Is General Theory of Relativity a Gauge Theory? Trajectories of Diffeomorphism.


Historically the problem of observables in classical and quantum gravity is closely related to the so-called Einstein hole problem, i.e. to some of the consequences of general covariance in general relativity (GTR).

The central question is the physical meaning of the points of the event manifold underlying GTR. In contrast to pure mathematics this is a non-trivial point in physics. While in pure differential geometry one simply decrees the existence of, for example, a (pseudo-) Riemannian manifold with a differentiable structure (i.e., an appropriate cover with coordinate patches) plus a (pseudo-) Riemannian metric, g, the relation to physics is not simply one-one. In popular textbooks about GTR, it is frequently stated that all diffeomorphic (space-time) manifolds, M are physically indistinguishable. Put differently:

S − T = Riem/Diff —– (1)

This becomes particularly virulent in the Einstein hole problem. i.e., assuming that we have a region of space-time, free of matter, we can apply a local diffeomorphism which only acts within this hole, letting the exterior invariant. We get thus in general two different metric tensors

g(x) , g′(x) := Φ ◦ g(x) —– (2)

in the hole while certain inital conditions lying outside of the hole are unchanged, thus yielding two different solutions of the Einstein field equations.

Many physicists consider this to be a violation of determinism (which it is not!) and hence argue that the class of observable quantities have to be drastically reduced in (quantum) gravity theory. They follow the line of reasoning developed by Dirac in the context of gauge theory, thus implying that GTR is essentially also a gauge theory. This then winds up to the conclusion:

Dirac observables in quantum gravity are quantities which are diffeomorphism invariant with the diffeomorphism group, Diff acting from M to M, i.e.

Φ : M → M —– (3)

One should note that with respect to physical observations there is no violation of determinism. An observer can never really observe two different metric fields on one and the same space-time manifold. This can only happen on the mathematical paper. He will use a fixed measurement protocol, using rods and clocks in e.g. a local inertial frame where special relativity locally applies and then extend the results to general coordinate frames.

We get a certain orbit under Diff if we start from a particular manifold M with a metric tensor g and take the orbit

{M, Φ ◦g} —– (4)

In general we have additional fields and matter distributions on M which are transformd accordingly.

Note that not even scalars are invariant in general in the above sense, i.e., not even the Ricci scalar is observable in the Dirac sense:

R(x) ≠ Φ ◦ R(x) —– (5)

in the generic case. Thus, this would imply that the class of admissible observables can be pretty small (even empty!). Furthermore, it follows that points of M are not a priori distinguishable. On the other hand, many consider the Ricci scalar at a point to be an observable quantity.

This winds up to the question whether GTR is a true gauge theory or perhaps only apparently so at a first glance, while on a more fundamental level it is something different. In the words of Kuchar (What is observable..),

Quantities non-invariant under the full diffeomorphism group are observable in gravity.

The reason for these apparently diverging opinions stems from the role reference systems are assumed to play in GTR with some arguing that the gauge property of general coordinate invariance is only of a formal nature.

In the hole argument it is for example argued that it is important to add some particle trajectories which cross each other, thus generating concrete events on M. As these point events transform accordingly under a diffeomorphism, the distance between the corresponding coordinates x, y equals the distance between the transformed points Φ(x), Φ(y), thus being a Dirac observable. On the other hand, the coordinates x or y are not observable.

One should note that this observation is somewhat tautological in the realm of Riemannian geometry as the metric is an absolute quantity, put differently (and somewhat sloppily), ds2 is invariant under passive and by the same token active coordinate transformation (diffeomorphisms) because, while conceptually different, the transformation properties under the latter operations are defined as in the passive case. In the case of GTR this absolute quantity enters via the equivalence principle i.e., distances are measured for example in a local inertial frame (LIF) where special relativity holds and are then generalized to arbitrary coordinate systems.

Agamben’s (Anti-)Utopia? Walter Benjamin Clues. Drunken Risibility.


The thought of Giorgio Agamben has been often accused of being utopian. Antonio Negri, for example, branded Agamben’s core concept, “naked” or “bare life,” as a “utopian escape” and then identified in (State of Exception) a “feverish utopian anxiety.” Agamben has also been accused that his notion of politics of dissolves “into an eschatological, utopian vision of social life,” infused with strong theological and messianic overtones, which would make of it a particular version of political theology. But, what of the quite common unease for a political project that is deemed unrealizable, empty, even impolitic. Is it Utopia or Anti-utopia? The clue is provided for through Walter Benjamin.

