# Can a theory have local Lorentz invariance but not diffeo invariance?

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This is related to the thread on the meaning of diffeomorphism invariance but is adressing a distinct point (at least I think so, but I may be proven wrong).

As Rovelli discusses in his book, the action of the Standard Model coupled to gravity has three types of invariance: under the gauge group of the SM, under local Lorentz transformations (which may be seen as a gauge group too) and under diffeomorphisms.

Presented this way, the issue of Local Lorentz invariance and of diffeomorphism invariance seem distincts. Is that really the case? Could we build a theory with Local Lorentz invariance but without diffeomorphism invariance (and vice versa)?
I thought (for some unclear reason) that the two issues were intimately linked but mathematically they are not, as far as I can tell. Let's say that Einstein had never thought of special relativity, but still had had his brillant insight that in free fall gravity is unobservable (up to tidal forces effect, as usual). What kind of theory would he have come up with in that case? He would still have been led to using tensors, right? Would this approach have necessarily forced him to special relativity or is it possible to make the whole theory diffeomorphism invariant without ever using Lorentz invariance?

It's probably a stupid question with a very obvious answer. If it is, feel free to tell me Related Beyond the Standard Model News on Phys.org
atyy
Let's say that Einstein had never thought of special relativity, but still had had his brillant insight that in free fall gravity is unobservable (up to tidal forces effect, as usual).
This is yet another principle - distinct from diff invariance of the Lagrangian ("no prior geometry") and local lorentz invariance. Some form of equivalence principle is also obeyed by Newtonian gravity (in Newton-Cartan form) and Nordstrom gravity (http://www.einstein-online.info/spotlights/equivalence_deflection [Broken] http://arxiv.org/abs/gr-qc/0405030) - both of which can be formulated geometrically - and free fall is then a geodesic in spacetime.

Incidentally, wrt Lorentz invariance, the equivalence principle, and "no prior geometry", Nordstrom gravity has the first two, but not the last - which is what distinguishes GR.

In GR, I tend to think of the equivalence principle as minimal coupling (http://www.blau.itp.unibe.ch/lecturesGR.pdf [Broken]) - also called comma to semicolon rule, or by Weinberg the "Principle of General Covariance" (not the same as Weinberg's "general covariance").

Can you have "no prior geometry" without local Lorentz invariance? Yes, by not imposing that the signature of the metric field be Lorentzian (-+++). There's a very interesting piece of speculation about a Euclidean signature (++++) and "What would “spacetime” be like if time did not exist?" in http://arxiv.org/abs/0711.1656

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This is yet another principle - distinct from diff invariance of the Lagrangian ("no prior geometry") and local lorentz invariance.
Ah yes. One of your references does say that, it had slipped my mind.

Some form of equivalence principle is also obeyed by Newtonian gravity (in Newton-Cartan form) and Nordstrom gravity (http://www.einstein-online.info/spotlights/equivalence_deflection [Broken] http://arxiv.org/abs/gr-qc/0405030) - both of which can be formulated geometrically - and free fall is then a geodesic in spacetime.

Incidentally, wrt Lorentz invariance, the equivalence principle, and "no prior geometry", Nordstrom gravity has the first two, but not the last - which is what distinguishes GR.

In GR, I tend to think of the equivalence principle as minimal coupling (http://www.blau.itp.unibe.ch/lecturesGR.pdf [Broken]) - also called comma to semicolon rule, or by Weinberg the "Principle of General Covariance" (not the same as Weinberg's "general covariance").

Can you have "no prior geometry" without local Lorentz invariance? Yes, by not imposing that the signature of the metric field be Lorentzian (-+++). There's a very interesting piece of speculation about a Euclidean signature (++++) and "What would “spacetime” be like if time did not exist?" in http://arxiv.org/abs/0711.1656

Very interesting. Thanks again for the interesting comments and references.

So there is

a) Invariance under local Lorentz transformation

b) Invariance under diffeomorphisms

c) The equivalence principle

The first and third ones are obviously rooted into physics. What about diffeomorphism invariance? Is there a physical motivation for it or is the motivation purely one of mathematical simplicity?

Usually invariance under general coordinate transformations is presented as an extension of invariance under local Lorentz transformations ( of the type "SR is invariant under Lorentz transformations, GR is designed to be invariant under not only Lorentz transformations but arbitrary coordinate transformations"). So this is conceptually misleading, right? I mean that the two considerations are totally divorced from one another, from a mathematical point of view. Is that a fair statement?

Thanks!

