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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 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.
As is often the case when confused about automorphisms, it clarifies the issue to talk about more general isomorphisms first. Thus, consider two different manifolds $X$ and $Y$ and a diffeomorphism $\psi:Y\cong X$, with metric $\mu$ on $X$ and pulled back metric $\mu' = \psi^*(\mu)$ on $Y$. Then $(X,\mu)$ and $(Y,\mu')$ represent the same physical reality, with $\psi$ as a witness of this fact. In particular, if $\psi(a)=b$, then the gravitational field at $a$ in $Y$ will coincide with the gravitational field at $b$ in $X$, as it should be since both represent the same point of spacetime.
Now suppose we have another diffeomorphism $\phi:Y\cong X$. Then in general $\mu'$ will be different from $\phi^*(\mu)$, and therefore there is no reason that the gravitational field at $a$ in $Y$ will coincide with the gravitational field at $\phi(a)$ in $X$. This should still be obvious. But the “hole paradox” now arises when we take $X$ and $Y$ to be the same and $\phi$ to be the identity map.
In other words, if we have used some isomorphism $\psi$ to witness that two manifolds represent the same reality, then we can’t “switch halfway through” our argument and start identifying them along some different isomorphism $\phi$. This is true even if the two manifolds are the same and $\phi$ is the identity.
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: when identifing isomorphic objects in a category, we have to remember which isomorphism we used. For a formulation of the situation in terms of homotopy type theory see the relevant section in general covariance.
For a treatment in terms of HoTT, see
John Dougherty, The Hole Argument, take n, Foundations of Physics (2019), (doi).
Wikipedia, hole argument
Last revised on September 29, 2019 at 11:16:01. See the history of this page for a list of all contributions to it.