When Should You Use Relativistic Equations Instead of Classical Ones?

[Blackthorn]

What is the defining moment when you use relativistic equations instead of classical ones? I have heard something as vague as "when it matters" and something about a ratio of rest energy before. I was hoping to know if there was a more concrete moment that defines when to use one or another. I of course use them anytime an object comes anywhere close to the magnitude of the speed of light.

 

[WannabeNewton]

Welcome to the forum mate! There are two major regimes. Say a system has a mass $M$, characteristic length scale $L$ and characteristic time scale $T$; for example, $L$ can be the size of a celestial orbit and $T$ the period of the orbit. We can form the two dimensionless parameters $\hat{c} = \frac{cT}{L}$ and $\hat{G} = \frac{GM T^2}{L^3}$ where $c$ is the speed of light and $G$ is Newton's constant. In essence, $\hat{c}$ is the velocity scale of our system and $\hat{G}$ is the scale of self-gravitation of our system.

The limit $\hat{c}\rightarrow \infty$ with $\hat{G}$ fixed gives us Newtonian gravity and the limit $\hat{G}\rightarrow 0$ with $\hat{c}$ fixed gives us special relativity. Imagine the two-dimensional parameter space of $(\hat{G},\hat{c})$; if you draw a graph using the $\hat{G}$ and $\hat{c}$ axes then you can label the regions where relativity is important. Loosely put, if $\hat{c}$ is small and/or $\hat{G}$ is large we will need relativity.

 

[Dale]

"When it matters" is the best answer. It depends on the precision needed. For any given precision you can convert that to the Lorentz factor and calculate the speed required.

 

[Blackthorn]

Awesome. Thanks for the quick replies folks.