and fiber optic gyroscopes, are capable of extreme levels of sensitivity. Time dilation The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels). Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by . To find the relation between the times between these ticks as measured in both systems, can be used to find: for events satisfying This shows that the time (Δt) between the two ticks as seen in the frame in which the clock is moving (S), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory. Length contraction The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage). Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S, in which the rod is moving, the distances x to the end points of the rod must be measured simultaneously in that system S. In other words, the measurement is characterized by , which can be combined with to find the relation between the lengths Δx and Δx: for events satisfying This shows that the length (Δx) of the rod as measured in the frame in which it is moving (S), is shorter than its length (Δx)