Scalars, Vectors, and Tensors

A student’s first physics class, often encountered in High School, will usually begin with a discussion of “vectors.” We assume these students are already schooled in simple mathematics such as arithmetic and geometry and even some trigonometry. They are quite familiar with the Pythagorean Theorem which gives values for the sides of a right triangle with the famous square of the hypotenuse is equal to the sum of the squares of the other two sides. I suspect all students, down through the ages, good student or bad student, it doesn’t matter, all students know this basic formula which was discovered by all the ancient cultures that existed before the modern Christian era.

For some reason, I don’t recall what it was. It may have been due to a broken bone or possibly a trip I took with my parents. In any case, I missed the first two weeks of Physics at Fergus County High School in my Junior Year. So when I showed up for class, all the other students knew about vectors and how to manipulate them, often using the ancient Greek theorem I described in the previous paragraph. But I didn’t!

While the rest of the class was learning how to add forces and manipulate the press of gravity and pushes and weights and measures, I was scrambling to add vectors to my sphere of understanding. It was a traumatic few weeks until I got caught up.

Now those who have taken the class know that things were divided into two categories or groups. One was called “scalars.” That is, measurements or values or numbers that had no direction. Scalars are physical measurements without any direction to them. For example, temperature, mass, and bank account balance. They can have a positive or a negative value, but, otherwise, they are simple integer or real numbers (decimal points).

In contrast, vectors have magnitude like a scalar, but also direction. They are like little pointy arrows or the hands on a clock face. They have a magnitude, often represented by the length of the vector in a geometrical drawing. But they also have direction like North or South, or like the previously stated hands on a clock, pointed at 12 noon or at 6.

In the physics class they could be represented by an angle from a baseline, which was usually horizontal like the floor or table top. So a vector pointing to the right, by convention, would be zero degrees and, pointing straight up, ninety degrees. If the vector points to the left it is 180 degrees, and so forth.

Then the fun begins. You could use a coordinate system such the common Cartesian coordinates, named after René Descartes, that have an “x” axis and a “y” axis … at least in two dimensions.

Now you could describe a vector as a set of numbers or coordinates. It was convenient to put the start of the arrow at the coordinate system origin, a location with x and y values of 0,0. Then you would describe the pointy end of the arrow with some value such as 2,2. That’s an arrow from 0,0 to 2,2. Some quick calculation would show the arrow has a length of the square root of (2)² + (2)², which is the square root of 8 or 2√ 2 , approximately 2.8.

Remember?

Of course, vectors can be in three dimensions too. A similar vector from 0,0,0 to 2,2,2 has a length of the square root of the three coordinates. (The Pythagorian theorem can be expanded to multiple dimensions.) So now the length would be √ (2)² + (2)² + (2)²  , almost 3.5.

Now suppose you expand it to more than three dimensions. Now it gets hard to visualize. Assuming that the vector starts at the origin, you can state a vector as a matrix with one row (or, more common, one column). For example, our vector on the flat plane would be [2,2] and if it was in five dimensions, it would be [2,2,2,2,2], where the square brackets are the symbol for a matrix or collection of numbers. And you would take the sum of the squares of the five numbers to calculate the length of the vector in 5-space.

As I pursued advanced mathematics, I learned to use matrices as a mathematical tool and coordinate systems could be represented by a matrix. Vectors and matrices are the common mathematical topics in all branches of modern physics.

Now I’m learning “Tensors.” What is a tensor, you ask? Well that’s an interesting question. They are related to scalars and vectors. They can describe the relationships between scalars and vectors and even between other tensors. Tensors are often represented by matrices with several rows and columns.

Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. That is one of their powerful capabilities and it is why Einstein chose tensors to represent space, which he supposed was not flat or Euclidean, but rather curved by the distortion of gravity and energy. That is the basis of his general relativity theory.

Both the concept of matrices, which were used to describe basic quantum theories, and tensors being used in the early 1920’s were new mathematics to physicists. At that time they were esoteric structures only studied by mathematicians. Now days, they are part of the basic math curriculum studied by every physics student.

Einstein actually had to seek help from mathematicians as he perfected his general theory field equations. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Tullio Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect.

(Levi-Civita was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus.)

So next time you are struggling with your school homework and asking for help, isn’t it comforting to know that even the great Einstein had to find a tutor to help him? As we always said at work, “It’s not what you know, but who you know.”