Motivation for usage of 4-vectors in special relativity I understand that if one considers a 4-dimensional space-time from the outset then 4-vectors are the natural quantities to consider (as opposed to 3-vectors as in Newtonian mechanics), since the tangent space of space-time at a given point will necessarily be 4-dimensional. [With the notion of space-time being motivated by the fact that in special relativity time is a frame-dependent quantity and as such is no longer independent on space, the two being inextricably linked since the time coordinate in one frame becomes a mixture of space and time coordinates in another frame. Since this space-time is 4-dimensional one requires 4 coordinates to specify the location of an event within it.]
However, is there a way to motivate the usage of 4-vectors before introducing the notion of a space-time? 
Is it something to do with the fact that both spatial and temporal coordinates transform non-trivially under so-called Lorentz transformations (in order to ensure that the speed of light is frame-independent), and as such the spatial interval $$d\tilde{s}^{2}=dx^{2}+dy^{2}+dz^{2}$$ is no longer invariant, but instead the so-called space-time interval $$ds^{2}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}$$ is invariant. As such, 3-vectors do not transform correctly under Lorentz transformations (their lengths are not Lorentz invariant) and we must consider objects that transform covariantly under such transformations such that their moduli are invariant, i.e. 4-vectors.
I'm unsure as to whether we are led to the usage of 4-vectors as a consequence of introducing space-time or whether there are other motivations for their usage prior to this?
 A: The set of transformations that leaves the speed of light unchanged is the Lorentz group. Representation theory enables us to investigate the irreducible representations of the Lorentz group. 
The lowest-dimensional representations act on


*

*scalars

*four-vectors


However, take note that usually we consider representations of the corresponding Lie algebra $\mathfrak{so}(3,1)$. Among the irreducible representations of this Lie algebra are additional representations which are not representations of the Lorentz group. These representations correspond to the double cover of the Lorentz group and among them is the famous spinor representation that describes spin $\frac{1}{2}$ particles. Similarly in particle physics, scalars describe spin $0$ particles and four-vectors particles with spin $1$.
A: In special relativity there are two major assumptions:
-the laws of physics are the same in all inertial frames
-the speed of light that you observe is always the same, (thus independent of the relative motion between the light source and the observer).
From this two assumptions follows the famous Lorentz transformations. In these Lorentz transformations time and space occur are intertwined, they don't appear in the set of equations independently.
In special relativity you cannot view location and time as two separate things anymore. 
So, if for example an event takes place then you need to provide three coordinates and a time to give the information. Positions and time by themselves make not much sense anymore. Therefore it is convenient to write this in one vector and develop a mathematical framework around it.
