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Warped extra dimension (Randall-Sundrum models) - 1999
Lisa Randall (Harvard)
Raman Sundrum
The Randall-Sundrum model considers a particular warped geometry bounded together by 2 branes. Proposed by Lisa Randall and Raman Sundrum: our universe is a 5-dimensional anti-de Sitter space. All of the particles, except for the graviton, are localized on a 4-dimensional brane.
The idea is that, if the universe is really a three-brane, floating in higher dimensional space, perhaps, that could explain why gravity is so much weaker than the other fundamental forces: electromagnetism and the strong and weak nuclear forces. The question of why gravity is such a feeble force, evaded even Isaac Newton and Albert Einstein. Why is gravity wildly different? In fact, Lisa Randall, when speaking of the difference in mass for the quarks with the scale of quantum gravity, said: "The discrepancy is not small; the two mass scales are separated by 16 orders of magnitude! Only theories that explain this huge ratio are likely candidates for theories underlying the Standard Model."
When we attempt to apply M-theory to the physical universe, our universe can be seen as a three-brane floating in a 5-dimensional space. This 5th dimension could perhaps be infinite. The vibrations on the surface of the three-brane correspond to the atoms that we see around us. However, these atoms cannot leave the three-brane: they are confined and cannot drift into the 5th dimension. If this is true, than, we are floating in the 5th dimension. However, we cannot go there because we are confined to the three-brane.
The task for these theorists was to show how the 5th dimension could be infinite, while preserving Newton's law of gravity. To achieve this, the three-brane would have to have it's own kind of gravitational attraction to prevent the gravitons from drifting away into the 5th dimensional bulk. The gravitons would cling to the three-brane. Thus, when we measure Newton's law: it would be correct for our universe. Gravity would be weakened as it drifts from the three-brane into the 5th dimension. However, this gravity does not get very far. Gravitons would still be attracted to the three-brane, as Newton's law still stays roughly intact.
These theorists will also exercise the possibility that there is another membrane parallel to ours. The weakness of gravity can be explained by adjusting the subtle interaction of gravity across the two membranes. If this idea is true, than gravity could be just as strong as the other fundamental forces. However, some of it leaks into the higher dimensional space.
In the Randall-Sundrum models, the spacetime geometry of the 5-dimensional bulk is very warped. There are two branes in this bulk: the Planck brane (or Gravitybrane), where gravity is strong and the Tevbrane (or Weakbrane), which is home to the Standard Model.
This warped geometry also brings about a proposed explanation to the hierarchy problem: objects moving from the Planckbrane to the Tevbrane, become lighter, hence, gravity becomes weaker. The vast range of masses in the Standard Model can be understood by the warped geometry of a 5-dimensional world. This vast array of masses would arise naturally from the physics of a warped 5-dimensional world. There are two proposed Randall-Sundrum models:
Two branes and the warped 5-dimensional bulk.
RS1: The bulk has a finite size and is an anti-de Sitter space. Anti-de Sitter space has negative curvature. There are also two branes, one at each side. In this 4-dimensional universe, there is warped geometry and two branes. However, the Standard Model, resides on the 4-dimensional weakbrane. The purpose of the Weakbrane, thus, is two-fold: to confine the particles of the Standard Model and to ensure that the 5th dimension is finite in size. There is a total of 5 spacetime dimensions in this set up. There are 4 spacetime dimensions. 3 of these dimensions extend across the branes and 1 that extends in between them. There are two branes with a 5th dimension inbetween them. All of the particles of the Standard Model, including the Higgs boson, are confined on one of these branes. Also, the only force that exists throughout the 5th dimension is gravity. Gravity is not restricted to the brane, however, propagates through the full 5-dimensional bulk. Gravity would be felt everywhere in the 5th dimension, however, not equally everywhere. The presence of energy in both the bulk and branes would result in spacetime curving dramatically. There are 2 branes, one at each boudary of the bulk. These 2 boundary branes are completely flat. This distance between the 2 branes would need to be only just a little larger than the Planck length. They need not be large, as in the ADD model.
This geometry is "warped": each slice is flat, however, are put together with an overall warp factor. This curvature would be everywhere. Each point in the 5th dimension would have a different curvature and clock speed. A way to visualize the curvature of warped geometry is by looking at the graviton probability function:
"The graviton probability function falls off exponentially as it moves from the Gravitybrane to the Weakbrane."
The probability function of the graviton tells us the probability of locating the graviton at any fixed position in space. Also, the larger the value of the function at any given point, the stronger gravity's interactions will be at that point. For flat spacetime, the graviton's probability function would be constant. Curvature corresponds to the shape of gravity. Thus, the graviton's probability function depends on it's position in the 5th dimension. The graviton probability function falls off exponentially as it moves from the Gravitybrane to the Weakbrane. The Gravitybrane carries positive energy and the Weakbrane carries negative energy. Thus, the graviton's probability function is much larger at the Gravitybrane. In fact, there is a very small chance that one would locate the graviton, the closer to the Weakbrane you get. Thus, the strength of gravity will depend on position in the 5th dimension! Gravity is strong on the Gravitybrane, where gravity is localized. However, the strength of gravity is feeble on the Weakbrane, where the Standard Model resides. Thus, the graviton probability function is lower at the weakbrane, where the graviton's interactions with the Standard Model is extremely weak. In this picture, the graviton is everywhere, however, it interacts far more weakly the closer to the Weakbrane you are. Gravity, could, on our brane, be weakened by warped geometry. This would solve the hierarchy problem!
