## 1 Introduction

“Hole argument” is the English translation of the German phrase “Lochbetrachtung,” used by Albert Einstein to describe his argument against the possibility of generally-covariant equations for the gravitational field, developed in late 1913 and accepted until late 1915. Einstein realized the desirability of general covariance, and showed that it was easily implemented for the rest of physics; but the hole argument purported to show why it could not be demanded of the gravitational field equations he was trying to formulate for the metric tensor.^{1}

This article is a historical-critical study, in Ernst Mach’s
sense.^{2}
It includes a review of the literature on the hole argument that concentrates on the interface between
historical, philosophical and physical approaches. Although recounting the history of the hole
argument, the primary purpose is to discuss its contemporary significance – in both physics and
philosophy – for the study of space-time structures. Like Mach, while presenting various other
viewpoints, I have not hesitated to advocate my own. In physics, I believe the main lesson of
the hole argument is that any future fundamental theory, such as some version of “quantum
gravity,” should be background independent, with basic elements obeying the principle of maximal
permutability. In the philosophy of space-time, this leads me to advocate a “third way” that I call
dynamic structural realism, which differs from both the traditional absolutist and relationalist
positions.

### 1.1 Why should we care?

One of the most crucial developments in theoretical physics was the move from theories dependent on fixed, non-dynamical background space-time structures to background-independent theories, in which the space-time structures themselves are dynamical entities. This move began in 1915 when Einstein stated the case against his earlier hole argument. Even today, many physicists and philosophers do not fully understand the significance of this development, let alone accept it in practice.

So it is of more than historical interest for physicists and philosophers of science to understand what initially motivated this move, as well as the later developments stemming from it. Einstein’s starting point was the search for a generalization of the special theory that would include gravitation. He quickly realized that the equivalence principle compelled the abandonment of the privileged role of inertial (i.e., non-accelerated) frames of reference, and started to investigate the widest class of accelerated frames that would be physically acceptable. His first impulse was to allow all possible frames of reference; since he identified frames of reference and coordinate systems, this choice corresponds mathematically to a generally-covariant theory. But he soon developed an argument – the hole argument – purporting to show that generally-covariant equations for the metric tensor are incompatible with his concept of causality for the gravitational field. The argument hinged on his tacit assumption that the points of space-time are inherently individuated, quite apart from the nature of the metric tensor field at these points. Only two years later, after other reasons compelled him to reconsider general covariance, did Einstein finally recognize the way out of his dilemma: One must assume that, in an empty region of space-time, the points have no inherent individuating properties – nor indeed are there any spatio-temporal relations between them – that do not depend on the presence of some metric tensor field.

Thus, general relativity became the first fully dynamical, background-independent space-time theory. Without some knowledge of this historical background, it is difficult to fully appreciate either the modern significance of the hole argument, or the compelling physical motives for the requirement of background independence.

### 1.2 Summary: Where we are headed

Einstein’s starting point in his search for a theory of gravitation was the theory we now call special
relativity. From a contemporary viewpoint, its most important feature is that it has two fixed, kinematical
space-time structures – the chrono-geometry embodied in the Minkowski metric tensor field
and the inertial field embodied in the associated flat affine connection – both of which are
invariant under the ten-parameter Lie group now called the Poincaré or inhomogenous Lorentz
group.^{3}

In Minkowski space-time, all dynamical theories must be based on geometric objects that form a representation (or more generally, a realization) of this group. There is a preferred class of spatial frames of reference in Minkowski space-time, the inertial frames. Einstein had shown how to define a class of physically preferred coordinate systems for each inertial frame of reference; in particular, he defined a clock synchronization procedure that provides a preferred global time for each frame. This enabled him to show how the principle of relativity of all inertial frames could be reconciled with the universal properties of light propagation in vacuum.