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atyy
The first and third ones are obviously rooted into physics. What about diffeomorphism invariance? Is there a physical motivation for it or is the motivation purely one of mathematical simplicity?
I don't think Einstein knew of any facts during his formulation of GR that would impose diffeomorphism invariance - there are theories with prior geometry that have the Schwarschild solution, which would have given him mecury's perihelion and Eddington's light bending. http://arxiv.org/abs/gr-qc/0611006

However, given that we know GR is a really good theory, I think the implication of diffeomorphism invariance is that there are no local observables - which is why nonlocality is a big theme in quantum gravity research:
http://motls.blogspot.com/2008/10/observables-in-quantum-gravity.html (as you probably know, Lubos can go on wild rants, but I think this piece of his is good.)
http://arxiv.org/abs/hep-th/0512200
http://arxiv.org/abs/hep-th/0205192
http://arxiv.org/abs/0801.0861
http://arxiv.org/abs/0905.3772
http://arxiv.org/abs/0907.2939

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atyy
Usually invariance under general coordinate transformations is presented as an extension of invariance under local Lorentz transformations ( of the type "SR is invariant under Lorentz transformations, GR is designed to be invariant under not only Lorentz transformations but arbitrary coordinate transformations"). So this is conceptually misleading, right? I mean that the two considerations are totally divorced from one another, from a mathematical point of view. Is that a fair statement?
Hmmm, in my understanding, once the metric field of GR has signature (-+++) we almost certainly get local Lorentz invariance. I know we certainly get that local coordinates in which the metric is diag(-1,1,1,1). I am not sure if that is sufficient to get local Lorentz invariance, or whether one also needs the equivalence principle (no curvature coupling).

Any physical theory is Lorentz invariant. Heck, you can write the non-relativistic heat equation in Lorentz-invariant form if you want. The problem is that you can't write such theories in diffeomorphism-invariant form; i.e., without introducing background geometrical data.

So to answer your question, Lorentz invariance is intimately related to diffeomorphism invariance. Starting with a manifestly Lorentz-invariant theory valid in flat space, through minimal substitution we obtain a diffeomorphism-invariant theory. If you hand me a theory which is not manifestly Lorentz invariant, then I can write it in a Lorentz-invariant form by introducing background geometrical data, but when the theory is coupled to gravity it will not be diffeomorphism invariant.

The phrase local lorentz invariance is very vague. Based on what I explained above, you could legitimately argue that any and every theory is locally lorentz invariant. Having not read Rovelli's book, however, I'm guessing he's actually referring to the freedom of choosing a frame (vielbein) in order to define the metric. This tetrad formalism becomes necessary when you consider spinors coupled to gravity and it certainly makes GR look like an SO(1,3) (or better yet SL(2,C)) gauge theory. As far as I am aware, there is no consensus on the correct way to do this. Recently I learned about a promosing approach to GR using the SL(2,C) group which leads to the Einstein Hilbert action plus a cosmological constant (cited in Ref.  of arXiv:hep-th/0309166v2).

Anyway, I doubt that this local SO(1,3) symmetry should really be counted as a symmetry in addition to diffeomorphism invariance. If it is, I would like to know what are the conserved Noether currents.

In summary, I would say that the only symmetries of the SM action coupled to gravity are the gauge symmetries
1)local SU(3)xSU(2)xU(1) symmetry
2)diffeomorphism invariance

I don't count local Lorentz invariance in this list because it is trivially true. On the other hand, diffeomorphism invariance in some sense follows from the fact that the matter Lagrangian in flat space is manifestly Lorentz invariant. This the essence of how GR knows about' Lorentz invariance.

I would say that the physical principles underlying GR are the equivalence principle and no background geometrical data (diffeomorphism invariance).

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atyy
Usually invariance under general coordinate transformations is presented as an extension of invariance under local Lorentz transformations ( of the type "SR is invariant under Lorentz transformations, GR is designed to be invariant under not only Lorentz transformations but arbitrary coordinate transformations"). So this is conceptually misleading, right? I mean that the two considerations are totally divorced from one another, from a mathematical point of view. Is that a fair statement?
I guess what I said in post #5 is wrong - apparently, one can have diffeomorphism invariance without Lorentz invariance! Jacobson, http://arxiv.org/abs/0801.1547

Mattingly http://relativity.livingreviews.org/Articles/lrr-2005-5/ [Broken] says in section 2.5 that one cannot break Lorentz invariance without breaking the EP. However, I don't understand this statement, since Newtonian gravity has some form of EP without Lorentz invariance.

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I guess what I said in post #5 is wrong - apparently, one can have diffeomorphism invariance without Lorentz invariance! Jacobson, http://arxiv.org/abs/0801.1547
I don't think the paper is correct. It is impossible to have `diffeomorphism invariant physics with preferred frame effects''. The physics discussed in the paper is trivially coordinate invariant as I discussed above.

i'm not a LCQ person, but diffeomorphisms are just differentiable bijective maps, which in physics terms means just coordinate transformations.

lorentz invariance means that the transition function taking you from one coordinate patch to another contains the lorentz group.

diffeomorphisms are a far more general idea. lorentz invariance, just tells us what the group of transformations glueing different patches together is. in this sense, of course you can have diffeomorphisms without having lorentz invariance. however if you require everything to obey relativity, then your transition functions should be lorentz invariant.