# Is general relativity resulted from diffeomorphism invariance?

Any action expressed as the integral of a 4-form in 4-dimensional spacetime is diffeomorphism invariant. For example the following 4-form topological (Pontryagin) action $$S = \int F\wedge F$$ is diffeomorphism invariant, where $F$ is the electromagnetic field strength 2-form. This action has nothing to do general relativity or gravity.

General relativity is usually alleged as the result of diffeomorphism invariance. But should we say that general relativity is resulted from local Lorentz gauge invariance in stead of diffeomorphism invariance?

Diffeomorphism invariance is rather a general condition satisfied by all actions pertaining spinors/fermions, Standard Model gauge fields, gravity spin connection (local Lorentz group) gauge field $\omega$, and Lorentz-covariant tetrad/frame field $e$.

For example, the massless Dirac spinor fermion action is a 4-form (thus diffeomorphism invariant) of exterior product of 3 frame field $e$ 1-forms and the co-derivative $(d+\omega)$ 1-form (for abbreviation, Lorentz indices are not shown here and only the Lorentz gauge interaction $\omega$ is included), $$S_{fermion} \sim \int{i\bar{\psi}e\wedge e \wedge e \wedge (d+\omega)\psi}.$$

The local Lorentz gauge action for gravity is a 4-form (thus diffeomorphism invariant) of exterior product of 2 frame field $e$ 1-forms and the Lorentz curvature 2-form $F_{\text{Lorentz}} = d\omega + \omega\wedge\omega$, $$S_{gravity} \sim \int e \wedge e \wedge F_{\text{Lorentz}}.$$

• Related/possible duplicate: physics.stackexchange.com/q/12461/50583, physics.stackexchange.com/q/346793/50583 and their linked questions. Also, this is the fourth question of yours in a few minutes where you have made a rather minor edit to the title. Please do not make minor edits to your posts simply to bump them on the active page. Jun 29, 2018 at 20:49
• I could be wrong but I'm pretty sure that the above actions are only invariant under diffeomorphisms if you assume your diffeomorphism is orthonormal (ie. orthogonal unitary), anything else will change your action I believe. This includes the Einstein Hilbert action as well. Nov 23, 2018 at 6:25
• Vladimir Fock wrote in his book that A. Einstein discovered in fact laws of gravity and that a "general relativity" is empty of sense since, cast in a tensor form, any equation is "covariant" under variable changes. Jan 27, 2019 at 17:18

General Relativity can be formulated as a locally Lorentz invariant theory, indeed. See Weinberg's book, for example.

The diffeomorphism invariance is a way of stating that the local laws are keeping their form everywhere on the manifold (kind of translational invariance). About all theories could be formulated in a way to be diffeomorphic invariant since their physical laws retain their form everywhere on their manifold : a law coulnd't be called a "law" if it isn't the same relation between local observables for all locations in spacetime! This is why diffeomorphism invariance is almost a triviality.

Take any manifold, especially a non-homogeneous one. An observer shouldn't be able to learn his local position on the manifold by making local measurements of local observables. A local law constraining these observables have the same form/shape/relation as for any other location on the manifold (or it's not a "law"!).

What is not clear about this interpretation of the diffeomorphism invariance, is its implicit and very subtle relation with Einstein's equivalence principle (EEP). I think there is some mixing with EEP and the idea that laws should stay invariant under local active translations (i.e. a "law" is the same relation everywhere). This implicit mixing isn't very surprising since EEP is also a "law" (or a "meta-law") that should be valid everywhere on the manifold.

• I think the content of EEP is in identifying the deviation in the form of the laws under a general diffeomorphism from its form within Poincare transformations with gravity. Would you agree?
– ACat
Jun 24, 2018 at 17:24

Diffeomorphism invariance is a nearly vacous statement in and by itself.

The thing that distinguishes GR from a field theory formulated in a fixed background is the lack of background structures.

Consider for example a field theory in Minkowski spacetime $$(M,\eta)$$, where $$\eta$$ is the flat metric. In this case there is a background structure, namely the flat metric. One can consider a diffeomorphism $$\phi:M\rightarrow M$$, and assuming no fields with gauge degrees of freedom are present (there is no natural representation of diffeomorphisms on fields with gauge degrees of freedom$$^\ast$$), if $$\psi$$ is any field that is a cross-section of a natural bundle (with natural bundle functor $$F$$), the diffeomorphism acts on $$\psi$$ by $$\psi^\prime=F(\phi)\circ\psi\circ\phi^{-1}.$$

In this case, if we also transform the metric tensor by $$\eta^\prime=\phi^{-1\ast}\eta\equiv (T^{\ast}\phi^{-1}\otimes T^{\ast}\phi^{-1})\circ\eta\circ\phi^{-1}$$ (along with all the fields), all equations stay valid. Thus, any field theory formulated on a manifold with all fields taking values in natural bundles can be formulated in a diffeomorphism-invariant manner.

However the presence of the background structure $$\eta$$ selects a "smaller" symmetry group, namely those for which $$\phi^\ast\eta=\eta$$ (which in the case of Minkowski space with the Minkowski metric is the Poincaré group), eg. they preserve the background structure.

What is usually conflated with diffeomorphism-invariance and what is special in GR is background independence, namely that there are no background structures in the theory (at least not on the level of subcategories of the smooth category - one can of course argue about whether the smooth structure or topology etc is a background structure), so there is really no analogue of what the Poincaré group is to a special relativistic field theory in GR.

There is a good discussion of this in Straumann: General Relativity, page 98 onwards.

$$^*$$ The case of fields with gauge degrees of freedom is more complicated because there is no natural way to lift diffeomorphisms into fibre bundle automorphisms for non-natural fibre bundles. It is likely that the "covariance group" of that theory should consist of fibre bundle automorphisms instead of diffeomorphisms of the base manifold, which also contain all small diffeomorphisms (the ones that are connected to the identity transformation) as projections, although there might be some "large" diffeomorphisms that do not admit lifts into automorphisms, depending on the topology of the fibre bundle in question. See Fatibene, Francaviglia: Natural and gauge natural formalism for classical field theories for more details.