13.3: Velocity and Momentum Four-Vectors
- Page ID
- 17447
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We already encountered the position four-vector in section 13.1. The four components of this four-vectors represent a 'position' in the four-dimensional spacetime of special relativity: three coordinates in space, one in time. We defined a dot product for these four-vectors in equation (13.2), and an associated length of a four vector (the square root of the dot product with itself). Positions are however not the only vector quantities we know - we encountered many others in part I, like velocity and momentum. Like position, these vector quantities have four-vector versions in special relativity. Also, like the position four-vector, we'll demand that their length should be invariant under Lorentz transformations.
Let us start with velocity. In classical mechanics, we calculate velocity as the time derivative of the position: \(\boldsymbol{v}=\mathrm{d} \boldsymbol{x} / \mathrm{d} t\). It is a straightforward exercise to check that the time derivative of the position four-vector \(\overline{\boldsymbol{x}}\) is not a four-vector itself, as its length is not invariant under Lorentz transformations. Perhaps this does not surprise you: after all, \(t\) is the time as measured by an outside observer, and we've argued extensively that time depends on the observer. Instead of taking the derivative to \(t\), we should therefore take the derivative with respect to time on a comoving clock, i.e., the proper time \(\tau\), in which case we do get the velocity four-vector:
\[
\overline{\boldsymbol{v}} \equiv \frac{\mathrm{d} \overline{\boldsymbol{x}}}{\mathrm{~d} \tau}=\gamma(v)\left(c, v_x, v_y, v_z\right),
\]
where \(v_x, v_y\) and \(v_z\) are the classical velocity components, and \(v\) is the speed. A straightforward calculation shows that the length of \(\overline{\boldsymbol{v}}\) equals \(c\), which is certainly invariant under Lorentz transformations.
Momentum in classical mechanics is defined as mass times velocity. We use the same definition in special relativity, with the only difference that the velocity is now given by the velocity four-vector. Because the mass is just a scalar, it simply multiplies the velocity four-vector, and is not affected by Lorentz transformations, so this procedure again yields a proper four-vector with length \(m c\) :
\[
\overline{\boldsymbol{p}} \equiv m \overline{\boldsymbol{v}}=m \gamma(v)\left(c, v_x, v_y, v_z\right)=\left(\gamma(v) m c, p_x, p_y, p_z\right)
\]
where \(p_x=\gamma(\nu) m \nu_x, p_y=\gamma(\nu) m \nu_y\) and \(p_z=\gamma(\nu) m \nu_z\) are the components of the 'three-momentum' \(\boldsymbol{p}=\) ( \(p_x, p_y, p_z\) ). Note that unlike the velocities, these components are not equal to their classical counterparts \({ }^4\), as they differ by a factor \(\gamma(\nu)\).