Examples of relativistic equations

Question

I am posting this question here because it is just reference request and I do not need a fully detailed answer.

Attending my physics class, we introduced two relativistic equations:

1. $$ \frac{d}{dt}\left(\frac{m \dot{x}}{\sqrt{1- |\dot{x}|^2/c^2}}\right) =-\alpha \frac{x}{|x|^3} \quad x\in\mathbb{R}^n\setminus\{0\},$$

where $n\ge 1$, $c>0$ is the speed of light, $\alpha = MmG$ with $M, m$ the mass of two bodies and $G$ the gravitational constant. We called it relativistic Kepler equation.

2. $$\frac{d}{dt}\left(\frac{m \dot{x}}{\sqrt{1- |\dot{x}|^2/c^2}}\right) = q(E(t, x) +\dot{x}\times B(t, q)),\quad x\in\mathbb{R}^3\setminus\{0\}$$ where $E$ and $B$ denote the electric and magnetic fields respectively. We called it relativistic Lorentz force equation.

As far as I understood, the term $\textbf{relativistic}$ comes from the relativistic nature of these equation which is exploited at LHS. During class, we talked about the relevance and importance of this kind of equations.

Thus, my question is: there exists other relevant relativistic equations or those are the unique ones? Could someone please provide some references?

Thank you in advance.

Answer 1

"Relativistic" means that the equation is covariant under the symmetry group of relativity, that is, either the Lorentz group in the special relativity case or some more complicated group in the presence of curved spacetime.

The fundamental example of (special) relativistic equation is the wave equation $$ \partial_t^2u=c^2\Delta u,$$ on Minkowski spacetime $\mathbb R^{1+3}$. (In the following I will set $c=1$).

This equation is indeed invariant under the Lorentz group and the quickest way to see this is to develop the solution $u$ in plane waves: $$ u(x_0, \boldsymbol{x})=\int_{\mathbb R^{1+3}} \tilde{u}(p_0,\boldsymbol{p})\exp(i xp)\, d^4p, $$ where $xp=x^0p^0-\boldsymbol{x}\cdot \boldsymbol{p}$. You see that $\tilde{u}$ must satisfy the equation $$ ((p^0)^2-\lvert\boldsymbol{p}\rvert^2)\tilde{u}=0, $$ which is manifestly Lorentz-invariant; by definition, Lorentz transformations are precisely the ones that preserve the quadratic form $(p^0)^2-\lvert\boldsymbol{p}\rvert^2$.

Answer 2

Recall that Newton's second law states that \begin{align} m\,\mathbf{a} = \mathbf{F}_{\mathrm{net}} \end{align} if the mass doesn't change. More generally, we have that \begin{align} \frac{d}{d t}\left(m\, \mathbf{\dot x}\right)=\frac{d}{d t}\left(m\, \mathbf{v}\right)= \mathbf{F}_{\mathrm{net}}. \end{align}

In particular, a particle experiencing gravitational attraction from the origin will satisfy the equation \begin{align} \frac{d}{d t}\left(m\, \mathbf{\dot x}\right)= \nabla\left(\frac{\alpha}{|\mathbf{x}|}\right) = -\frac{\alpha}{|\mathbf{x}|^2}\frac{\mathbf{x}}{|\mathbf{x}|}. \end{align}

For a relativistic particle, you need to account for the correction to its velocity by $\frac{\mathbf{\dot x}}{\sqrt{1-|\mathbf{\dot x}|^2/c^2}}$. Hence the general equation of motion is given by \begin{align} \frac{d}{d t}\left(\frac{m\,\mathbf{\dot x}}{\sqrt{1-|\mathbf{\dot x}|^2/c^2}}\right) = \mathbf{F}_{\mathrm{net}}. \end{align}

In general, if you know how forces act on a relativistic particle in space, then the above equation is meaningful.

A class of important forcing will be given by $\mathbf{F}_{\mathrm{net}}= - \nabla V$ for some $V(\mathbf{x})$ called the potential function.