This Quantum World/Appendix/Relativity/4-vectors

4-vectors
3-vectors are triplets of real numbers that transform under rotations like the coordinates $$x,y,z.$$ 4-vectors are quadruplets of real numbers that transform under Lorentz transformations like the coordinates of $$\vec{x}=(ct,x,y,z).$$

You will remember that the scalar product of two 3-vectors is invariant under rotations of the (spatial) coordinate axes; after all, this is why we call it a scalar. Similarly, the scalar product of two 4-vectors $$\vec{a}=(a_t,\mathbf{a})=(a_0,a_1,a_2,a_3)$$ and $$\vec{b}= (b_t,\mathbf{b})=(b_0,b_1,b_2,b_3),$$ defined by



(\vec{a},\vec{b})=a_0b_0-a_1b_1-a_2b_2-a_3b_3,$$

is invariant under Lorentz transformations (as well as translations of the coordinate origin and rotations of the spatial axes). To demonstrate this, we consider the sum of two 4-vectors $$\vec{c}=\vec{a}+\vec{b}$$ and calculate



(\vec{c},\vec{c})=(\vec{a}+\vec{b},\vec{a}+\vec{b})= (\vec{a},\vec{a})+(\vec{b},\vec{b})+2(\vec{a},\vec{b}).$$

The products $$(\vec{a},\vec{a}),$$ $$(\vec{b},\vec{b}),$$ and $$(\vec{c},\vec{c})$$ are invariant 4-scalars. But if they are invariant under Lorentz transformations, then so is the scalar product $$(\vec{a},\vec{b}).$$

One important 4-vector, apart from $$\vec{x},$$ is the 4-velocity $$\vec{u}=\frac{d\vec{x}}{ds},$$ which is tangent on the worldline $$\vec{x}(s).$$ $$\vec{u}$$ is a 4-vector because $$\vec{x}$$ is one and because $$ds$$ is a scalar (to be precise, a 4-scalar).

The norm or "magnitude" of a 4-vector $$\vec{a}$$ is defined as $$\sqrt{|(\vec{a},\vec{a})|}.$$ It is readily shown that the norm of $$\vec{u}$$ equals $$c$$ (exercise!).

Thus if we use natural units, the 4-velocity is a unit vector.