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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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It all depends on what you are considering. Suppose you are timing a sports event like a sprint. Surely, relativistic corrections will be irrelevant since the time scales we are considering are much bigger than the relativistic corrections to any time measurement.

On the other hand, when we are considering GPS, which is based on sending signals to and from faraway satellites, then relativistic corrections even if they are small do have their importance, especially as they are cumulating over time.

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Say I have a particle travelling at v=kc, then what order of magnitude should k be such that the relativistic effects are negligible? – Gabe Oct 15 '11 at 8:52
Makes no sense. It might very well be that the particle is travelling at 1% of c in one case and that you need to account for relativistic effects, where as the same particle travelling at the same speed but in another context doesn't require you to add that level of detail in the computations. – Raskolnikov Oct 15 '11 at 9:01
"Negligibility" depends entirely on how much precision you require. Even the results of a 100 meter dash exhibit "relativistic effects" if, for some reason, you are observing the runners to picometre precision in spatial extent and attosecond precision on a time scale. – Niel de Beaudrap Oct 15 '11 at 11:28

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