Most string theory proposals have been based on the concept that the extra dimensions required by the theory are curled up so small that they can’t be observed. With M-theory and brane worlds, it may be possible to overcome this restriction.
A few scenarios have been proposed to try to describe a mathematically coherent version of M-theory, which would allow the extra dimensions to be extended. If any of these scenarios hold true, they have profound implications for how (and where) physicists should be looking for the extra dimensions of string theory.
Measurable dimensions
One model that has gotten quite a bit of attention was proposed in 1998 by Savas Dimopoulos, Nima Arkani-Hamed, and Gia Dvali. In this theory, some of the extra dimensions could be as large as a millimeter without contradicting known experiments, which means that it may be possible to observe their effects in experiments conducted at CERN’s Large Hadron Collider (LHC). (This proposal has no unique name, but I call it MDM for millimeter dimension model. Who knows, maybe it’ll catch on!)
When Dimopoulos introduced MDM at a 1998 supersymmetry conference, it was actually something of a subversive act. He was making a bold statement: Extra dimensions were as important, if not more so, than supersymmetry.
Many physicists believe that supersymmetry is the key physical principle that will prove to be the foundation of M-theory. Dimopoulos proposed that the extra dimensions — previously viewed as an unfortunate mathematical complication to be ignored as much as possible — could be the fundamental physical principle M-theory was looking for.
In MDM, a pair of extra dimensions could extend as far as a millimeter away from the 3-dimensional brane that we reside on. If they extend much more than a millimeter, someone would have noticed by now, but at a millimeter, the deviation from Newton’s law of gravity would be so slight that no one would be any the wiser. So because gravity is radiating out into extra dimensions, it would explain why gravity is so much weaker than the brane-bound forces.
The way this works is everything in our universe is trapped on our 3-dimensional brane except gravity, which can extend off of our brane to affect the other dimensions. Unlike in string theory, the extra dimensions wouldn’t be noticeable in experiments except for gravity probes, and in 1998, gravity hadn’t been tested at distances shorter than a millimeter.
Now, don’t get too excited yet. Experiments have been done to look for these extra millimeter-sized dimensions and, it turns out, they probably don’t exist. Experiments show that the dimensions have to be at least as small as a tenth of a millimeter, but that’s still far larger than in most other string theory scenarios. Instead of requiring the 1019 GeV (giga-electronvolts, a unit of energy) needed to explore the Planck length, exploring a millimeter would require only 1,000 GeV — still within the range of CERN’s LHC!
Infinite dimensions: Randall-Sundrum models
If a millimeter-sized dimension turned heads, the 1999 proposal by Lisa Randall and Raman Sundrum was even more spectacular. In these Randall-Sundrum models, gravity behaves differently in different dimensions, depending on the geometry of the branes.
Yet another string theory: F-theory
Another theory that sometimes gets discussed is called F-theory (the name is a joking reference to the idea that the M in M-theory stands for mother). Cumrun Vafa proposed F-theory in 1996 after noticing that certain complicated solutions of Type IIB string theory could be described in terms of a simpler solution of a different theory with 12 dimensions, up from the 10 dimensions of superstrings or the 11 dimensions of M-theory. Unlike M-theory, where all the dimensions of space-time are treated on equal footing, two of the dimensions of F-theory are fundamentally different than the rest: They always have to be curled up. So now to get to three space dimensions, we have eight small dimensions instead of six!
This makes it seem as though the theory is getting more complicated, but in fact the
F-theory description is often simpler. These eight dimensions include not only all the information from the previous six, but also information about what branes exist in the solution (those setups could get complicated). This is an example of a common theme in the development of string theory; more and more of the theory’s details, such as what particles exist and how they interact, or what branes live where, can be described simply in terms of the geometry of the extra dimensions. This geometry is often easier to understand and analyze.
F-theory has been receiving more attention in the past few years because its rich structure allows solutions that reproduce many of the phenomena of the Standard Model and GUT theories (see topic 12 for more on those).
In the original Randall-Sundrum model, called RS1, they propose a brane that sets the strength of gravity. In this gravitybrane, the strength of gravity is extremely large. As you move in a fifth dimension away from the gravitybrane, the strength of gravity drops exponentially.
An important aspect of the RS1 model is that the strength of gravity depends only on the position within the fifth dimension. Because our entire 3-brane (this is a brane world scenario, where we’re trapped on a 3-brane of space) is at the same fifth-dimensional position, gravity is consistent everywhere in the 3-brane.
In a second scenario, called RS2, Randall and Sundrum realized that the 3-brane that we’re stuck in could have its own gravitational influence. Though gravitons can drift away from the 3-brane into other dimensions, they can’t get very far because of the pull of our 3-brane. Even with large dimensions, the effects of gravity leaking into other dimensions would be incredibly small. Randall and Sundrum called the RS2 model localized gravity.
In both of these models, the key feature is that gravity on our own 3-brane is essentially always the same. If this weren’t the case, we’d have noticed the extra dimensions before now.
In 2000, Lisa Randall proposed another model with Andreas Karch called locally localized gravity. In this model, the extra dimension contained some
negative vacuum energy. It goes beyond the earlier models, because it allows gravity to be localized in different ways in different regions. Our local area looks 4-dimensional and has 4-dimensional gravity, but other regions of the universe might follow different laws.
