Four vectors in spacetime

A vector is nothing but a straight line with a specific direction. Any event in spacetime can be represented by a four vector. If you know what spacetime is, you will agree that the “spacetime” is made of 4 dimensions – three spatial dimensions and the time dimension(If you don’t have any idea about “spacetime” then you can go through our article on spacetime – “What is spacetime?”). So, a vector in spacetime has four coordinates and so its name. As like the vectors in three dimensional Euclidean space, a four vector is something whose components transform as do the coordinates under a coordinate transformation. It means that if you change the coordinate system then the components of the four vector measured by you will be different from the previous measurements. But the vector itself does not change. It remains the same i.e. its magnitude and direction does not change. A four vector is the displacement vector which points from one point to another and its components are equal to the coordinate differences. If the coordinate system is changed then the coordinates of the two points will change and so the components of the vector. Four vector is denoted by the following notation

An arrow over says that  is a vector and the arrow after  says that “it has components”. O below the second arrow says that the components after the second arrow are measured “in the O frame”(It is important to know in which frame you are doing the measurements because the coordinates change as the coordinate system changes). The coordinates of a four vector is written in the same order as shown in (1).  Don’t forget that the vector  is between the two events so changing the coordinate system will not affect the vector( particularly the magnitude and the direction as I said earlier). The notation (1) can also be written like the following

Where 𝛼 can take values from the set {0,1,2,3}. It is the shorthand for the notation (1). is equivalent to . If the observer is changed(i.e. the coordinate system is changed) then the above notation can be written as –



The bar over O and 𝛼 says that the components of the vector are measured in  frame. Remember that the vector  itself does not change, it is the same vector and that’s why it does not need any bar over it. Note that every coordinate system in spacetime (not only in spacetime but also in two dimensional and three dimensional spaces) is somehow related with each other. There are always some equations with which you can relate the two different systems or frames. In four dimensional spacetime the frames are related by their relative speed with each other. We can determine the relation between the two frames using the Lorentz transformation rule.




 which can be written as





where   and similarly for other components in the frame  . Note that we have used Einstein summation notation where

The general vector is defined by a collection of numbers like the following

Four vectors obey the triangle rule of vectors i.e.

A four vector multiplied with a constant number is itself a new vector.

The magnitude of a four vector is equal to the interval between the two events.