At the end of his essay on Surrealism (Surrealism The Last Snapshot of the European Intelligentsia), Walter Benjamin writes:

For what is the program of the bourgeois parties? A bad poem on springtime, filled to bursting with metaphors. The socialist sees that ‘finer future of our children and grandchildren’ in a society in which all act ‘as if they were angels’ and everyone has as much ‘as if he were rich’ and everyone lives ‘as if he were free.’ Of angels, wealth, freedom, not a trace – these are mere images.

In a piece written two years later, Der destruktive Charakter, he insists that a radical, revolutionary politics must renounce optimistic, metaphoric contemplation:

The destructive character sees no image hovering before him.

Benjamin does not employ the term utopia; it is clear nonetheless that a political project founded on mere – and optimistic – images of the future is the target of his harsh criticism. If political utopianism in fact originated – strongly influenced by world travels and discoveries of new lands – by situating a political alternative in a spatial displacement (a nonplace that is, however, another place), at least from the Enlightenment it assumed the character of a “better future” toward which a progressive politics should strive. If, following Benjamin, we define “utopia” as a political project construed around images of the future and rhetorically based on the syntagma “as if ” (als ob), then Agamben’s project exudes an intrinsic and intense anti-utopianism. It is true that there is no explicit attack on utopia in his work and even that, in the preface to (Stanzas: Word and Phantasm in Western Culture), he defines the “philosophical topology” presented as his method as “constantly oriented in the light of utopia”; however, the messianism in which his philosophical project is steeped constitutes an implicit but evident rejection of utopianism.

Sellarsian Intentionality. Thought of the Day 59.0


Sellars developed a theory of intentionality that seems calculated to so construe intentional phenomena as to make them compatible with developments in the sciences.

Now if thoughts are items which are conceived in terms of the roles they play, then there is no barrier in principle to the identification of conceptual thinking with neurophysiological process. There would be no “qualitative” remainder to be accounted for. The identification, curiously enough, would be even more straightforward than the identification of the physical things in the manifest image with complex systems of physical particles. And in this key, if not decisive, respect, the respect in which both images are concerned with conceptual thinking (which is the distinctive trait of man), the manifest and scientific images could merge without clash in the synoptic view. (Philosophy and the Scientific Image of Man).

The first thing to notice is that Sellars maintains that intentionality is irreducible in the sense that we cannot define in any of the vocabularies of the natural sciences concepts equivalent to the concepts of intentionality. The language of intentionality is introduced as an autonomous explanatory vocabulary tied, of course, to the vocabulary of empirical behavior, but not reducible to that language. The autonomy of mentalistic discourse surely commits us to a new ideology, a new set of basic predicates, above and beyond what can be constructed in the vocabularies of the natural sciences. What we get from the sciences can be the whole truth about the world, including intentional phenomena, then, only if there is some way to construct, using proper scientific methodology, concepts in the scientific image that are legitimate successors to the concepts of intentionality present in the manifest image. That there is such a rigorous construction of successors to the concepts of intentionality is, a clear commitment on Sellars’s part. The only real alternative is some form of eliminativism, an alternative that some of his students adopted and some of his critics thought Sellars was committed to, but which never held any real attraction for Sellars.

The second thing to notice is that the concepts of intentionality, especially the concepts of agency, differ in some significant ways from the normal concepts of the natural sciences. Sellars puts it this way:

To say that a certain person desired to do A, thought it his duty to do B but was forced to do C, is not to describe him as one might describe a scientific specimen. One does, indeed, describe him, but one does something more. And it is this something more which is the irreducible core of the framework of persons.

Here the focus is explicitly on the language of agency, but the point is fundamentally the same as in Sellars’s well-known dictum from Empiricism and Philosophy of Mind:

in characterizing an episode or a state as that of knowing, we are not giving an empirical description of that episode or state; we are placing it in the logical space of reasons, of justifying and being able to justify what one says.

In both epistemic and agential language something extra-descriptive is going on. In order to accommodate this important aspect of such phenomena, Sellars tells us, we must add to the purely descriptive/explanatory vocabulary of the sciences “the language of individual and community intentions”. He points to intentions here because the point is that epistemic and agential language – mentalistic language in general – is ineluctably normative; it always contains a prescriptive, action-oriented dimension and engages in direct or indirect assessment against normative standards. In Sellars’s own theory, norms are grounded in the structure of intentions, particularly community intentions, so any truly complete image must contain the language of intentions.