# Is local Lorentz + diffeomorphism invariance equivalent to full local Poincaré invariance?

Consider classical General Relativity without the torsion field (the affine connection is already assumed to be symmetric from the start). It is well known that this theory is independent of the coordinates used. The physics stay the same when we perform a passive coordinates change. This is known as general covariance, and any theory could be formulated in this way. It is not a fundamental characteristic of GR.

It is also known (but there's lot of confusion out there about this) that GR is also invariant under active coordinates transformations (also known as diffeomorphisms), which could be interpreted as a kind of gauge transformations, and not just as simple changes of local coordinates. This property describes physics: an active local coordinates transformation "pushes" the spacetime point to another place of the same manifold. Invariance of the theory under such gauge transformations is telling that the physics is the same at any place on the manifold. Any spacetime point is equivalent, to describe the physics, and the theory is background independent. I believe that this is a subtle formulation of translation invariance, in GR.

Please, don't confuse the general property above with some symmetry of the metric (i.e. isometry). I'm considering general spacetimes, not particular solutions with some special symmetry (isotropy, homogeneity, etc).

GR is also incorporating local Lorentz invariance (at any spacetime point), which says that the physics is independent of the local frame used by the observer (accelerating, rotating axes, free fall, ...).

Usually, the full Poincaré group (Lorentz + translations) isn't made local in standard GR: only the homogeneous Lorentz part is made local. I'm wondering if the "missing" translation part is actually subtly included through the diffeomorphism invariance.

So the question is this: Is the local Lorentz invariance + diffeomorphism invariance (active coordinates changes) of classical GR equivalent to the full local Poincaré invariance?

I'm expecting that some will say "NO" to the question above, because full Local Poincaré invariance is supposed to bring torsion into GR (I never saw any convincing proof of this). Torsion (i.e. the antisymmetric part of the affine connection) is usually assumed to vanish trivially from the start in GR, but there is no contradiction in letting it enter the classical GR formulation. I don't see why we need to explicitly "add" the local translation invariance to get torsion. It can already be there in classical GR, and I suspect this is because of the diffeomorphism invariance -- interpreted as a formulation of the local translation invariance.

I never saw that interpretation before, so I need opinions on it. Maybe I'm getting it all wrong!

If the answer is really a big "NO", then how do we describe local translation invariance to explicitly bring torsion into the theory? As far as I know, torsion could be added directly to classical GR, without the need to talk explicitly about local translation invariance.

• Qmechanic's answer from physics.stackexchange.com/q/175297 appears to agree with my interpretation. Am I wrong? – Cham Apr 16 '18 at 0:52
• This paper appears to support my interpretation : arxiv.org/abs/1705.00297. I would like to get access to a PDF version of this paper from Kibble : T. W. B. Kibble. Lorentz invariance and the gravitational field. J. Math. Phys. 2, 212–221 (1961) – Cham Apr 17 '18 at 0:20
• I started a new question related to the question above : physics.stackexchange.com/q/401696. – Cham Apr 23 '18 at 23:27
• See pages 5 and 6 of this paper : arxiv.org/abs/gr-qc/0603087. It appears to show that the diffeomorphisms group of GR is really a "symetry" group which imposes a strong contraint on the acceptable "physical laws". – Cham Apr 28 '18 at 15:09

I cannot give you a full answer, that honor has to pass on to someone else, but nontheless I can give you some info you might value.

It is also known (but there's lot of confusion out there about this) that GR is also invariant under active coordinates transformations (also known as diffeomorphisms), which could be interpreted as a kind of gauge transformations, and not just as simple changes of local coordinates.

I honestly think this is wrong. The point is, if you choose the local coordinate formalism, active diffeos cannot be told apart from passive diffeos. These are all maps of the form $$y^\mu=\Phi^\mu(x^1,...,x^n).$$ If your theory is invariant under one, it is invariant under both.

Background independence is a statement that is independent of diffeomorphism invariance. Or at least it is independent to the degree that diffeomorphism-invariance is necessary for background independence, but diffeomorhpism invariance, active or passive, does not imply background independence.

Background independence, at least as long as we consider only local behaviour and not global topological aspects is a consequence of the fact that the metric is dynamical and subject to the EFE $G_{\mu\nu}=8\pi G T_{\mu\nu}$.

Consider and arbitrary spacetime $(M,g)$ where $g$ is a flat metric in the sense that $\text{Riem}[g]=0$. Any theory you build on this spacetime will not be background-independent, because the flatness condition at least locally restricts spacetime geometry to be Minkowskian. However if $S=0$ is a tensor equation, and $\phi:M\rightarrow M$ is a diffeomorphism, then $$\phi^\ast S=0$$ is also satisfied, hence, it is diffeomorphism invariant.

