Lorentz or Poincare invariant?

[xiaomaclever]

Generally we say GR is local Lorentz invariant. Does it mean the action or field equation?
Why not Poincare invariant? Thanks!

 

[javierR]

The action (not only in GR but in the standard model) has local Lorentz symmetry, while the field equations, with spacetime tensors for its terms, are covariant (when we write a solution to the field equations, we have to pick a reference frame to write it in, so they must not be *invariant* equations, right?). The Poincare group includes translations, but spacetime does not "look the same" at different points...just from different local Lorentz frames. So no Poincare symmetry. However, a special solution to the field equations does have Poincare invariance: Minkowski.

 

[Entropia]

The concept of Lorentz invariance refers to the invariance of physical laws under transformations of space and time, known as Lorentz transformations. These transformations preserve the speed of light and are a fundamental principle of special relativity. In general relativity, the local Lorentz invariance means that the laws of physics are the same for all observers in free-fall in a gravitational field. This is because the equations of general relativity are formulated in terms of tensors, which are invariant under coordinate transformations, including Lorentz transformations.

On the other hand, Poincare invariance refers to the invariance of physical laws under the full group of Poincare transformations, which include translations, rotations, and boosts in addition to Lorentz transformations. In general relativity, the equations of motion (field equations) are Poincare invariant, but the action principle is not. This means that while the equations of motion are the same for all observers, the action (which describes the dynamics of the system) may have different forms for different observers.

In summary, general relativity is generally said to be locally Lorentz invariant because the equations of motion are Lorentz invariant, but the action is not. This is because the action is a global concept, while the equations of motion are local. Poincare invariance is a more general concept that includes Lorentz invariance but also includes other transformations.