Contents

Idea

When the theory of gravity in the form of general relativity was developed at the beginning of the 20th century, the abstract notion of a smooth manifold independently of its coordinate charts was still unfamiliar. Theories like that of electromagnetism were traditionally all formulated with respect to what today would be called a choice of coordinate patch. The development of general relativity by Albert Einstein went through a phase in which what is now called the principle of general covariance was only gradually understood.

The “hole argument” or “hole paradox”, which was put forward in (Einstein-Grossmann), is the observation that given a pseudo-Riemannian metric $g$ on some spacetime $X$ – hence a field configuration of gravity, and given a point $x : * \to X$ and a diffeomorphism $\phi : X \stackrel{\simeq}{\to} X$, in general the value of $g$ around $x$ is entirely unrelated to the value of the pullback? metric $\phi^* g$ at $x$.

This was regarded as a source of puzzlement for a while – a “paradox” – because if one thinks that the point $x$ in $X$ is absolutely fixed (the locus of an “observer” in spacetime) and that two metrics related by a diffeomorphism as above are equivalent field configurations, it would seem that the field of gravity “at $x$” is undetermined and undeterminable. For a while this reasoning was regarded as an obstacle to the idea of general covariance and indeed in (Einstein-Grossmann) the notion of general covariance was still rejected in view of this.

But of course the value of the metric $g$ at $x$ does coincide with that of $\phi^* g$… at the transformed point $\phi(x)$. An attempt at more words on this is below in Discussion.

The “hole argument”

The original argument was formulated more in detail along the following lines.

In general relativity, every chart of the atlas of the given spacetime represents the viewpoint of a physical observer.

Let $(X,\mu)$ be a spacetime. Let $\phi: V \subset X \to \mathbb{R}^4$ be a chart of $X$ such that there is an open, simply connected subset $U \subset \V$ with vanishing stress-energy tensor $T$ in $U$.

Assume that there are two points $a, b \in U$ such that the curvature vanishes in a neighborhood of $a$ and does not vanish near $b$. This means in terms of the differential geometry describing the situation that the pseudo-Riemannian metric $\mu$ on $X$ is Ricci flat? when restricted to $U$ and flat when restricted to a neighborhood of $a$.

Given these assumptions, there is a diffeomorphism $\psi: X \stackrel{\simeq}{\to} X$ (an isomorphism in Diff) reducing to the identity outside of $U$, with $\psi(a) = b$. Let $\mu' = \psi^*(\mu)$ be the pullback of $\mu$ along $\psi$. Since the field equations of general relativity are covariant (respect isomorphisms in the category of pseudo-Riemannian manifolds), both $\mu$ and $\mu'$ are solutions to the field equations. So one observer will say that there is no gravitation at the spacetime point $a$, while another will say there is.

Discussion

In fact, strictly speaking no observer will say anything like this, because it is impossible to characterize a single point $a$ in a spacetime in particular and in any manifold in general in an intrinsic way, without referring to extra structure: given a bare manifold $X$, a point $a$ in it is a morphism $x : {*} \to X$ in Diff, a generalized element of $X$. But under an automorphism $X \stackrel{\simeq}{\to} X$ of $X$ in Diff, this point is taken to any other point $b : * \stackrel{a}{\to} X \stackrel{\simeq}{\to} X$. So when regarded just by its probes by $*$, a manifold appears just as a bare set of points, with no interrelation. It is evil to try to distinguish these points, because in the slice category $(*,X)$ of elements, they are all isomorphic.

Rather, what really does characterize the manifold underlying a spacetime is its collection of all probes by the test spaces $\mathbb{R}^n$, i.e. by all morphisms $\mathbb{R}^n \to X$ in Diff and their relation among each other. The collection of information encoded by these probes yields the sheaf $\bar X := Hom_{Diff}(-,X) :$ CartSp${}^{op} \to Set$, and this does characterize the full structure of the manifold. For more on this see diffeological space.

On the other hand, if there is extra structure available on the manifold $X$, such as, for instance, a “scalar field”, i.e. a smooth function $\phi : X \to \mathbb{R}$, and if we take morphisms of such manifolds to respect this extra structure, then it is possible in an non-evil way to characterize intrinsically, say, the subset of points on which $\phi$ takes a fixed value. The scalar curvature invariants of a pseudo-Riemannian metric on $X$ do play such a role, and thus induce intrinsic observable structure. So what matters in the above example is not that one point is called $a$ and one is called $b$, but that at one point the Ricci curvature function vanishes, and at the other not. As a diffeomorphism is applied to the manifold, the points may be re-identified, but if the Ricci curvature vanished at one point before, it will vanish at the re-identified point after the re-identification, and hence still characterizes that as one of the points where the Ricci curvature vanishes. So the apparent paradox in the above arises from insisting that $a$ is $a$ and $b$ is $b$ even after applying a diffeomorphism. This is evil. The diffeomorphism identifies $a$ with $b$ and $b$ with $a$ instead.

Notice that this argument has really nothing specifically to do with physics or general relativity. The analogue of what is said here about pseudo-Riemannian manifolds could be said about, say, symplectic manifolds or whatever.

So from the nPOV there is no mystery here, but the above argument originally troubled Einstein, because at his time it was felt that it violates the demand that the statement “at a certain region in time and space, there is (or isn’t) gravitation” should have an objective, observer independent meaning, whether or not there is matter present that “feels” the influence of gravitation. This assumption is based on the Newtonian notions of the absolute, objective existence of space and time. For Newton’s physics, space and time exist independently of any observers and of any objects that are present in time and space. If one adds a structure that models gravitation to space and time in this sense, the statement “at a certain region in time and space, there is (or isn’t) gravitation” is independent of observers and of the presence of further content (or structures) like matter.

The conclusion of Einstein and therefore of general relativity was however that the statement “at a certain region in time and space that contains no matter, there is (or isn’t) gravitation” is not independent of the observer. This means that the physical notions of space and time do not have the same objective meaning as in Newton’s physics.

On the other hand, the conclusion that one draws from the nPOV is a simpler and much more general one: it is evil to try to identify objects in a category in a way that does not respect their isomorphisms.

References

• Albert Einstein, M. Grossmann, Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation Zeitschrift für Math. Phys. 62, 225–259 (1914)

Revised on October 8, 2013 06:40:34 by David Corfield (87.115.31.43)