Difference between general covariance and Poincaré invariance

Question

As far as i understand, Poincaré invariance is invariance under spacetime rotations, translations and boost which is encoded in the group $ \operatorname{ISO}^+(n) $ which in 3+1D spacetime would be $\operatorname{ISO}^+(1, 3)$. This is the most general extension of Lorentz invariance and the one we use in particle physics and special relativity. However there's also the concept of general covariance which is the base idea from which we construct general relativity. It is the idea of invariance under coordinate transfromations and that physics doesn't depend on our choice of coordinates, which is encapsulated in the $ \operatorname{GL}(n) $ group which is the group of general coordinate transformations.

My question is: what are the differences and/or relations between these two concepts? Isn't it possible to reach all coordinate systems via Poincaré transformations? Because if that were the case, they should be the same right?

Answer 1

When it comes to general covariance, it covers not only linear(-affine) transformations which make up the group $GL(n)$, but also non-linear transformations. A very basic example of it for instance is the choice of spherical coordinates:

$$t = t'$$ $$x = r \sin\theta \cos\phi$$ $$y = r \sin\theta \sin\phi$$ $$z = r \cos \theta$$

Of course the invariant line element no longer looks like a Minkowski-metric, but this is the essence of general covariance, that it deals with any kind of metric.

$$ds^2 = dt'^2 -dr^2 -r^2(d\theta^2 + \sin^2(\theta) d\phi^2)$$

(Of course linear coordinate transformations which are not Poincare' can also already change the metric --- for instance $u=t+x$ and $v=t-x$). The given example of course does not change the property of the given space being flat. Actually a flat space cannot be become curved by a general coordinate transformation.