Warped Passages (67 page)
A Reflective and Expansive Passage
Someday girl I don’t know when
We’re gonna get to that place
Where we really want to go.
Bruce Springsteen
Ike XLII was ready to live large. He wanted to test the Alicxvr’s ultra-high settings of many megaparsecs, with which he could explore places beyond the Galaxy and the known universe and experience distant regions no one had ever seen before.
Ike was thrilled when the Alicxvr took him to distances 9, 12, and even 13 billion light-years away. But his excitement diminished when he tried to go farther and his signal strength fell precipitously. When he aimed for 15 billion light-years, his exploration aborted completely: he no longer received any information at all. Instead, he heard, “Message 5B73: The Horizon customer you are trying to reach is beyond your calling area. If you need assistance, please contact your local long-distance operator.”
Ike couldn’t believe his ears. It was the thirty-first century, yet his Horizon service still provided only limited coverage. When Ike tried to contact the operator, a recording said, “Please stay on the brane. Your call will be answered in the order in which it was received.” Ike suspected that the operator would never respond, and was wise enough not to wait.
The previous chapter explained why warping can emancipate an extra dimension and allow it to be infinite, yet unseen. But the infinite extra dimension is not the end of the physics story: things get even more bizarre. This chapter will explain how four-dimensional gravity (that is, with three dimension of space and one of time) can be truly a local phenomenon—gravity might look very different far away. We’ll see that not only could space appear to be four-dimensional when there are truly five dimensions, but we might be living in an isolated pocket with four-dimensional gravity inside a five-dimensional universe.
The model we’ll now consider demonstrates that, remarkable as it might seem, different regions of space can appear to have different numbers of dimensions. The physicist Andreas Karch and I found a model for spacetime in which this was the case in the course of investigating some perplexing features of localized gravity. The new and radical scenario we ended up with suggests that the reason we don’t see additional dimensions could be much more peculiar to our environment than anyone had ever believed. We could be living in a four-dimensional sinkhole in which three spatial dimensions is merely an accident of location!
Reflections
When I look back at the e-mail record from the time during which Raman and I had been collaborating, I find it a little mind-boggling how we completed our work in the midst of so many other distractions. When we began our research, I was in the process of moving from MIT to Princeton, where I was about to take a position as a professor, and I was also planning a six-month workshop in Santa Barbara for the following year. Raman, who had had several postdoctoral positions, was concerned about getting a faculty offer, so he was busy preparing talks and job applications. It was difficult to believe. He had done great work, yet I and others were trying to convince him that it would work out in the end and he shouldn’t abandon physics and search for another career. Raman was clearly meant to continue with physics and strongly deserved an excellent faculty position, yet he was having trouble finding a job.
The e-mails from the time illustrate the chaos: interesting physics issues alternate with requests for letters of recommendation, scheduling talks, arranging my Princeton housing, and some Santa Barbara conference organization. There were also a few e-mail exchanges with other physicists about our work. But not many. Although the RS2 paper was ultimately cited thousands of times and became well accepted, the work’s initial reception was mixed. It took a while before most physicists understood and believed us. A colleague tells me that at first people were waiting for someone else to find the loophole so they wouldn’t have to pay attention. Certainly at Princeton, the reaction to a talk Raman gave could at best be described as tepid.
Even those who did listen didn’t necessarily believe us right away. A conversation we had with the string theorist Andy Strominger was very enlightening, even though he now laughs at how he initially didn’t believe a word we said. Fortunately, he hadn’t been too skeptical to listen and talk.
In the physics community, there were a few who understood and believed what we were doing right from the start. We were lucky that Stephen Hawking was among them, and that he did not hesitate to share his enthusiasm with physics audiences. I remember Raman excitedly telling me how Hawking’s prestigious Loeb Lectures at Harvard concentrated heavily on our work.
Several others also worked on some related ideas. But the following fall, several months after our paper was published (and many months after we had begun talking about it), was when the theoretical physics community at large started paying attention. It turned out to be good fortune that David Kutasov, a University of Chicago physicist from Israel, and Misha Shifman, a Russian-born particle theorist from the University of Minnesota, and I had organized a six-month workshop in the fall of 1999 at the Kavli Institute for Theoretical Physics in Santa Barbara. The original goal of this workshop had been to bring together string theorists and model builders, and profit from an incipient convergence of research interests on topics such as supersymmetry and strongly interacting gauge theories. We had started planning the workshop well in advance, before the concept of branes and extra dimensions had created such a stir. Although we had hoped for some
positive synergy between string theorists and model builders, we didn’t know at the time we started the organization that we’d be thinking about extra dimensions when the conference actually happened.
But the timing proved to be fortuitous. The workshop provided an excellent opportunity to flesh out ideas about extra dimensions, and for model builders, string theorists, and general relativists to share expertise. Many exciting discussions took place, and warped geometry was one of the chief topics. In the end, both model builders and string theorists took warped five-dimensional geometry seriously. In fact, the distinction between the two fields blurred as people teamed up to work on similar problems on warped geometry and other ideas.
Many physicists later worked on other aspects of warped geometries, establishing connections and exploring subtleties that made localized gravity even more interesting. Although string theorists had originally dismissed RS1 (the warped geometry with two branes) as just a model, once they began to search they found ways to realize the RS1 scenario in string theory. Questions about black holes, time evolution, related geometries, and the connections to ideas from string theory and particle physics have also been fertile areas of research. Localized gravity has now been investigated in various contexts, and new ideas continue to emerge.
