Lorentz transformation under symmetric configuration
Encyclopedia
In physics
, the Lorentz transformation
converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other.
Assume there are two observers
and 
, each using their own Cartesian coordinate system
to measure space and time intervals.
uses 
and 
uses 
. Assume further that the coordinate systems are oriented so that the 
-axis and the 
-axis overlap but in opposite directions. The 
-axis is parallel to the 
-axis but in opposite directions. The 
-axis is parallel to the 
-axis and in the same direction. The relative velocity between the two observers is 
along the 
or 
axis. 
is defined as a positive number when 
sees 
sliding in the direction of 
. Also assume that the origins of both coordinate systems are the same. If all this holds, then the coordinate systems are said to be in symmetric configuration.
In this configuration, frame
appears to 
in the identical way that frame 
appears to 
. However, in the standard configuration, if 
sees 
going forward then 
sees 
going backward. This symmetric configuration is equivalent to the [Lorentz transform#Lorentz transformation for frames in standard configuration|standard configuration]] followed by a mirror reflection of the x and y-axes. For the stationary 
case, this reduces to only the reflections, whereas the standard form reduces to the identity transformation.
The Lorentz transformation for frames in symmetric configuration is:
where
is the Lorentz factor
.
The inverse transformation is:
The above forward and inverse transformations are identical. This offers mathematical simplicity.
In matrix
form the forward symmetric transformation is:
where
.
The inverse symmetric transformation is:
A single transformation matrix is used for both the forward and the inverse operation.
As expected:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other.
Assume there are two observers
and , each using their own Cartesian coordinate systemCartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
to measure space and time intervals.
uses and uses . Assume further that the coordinate systems are oriented so that the -axis and the -axis overlap but in opposite directions. The -axis is parallel to the -axis but in opposite directions. The -axis is parallel to the -axis and in the same direction. The relative velocity between the two observers is along the or axis. is defined as a positive number when sees sliding in the direction of . Also assume that the origins of both coordinate systems are the same. If all this holds, then the coordinate systems are said to be in symmetric configuration.In this configuration, frame
appears to in the identical way that frame appears to . However, in the standard configuration, if sees going forward then sees going backward. This symmetric configuration is equivalent to the [Lorentz transform#Lorentz transformation for frames in standard configuration|standard configuration]] followed by a mirror reflection of the x and y-axes. For the stationary case, this reduces to only the reflections, whereas the standard form reduces to the identity transformation.The Lorentz transformation for frames in symmetric configuration is:
where
is the Lorentz factorLorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
.
The inverse transformation is:
The above forward and inverse transformations are identical. This offers mathematical simplicity.
In matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
form the forward symmetric transformation is:
where
.The inverse symmetric transformation is:
A single transformation matrix is used for both the forward and the inverse operation.
As expected: