Warped Passages (64 page)

But it turns out that our model is safe. While sitting on a bench surrounded by gorgeous mountain scenery, I worked out the interactions of the KK particles (Joe did the same calculation, but I think he was inside his office at the Center). We calculated a result that told us that, although the KK particles’ interactions would be big enough to be interesting for future experiments, they would not be so big as to have already been seen.

In the future, the LHC will have a good chance of producing the KK particles of this model, should they exist. These particles won’t look like the ones from the finite-sized extra-dimensional warped model. Instead of nice KK particles that decay inside the detector, the KK particles in this model with an infinite extra dimension will escape into the extra dimension (similar to the KK particles’ behavior when there are large dimensions). So if there’s an infinite extra warped dimension and a Weakbrane that solves the hierarchy problem, experiments could only hope to discover events with missing energy. Even so, at sufficiently high energies, missing energy should be a sufficiently telling signal that something new is out there.

Black Holes, Strings, and Other Surprises

In addition to KK particles, other remarkable signals of extra dimensions could turn up when the LHC turns on. Although the effects of five-dimensional gravity are minuscule at ordinary energies, five-dimensional gravity will be a major player when colliders create high-energy particles. In fact, when energies reach about a TeV, the effects of five-dimensional gravity would be enormous—they would overwhelm the interactions of the feebly interacting four-dimensional graviton, which has a small probability function on the Weakbrane where we live (and experiments take place).

The enormous strength of five-dimensional gravity means that five-dimensional black holes might be produced, as well as five-dimensional strings. Furthermore, once energies reach about a TeV,
everything located on or near the Weakbrane would interact strongly with everything else. That is because the effects of gravity and the extra KK particles would be enormous at TeV energies, and they would conspire to make everything interact with everything else. Such strong interactions among all known particles and gravity wouldn’t occur in a four-dimensional scenario; they would be a definite signal of something new. As was the case with large extra dimensions, we don’t yet know if the energy will be high enough to see these new objects. But if interactions are strong at energies not too much greater than a TeV, experiments won’t miss it.

Coda

The connection between a solution to the hierarchy problem and experimental consequences at TeV energies is robust, but the details of what we will see depend on the model. Different models have distinctive experimental consequences, and that is very reassuring. These distinctive signatures mean that once the LHC is up and running, we have a good chance of identifying which—if any—of these models applies to our world.

What’s New

• Spacetime can be dramatically curved in the presence of bulk and brane energy, even when the brane itself is completely flat.
  
• The model we considered in this chapter has two branes, the Gravitybrane and the Weakbrane, at each of the ends of a finite-sized fifth dimension. Energies in the bulk and on the branes warp spacetime.
  
• A single extra dimension introduces an entirely new way to solve the hierarchy problem. The fifth dimension in this model is not big, but it is very warped. The strength of gravity depends strongly on where you are in the fifth dimension. Gravity is strong on the Gravitybrane and is feeble on the Weakbrane, where we are located.
  
• From the perspective of an observer who thinks he is in four dimensions, objects should have different sizes and masses if they originate from different places in the fifth dimension. Objects that are confined to the Gravitybrane should be very heavy (with mass of about the Planck scale mass), whereas objects confined to the Weakbrane should have much lower masses, about a TeV.
  
• All forces can unify and the hierarchy problem can be solved if the Higgs particle (but not the gauge bosons) is confined to the Weakbrane.
  
• The Kaluza-Klein partners of the graviton should give rise to very distinctive particle collider events in which they decay into Standard Model particles within the detector.
  
• In models in which Standard Model particles are in the bulk, other KK particles can be produced and observed.
  

21

The Warped Annotated “Alice”
^*

Go ask Alice.

When she’s ten feet tall.

Jefferson Airplane

Athena stepped out of the dreamworld’s elevator into the warped five-dimensional world and was astonished to see only three spatial dimensions. Was the Rabbit playing games, pretending to take her to a world with four spatial dimensions when in fact there were only three? What a funny way to travel to what looked like an ordinary world!
^†

With great gallantry, a local received the puzzled new arrival. “Welcome to Branesville,
^‡
our glorious capital. Permit me to show you around.” Athena, who was tired and confused, blurted out, “Branesville doesn’t look all that special. Even the mayor looks completely normal,” although, she had to confess, she wasn’t entirely sure as she had never seen a mayor before.

The mayor to whom Athena referred had arrived accompanied by the Cheshire Fat Cat, his Chief Advisor. The Cat’s job was keeping tabs on everything in the city, which was greatly facilitated by his skill at catching people unawares—especially surprising in light of the Cat’s enormous bulk. The Cat loved to explain that he owed this skill to his ability to disappear into the bulk, but no one ever understood what he meant.
^*

The Cat materialized next to Athena and asked if she would like to accompany him as he made his rounds. He warned her that she had better be comfortable with bulk, to which Athena eagerly responded that her favorite uncle was in fact very, very fat. The Cat looked skeptical, but agreed to take her along. He offered Athena cream cake with butter frosting, in which she happily indulged. And off they went.