After our theory was accepted and no longer thought incorrect, some physicists actually went overboard in a different direction, claiming our theory was nothing new. One string theorist even went so far as to conclude that a string theory calculation of the impact of Kaluza-Klein modes was the “smoking gun” that proved our theory was the same as a version of string theory that string theorists had already been studying. This conformed to the joking adage in science that a new theory goes through three stages before being accepted: first it’s wrong, then it’s obvious, and finally somebody claims that someone else did it first. In this case, however, the smoking gun went up in smoke when physicists realized that the string theory calculation was subtler than they had thought, and the purported string theory answer actually hadn’t been right.
The truth was that the intersection with work in string theory was very exciting to all of us, and led to important new insights. Localized gravity turned out to have strong overlaps with the most important
string developments of the time: both our work and the research of string theorists involved a similarly warped geometry. In fact, perhaps because our research didn’t directly challenge string theory models, the string theory community actually accepted and recognized the significance of our work sooner than the model building community. Although it had initially seemed coincidental, maybe this was some indication that we were all on the right track. And happily, Raman had no trouble getting job offers afterward. (He’s now a professor at Johns Hopkins.)
However, some skeptics remained. The precise model Raman and I considered led to interesting questions that no one could answer right away. Did localization depend on the form of spacetime very far away? When people tried to find examples of the type of geometry that Raman and I had suggested in supergravity theories, the form of gravity far from the localizing brane seemed to be the stumbling block. But were those conditions essential? Another question we wanted to answer was, did spacetime necessarily look four-dimensional everywhere? Localized gravity made the entire five-dimensional universe behave as if there were four-dimensional gravity. Did this always happen, or could some regions look four-dimensional and some regions behave differently? And what happens when the Gravitybrane isn’t completely flat? Does localization work the same way for a brane with a different geometry? These are some of the questions that
locally localized gravity
, the theory that Andreas and I developed, could address.
Locally Localized Gravity
How many dimensions of space are there? Do we really know? By now, I hope you will agree that it would be overreaching to claim that we know for certain that extra dimensions do not exist. We see three dimensions of space, but there could be more that we haven’t yet detected.
You now know that extra dimensions can be hidden either because they are curled up and small, or because spacetime is warped and gravity so concentrated in a small region that even an infinite dimen
sion is invisible. Either way, whether dimensions are compact or localized, spacetime would appear to be four-dimensional everywhere, no matter where you are.
This might be a little less obvious in the localized gravity scenario, in which the graviton’s probability function becomes smaller and smaller as you go out into the fifth dimension. Gravity acts as it does in four dimensions if you’re near the brane. But what about everywhere else?
The answer is that in RS2, the influence of four-dimensional gravity is inescapable, no matter where you are in the fifth dimension. Although the graviton’s probability function is largest on the Gravitybrane, objects everywhere can interact with one another by exchanging a graviton, and therefore all objects would experience four-dimensional gravity, independently of location. Gravity everywhere looks four-dimensional because the graviton’s probability function is never actually zero—it continues on for ever. In the localized scenario, objects far from a brane would have extremely weak gravitational interactions, but weak gravity would nonetheless behave in a four-dimensional manner. So, for example, Newton’s inverse square law would hold, no matter where you were along the fifth dimension.
The small but nonzero graviton probability function away from the Gravitybrane was essential to the solution to the hierarchy problem I presented in Chapter 20. The Weakbrane, located away from the Gravitybrane in the bulk, experiences gravity that appears to be four-dimensional, even if it experiences that gravity only extremely feebly. Like water far from your own garden in the sprinkler analogy, there is always some water, just not a lot.
But suppose we reflect even further and ask what we really know with certainty about the dimensions of space. We do not know that space everywhere looks three-dimensional, only that space
near us
looks three-dimensional. Space appears to have three dimensions (and spacetime to have four) at the distances
that we can see
. But space might extend beyond that, into inaccessible territory.
After all, the speed of light is finite, and our universe has existed for only a finite amount of time. That means that we can only possibly know about the surrounding region of space within the distance that light could have traveled since the universe’s inception. That
is not infinitely far away. It defines a region known as the
horizon
, the dividing line between information that is and is not accessible to us.
Beyond the horizon, we don’t know anything. Space needn’t look like ours. The Copernican Revolution is repeatedly updated and revised as we see further into the universe and realize not everywhere is necessarily the same as what we see. Even if the laws of physics are the same everywhere, that doesn’t mean that the stage on which they are played out is the same. It could be that nearby branes induce a different gravitational force law in our vicinity than would be seen elsewhere.
How can we claim to know the dimension of the universe outside our purview? There would be no contradiction if the universe beyond exhibited more dimensions—maybe five, maybe ten, maybe more. By thinking about the bare essentials, rather than assuming that everywhere, even inaccessible regions, is made up of spacetime that looks like ours, we can deduce what is really fundamental and what is ultimately conceivable and legitimate.
All we know is that the space we experience appears to be four-dimensional. It might be overstepping the mark to assume that all other regions of the universe must be four-dimensional as well. Why should a world extremely far from ours, which might not interact with us at all—or perhaps only via extremely weak gravitational signals—have to see gravity and space the way we do? Why can’t it have a different type of gravity?
The marvelous thing is that it can. Our braneworld could experience three-plus-one dimensions, while outside regions do not. To our amazement, in 2000, Andreas Karch and I developed a theory in which space looks four-dimensional on or near the brane, but most of the space far from the brane appears higher-dimensional. This idea is schematically illustrated in Figure 90.
