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showed that this was possible while he was working on a research project with one of my former MIT colleagues, who had intended to challenge Raman’s and my work. (Happily for us, their collaboration did a beautiful job of showing that our work was right.) In the course of his project, Andreas identified a model that was closely related to the one Raman and I had developed, but which had some very peculiar properties. When Andreas visited Princeton, he came to talk to me about it. Eventually we figured out that this model has some startling implications. At first, Andreas and I collaborated via e-mail and on visits to each other’s institutions, and afterward, more easily, when I was back in Boston. And what we found was quite remarkable.

This model was very similar to the one I had studied with Raman; it had a single brane in five-dimensional warped space. But the difference in this case was that the brane was not exactly flat. This was because it carried a tiny amount of negative vacuum energy. In general relativity, as we have seen, not only relative energy but also the total amount of energy is meaningful. The total energy tells spacetime how to curve. For example, constant negative energy in five-dimensional spacetime gives rise to the warped spacetime that we have been discussing in the last few chapters. However, in that case the branes themselves were flat. Here, negative energy on the brane makes the brane itself slightly curved.

The negative energy on the brane leads to an even more interesting theory. However, we weren’t actually interested in the negative energy itself—if we live on a brane, our brane should actually have a tiny positive energy to agree with observations. Andreas and I decided to study this model solely because of its fascinating implications for dimensionality.

To understand what we found, let’s briefly return to a setup with two branes, with the understanding that afterward we will remove the second one. When the second brane was sufficiently far away, we found that there were two different gravitons, one localized near each of the two branes. Each of the graviton probability functions peaked near one of the two branes, and decreased exponentially quickly as you left it.

Neither of the gravitons was responsible for four-dimensional gravity over the entire space. They produced four-dimensional gravity only in the region adjacent to the brane on which they were localized. The gravities experienced on the different branes were different. They could even have very different strengths. And objects on one brane didn’t interact gravitationally with objects on the other.

The setup with two widely separated branes can be compared to a situation in which someone on the opposite, very distant shore is also feeding ducks. Those ducks could even be of a different type; perhaps you are attracting mallards but, on the opposite shore, someone is attracting wood ducks. In that case, there would be a second concentration of ducks along the opposite shoreline, analogous to the second graviton probability function that is localized near a second brane.

The appearance of two different particles that both look like the four-dimensional graviton was a big surprise to us. General physical principles were supposed to ensure that there is only a single theory of gravity. And indeed, there is a single five-dimensional theory of gravity. However, five-dimensional spacetime turns out to contain two distinct particles that each communicate a gravitational force that acts as if it is four-dimensional, each in a distinct region of five-dimensional space. Different regions of space look like they both contain four-dimensional gravity, but the graviton communicating the four-dimensional gravitational force in those theories is different.

But there was a second surprise as well. According to general relativity, the graviton is massless. Like the photon, it should travel at the speed of light. But Andreas and I discovered that one of the two gravitons has a nonzero mass and didn’t travel at this speed. This was truly surprising