### Calculating Time Dilation using the Lorentz Transformation

Here's a derivation of the formula used to calculate relativistic time dilation.

To determine the relationship between times in two different reference frames, imagine that a person J (for Joe) is travelling in a straight line at velocity v, relative to an observer O (for Oscar) some distance away. When Joe reaches a point we will call A, he releases one photon from his special flashlight, which he is pointing at right-angles to his direction of travel. To Joe, this photon appears to travel straight out away from him - although since he is moving, the photon moves with him. For Oscar, this photon appears to travel in a diagonal line. As it happens, Oscar is standing directly in the photon's line of travel. Here's a diagram:

The dotted line shows the way Joe perceives the motion of the photon, i.e. travelling out at right angles to his direction of travel, although it doesn't actually travel along this line. The line from A to O represents the actual motion of the photon, which appears to the moving Joe as a straight line, perpendicular to the path he's travelling.

Since we know the velocity of the photon and of Joe, we can work out the lengths of the sides of this triangle as follows (remembering that Rate multiplied by Time equals Distance):

The distance that O observes the photon travelling, equal to the speed of light c multiplied by the time T spent travelling to O. T is the elapsed time in O's reference frame. | ||

The distance that J observes the photon travelling, equal to the speed of light c multiplied by the time t spent travelling to O. t is the elapsed time in J's reference frame. | ||

The distance travelled by J, as observed by O, equal to J's velocity v (relative to O) multiplied by the travel time T. |

Pythagorus says that the square of the hypotenuse of a right-triangle is equal to the sum of the squares of the two shorter sides, i.e.

Substituting the above values for the lengths of the triangle's sides gives:

Then using a bit of algebra, we can solve for T as
follows:

Get all the T's on the same side of the equal sign | |||

Square everything to get rid of the parentheses | |||

Factor out T squared | |||

Get T squared on its own | |||

Factor out c squared on the bottom, so we can have it cancel with the top | |||

Cancel c squared | |||

Take the square root of it all to get T |

And that's it! This gives
us a formula for relating time T in a static reference frame to time t in a
frame which is moving at velocity v relative to the static frame.

As an example, if a rocket travels away from earth at
0.99 of the speed of light for one year (t=1yr), the
time T elapsed on earth will be:

This formula is part of what's known as the Lorentz
transformation, which Lorentz (and others, independently) developed to handle
transformation of coordinates between different frames of reference. They are
independent of relativity and merely describe how values must be transformed
from one reference frame to another. One of Einstein's unique contributions was
to realize that the Lorentz transformations could and should be applied to time
and space, with the speed of light as an invariant.