, as well as these equations, through the gravitational analog of Gauss’ Law. Gauss’ Law takes this form when we talk of gravitational flux:
Although most of my calculations do not use this explicitly, when using Gauss’ Law I take g to be uniform throughout, so the best of showing the propagation of flux lines equally in all directions is to assume that my surface is a sphere. Just to show that the Gauss’ Law model of gravitation is consistent with common sense understanding, we shall consider a sphere of matter producing gravitational flux lines. If I rewrite this equation simply as:
I can take the ratio of flux divided by area being equivalent to the gravitational field at that point in our spherical surface.
Thus, if we take the surface area to be very large, g ( or the gravitational field) gets very small, and the number of flux lines per unit surface area gets very small. This appeals to our normal sense – if we move farther and farther away from the mass, the gravitation field gets weaker and weaker. Here we use a different notion – here we are expanding the spherical surface that we consider flux lines within. The farther away we get, the less flux lines we encounter, and thus the weaker the field. Here is a diagram that shows this – the outer circle is a big Gaussian surface, while the smaller circle is a miniature version nearer to the center mass. Note how in the smaller one has more flux lines per unit surface area – the gravitational field will be larger here than at the larger surface farther away.
If we consider Gauss’s law again, for the case of attraction between two masses, we see that the term M will represent either mass in reality, but we consider in this case that one mass is producing the gravitational field while the other is being attracted by it. Therefore, we will assume, for the sake of our argument, that m1 is the mass that produces the gravitational field and that m2 is attracted by it. Therefore, our Gauss’s law equation becomes:
Since g is uniform throughout, and dA is simply a generic surface area, we can write:
We multiply both sides by the other mass, m2, to get the force expression, with A ( at least for the case of a 3 dimension world) to be equal to r^2 times some factor pertaining to the surface, we get the classical expression for gravitation:
However, it important to remember the previous version of g being given in terms of area. This is important in forming our understanding of how extra dimensions affect the propagation of gravitational flux. The absence of “G” in the equations given by Dimopoulos concern the fact that he has expressed “G” in terms of the Planck mass, which is essential to solving the problem that I set forth at the beginning. However, at this time I will concentrate on the derivation of all parts of the equation save the Planck Mass term, which I will discuss after expounding on the other parts. Examining the Dimopoulos equations again,
It becomes clear that the extra factors of R and r come from the area factor that we have been discussing. However, the question is, how does this come about, and why are there two distinct cases, one for objects deep within the extra dimension(s) and one for very far away from them?
When considering the small case, all sizes are within the dimension, so r (the distance between the two objects) and R (the size of the dimension) are all within the same boundary, and form a “spherical surface”, or whatever the dimension equivalent of that is. I have split up the equations into the following form before we continue, as it illustrates the regular and dimensional components of the two equations separately, at least when we are talking about the area:
Equation 1 can be derived in the following way. Gravitational potential is the integral over the Force for a certain distance ds, or in this case dr. We can define Force here as:
Both are small r since we are in the domain of the dimension – and thus we are able to extend in all directions with the same r – essentially we are able to create a spherical surface. Therefore, it’s surface area is a 2+n dimensional one. Depending on how many dimensions there are, there will be that many factors of r. Now we can integrate this with respect to dr and obtain gravitational potential.
This is exactly the same as equation 1 , except that we have not expressed G in terms of the Planck mass. As I mentioned, I will explain this after I explain the derivation of the rest of the 2 equations.
The derivation for the second is equation is roughly the same thing. Except this time the extra dimension are much smaller than the r between the two masses. In this case, the area brought about by the dimensions must be a constant factor, a factor of R^n, where R is the size of the dimension.
This surface is no longer spherical, but has a very odd shape. An interesting thing to note is that while the dimensions still affect the gravitation between the two masses, they contribute very little because of the small surface area that it is contributing. Now, if we integrate again with respect to dR:
This equation is again the same thing as equation two, save for the G factor.
There are several interesting consequences that arise from these calculations in terms of gravitation. I mentioned briefly before that a weakness of gravity was built into the theory, and these calculations are the mathematical proof. As we have seen before, if we increase the surface area of the Gaussian surface in which the flux lines are being considered, you decrease the gravitational field at that point. If we extend into extra dimensions as a sphere, we are definitely increasing our area, and thus gravitation will be weaker. In the smaller dimensions this in most evident. In the larger dimensions, the surface we are considering along with the smaller dimensions cannot be a perfect sphere. In fact, the contribution to the Volume from the smaller dimensions at distances very for away from the size of the dimension are almost negligible. The following diagrams demonstrate this point:
(See next page )
These two figures are an attempt to bring the idea of this expansion into an extra dimension and the consequences for the two equations into a familiar light. In the first illustration, the by extending the two-dimensional figure up into the third dimension, we have increased the surface area. If we were inside the bounds of the dimensions (assuming that our h was bounded in the sense that R is bounded), this would work perfectly. However, in the second illustration, the length and width are not bounded but the height is. If this height approaches a very small size as I have drawn, volume contributed is near zero and is of no consequence.
What follows now is what I have been referring to – the derivation of the Planck Mass term. To derive the term we need to do dimensional analysis using other known quantities. The following formulas were used to derive the Planck term:
These equations were solved as simultaneous equations by the following methods:
Multiplying the second and third equations together, we get:
Solving the previous equation for L^3 and substituting into the Newton’s Gravitational relation, we get
By virtue of the fact that following from the original 3 equations,
We can therefore combine this t with L=ct and get
This allows us to substitute back into our recently modified version of Newton’s Gravitational constant, and get the following, which through the substitutions shown allows us to solve for an expression for the Planck mass:
This result ties in with solving the problem of the “too large scale” that I went over extensively in the beginning. As the number of extra dimensions goes up, the value for the Planck mass goes down, becoming less and less.
It is interesting to note that the equations we have been discussing do account for what we consider to be reality. In our everyday lives, or our approximations of planetary motions and such, the normal equation for gravitational potential applies. It is important to see that when looked at properly, these equations do cancel to their “proper” forms.
For instance, let us rewrite the first major equation, substituting the Planck term for what we have just calculated.
Taking n=0 dimensions (since in “regular” observation we don’t take these extra dimensions into account), and taking h(bar)=1 and c =1, we obtain the normal expression for the potential:
Thus, in conclusion to this paper, we have delved into a small part of what Dimopoulos has done with this concept. He taken it further, whereas I have expounded one one tiny portion of it. It is said that these extra dimensions, called branes, although we have said they experience weak gravity – at very short distances the gravity becomes stronger than you would expect. This is the same effect as you have if you start putting fractions into the “regular” gravitation equation – the inverse square proportionality makes the gravity become stronger. This is especially true of the branes, since the factors depend on the dimension – some even having r^(n+1) for the smaller distances. Studies are being conducted right now on the smaller scale to test whether this theory actually holds any weight. Previously people had been concerned only with large-scale tests of the theory of gravitation – such as the one Cavendish originally did to calculate G. Branes are being hailed as possibly a “ new kind of dark matter” – since the branes could contain mass, and this could affect galaxies and not be seen since the dimensions are so tiny. With all the tests that are going to to either validate or destroy the theory – I would be very anxious to see what the results are from the tests. If the theories hold it could lead to a whole new way in which we interpret the universe- and perhaps reality itself.