Time dilation and Length contraction.

Time Dilation

Definition: Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

A closely related phenomenon predicted by special relativity is the so-called twin paradox. Suppose one of two twins carrying a clock departs on a rocket ship from the other twin, an inertial observer, at a certain time, and they rejoin at a later time. In accordance with the time-dilation effect, the elapsed time on the clock of the twin on the rocket ship will be smaller than that of the inertial observer twin—i.e., the non-inertial twin will have aged less than the inertial observer twin when they rejoin.

The time-dilation effect predicted by special relativity has been accurately confirmed by observations of the increased lifetime of unstable elementary particles traveling at nearly the speed of light. The clock paradox effect also has been substantiated by experiments comparing the elapsed time of an atomic clock on Earth with that of an atomic clock flown in an airplane. The latter experiments, furthermore, have confirmed a gravitational contribution to time dilation, as predicted by the theory of general relativity.

It turns out that as an object moves with relativistic speeds a strange thing seems to happen to its time as observed by "us" the stationary observer. What we see happen is that the clock in motion slows down according to our clock, therefore we read two different times. Which time is correct? well they both are because time is not absolute but is relative, it depends on the reference frame.

 

t = t0/(1-v2/c2)1/2

where: t = time observed in the other reference frame

t0 = time in observers own frame of reference (rest time)

v = the speed of the moving object

c = the speed of light in a vacuum

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances  between each other, then express each distance in terms of the time of travel (respectively either  Δt  or  Δτ ) of the pulse in the corresponding reference frame. The resulting equation can then be solved for  Δt  in terms of  Δτ .

Length contraction:

Definition :Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame.

 It is also known as Lorentz contraction or Lorentz–FitzGerald contraction (after Hendrik Lorentz and George Francis FitzGerald) and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

The deviation between the measurements in all inertial frames is given by the formulas for Lorentz transformation and time dilation (see Derivation). It turns out that the proper length remains unchanged and always denotes the greatest length of an object, and the length of the same object measured in another inertial reference frame is shorter than the proper length. This contraction only occurs along the line of motion, and can be represented by the relation

where

L is the length observed by an observer in motion relative to the object

L₀ is the proper length (the length of the object in its rest frame)

γ(v) is the Lorentz factor, defined as

where

v is the relative velocity between the observer and the moving object

c is the speed of light

Replacing the Lorentz factor in the original formula leads to the relation

We do not notice length contraction in everyday life.The distance to the grocery store does not seem to depend on whether we are moving or not. Examining Equation we see that at low velocities (  v≪c ), the lengths are nearly equal, which is the classical expectation. However, length contraction is real, if not commonly experienced. For example, a charged particle such as an electron traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer.