…More on Vectors…

…AND SOME GR!

Image courtesy of https://en.wikibooks.org/wiki/Special_Relativity/Mathematical_approach2

As stated previously, it is my intention to make this page as mathematically un-intense as possible. But there are some things it would just feel wrong to leave out when discussing general relativity. For this, I will cover the concept of a four-vector. You may already know by this point, but we will be dealing with four-dimensional space quite a bit on this page. Four-vectors are perfect for this, as they deal in just that dimensionality. A four-vector is often used when discussing space-time coordinates or the energy of a particle.


Vector Representations

One will often see vectors represented as a vertical column, as shown below.

Personally created image.

This vector is three dimensional, and uses the x, y, z Cartesian coordinates. Remember, we can represent a vector in any coordinate system we want. What this representation tells us is that any value in the “X” place will be a projection in the X dimension, Y for Y, and Z for Z. Pretty straight forward stuff. In fact, this representation helps us when doing operations with vectors on a matrix (columns and rows matching up).

 


Space-Time Four-Vector

We can do the same representation with a four-vector, but with some different notation. Below is an image of the space-time four-vector, which represents a location.

Personally created image.This one looks a little different, as mentioned. The spatial portions of the vector have been shifted downwards. Here, we have a dimension of time “t” first, but multiplied by the speed of light “c”. In the discussion on space-time diagrams, we will see why multiplying t by c makes for simplicity. For now, understand that multiplying c by t gives us this top term in units of space, just like the x, y, and z components possess.

 

Nevertheless, this is a four-vector, but don’t be alarmed. It can still operate on matrices just like a three-dimensional vector. If we really want, we can simplify this vector down even further. By labelling the x, y, and z components as the vector r, the four-vector turns into this:

Personally created image.


Energy-Momentum Four-Vector

The next vector to discuss is the energy-momentum four-vector, shown below. In this image, we have some variables to clarify.

Personally created image.

-E represents the energy of the particle, both it’s rest and kinetic energies.

-px represents the x-component of the particle’s momentum.

-py represents the y-component of the particle’s momentum.

-pz represents the z-component of the particle’s momentum.

 

As always, c represents the speed of light. But why must we multiply momentum by c as well? The answer is to get all of our units here into units of mass. As I am sure you are aware, energy and mass are interchangeable thanks to E=mc2. Energy can be expressed in units of energy, or units of mass. So when we multiply the momentum by the speed of light, we get something like this below:

Image courtesy of myself.

We see the familiar equation E=mc2. In reality, the E in the four-vector is a combination of the particle’s rest energy and its kinetic energy. That looks quite different. But for this discussion, we need only know that the units of kinetic energy come out to be the same as for E.

Image courtesy of myself.

For momentum, we see that the units are different than those for Energy. To solve this issue, we multiply by the speed of light.

Image courtesy of myself.

 

When the speed of light is multiplied, we get the term p*c (as seen in the four-vector for energy-momentum). This term has the same units as E, which allows calculations to go through without                                                                                            any hiccups.

 

*A key concept in physics if you don’t know it: we love for units to be the same*


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Tensors