What is an example of a vector law which is not covariant under Lorentz transformations? I'm currently studying relativity and the covariant formalism. From what I understood the main reason why this formalism has been introduced is because in this way the laws of physics are covariant under the Lorentz transformations, which are the correct transformations of space and time coordinates. 
Now I'd like to have an example of a vector law which is not covariant under Lorentz transformations if written in the 3D-vector formalism and which is instead covariant under the 4D-vector one.
 A: Probably the simplest example of this is the first vector law you ever learned,
$$\mathbf{F} = m \mathbf{a}.$$
If you define these quantities in the usual way, 
$$\mathbf{F} = \frac{d\mathbf{p}}{dt}, \quad \mathbf{a} = \frac{d \mathbf{v}}{dt}$$
then this equation is simply not true in special relativity. It's not even true if you artificially allow the coefficient $m$ to vary, because in situations where the force is not parallel to the velocity, the force and the acceleration aren't even parallel!
Instead, in special relativity you use the four-vector equation
$$f^\mu = m a^\mu$$
where the quantities are defined as 
$$f^\mu = \frac{dp^\mu}{d\tau}, \quad a^\mu = \frac{du^\mu}{d\tau}$$
where $\tau$ is the proper time and $u^\mu = dx^\mu/d\tau$ is the four-velocity. This is a four-vector equation that reduces to $\mathbf{F} = m \mathbf{a}$ at low velocities, but which is true in general. 
A: The notation or formalism used to express a physical law doesn’t affect whether that law is covariant under some group of transformations. Maxwell’s equations, for example, are covariant under Lorentz transformations whether written in terms of $\mathbf{E}$ and $\mathbf{B}$ or in terms of $F^{\mu\nu}$.
But using Lorentz four-vectors and four-tensors makes a law obviously (or “manifestly”) Lorentz-covariant because of their simple Lorentz transformation properties, in the same way that three-vector equations are manifestly covariant under Euclidean rotations.
It is not obvious that Maxwell’s equations are Lorentz-covariant when written in terms of $\mathbf{E}$ and $\mathbf{B}$, but it is when they are written in terms of $F^{\mu\nu}$.
The formalism of Lorentz four-vectors and four-tensors actually makes it impossible to write laws which are not Lorentz-covariant, when you respect the rules of the formalism.
