There is a deep difference between Lorentz invariance and Poincaré invariance in GR. The former is exact, whereas the latter is only approximated. Mathematically speaking, the reason is that Lorentz invariance can be completely described referring to the tangent space only which is unaffected from curvatures. Conversely Poincarè invariance concerns movements of points in the manifold. Alternatively it can be discussed in the tangent space, where it is exact, but the tangent space is just an approximate model of the underlying manifold (using the exponential map for instance).
Local Poincaré invariance pratically means that in a neighbourhood of a timelike geodesics $\gamma$ you can build up a coordinate system (a so-called geodesically normal coordinate system, whose existence is guarnteed form differential geometry) where the metric takes the Minkowskian expression and the first derivative of the metric vanish, both exactly on $\gamma$ (consequently the connection coefficients $\Gamma_{\mu\nu}^\alpha(\gamma)$ also vanish on $\gamma$). So, translations along the four coordinates, starting from $\gamma$, are approximated symmetries: the components of the metrics do not change, keeping the Minkowskian form, at the first order of Taylor expansion around $\gamma$. In this way, around $\gamma$ there is an approximeted Poincaré symmtery in addition to the exact Lorentz symmetry already present separately at each point (each tangent space) of the manifold.
Physically speaking this geometric result also implies a, quite precise, mathematical formulation of Einstein's equivalence principle.
To see this fact consider a second timelike geodesic $\alpha$ crossing $\gamma$ exactly at the event $p$. This second geodesic describes the story of a point of matter freely falling with the observer $\gamma$, with some non vanishing initial velocity with respect to the observer.
Let us describe $\alpha$ in the said normal coordinates system around $\gamma$, as $x^\mu= x^\mu(\lambda)$, assuming to fix the origin of both the normal coordinates and of the affine parameter $\lambda$ just at the crossing point $p$. One has the Taylor expansion which makes sense as soon as $\lambda \to 0$, namely when the point of matter is close to the observer:
$$x^\mu(t) = 0 + \lambda \frac{dx^\mu}{d\lambda}|_{\lambda=0} + \frac{\lambda^2}{2!}\frac{d^2x^\mu}{d\lambda^2}|_{\lambda=0}+ O^\alpha(\lambda^3)\:.\quad (1)$$
However, since $\alpha$ satisfies the geodesic equation:
$$\frac{d^2x^\mu}{d\lambda^2} = - \Gamma^\mu_{\nu\sigma}\frac{dx^\nu}{d\lambda}\frac{dx^\sigma}{d\lambda}$$
and since $\Gamma^\mu_{\nu\sigma}(\rho)=0$ as said above, we also have:
$$\frac{d^2x^\mu}{d\lambda^2}|_{\lambda=0} = - \Gamma^\mu_{\nu\sigma}(\rho)\frac{dx^\nu}{d\lambda}|_{\lambda=0}\frac{dx^\sigma}{d\lambda}|_{\lambda=0} = 0\:.\qquad (2)$$
We conclude that, in normal coordinates around the free falling observer $\gamma$, the motion of the free falling point of matter $\alpha$ is described by (1), taking (2) into account:
$$x^\mu(\lambda) = \lambda \frac{dx^\mu}{d\lambda}|_{\lambda=0} + O^\alpha(\lambda^3)\:. \qquad (3)$$
Since both geodesics are timelike, it must hold $\frac{dx^0}{d\lambda}|_{\lambda=0}\neq 0$. Thus we can use the time coordinate of the normal coordinate system, $t=x^0$, as a new, more natural parameter along $\alpha$. If $v^i$, $i=1,2,3$, denote the components of the spatial $3$-velocity of the geodesic $\alpha$ at $t=0$ (when it crosses $\gamma$), computed with respect to $\gamma$ its-self
$$v^i = \frac{\frac{dx^i}{d\lambda}|_{\lambda=0}}{\frac{dx^0}{d\lambda}|_{\lambda=0}}\,$$
(3) can be re-arranged to, for $i=1,2,3$:
$$x^i(t) = t v^i + O^i(t^3)\:. \qquad (3)$$
The fact that no terms of order $t^2$ show up in the right-hand side is the mathematical expression of Einstein's equivalence principle: Around a free falling observer, the gravitational acceleration disappears exactly at the origin of coordinates and, for short intervals of time and small regions around it, the gravitational field is suppressed and free falling bodies are seen to be in inertial motion.
The existence of this natural approximated Poincaré simmetry around each timelike geodesic, permits to export some physical laws from special relativity to general relativity. It can be done for laws which (1) are local and (2) do not involve spacetime derivatives of order $>1$. For instance $\partial_\mu F^{\mu\nu} = J^\nu$ is an equation valid in special relativity for the electromagnetic field. Exactly on $\gamma$, since the connection coefficients vanish, the expression of first derivative of tensors in coordinate and the expression in coordinate of the first covariant derivative coincide. So, assuming that, for free falling observers, physics is (locally) described by the same laws as in special relativity, at least for law involving first derivatives at most, we can conclude that $\partial_\mu F^{\mu\nu} = J^\nu$ holds even in general relativity, in normal coordinates around $\gamma$. Passing to generic coordinates, the same law can be written:
$$\nabla_\mu F^{\mu\nu} = J^\nu$$
where we have introduced the covariant derivative $\nabla$.
Since we are free to suppose that every event of the spacetime can be crossed by a free falling observer, we can eventually conclude that $\nabla_\mu F^{\mu\nu} = J^\nu$ is valid everywhere in general relativity.
This is another stronger mathematical version of the equivalence principle, again relying upon the appearance of this approximated Poincaré symmetry due to the existence of normal coordinates.
Regarding the choice between full Lorentz group/ orthochronous one, the point is that, mathematically you can describe and use both. Physics decides which is the appropriate symmetry for a given physical system.
Q&A (4871)
