TITLE: Relativistic corrections to classical physics formulae
QUESTION [1 upvotes]: How are classical formulas in physics (such as p = mv, or kinetic energy, or maxwell distribution of speeds) treated with the appropriate relativistic correction/modification? Is it done by using the Lorentz transformation equations? Could anyone give me a few examples of relativistic corrections to classical formulae? Thanks.

REPLY [1 votes]: Alfred Centauri's wiki link certainly gives you a nice list of such things, but that entry doesn't exactly show you how such quantities reduce to their non-relativistic counterparts.  
In particular, in the context of classical mechanics, one can generically show that relativistic expressions reduce to their non-relativistic counterparts in the limit of velocities $v$ that are small compared to the speed of light $c$.
The key to seeing this for many such quantities is to notice that the parameter $\gamma$ which appears everywhere in relativistic expressions is close to $1$ when $v\ll c$.  Specifically, note that $\gamma$ is defined in terms of the ratio $\beta = v/c$ as follows:
\begin{align}
  \gamma = \left(1-\beta^2\right)^{-1/2}
\end{align}
When $\beta$ is small, namely when we are doing physics for velocities $v$ much smaller than $c$, the following Taylor expansion of $\gamma$ about $\beta = 0$ becomes relevant;
\begin{align}
  \gamma = 1 + \frac{1}{2}\beta^2+ O(\beta^4)
\end{align}
In particular, the first two terms are a very good approximation at low speeds.  Here's an example:
Kinetic Energy.
The relativistic definition of the kinetic energy of a particle of mass $m>0$ is it's total energy minus its rest energy;
\begin{align}
  K = E-mc^2
\end{align}
where the total energy is defined as
\begin{align}
  E = \gamma mc^2
\end{align}
Combining these facts, we see that the kinetic energy is given by
\begin{align}
  K = (\gamma - 1)mc^2 = \frac{1}{2}\beta^2 mc^2 + O(\beta^4) = \frac{1}{2}mv^2 + O(\beta^4)
\end{align}
where I used the Taylor expansion of $\gamma$ about $\beta = 0$ in the first step, and the definition of $\beta$ in the second step.  In other words, to lowest non-vanishing order in $\beta$, the relativistic expression for the kinetic energy of a massive particle is the non-relativistic expression.
{"set_name": "stack_exchange", "score": 1, "question_id": 71943}