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.

 

[sardar]

The defining moment to use relativistic equations instead of classical ones is when the speed of the object is a significant fraction of the speed of light, typically greater than 10% of the speed of light. This is because at these speeds, the effects of special relativity become significant and cannot be accurately described by classical equations.

The ratio of rest energy is also a good indicator, as objects with a rest energy greater than 1 MeV, such as subatomic particles, must be described using relativistic equations.

In general, it is important to use relativistic equations whenever the object's speed is close to the speed of light or its energy is significant in relation to the rest energy. This ensures that the calculations are accurate and take into account the effects of special relativity.