Editing Time dilation (section)

===Derivation and formulation===
[[File:Time dilation.svg|thumb|upright=1.4|[[Lorentz factor]] as a function of speed (in natural units where ''c'' = 1). Notice that for small speeds (as v tends to zero), γ is approximately 1.]]

In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of the [[Lorentz transformation]].<ref name="born">{{Citation |last=Born |first=Max |title=Einstein's Theory of Relativity |url=https://archive.org/details/einsteinstheoryo0000born |year=1964 |publisher=Dover Publications |isbn=978-0-486-60769-6 |author-link=Max Born |url-access=registration}}</ref> Let there be two events at which the moving clock indicates <math>t_{a}</math> and <math>t_{b}</math>, thus:
:<math>t_{a}^{\prime}=\frac{t_{a}-\frac{vx_{a}}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ t_{b}^{\prime}=\frac{t_{b}-\frac{vx_{b}}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}</math>

Since the clock remains at rest in its inertial frame, it follows <math>x_{a}=x_{b}</math>, thus the interval <math>\Delta t^{\prime}=t_{b}^{\prime}-t_{a}^{\prime}</math> is given by:
:<math> \Delta t' = \gamma \, \Delta t = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} \,</math>
where Δ''t'' is the time interval between ''two co-local events'' (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on their clock), known as the ''[[proper time]]'', Δ<var>t′</var> is the time interval between those same events, as measured by another observer, inertially moving with velocity ''v'' with respect to the former observer, ''v'' is the relative velocity between the observer and the moving clock, ''c'' is the speed of light, and the [[Lorentz factor]] (conventionally denoted by the Greek letter [[gamma]] or &gamma;) is:
:<math> \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \,</math>

Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, where {{nowrap|1=''v'' ≪ ''c'',}} even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. As an approximate threshold, time dilation of 0.5% may become important when an object approaches speeds on the order of 30,000&nbsp;km/s (1/10 the speed of light).<ref>{{Cite book |last=Petkov |first=Vesselin |url=https://books.google.com/books?id=AzfFo6A94WEC |title=Relativity and the Nature of Spacetime |publisher=Springer Science & Business Media |year=2009 |isbn=978-3-642-01962-3 |edition=2nd, illustrated |page=87}} [https://books.google.com/books?id=AzfFo6A94WEC&pg=PA87 Extract of page 87]</ref>