Alice 3

We can imagine that students learning Tensor Calculus want to see a “real live one”.

We I’m sure you a simple Vector with three components that provides a displacement in a 3D space. A student might say, “Oh that’s nothing, we saw those back in grade school!”

We get it–you want to see something with a lot more “cool-ness”.

Below is a boost matrix that performs a Lorentz transformation. The \gamma[latex] is a multiplication factor that exists because velocity causes a change to the flow of time.</p> <p class="has-text-align-center">[latex] \begin{bmatrix} \gamma & -\dfrac {\gamma v_x}{c} &-\dfrac {\gamma v_y}{c} & -\dfrac {\gamma v_z}{c} \\ \\ - \dfrac {\gamma v_x}{c} & \gamma(1 - \gamma) \dfrac {v^2_x}{v_x} & \gamma(1 - \gamma) \dfrac {v_xc_y}{v_x} & \gamma(2 - \gamma) \dfrac {v_xv_z}{v_x} \\ \\ - \dfrac {\gamma v_y}{c} & \gamma(1 - \gamma) \dfrac {v_yv_x}{v_y} & \gamma(1 - \gamma) \dfrac {v_y^2}{v_y} & \gamma(1 - \gamma) \dfrac {v_yv_z}{v_z} \\ \\ - \dfrac {\gamma v_z}{c} & \gamma(1 - \gamma) \dfrac {v_zv_x}{v_z} & \gamma(1 - \gamma) \dfrac {v_zv_y}{v_z} & \gamma(1 - \gamma) \dfrac {v_z^2}{v_z} \end{bmatrix}

The above math object exists in a 4D space where three dimensions are spacial and one dimension is temporal- temporal means time.

It is not a tensor. It is a transformation that we can think of as a tool that changes tensors. Elsewhere we will tell you that when you see an array of numbers that is a sensor, you won't know just by looking at the numbers, which tensor it is. To this we add that a group of numbers put together might not be a tensor, it might be a transformation.

A tensor that resembles a 4x4 matrix can do something to a vector with 4 components. We will leave it at that first now.

Lorentz T

Abstract Index Notation

Subscripts depict arrows coming out of a symbol (think of water flowing from a tank out through a pipe below). Superscripts indicate arrows going in. 

We found the following quote:

"Abstract index notation evolved out of an earlier one called the Einstein summation convention, in which superscripts and subscripts referred to specific coordinates."

Research suggests that Roger Penrose should be given credit for the index notation that we use today. Einstein's contribution was the elimination of the Summation sign (and possibly more--students are encouraged to go digging, you will learn it better for so doing).

Physi Libretexts