Relativity 4

Example 2: How Good is the Approximation?

The Question:

Remember that the expansion of the gamma function that was used was $$

\[\begin{align*} \gamma &=\frac{1}{\sqrt{{1-\beta^2}}} \\ \text{ the approximation is: } \gamma &\approx 1+\dfrac{1}{2}\beta^2 \end{align*}\]

\[ \begin{align}\begin{aligned} How far out in speed must we go with the approximate expansion of the gamma function to deviate from the real value by 0.1?\\ ------\\**The Answer:** \\Let's look at some values of $\beta$ and use the interactive $\gamma$ in the text and compare to the calculated approximation. For example:\\for $\beta = 0.2$, \end{aligned}\end{align} \]

\gamma \approx 1+ \frac{1}{2} (0.2)^2 = 1.02 \nonumber $$

Here’s a collection of comparisons:

\(\beta\)	actual \(\gamma\)	approx. \(\gamma\)
0.2	1.021	1.02
0.3	1.048	1.045
0.4	1.091	1.08
0.5	1.155	1.125
0.6	1.25	1.18

So by the time the speed has reached 60% of the speed of light, the approximation is no longer valid by about 0.1. One would then add another term in the expansion which would make the approximation: $\( \gamma \approx 1+ \frac{1}{2} \beta^2 + \frac{3}{8}\beta^4 \nonumber \)$ which would add 0.049 to the value in the table.