Sink or float: a relativistic submarine

Imagine an underwater submarine moving horizontally at near the speed of light. While not entirely practical, the situation makes for a fascinating example of special and general relativity.

Lorentz contraction makes fast objects look shorter in the direction of motion. In the water’s reference frame, the submarine appears to contract. Its mass remains the same, making it denser and causing it to sink. In contrast, in the submarine’s reference frame, the water has a large horizontal velocity and therefore looks denser. In that case, the submarine should float upwards.

However, both perspectives cannot be true. Hrvoje Nikolic addressed this submarine paradox by replacing the ship with a simplified model.

“Since the problem involves gravity, one should use not only special relativity but also general relativity,” said Nikolic. “On the other hand, an analysis of a solid body in general relativity is very complicated. Hence, to simplify the problem, I replaced the solid submarine with a fluid drop.”

The paradox is resolved by considering the motion relative to Earth. All observers can agree that the submarine moves relative to the planet. Sinking or floating depends on buoyancy, which is a consequence of gravity. Therefore, the relevant velocity of the submarine is not the velocity relative to water, but the velocity relative to the gravitational field and its source.

This work proves the paradox does not depend on the assumption that the submarine is a rigid body. Nikolic hopes it will make a useful tool for understanding general relativity in the classroom and beyond.

“There are many other paradoxes in physics that I want to think about,” he said.

Source: “Submarine paradox softened,” by Hrvoje Nikolić, American Journal of Physics (2022). The article can be accessed at https://doi.org/10.1119/5.0084185 .