The lesson he drew was the need to find a physical interpretation of the coordinates associated with an
inertial frame of reference – a lesson that had to be painfully unlearned in his search for a
generalized theory of relativity. In large part, the history of the hole argument is the story of that
unlearning process. The end result was the formulation of the general theory of relativity, the first
background-independent physical theory turning all space-time structures into dynamical fields. This was
such a revolutionary break with all previous physical theories, in which space-time structures
constitute a fixed, non-dynamical background, that its ultimate significance is still debated by
physicists.^{4}

Understanding the hole argument in both its historical and contemporary aspects can help to clarify the issues at stake in this debate. The basic issue can be stated as follows: Given a physical theory, when should an equivalence class of mathematically distinct models of the theory be identified as corresponding to single, unique physical model? The hole argument shows that, for any theory defined by a set of generally-covariant field equations, the only way to make physical sense of the theory is to assume that the entire equivalence class of diffeomorphically-related solutions to the field equations represent a single physical solution. As will be seen later, mathematically this result can best be stated in the language of natural bundles.

But a similar result holds for the even broader class of all gauge-invariant field theories, notably Yang–Mills theories: an equivalence class of gauge-related models of any such theory must be physically identified. Mathematically, broadening the question in this way requires the language of gauge-natural bundles. General relativity itself may also be treated by the use of gauge-natural bundle techniques: its similarities to and differences from gauge theories of the Yang–Mills type will also be discussed.

This move to natural and gauge-natural formulations of field theories also has important implications for the philosophy of space and time. The old conflict between absolute and relational interpretations of space and then space-time has been renewed on this new ground. But I shall argue that this reformulation of the question suggests a third position, around which a consensus is forming. This position has been given various names, but I prefer dynamic structural realism.

### 1.3 Outline of the article

Sections 2.1 – 2.5 recount the developments leading up to Einstein’s adoption of the hole argument against general covariance in 1913, how it misled him for over two years, the reasons for his rejection of it in late 1915, and its replacement by the point-coincidence argument for general covariance.

Section 2.6 discusses Kretschmann’s 1917 critique of the concept of general covariance and Einstein’s 1918 reply; decades later this debate led Komar to propose the use of what are now called Kretschmann–Komar coordinates as a way of resolving the hole argument.

Finally, Section 2.7 discusses Hilbert’s 1917 reformulation of the hole argument: He replaced the four-dimensional hole in space-time with a space-like hypersurface, on which he posed an initial value problem for the field equations; this was the first step in a series of developments culminating a decade later in a fully satisfactory formulation by Darmois of the general-relativistic Cauchy problem.

Section 3 discusses the revival of interest in the hole argument in the 1970s, which grew out of an attempt to answer a historical question: Why did three years (1912 – 1915) elapse between Einstein’s adoption of the metric tensor to represent the gravitational field and his adoption of what are now called the Einstein equations for this field? Some highlights of this discussion are recalled, from the post-World War II revival of interest in general relativity up to the present.

Section 4 presents a modern version of the hole argument in general relativity, and its generalization from metric theories of gravitation to gauge-natural field theories. By abstraction from continuity and differentiability, a formulation general enough to include theories with discrete fundamental elements is obtained. The concept of general covariance of a field theory is similarly extended to general permutability, a concept wide enough to include theories based on relations between the elements of any set.

Sections 5 and 6 focus on current discussions of philosophical and physical implications of the hole argument, respectively; no attempt is made to rigidly separate issues that overlap both areas.

Section 5 discusses such issues as: the range of applicability of the hole argument, the correct mathematical definition of general covariance and its physical significance, the controversy between relationalists and substantivalists in discussions of space-time structures. The arguments of Earman, Pooley and Stachel are reviewed, and their convergence on a third alternative, which I call dynamic structural realism, is stressed.

Section 6 discusses such issues as partially background-independent theories, including mini- and midi-solutions to the Einstein field equations; the reformulation of general relativity as a gauge natural theory; and some implications of the hole argument for attempts to formulate a quantum theory of gravity.