Invariants in general relativity

[eljose]

do invariants in general relativity exist? i mean quantity J so $$\frac{dJ}{dt}=0$$...

another question let suppose we take the Lie gorup of $$g_ab,\pi_ab$$ being g_ab and Pi_ab the metric and momentum density could we obtain the Casimir invariant of this group?...

the last question given the lagrangian of special relativity $${g^1/2}Rdx^4$$ how do you calculate the momentum $$\pi_ab$$ ?...

 

[selfAdjoint]

Any tensor that is identically zero is invariant in GR.

 

[dextercioby]

What do you mean by this operator $$\frac{d}{dt}$$ in GR...?Who's "t"...?

For the second question,how would you build the Lie group...?How do you define the Lie product...?

As for the last,things are not that easy.Hamiltonian formalism for GR needs other variables,for example working with $g_{\mu\nu}$ and $\pi_{\mu\nu}$ is not easy and usually,there are different notations of the elements of the metric.

Anyway,the definition is still the same:

$$\pi_{\mu\nu}=:\frac{\partial \mathcal{L}}{\partial^{0}g^{\mu\nu}}$$

Daniel.