You can read a very good discussion of the relationship between general covariance, general invariance (this terminology is nonstandard I think) and background independence in Straumann's General Relativity.

because full Local Poincaré invariance is supposed to bring torsion into GR (I never saw any convincing proof of this)

A convincing proof, which I will only sketch/explain here but not actually do can be found in Kobayashi & Nomizu (Foundation of Differential Geometry vol 1).

The point is that local transformations act on the individual tangent spaces $T_pM$ for all $p\in M$. In a tangent space, you can easily interpret a homogenous/linear transformation, but what does shifts of the form $v^\mu +a^\mu$ mean? After all $T_pM$ is usually not interpreted as a space of points.

However one can construct a fiber bundle called an affine bundle, which is essentially a fiber bundle whose local trivializations are of the form $U\times A$ where $U\in \tau_M$ is an open subset of $M$ and $A$ is an affine space. A connection on an affine bundle as called an affine connection. There is a subtle relationship between affine connections and linear connections (connections on vector bundles) and in the vast majority of the cases, an affine connection induces a linear connection in an associated vector bundle (which can basically be obtained from an affine bundle by choosing a zero section), so affine connections are somewhat more general, but for pretty much all relevant cases, they are essentially the same (hence why usually less precise sources use the two terms synonymously).

One may also construct a bundle of affine frames as a principal bundle associated to an affine bundle. If the most general vector bundle of rank $k$ admits an associated principal bundle whose structure group if $\text{GL}(k,\mathbb R)$ then the most general associated principal bundle to an affine bundle has $\text{GL}(k,\mathbb R) \rtimes \mathbb R^k$ as its structure group. The Lie algebra of this group is then isomorphic to $\mathfrak{gl}(k,\mathbb R)\oplus \mathbb R^k$, so any connection on this principal bundle can be written as a pair $$(\theta^a,\omega^a_{\ b}),$$ where the first is an $\mathbb R^k$-valued 1-form and the second is a $\mathfrak{gl}(k,\mathbb R)$-valued 1-form. The latter can be identified as the usual connection form of a $\text{GL}(k,\mathbb R)$-connection, while the former can be identified with the tautological/soldering 1-form on the usual frame bundle. It is known that the torsion of a usual linear connection is given by $$T=d_\omega \theta,$$ and the curvature is given by $$\Omega=d_\omega\omega.$$ However from the point of view of the affine bundle, both $\theta$ and $\omega$ are connection forms, so one can essentially see that the torsion is the "translational" part of the curvature and $\Omega$ is the "linear" part of the curvature (rotational in case of metric compatibility).

However note that once again, local translation invariance (represented by the affine bundle) is not sufficient for nonvanishing torsion, it just allows it. And since as I have said, the majority of the time there are strong isomorphisms between linear and affine connections, this whole formalism is not necessary, it just gives an interpretation of torsion in terms of the translational part of a connection. You can, however simply incorporate torsion into your theory via more pedestrian means fine.

• I think that H. Ohanian describes very well the fundamental difference between passive coordinates transformations and active ones. According to him, the active transformations can be told apart from the passive transformations. See his book Gravitation and Spacetime, chapter 7 (Geometrodynamics), especially section 7.1 on pages 251-259. – Cham Apr 16 '18 at 10:58
• @Someone I have just checked the chapter in the book you mentioned. There is absolutely nothing there about passive Vs active diffeos. There is a discussion about fixed and dynamical objects, though, which is basically a baby version of what I referred to in Straumann. – Bence Racskó Apr 16 '18 at 11:44
• @Someone I see. So if you are calculating the dynamics of particles in a background EM field $A_\mu$, then you cannot subject $A_\mu$ to the transformation $A_\mu\mapsto A_\mu+\partial_\mu\chi$? After all it is a fixed object then :) . You might wanna update some textbooks with that. No, seriously, go read Straumann. None of the reputable, mathematically precise GR books (Wald, Hawking/Ellis, Straumann, Penrose) ever deal with this active vs passive bs. What the author of your book says is correct, but it is stated in a confusing and imprecise manner and you are misinterpreting it. – Bence Racskó Apr 16 '18 at 17:44
• I don't think that I'm misinterpreting. As a side note : It is well known that the invariance of the matter action alone under active coordinates transformations (i.e diffeomorphisms) implies local conservation of the energy-momentum : $\nabla_{\mu} \, T^{\mu \nu} = 0$. This is shown in Weinberg's book, for example. Yet, energy-momentum is the conserved Noether charge under translations in Minkowski spacetime, so this is a support to the idea that diffeomorphisms are a subtle representation of translations in GR. – Cham Apr 16 '18 at 17:56
• The gauge transformations we're talking about in GR aren't exactly of the same type as Yang-Mills-like gauge transformations. In GR, the "gauge" transformations are to be taken in a more general sense, not acting on some "internal" space. But still, it's a kind of "gauge" anyway since it's related to the observable variables (i.e. dynamical fields). – Cham Apr 16 '18 at 18:22