Four-vectors and notation

Why four-vectors are useful

Special relativity mixes space and time. A mathematical object that keeps track of this mixing is a four-vector. Four-vectors transform under Lorentz transformations in a way that preserves spacetime structure.

Just as ordinary vectors have components that change under rotations while the vector itself has invariant meaning, four-vectors have components that change between inertial frames while their spacetime magnitude remains invariant.

Position four-vector

The position four-vector is often written

$x^mu=(ct,x,y,z)$

where the index $mu$ runs over 0, 1, 2, 3. The time component is $x^0=ct$, and the spatial components are $x^1=x$, $x^2=y$, $x^3=z$.

Using $ct$ gives all components units of length.

Index notation

Greek indices usually represent spacetime components:

u=0,1,2,3$$ Latin indices often represent spatial components: $$i,j=1,2,3$$ This notation makes relativistic equations compact and frame-independent. ## Lorentz transformation as matrix A Lorentz transformation can be written as $$x'^mu=Lambda^mu*{ u}x^ u$$ For motion along x, the transformation includes $$ct'=gamma(ct-eta x)$$ $$x'=gamma(x-eta ct)$$ where $$eta=rac{v}{c}$$ and $$gamma=rac{1}{sqrt{1-eta^2}}$$ ## Four-velocity The four-velocity is the derivative of position four-vector with respect to proper time: $$U^mu=rac{dx^mu}{d au}$$ For a particle moving with ordinary velocity $ec{v}$, $$U^mu=gamma(c,ec{v})$$ Its invariant magnitude is fixed: $$U^mu U*mu=-c^2$$ using the $(-,+,+,+)$ metric convention. ## Four-momentum The four-momentum is rest mass times four-velocity: $$p^mu=mU^mu$$ Thus $$p^mu=left(rac{E}{c},ec{p} ight)$$ where $$E=gamma mc^2$$ and $$ec{p}=gamma mec{v}$$ Four-momentum combines energy and momentum into one relativistic object. ## Invariant magnitude The invariant magnitude of four-momentum is $$p^mu p_mu=-m^2c^2$$ This is equivalent to $$E^2=p^2c^2+m^2c^4$$ Four-vector notation makes the invariant structure obvious. ## The big idea Four-vectors are the natural language of special relativity. They combine time with space and energy with momentum in ways that transform consistently between inertial frames. Once physical laws are written in four-vector form, their relativistic invariance becomes clearer.