# Local Lorentz invariance or local Poincaré invariance?

Fast question.

I know that the group of all the isometries in Minkowski's space-time is the Poincaré group that is just Lorentz's group (rotations and boosts) and translations in space-time.

Now, in general relativity texts I often read local Lorentz invariance. I figure that in GR since we have that locally SR is fulfilled what is really meant here is local Poincaré invariance.

So, is this terminology abuse or is it really local Lorentz? And if it is Local Lorentz, is it the full Lorentz group, just the proper orthocronous or any other thing?

There is a risk of confusion here because in some sense Minkowski space is too nice. What I mean by this is that in the setting of Minkowski space, because of its simple structure, there are identifications and globalizations possible that do not make sense in a general spacetime.

In a general spacetime at every point you can define the tangent vector space. It roughly has one direction for every direction you can move in. This does not mean that the space is an affine space (an affine space is like a vector space without a preferred origin), it could be a sphere for example. But for Minkowski space, the spacetime is indeed an affine space and this leads you confusion of the whole space with the tangent space.

Let us talk about relativity. An observer in spacetime can find three spacelike curves and one timelike curve through his or her time and position. The principle of Lorentz invariance is that any choice is fine! For the space part this is just that you can rotate your laboratory and get the same results. That you are allowed to mix time and space comes from that the speed of light should be the same for observers in relative motion.

So really, what local Lorentz invariance means is that you can rotate your laboratory without changed results, and observers moving relative to it see the same physics. This is an expression of symmetry in the tangent space.

Now in Minkowski spacetime pick an arbitrary origin. Then Minkowski spacetime has the same structure as the (1+3) tangent spaces of general relativity, so the local Lorentz invariance can be made global. Since the origin was arbitrary you have also four translation symmetries. This is the Poincare symmetry.

Local Lorentz invariance is a statement about how your local choice of time and space axis is unimportant. Global Lorentz and Poincare invariance is a much stronger statement about the symmetries of spacetime itself. In particular, a spacetime need not have any symmetries at all (and there are many known examples of solutions to Einstein's equations that don't).

• so, if i understood right, you say that it is just Lorentz (with parity and everything) but not Poincaré, right? – Yossarian Mar 31 '14 at 14:05
• Not necessarily with parity and time inversion. The standard model has violations of both (But CPT is always a symmetry.) – Robin Ekman Mar 31 '14 at 15:15
• I see that Poincaré trans are not ok because they involve spacetime translations. BUT if we considered a DIFFERENTIAL Poincaré trans, wouldn't the metric remain also invariant (at least to first order)? so could we say in GR we have local infinitesimal Poincaré symetry? – Yossarian Apr 1 '14 at 21:41
• An infinitesimal transformation that preserves the metric is called a Killing vector, and if you integrate that, you get a symmetry of the spacetime. The symmetry does not have to be a translation, it can also be a rotation. For example the Schwarschild metric has a translation symmetry in time and three rotational symmetries in space. The Big Bang models have three of each. In fact, if you have three space-translations and a time-translation and they commute, the spacetime has to be flat, because for each commuting symmetry, you can eliminate one coordinate from the metric . – Robin Ekman Apr 1 '14 at 22:19
• you are absolutely right – Yossarian Apr 2 '14 at 1:13

Local Lorentz invariance is usually invoked when talking about some aspects of the equivalence principle: that different observers correspond to different inertial reference frames. Inertial frames are related to one another by local Lorentz transformations.

In addition, the metric is invariant under a local Lorentz transformation. In formulations of gravity using tetrads, the rotation coefficients obey a gauge transformation under local Lorentz transformations, but the Riemann tensor is still Lorentz covariant (hence why it's called a gauge transformation, after all).

These notions don't really make sense for the full Poincare group, as translations don't relate to the notion of observers in different reference frames. So I don't think wikipedia is correct on this point.

wikipedia says that local Lorentz invariance can be generalized to the Poincare group. From what I know of gravity using tetrads, I think this corresponds to the transformation law of the tetrad: that under a translation (or a general coordinate transformation), the tetrad must transform in a certain manner so that the inner prouducts of vectors and covectors remain invariant. This too can be considered a gauge transformation. Tetrads transform under local Lorentz transformations as well, so this aspect of coordinate transformation in addition to LLTs may correspond to full "local Poincare covariance," but I would investigate further to be sure.

• my knowledge on tetrads is quite limited so I don't really follow your last paragraph, but thanks for the answer. I'll try to check what you say – Yossarian Mar 31 '14 at 14:08

All the other answers are very thorough, I just wanted to add some physical intuition:

When restricting to "local" you can think about it as studying the physics at a single point in spacetime. We assume different observers are all at that point and this means they can only be boosted with respect to each other. Hence the importance of local Lorentz invariance. To discuss invariance under translations you need two different points and how this works is described in the other answers...