HPS 0410 Einstein for Everyone

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Philosophical Significance of the General Theory of Relativity
What does it all mean, again?

Ontology of Space and Time:
The Hole Argument

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

In the last chapter, we saw a migration in Einstein's understanding of the foundations of his developing general theory of relativity. His initial hope in 1907 was for a theory that would eliminate the absoluteness of acceleration that still remained in his special theory of relativity. By 1915, when the theory was complete, those aspirations had migrated into the requirement that his spacetime coordinate systems lose their independent existence. This, he achieved, through the general covariance of his final theory.

It was not so clear to him initially just what that loss of independence meant when carried through to the fullest extent. Its full import was revealed only after Einstein had taken an embarrassing detour through his "hole argument," which he initially believed renderred all generally covariant theories physically uninteresting. With its repudiation, Einstein felt he had now established the loss of independent existence for spacetime coordinate systems. That loss became the revised version of Einstein's generalized principle of relativity.

This chapter tells the story. We will use some of the technical details developed in the last chapter.

The Disaster of 1913

We saw in an earlier chapter that Einstein's investigations on gravity and spacetime did not proceed smoothly. In 1913, together with the mathematician, his friend Marcel Grossmann, Einstein published his "Entwurf..." paper. It contained all the essential  components of the general theory of relativity but one. That one was the most important. It was the generally covariant gravitational field equations of the final theory. In spite of all of Einstein's efforts, the theory of 1913 was not generally covariant.

This failure was a source of continuing trouble for Einstein. He alternated between concern that something was wrong in the theory; and confidence that all was right. A major contribution to this latter confidence was an argument he hit upon shortly after completing the "Entwurf..." and which he ended up publishing four times in 1914. This argument purported to establish that Einstein's failure to find a generally covariant theory was no failure at all, for a generally covariant theory would be physically uninteresting.

More specifically, what this "hole argument" purported to show was that a generally covariant theory would contradict what Einstein called the "law of causality." In more modern terms, what it purported to show was that such a theory suffered a severe form of indeterminism.

A common form of indeterminism arises in quantum mechanics. In standard accounts, we can specify the present state of the quantum system. However that will not fix what later measurements might return on the system. The best we can have are probabilities for different measurement outcomes.

The form of indeterminism of the hole argument was more severe. We specify the metric of spacetime everywhere in the past; and everywhere in the future, excepting a tiny region of spacetime: "the hole." This near complete specification is insufficient to determine what the metric will be in the hole. Worse, there are no probabilities supplied for the different possibilities.

This failure of determinism was so calamitous a defect that, Einstein felt, it made generally covariant theories physically uninteresting.

The Hole Argument

The machinery of the hole arguments depends only on the most basic facts of a generally covariant theory. The metric tensor in a generally covariant theory will be governed by gravitational field equations, such as the Einstein gravitational field equations we saw in an earlier chapter. What this means is that we can apply the field equations in any coordinate system we choose.

For example, the field equations might tell us that some metric g can arise in some coordinate system x. That means that, for each event with coordinate x in the spacetime, there is a particular metric g. We write this as g(x).

Because the theory is generally covariant, we can also solve the field equations in another coordinate system x' for the same physical case. We would then recover a metric g'(x'), where the tables of values in g' assigned to the coordinates will, in general differ from those assigned by g. If we transform from the coordinate system x to the coordinate system x', then the rules for transforming tensors will transform g to g'.

For more on transforming tensors between coordinate systems, see the Technical Appendix of the last chapter.

These two assignments of tables of numbers g(x) and g'(x') to the coordinates are merely two different coordinate-based representations of the same set of facts about proper times and distances in spacetime.
To recall: the metric g is just a compact way of writing the 4x4 table of numbers gμν that comprise the metric tensor in this coordinate system.

The quantity x represents the four coordinates assigned to an event in spacetime: x = xμ = (x0, x1, x2, x3).

The quantity "g(x)" represents the assigning of a 4x4 table to each value of the quadruple(x0, x1, x2, x3).

In special cases, the same table will be assigned to every quadruple of coordinates. Here's an example in the last chapter (Minkowski spacetime metric in inertial coordinates).

In general, there might be a different table for each value of the quadruple of coordinates. Here's an example in the last chapter. The tables are different according to the value of x = x1 in the quadruple. (Minkowski spacetime metric in uniformly accelerated coordinates.)

