General Relativity/Comma derivative

< General Relativity

In General Relativity we write our (4-dimensional) coordinates as $$(x^0,x^1,x^2,x^3)$$. The flat Minkowski spacetime coordinates ("Local Lorentz frame") are $$x^0=ct$$, $$x^1=x$$, $$x^2=y$$, and $$x^3=z$$, where $$c$$ is the speed of light, $$t$$ is time, and $$x$$, $$y$$, and $$z$$ 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. $$T^\alpha_{\ \beta, \gamma} = \frac {\partial T^\alpha_{\ \beta}} {\partial x^\gamma}$$

2. $$f_{, \mu} = \frac{\partial f} {\partial x^\mu}$$

3. $$w^\mu_{\, \nu} = \frac{\partial w^\mu} {\partial x^\nu}$$

4. $$\Gamma^\alpha_{\ \beta \gamma ,\mu} = \frac {\partial \Gamma^\alpha_{\ \beta \gamma}} {\partial x^\mu}$$

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

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

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

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

Finally, we present the following important theorem:

Theorem: $$x^\alpha_{\, \mu} x^\mu_{\ , \beta} = \delta^\alpha_\beta$$

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

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