Albert Einstein, after presenting his special theory of relativity in 1905, spent years grappling with the challenge of developing a theory of gravity. He aimed to demonstrate that gravity results from the bending of space-time geometry due to the presence of matter. However, he faced a significant obstacle in reconciling the relative nature of time and distance, which vary depending on one’s frame of reference. The concept that movement can cause distances to contract and time to slow down posed a dilemma in objectively defining gravity, irrespective of one’s state of motion.
Einstein found the solution to this predicament in a novel geometric theory advanced by Italian mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita. This theory laid the groundwork for what later became known as a “tensor.” Tensors have since played a pivotal role not only in Einstein’s general theory of relativity but also in diverse fields such as machine learning, quantum mechanics, and biology. They serve as an efficient tool for organizing equations, serving as the natural language for geometric objects.
Defining tensors can be challenging, as they hold different meanings depending on the perspective. While computer scientists may view tensors as arrays of numbers storing crucial data, physicists and mathematicians perceive them as embodying a deeper geometric significance. A tensor can be represented by an array of numbers, but its essence lies in its geometric interpretation.
To comprehend the geometric essence of tensors, one can start with vectors, visualizing them as arrows in space with a length and direction. Despite being represented by a list of numbers, vectors retain their intrinsic properties regardless of changes in the coordinate system. Tensors extend this concept, encapsulating more intricate geometric information.
For instance, envision a block of steel and the multitude of forces acting upon it. A rank 2 tensor, depicted as a matrix, can describe these forces comprehensively. Each face of the block experiences forces in different directions, necessitating a matrix of nine numbers to represent the forces acting on the block.
Mathematicians often conceptualize tensors as functions that take vectors as inputs and yield another vector or a number as output. Unlike general functions, tensors’ outputs remain independent of the coordinate system, highlighting their unique nature. By understanding tensors, one can appreciate their significance in diverse fields and their pivotal role in elucidating complex phenomena.
