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Relativistic Submarine Sinks

Think twice before you crank up the propellers on your submarine. If you travel too close to the speed of light, you will wind up in Davy Jones's locker--according to the theory of general relativity. In the July issue of Physics Review D, a physicist has extended Archimedes' law of buoyancy to extremely fast objects, thereby solving a longstanding paradox. This is more than a mere intellectual exercise; physicists hope that the relativistic Archimedes' principle might yield insight into the laws of thermodynamics, the behavior of black holes, and even the growth of crystals.

A stationary submarine that's exactly the same density as water will neither float nor sink. But as the submarine begins to move very fast a paradox arises. Objects moving close to the speed of light get more massive and shrink, so from the perspective of an observer at rest, a relativistic submarine will sink. On the other hand, for Captain Nemo aboard the submarine, the sub is at rest while the water rushes by at great speed and becomes denser. So the sub should begin to float. Obviously, it can't sink and float at the same time.

When a student approached George Matsas, a physicist at the Universidade Estadual Paulista in São Paulo, Brazil, and asked how to resolve the paradox, Matsas was at a loss. "It was embarrassing," says Matsas. "I thought it was a scandal that there was no answer to this paradox, so I decided to waste time on it."

Matsas plugged the superfast submarine into the equations of general relativity to find out what actually happens. After juggling forces and equations, he had the answer: The sub sinks. Why? Buoyancy is a function of gravity, and gravity, like size and time, is affected by rapid motion through space. As the sub zooms through the water, Earth exerts more force on the moving sub, causing it to sink.

In solving the paradox, Matsas has created a relativistic extension of Archimedes' principle that might be of some use in understanding the inflow of fluids around neutron stars or black holes. And other scientists are intrigued. Yale's John Wettlaufer, who studies the thermodynamics of crystallizing materials, sees a similarity between relativistic buoyancy and an equation that describes forces at the interfaces between solids, liquids, and gases. "They bear a lot in common mathematically, so I want to look at what's under the hood," he says.