# 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?

• "The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" H.MINKOWSKI in SPACE AND TIME, 21 September 1908. – Frobenius May 3 '16 at 15:10

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 $1$.

• Could one argue for the usage of 4-vectors as follows. From Einstein's postulates we are naturally led to the Lorentz transformations, which relate the coordinates of one inertial reference frame to another. In doing so one finds that time is in fact a frame dependent quantity and furthermore, spatial and temporal coordinates are mixed together under such transformations, showing that space and time are not in fact independent but should be considered as a 4-dimensional continuum, which we call spacetime... – user35305 May 3 '16 at 18:30
• ... We are then naturally led to consider 4-dimensional vectors, since these span the entire space and furthermore they transform under Lorentz transformations in such a way that the equations describing physical phenomena are Lorentz invariant, a requirement of Einstein's postulates. An additional argument for their usage is that in special relativity it is the spacetime interval that is an invariant quantity and not the traditional Pythagoras line element as in Classical mechanics... – user35305 May 3 '16 at 18:31
• ... It is seen that the lengths of 4-vectors are preserved in this case, whereas the lengths of 3-vectors are not, hence we should construct physical equations out of 4-vector quantities (also scalars and tensors). Would this be a correct assessment at all? – user35305 May 3 '16 at 18:31
• @user35305 Yes, that sounds correct to me – jak May 4 '16 at 9:09

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.

• So the usage of 4-vectors follows purely from the fact that time and space are intertwined by the Lorentz transformations then? Why does one only use 3-vectors in Newtonian mechanics then, is it simply because time is absolute in this case and so is independent of space, allowing us to describe spatial locations of objects and parametrising them by time? – user35305 May 3 '16 at 13:01
• That is essentially correct. The choice to use 4-vectors is also simply a matter of convenience. Maxwell originally worked with the field equations in terms of quaternions, which is really just horrendous. 4-vectors are actually defined as "4-vectors" only when we prescribe how they transform under some operation. For Newtonian mechanics, we assume Galilean invariance, in which velocities add linearly (which depend on a fixed notion of time). In special relativity, we assume Lorentz invariance. – MSha May 4 '16 at 0:53