I recently had someone ask me why we use 4-vectors in special relativity and what is the motivation for introducing them in the first place. This is the response I gave:(adsbygoogle = window.adsbygoogle || []).push({});

From Einstein's postulates( i.e. 1. the principle of relativity - the laws of physics are identical (invariant) in all inertial frames of reference, and 2. the speed of light in vacuum is the same for all inertial observers, regardless of the motion of the observer or the source) 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 callspacetime.

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 Pythagorean line element as in Classical mechanics. 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 ( and of course, in general, scalars and tensors).

I'm now starting to doubt my own understanding a bit and I'm worried I may have conveyed incorrect information. Would someone be able to clarify whether what I've written is correct or not (and if it's not correct, explain why)?

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# I Motivation for the usage of 4-vectors in special relativity

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