Understanding Special Relativity: The Lorentz Transformation, Time Dilation, and Length Contraction

​
velocity/
c
0.5
proper time/lengthfor moving observer
0.4
(x, t)
gridlines
(
′
x
,
′
t
)
gridlines
axis labels
plot choice
space-time event
time dilation
length contraction
The Lorentz transformation describes how two observers in relative motion can relate their measurements of space and time. This Demonstration illustrates the space and time axes for two observers, one at rest (black axes, red gridlines) and one in motion (blue axes and gridlines) for a single space-time event, time dilation, or length contraction. The coordinate axes can be shown for each observer, and the relative velocity can be changed at will.
You can drag the space-time event (black dot) and see its
(x,t)
and
(x',t')
coordinates by mousing over it.
Time dilation is the phenomenon whereby two observers disagree on the time interval between two events: moving clocks run slowly. Here the proper time of a moving observer (in blue) corresponds to a longer (dilated) time for an observer at rest (in red). You can change the proper time.
Length contraction (also known as Lorentz–FitzGerald contraction) is the phenomenon whereby two observers disagree about the length of an object: moving objects shrink. Here the proper length of a moving object (worldlines in blue) corresponds to a shorter (contracted) length for an observer at rest (in red). You can change the proper length.

Details

The Lorentz transformation between two reference frames with coordinates
(x,t)
and
(x',t')
, in relative motion with velocity
v
, can be represented by

x'
t'
=
γ
-γv
-γv
γ

x
t

, or the inverse

x
t
=
γ
γv
γv
γ

x'
t'

, where
γ=1
1-
2
v
and
c=1
. The inverse transformation is used to plot gridlines of constant
x'
and
t'
.
See also:
http://en.wikipedia.org/wiki/Lorentz_transformation
http://en.wikipedia.org/wiki/Time_dilation
http://en.wikipedia.org/wiki/Length_contraction

Permanent Citation

Frederick W. Strauch
​
​"Understanding Special Relativity: The Lorentz Transformation, Time Dilation, and Length Contraction"​
​http://demonstrations.wolfram.com/UnderstandingSpecialRelativityTheLorentzTransformationTimeDi/​
​Wolfram Demonstrations Project​
​Published: August 15, 2011