Athena wondered what it was she’d eaten. She now appeared to be on a four-dimensional slice of a five-dimensional world, and as far as she could tell, she was no thicker than this thin four-dimensional slice. She exclaimed, “I am like my paper doll! But whereas Dolly has two spatial dimensions in a three-dimensional world, I have three spatial dimensions in a four-spatial-dimensional world.”

The Cat grinned sagely and explained, “You are now conscious of what I like to call The Bulk. You are still in Branesville, but will be leaving (and growing) momentarily. Branesville is in reality part of a five-dimensional universe, but the fifth dimension is warped so discreetly that Branesville residents are completely unaware of its existence. They have no idea that Branesville is the border of a five-dimensional state. You too mistakenly concluded on your arrival that there are only three spatial dimensions. The new Athena, untethered from the brane, is free to travel out into the fifth dimension. May I suggest for our destination another village called Weakbrane, at the other edge of the five-dimensional universe?”

What a strange five-dimensional journey it turned out to be. After leaving Branesville, Athena found herself moving in another dimension, and growing as she did so (as shown in Figure 85).
^†
When the observant Cat noticed the confused look on Athena’s face, he reassuringly explained, “Weakbrane is close by and we will be there very soon.
^*
It’s lovely, but don’t be alarmed when you see that, like the Branesville residents you encountered, Weakbrane residents scoff at the notion of four spatial dimensions. You, who can see out into the bulk, will see a huge shadow on Branesville, ten million billion times bigger than the one with which you started. Almost everything else will seem to you and to them to be entirely normal.”

Figure 85.
Alice got bigger as she moved through the bulk from the Weakbrane to the Gravitybrane.

But upon her arrival in Weakbrane, Athena noticed one other thing. The four-dimensional graviton had quietly accompanied the travelers on their journey and was softly tapping on her shoulder. He touched her so extremely gently that she had barely noticed.
^†

But she couldn’t ignore the graviton when he launched into a litany of complaints. “Weakbrane would be so exciting, were it not for the superior influence of the entrenched hierarchy. The strong, weak, and electromagnetic armed forces on the Weakbrane permit me only the most feeble strength.” The graviton whined how everywhere else he was a force to be reckoned with, especially in Branesville, which is ruled by an oligarchy with comparably strong forces.
^*
Weakbrane, where gravity was the most suppressed, was the graviton’s least favorite place.
^†
The graviton turned to Athena in the hope of enlisting her in his plan to wrest power from the reigning authorities.

Athena thought she had better leave immediately and looked around for the rabbit hole, but couldn’t find it. She did find a white rabbit, whom she expected to be an efficient guide. But the Weakbrane rabbit had an alarmingly sluggish gait, and kept repeating how happy he was that his date would wait.
^‡
Athena realized that this rabbit wasn’t going anywhere, so she found a more anxious rabbit she could follow, and worked her way back home. Once she understood the physics implications, Athena enjoyed her dream enormously—though it should be noted that she never again ate cream cake.

22

Profound Passage: An Infinite Extra Dimension

From another dimension,

With voyeuristic intention,

Let’s do the Time Warp again

Vanessa (
The Rocky Horror Picture Show
)

Athena woke up with a start. Her recurring dream had once again taken her down the rabbit hole. This time, however, she asked the Rabbit to take her straight back to the warped five-dimensional world.

Athena arrived back in Branesville (or so she thought). The Cat soon appeared, and she eagerly turned to him, anticipating her dream cake and a delightful excursion to Weakbrane. She was sorely disappointed when the Cat told her there was no such thing as Weakbrane in this particular universe.
^*

Athena didn’t believe the Cat and thought there must be another brane further away. Proud of herself for understanding how, in the warped geometry, further-away branes had weaker gravity, she decided it was probably called “Meekbrane” and asked the Cat whether she could go there.

But once again she was in for a disappointment. The Cat explained, “There is no such place. You are on the Brane; there are no others.”

“Curiouser and curiouser,” thought Athena. This clearly wasn’t exactly the same space as before, since it had only a single brane. But Athena wasn’t ready to give up. “May I see for myself that there is no other brane?” she asked in her sweetest tone.

The Cat strongly advised her against it, warning, “Four-dimensional gravity on the Brane is no guarantee of four-dimensional gravity in the bulk. Once I nearly lost everything but my smile there.”

Athena was a cautious girl, despite her many adventures, and she took the Cat’s warning to heart. But she often wondered what the Cat meant. What did lie beyond the Brane, and how would she ever know?

Curved spacetime has remarkable properties. We explored some of these in Chapter 20, including how mass and size and the strength of gravity can all depend on location. This chapter presents an even more extraordinary feature of curved spacetime: it can appear to have four dimensions, even when there are truly five. By examining the warped spacetime geometry more carefully, Raman and I realized, to our astonishment, that even an infinite extra dimension can sometimes be invisible.

The spacetime geometry that we will consider in this chapter is almost the same as the one described in Chapter 20. But as the above story should suggest, this geometry has a single distinguishing feature: it has only a single brane. But this is a tremendously important distinction: because there is no second boundary brane, a single brane means that the fifth dimension is infinite (see Figure 86).