I mentioned that objects moving from the Weakbrane to the Gravitybrane, become lighter, hence, the force of gravitation becomes weaker. Objects get bigger and become lighter as you move from the Gravitybrane to the Weakbrane. Time also moves slower. The closer to the Weakbrane, the more distance and time increase and the more mass and energy shrink. Sizes increase (by about 16 orders of magnitude!) and masses or energies decrease as one moves from the Gravitybrane to the Weakbrane. We must understand that gravitational attraction is proportional to mass. Thus, mass must be different at different points along the 5th dimension. All of these properties: energy, mass, time and size, depend on location in the 5th dimension. The warp factor will determine how much mass and energies will shrink. This is a rescaling in a way proportional to the amplitude of the graviton's probability function. Mass and size and the strength of gravity can depend on location.
It also should be noted that other forces, not just gravity can inhabit the 5-dimensional bulk. In string theory, gauge bosons could be stuck on a higher dimensional brane or in the bulk along with gravity. Gravitons arise from closed strings. However, the gauge bosons must be either an open or closed string. Subatomic particles can either be confined to a membrane or free to move through the bulk. This is not a problem in warped geometry, as it would be in the ADD model. These gauge bosons in the bulk could experience the entire energy range. This makes the unification of forces possible, since, they could, theoretically, operate at high enough energies. Forces (gravity, electromagnetism, and the strong and weak nuclear forces) can be unified at high energies, even in theories with extra dimensions. Another implication of the idea has to do with the Higgs. For the hierarchy problem to be solved, in the warped extra dimension proposal, the Higgs particle must be on the Weakbrane. The mass of the Higgs particle must be low to achieve this. The reason: the Higgs field is responsible for the spontaneous symmetry breaking, that is the source of all elementary particles masses. If the weak symmetry is not broken, than elementary particles don't have mass. That being said, if the Higgs particle is on the Weakbrane and the gauge bosons are in the bulk, the problems of unification and of the hierarchy of masses can both be addressed!
RS2: This was the 2nd paper written on warped geometry by Randall and Sundrum. The title is misleading because it does not have two branes, as in the case of the RS1 model. However, there is only one brane in the RS2 model! One of the branes (Weakbrane) is placed infinitely far away, thus is not included in the model. The Gravitybrane, becomes home to the particles of the Standard Model. It is also, according to the graviton probability function, the place where gravity is localized, as we will see.
In the RS2 approach, there is only one 4-dimensional brane in an infinite 5-dimensional bulk.
In this approach, this infinite extra dimension can appear invisible. There will appear to be 4 dimensions, even though, there are really 5! The RS2 approach is different from the RS1 approach because of a distinguishing feature: there is only 1 brane in the model. Since there is no 2nd boundary brane, than, the size of the 5th dimensional bulk is infinite. This is infinite warped geometry with a single brane. There is one 4-dimensional brane in a 5-dimensional universe. All of the particles of the Standard Model live on this 4-dimensional brane. The source of gravitation would not be lost throughout the 5th dimension, however, it would be concentrated around the brane. Gravitons would be localized near the brane, having unrestricted access to the infinite 5th dimension. The further from the brane, the lower the concentration of the force of gravitation. The graviton remains close by to the brane, despite the infinite size of the 5th dimension. The particles of the Standard Model are also located on this brane. This is different from the RS1 approach, where the Standard Model is located on the Weakbrane, since it is now infinitely far away and not included in the RS2 model. The idea behind the RS2 model is that a 2nd brane is not necessarily required in a warped spacetime geometry to reproduce the effects of gravity that we observe. The interactions of the 4-dimensional graviton would be independent from the size of the 5th dimension.
This is the graviton probability function in the RS2 model where there is only one brane embedded in an infinite 5th dimension. As you can see, the graviton probability function, is most large near the Gravitybrane. This is despite the fact that the size of the 5th dimension is infinite. The function appears to keep going indefinitely. Distant regions from the Gravitybrane, perhaps can be ignored, since the likelihood of finding a graviton there get lower and lower as you delve deeper and deeper into the 5th dimension. However, theoretically, a graviton can be anywhere along the 5th dimension. However, the graviton probability function is much more concentrated near the Gravitybrane. It should be noted: the extent of the 5th dimension is irrelevant, since, it is gravity, that is attracting the gravitons to the Gravitybrane. The gravitational field lines are spread out mostly, along the brane. When gravitational field lines come off the brane, they bend back and become parallel to the brane.
Key points on localized gravity.
In the RS2 model, the gravitational force is localized on the brane, as if the size of the 5th dimension were finite. This phenomenon is called localized gravity and I will explain it now. Despite the infinite size of the 5th dimension, the gravitational field is localized near the brane. Furthermore, the field lines resemble what would be expected if they were interacting in only four spacetime dimensions. The size of the 5th dimension is irrelevant to the strength of gravity near the brane. Gravity can still look 4-dimensional, if there is only one brane and the 5th dimension is infinite. This phenomenon is known as localized gravity, becuase the graviton probability function is localized near the Gravitybrane. In other words: gravity can leak out into the 5th dimension. This is because the 5th dimension is infinite. However, it doesn't: the graviton probability function assigns a low probability of there being any gravitons being found far away.
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