View source for General Relativity/Comma derivative

< [[General Relativity]]

In General Relativity we write our (4-dimensional) coordinates as <math>(x^0,x^1,x^2,x^3)</math>.  The flat Minkowski spacetime coordinates ("Local Lorentz frame") are <math>x^0=ct</math>, <math>x^1=x</math>, <math>x^2=y</math>, and <math>x^3=z</math>, where <math>c</math> is the speed of light, <math>t</math> is time, and <math>x</math>, <math>y</math>, and <math>z</math> are the usual 3-dimensional Cartesian space coordinates.

A '''comma derivative''' is just a convenient notation for a partial derivative with respect to one of the coordinates.  Here are some examples:


1.  <math>T^\alpha_{\ \beta , \gamma} = \frac {\partial T^\alpha_{\ \beta}} {\partial x^\gamma}</math>


2.  <math>f_{, \mu} = \frac{\partial f} {\partial x^\mu}</math>


3.  <math>w^\mu_{\ , \nu} = \frac{\partial w^\mu} {\partial x^\nu}</math>


4.  <math>\Gamma^\alpha_{\ \beta \gamma ,\mu} = \frac {\partial \Gamma^\alpha_{\ \beta \gamma}} {\partial x^\mu}</math>


If several indices appear after the comma, they are all taken to be part of the differentiation.  Here are some examples:


1.  <math>S_{\alpha \ ,\mu \nu}^{\ \beta} 
= \left( S_{\alpha \ ,\mu}^{\ \beta} \right)_{, \nu}
= \frac {\partial} {\partial x^\nu} \left( \frac{\partial S_\alpha^{\ \beta}} {\partial x^\mu} \right)
=\frac {\partial^2 S_\alpha^{\ \beta}} {\partial x^\nu \partial x^\mu}
</math>


2.  <math>f_{, \alpha \beta \beta}
=\left[ \left( f_{, \alpha} \right)_{, \beta} \right]_{, \beta}
= \frac{\partial^3 f} {\partial^2 x^\beta \partial x^\alpha}
</math>


Now, we change coordinate systems via the '''Jacobian''' <math>x^\mu_{\ , \nu}</math>.  The transformation rule is <math>x^{\bar \mu} = x^\mu x^{\bar \mu}_{\ , \mu}</math>.


Finally, we present the following important theorem:


''Theorem:''  <math>x^\alpha_{\ , \mu} x^\mu_{\ , \beta} = \delta^\alpha_\beta</math>


''Proof:''  <math>x^\alpha_{\ , \mu} x^\mu_{\ , \beta} = \sum_{\mu=0}^3 
\frac {\partial x^\alpha} {\partial x^\mu}
\frac {\partial x^\mu} {\partial x^\beta}</math>, which by the chain rule is <math>\frac {\partial x^\alpha} {\partial x^\beta}</math>, which is of course <math>\delta^\alpha_\beta</math>.  <math>\square</math>


So, as a matrix, the matrix inverse of <math>\left( x^\alpha_{\ , \mu} \right)</math> is <math>\left( x^\mu_{\ , \beta} \right)</math>.

{{BookCat}}