The concept of Lorentz covariance is important because it allows us to quickly determine the transformation properties of expressions under changes of inertial reference frames. The principle of relativity requires that all laws of physics are unchanged as seen by different inertial observers. Hence, we need to ensure that expressions reflecting statements of a law of physics are Lorentz covariant, i.e., that they retain their structural
form under Lorentz transformations. A useful application of this comes from the modified second law of dynamics,

\[f_{\mu}=\frac{d p^{\mu}}{d \tau}\]

Forces that we insert on the left hand side of this equation must be Lorentz covariant expressions that transform as four-vectors. This ensures that observer \(\mathcal{O}^{\prime}\) can write simply

\[f_{\mu^{\prime}}=\frac{d p^{\mu^{\prime}}}{d \tau}\]

For example, we could write \(f_{\mu}=K_{\mu}\) with a constant four-vector \(K_{\mu}\). (a) Is a "relativistic spring law" \(f_{\mu}=-(0, k \mathbf{r})\) for some constant \(k\), a Lorentz covariant expression? (b) What about a modified spring law \(f_{\mu}=-K r^{\mu}=-k(c t, \mathbf{r})\) ? (c) What about Newtonian gravity \(\mathbf{F}=-\left(k / r^{3}\right) \mathbf{r}\) ? Is such a force covariant?