Fabric of the Universe

UNC-TV Science: March 20, 2014
From Einstein, to Hawking, to Geometry You Can’t Do with a Protractor

This is a story of math, the very fabric of the universe and two Duke mathematicians’ work to back up some of the most brilliant minds of the 20th century. But in order to really make sense of it, we need to start with a little history.

On a medium-sized scale (everything bigger than a cell and smaller than a planet), physics is pretty easy. Isaac Newton invented calculus in a summer, and over the next 20 years or so polished off his laws of motion, classical gravity and basic optics. James Maxwell wrote the defining equations of electricity and magnetism in 1861 and by 1884, two other physicists pretty much closed the book on that subject through Maxwell’s work.

In fact, by the early 1900s, many scientists thought that they had pretty much figured out everything there was to figure out in the world of physics – that is until a German scientist with extraordinarily bad hair showed that for really small and really large things, classical physics fell short.

Einstein’s work on the photoelectric effect (which makes electrons fly off a metal surface when you shine light at it) laid the foundation for quantum mechanics, which dominates small things. But he was more satisfied with general relativity, his theory of gravity on an astronomical scale, a theory which is still broadly accepted today.

General relativity hinges on the idea of equivalence, which essentially says that it’s impossible to tell the difference between acceleration and gravity. So if you’re sitting in a space ship and you feel yourself being pulled toward your seat-back, you wouldn’t be able to tell whether that’s the space ship hitting the boosters and your body resisting the motion or extra gravity being applied to the space ship pulling it forward.

Seems simple enough, but it opens the door to an idea of gravity that can be a little hard to visualize.

You have to start with space-time, which is combination of the three dimensions of space and one more of time. General relativity says that matter bends space-time and that objects flying through space-time tend to follow these bends.

Think of space-time like a trampoline. If it’s super tight, you can roll a marble across and the marble won’t change course. But if you sit in the center, you create a divot. Now if you try to roll the marble across it will probably curve with the trampoline and wind up in the divot with you. Repeat the experiment with your dog in the center and the marble might not curve as much because your dog makes a smaller divot.

Zoom way out and you can picture Einstein’s theory at work. Imagine Earth making the divot and a comet passing by. The comet may not wind up in Earth’s divot but the bend in the trampoline of space-time will cause it to turn. Through general relativity, Einstein gives us a picture of how gravity attracts one object to another.

But this picture comes at a price. Newton’s model involves mass, acceleration, and distance, so if you wanted to know something’s mass, you could figure it out using its acceleration, the mass of the objects attracting it, and how far those objects are away. This is tough, but doable. In Einstein’s picture of warped space-time, determining the mass of a region of space becomes incredibly complex.

Some physicists make entire careers out of trying to solve this problem, writing equations upon equations upon equations to describe the mass of a particular region of space. The trouble is, the most accurate systems of equations border on impossible to compute.

Stephen Hawking’s model from the 1960s has the opposite issue. His formula for the mass of a region of space, called the Hawking mass or Hawking energy, is “easy” to compute but it is not as accurate as the more complex models.

Enter Duke University physicist Hubert Bray and one of his Ph.D. students, Jeffery Jauregui. Using a branch of mathematics called differential geometry, they set out to prove that in certain cases, Hawking mass is almost dead on.

In a series of two papers published on arXiv.org, the two researchers provide three separate geometrical proofs that Hawking mass is an accurate approximation of the mass of a region of space enclosed in what are called time-flat surfaces.

Remember that space-time is a continuum combining our three dimensions of space and one of time. In the space-time continuum, you can travel through the dimensions of space just like we move every day. But space-time makes no distinction between the space dimensions and the time dimension, so by standing perfectly still you would be travelling forward in the time dimension while staying still in the space ones. A time-flat surface is a boundary that doesn’t stretch in the time dimension. It can occupy any region of space but it exists only in the present, not in the past or the future.

Bray and Jauregui were able to show through their proofs that for regions bound by time-flat surfaces, mass will increase as the area grows in size, and that the Hawking mass works very well to describe this particular class of space regions.

While modern physics is still a long way off from understanding the wholesale mechanics of the universe, advances in mathematical proofs like this could slowly fill in gaps of our understanding.

- Daniel Lane

Daniel Lane covers science, medicine and the environment as a reporter/writer. He is currently pursuing a master's degree in medical and science journalism at UNC - Chapel Hill.