# Relativistic effects

When are relativistic effects justifiably negligible? (I know that that is true for 'small velocities', but how small is 'small enough'?) 0.1c, 0.01c, etc.? And how does one properly justify that? I reckon by Taylor expansion of the relativistic formula, but still, how do you set the threshold speed? Thanks.

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If you had infinite measuring precision, for no speed would relativistic effects be negligible. –  Benjamin Horowitz Oct 15 '11 at 20:22

If $v = kc$ with $k$ small then most special relativistic effects such as time dilation and length contraction are proportional to $$\gamma = \frac{ 1}{ \sqrt{1 - k^2}} \approx 1 + \frac{1}{2} k^2$$ or it's reciprocal. See Wikipedia here for a more complete discussion. If you consider, for example, $\gamma \le 1.0001$ (a $0.01\%$ effect) to be negligible then this is true if $k \le 0.014$ giving $v \le 9,400,000 MPH = 4,200,000\frac{m}{s}$.

GPS satellites not only need to consider the effects of special relativity but also the effects of general relativity to give accurate positioning.

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