Difference between revisions of "4-gradient"

From New IAC Wiki
Jump to navigation Jump to search
Line 48: Line 48:
 
\begin{bmatrix}
 
\begin{bmatrix}
 
\frac{\partial}{\partial x_0}  \\
 
\frac{\partial}{\partial x_0}  \\
 +
\\
 
\frac{\partial}{\partial x_1}  \\
 
\frac{\partial}{\partial x_1}  \\
 +
\\
 
\frac{\partial}{\partial x_2}  \\
 
\frac{\partial}{\partial x_2}  \\
 +
\\
 
\frac{\partial}{\partial x_3}  
 
\frac{\partial}{\partial x_3}  
 
\end{bmatrix}=
 
\end{bmatrix}=
 
\begin{bmatrix}
 
\begin{bmatrix}
 
\frac{\partial}{\partial t}  \\
 
\frac{\partial}{\partial t}  \\
 +
\\
 
\frac{\partial}{\partial x}  \\
 
\frac{\partial}{\partial x}  \\
 +
\\
 
\frac{\partial}{\partial y}  \\
 
\frac{\partial}{\partial y}  \\
 +
\\
 
\frac{\partial}{\partial z}  
 
\frac{\partial}{\partial z}  
 
\end{bmatrix}=
 
\end{bmatrix}=
Line 72: Line 78:
 
\begin{bmatrix}
 
\begin{bmatrix}
 
\frac{\partial}{\partial x_0}  \\
 
\frac{\partial}{\partial x_0}  \\
 +
\\
 
\frac{\partial}{\partial x_1}  \\
 
\frac{\partial}{\partial x_1}  \\
 +
\\
 
\frac{\partial}{\partial x_2}  \\
 
\frac{\partial}{\partial x_2}  \\
 +
\\
 
\frac{\partial}{\partial x_3}
 
\frac{\partial}{\partial x_3}
 
\end{bmatrix}=
 
\end{bmatrix}=

Revision as of 14:39, 10 July 2017

\underline{Navigation}

4-gradient

From the use of the Minkowski metric, converting between contravariant and covariant


xμημμxμ


Where we have already defined the covariant term,

xμ=[x0x1x2x3]

and the contravariant term

xμ=[x0x1x2x3]


From quantum mechanics we know that partial differential is a linear operator. Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector, and the derivative with respect to a covariant coordinate transforms as a contravariant vector.

μ=xμ


μ[x0x1x2x3]=[txyz]=[t]


μ=xμ


μ[x0x1x2x3]=[txyz]=[t]


Since it is an operator, the dot product of two partial differentials yields an operator known as the D'Alembert operator.

μμ=[x0x1x2x3][x0x1x2x3]=2t22


\underline{Navigation}