This Quantum World/Appendix/Relativity/4-vectors

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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


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


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.