From Quanta Magazine, August 12:
Tensors are used all over math and science to reveal hidden geometric truths. What are they?
After Albert Einstein published his special theory of relativity in 1905, he spent the next decade trying to come up with a theory of gravity. But for years, he kept running up against a problem.
He wanted to show that gravity is really a warping of the geometry of space-time caused by the presence of matter. But he also knew that time and distance are counterintuitively relative: They change depending on your frame of reference. Moving quickly makes distances shrink and time slow down. How, then, might you describe gravity objectively, regardless of whether you’re stationary or moving?
Einstein found the solution in a new geometric theory published a few years earlier by the Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita. In this theory lay the mathematical foundation for what would later be dubbed a “tensor.”
Since then, tensors have become instrumental not just in Einstein’s general theory of relativity, but also in machine learning, quantum mechanics and even biology. “Tensors are the most efficient packaging device we have to organize our equations,” said Dionysios Anninos, a theoretical physicist at King’s College London. “They’re the natural language for geometric objects.”
They’re also tough to define. Talk to a computer scientist, and they might tell you that a tensor is an array of numbers that stores important data. A single number is a “rank 0” tensor. A list of numbers, called a vector, is a rank 1 tensor. A grid of numbers, or matrix, is a rank 2 tensor. And so on.
But talk to a physicist or mathematician, and they’ll find this definition wanting. To them, though tensors can be represented by such arrays of numbers, they have a deeper geometric meaning.
To understand the geometric notion of a tensor, start with vectors. You can think of a vector as an arrow floating in space — it has a length and a direction. (This arrow does not need to be anchored to a particular point: If you move it around in space, it remains the same vector.) A vector might represent the velocity of a particle, for example, with the length denoting its speed and the direction its bearing....
....MUCH MORE
