Variation of the metric tensor

[unchained1978]

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a related note, does the covariant derivative of the metric always vanish, regardless of the metric chosen?

 

[quasar987]

There are various notions of derivative in differential geometry. There is the covariant derivative. Given a metric, there is a unique connection "compatible with it" and by definition or characterisation, this compatibility condition is presicely the fact that the covariant derivative of the metric vanishes identically. (Note that this answers your second question: given a fixed connection D, a metric g is such that Dg=0 iff g is compatible with D.)

There is also the notion of the Lie derivative.. which differentiates a tensor with respect to some vector field. For instance, the flow of some vector field X is a 1-parameter family of isometries iff L_Xg=0. In this case, X is called a Killing vector field.

Or, if your metric depends on a real parameter t, you can just differentiate it in the usual manner wrt t and you get a new symmetric tensor.

 

[unchained1978]

I'm not talking about the covariant derivative of the metric, I simply mean how does one define this quantity: $\delta$$g^{\mu\nu}$? I've seen it used countless times in many derivations, i.e. deriving the Einstein equations from the appropriate action relies heavily on this, but I've never seen it properly defined. I understand the geometric concept, but have no way of mathematically defining it.

 

[quasar987]

Oh! Then I don't know. I've read a bit on wikipedia but don't really get it.

 

[holy_toaster]

Hi there. These $\delta$ quantities are commonly used in Physics literature to be "small" changes in calculus of variations. It is indeed not so easy to define them in a mathematically rigorous way, because they can sometimes be treated like ordinary differentials in calculus, but they are something different.

I think they appear in context of the variation of the Einstein-Hilbert action in General Relativity. The action is a functional, i.e. a map depending on the metric $g_{ab}$ and its first and second partial derivatives $g_{ab,c}$, $g_{ab,cd}$ with values in $\mathbb{R}$. Now if one calculates the functional derivative of such an object one looks how the functional changes if one changes the metric by "small" amounts. These small amounts are then called $\delta g_{ab}$.

This can be defined mathematically by using the Gateaux-derivative: http://en.wikipedia.org/wiki/G%C3%A2teaux_derivative" . In this Wikipedia article the $\delta g_{ab}$ correspond to the $\psi$ used in the definition of the Gateaux-derivative.

 

[unchained1978]

Thanks a lot, that helped very much.

 

[samalkhaiat]

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a related note, does the covariant derivative of the metric always vanish, regardless of the metric chosen?

In the calculus of variation, we subject the relevant function (or in your case the metric) to a small change and write
$$g(x) \rightarrow \bar{g}(x) = g(x) + \delta g(x)$$
Now, if this small change is brought about by infinitesimal coordinate transformation
$$x \rightarrow \bar{x} = x + f(x),$$
then one can define two type of variations :
$$\delta g(x) = \bar{g}(\bar{x}) - g(x)$$
This compares the transformed metric with original one at the same geometrical point P which has two different coordinate values $x$ and $\bar{x}$. Some people call this type of variation “local variation” other call it “total variation”.
The other, more important, variation is defined by
$$\bar{\delta}g(x) = \bar{g}(x) - g(x)$$
Here, we compare the transformed metric $\bar{g}$ at point $\bar{P}$ (with coordinate value equal to $x$) with the original metric $g$ at point $P$ with coordinate value $x$. This means that $\bar{\delta}$ refers to two different points having the same coordinate values, i.e., it is the Lie derivative along the vector field $f(x)$ which generates the coordinate transformation. Notice that the two variations are related by
$$\bar{\delta}g = \delta g - f(x)\partial_{x}g$$

regards

sam