Extra dimensions

"Extra spatial dimensions could be a good thing. ... In a nongravitational theory, the spacetime geometry is a rigid background on which the dynamics takes place. In that setup, the fact that we observe four-dimensional Minkowski spacetime is a compelling argument to formulate the theory in that background geometry. As you know very well, this is part of the story of the Standard Model. However, in a gravitational theory that abides by the general principles laid out by Einstein, the spacetime geometry is determined by the dynamical equations. In such a setup extra dimensions can make sense provided that the equations of the theory have a solution for which the geometry is the product of four-dimensional Minkowski spacetime and a compact manifold that is sufficiently small to have eluded detection. It turns out that there are many such solutions. Moreover, the details of the compact manifold play a crucial role in determining the symmetries and particle content of the effective low-energy theory in four dimensions, even when the compact dimensions are much too small to observe directly."

"Why try to unite the four forces in a single theory? Why not simply use Einstein’s theory of general relativity to govern big things and quantum mechanics for little ones? Some concepts, such as the Big Bang or how black holes form, live in both domains. When we combine equations of the four forces to describe these ideas, our answers usually end up being either zero or infinity. … Here’s where string theory comes to the rescue. By adding seven hidden dimensions to the familiar three and another for time, plus antiparticles and a mirror set of particles called superparticles, the math starts to make sense. The force of gravity is diluted because it permeates into one or more of the hidden dimensions. Dark matter and dark energy also may invisibly shape our universe from these phantom dimensions."