Einstein applies these basic facts of a generally covariant theory to a quite particular pair of coordinate systems x and x'. The two systems agree everwhere except in some very small region of spacetime where they come smoothly to differ.

An example of  two such systems of coordinates is shown in the figures. For simplicity, only the t=x0 and x=x1 coordinates of x are shown. Here is the spacetime with this coordinate system and the hole marked.

The metric g induces spatial distances and proper times elapsed onto the spacetime as well all distinguishing the trajectories of light signals. These trajectories form the light cone structure of the spacetime, as represented below. We will track the alterations in the metric tensor g visually by tracking changes in these light cones.

We now introduce a transformation to a new coordinate system (t', x') that is the same everywhere outside the hole but comes smoothly to differ from it inside the hole. In this example, the transformation leaves t unchanged, since t = t'. The coordinate values of x' however are smoothly shifted towards the -x direction of the original coordinate system, so that, at t=0, the origin of the x' coordinate system is located in x at x = -1.

 The metric in this new coordinate system, g'(x'), will consist of an assignment of table of numbers to the coordinates that differ from those in the original coordinate system, g(x). However nothing has been done, so far, to alter the physical facts about times and spaces represented. In the figure, this is seen in the way the light cones are unaltered. All that has happened is that we have labeled the events of the spacetime with different numbers. The event that was labeled t=0, x=0 in the original coordinate system, for example, is now labeled t'=0, x'=1. To preserve the physical facts about spaces and times, we have had to make adjustments to the tables assigned in g(x), which becomes those assigned in g'(x').

It now follows that the g(x) and g'(x') agree everywhere in spacetime except when the coordinates x and x' are those of events within the hole. For when we transform to a new coordinate system, we must also transform the table of numbers comprising g.

g(x) ≠ g'(x')   when  x = x' for events within the hole
g(x) = g'(x')   when  x = x'

That they differ within the hole is no special cause for concern. g(x) and g'(x') represent the same spacetime metric, but in different coordinate systems, so there is no necessity for them to be the very same table of numbers. Indeed, for events within the hole, when x = x', the two sets of coordinates x and x' are, by construction, labeling different events.

Now comes the most delicate point in Einstein's argument. It is a point that was overlooked in later commentaries, where Einstein's argument was initially dismissed incorrectly as trivially flawed. When generally covariant field equations allow some metric in some coordinate system, all that matters to the field equations is the particular functional dependence of the g on x or the g' on x'. The field equations are satisfied by any pair of g* and x* that stand in the right functional relationship. There is no extra condition that the coordinate system x* must satisfy.

This means that we can take the functional dependence of g' on x' and recreate it in the first coordinate system x. That is, we form the metric tensor g'(x) in the first coordinate system x. We do this by taking the table assigned by g'(x') to coordinate  x' and reassigning it to the coordinate x in the old system, where the coordinate x has the same numerical values as x'. For example, imagine that g'(x') assigns some particular table of values "G" to the origin of its coordinates x' = (0, 0, 0, 0). The new assignment g'(x) will now assign that same table of values G to the origin x = (0, 0, 0, 0) of the first coordinate system.

We now have a set of conditions that look quite like the first set, but differ in an important way:

g(x) ≠ g'(x for events with coordinate x within the hole
g(x) = g'(x)   for events with coordinates x outside the hole

This new inequality is troubling. For it tells us that there are two solutions of the gravitational field equation in the same coordinate system x such that the two solutions g(x) and g'(x) differ within the hole, but agree everywhere outside it.

The figure below is an attempt to show how the creation of this new metric g'(x) appears. The effect of the creation is equivalent to taking the original metric g(x) and smoothly dragging to the right, that is, to the +x direction, all its physical content. For example, the new coordinate system assigns t'=0, x'=+1 to the origin event of the old coordinate system t=0, x=0. So the dragging takes the metrical structure at t=0, x=0 and drags it one coordinate unit to the right, leaving it a t=0, x=+1. The lines depicting the coordinate system x are also shown dragged to help visualize the dragging.

To see these effects a little more clearly, we should restore the dragged lines of the original coordinate system x  to their correct disposition.

The light cones appear distorted by this dragging. This distortion of the light cones is no longer an artifact of being in a different coordinate system. The distortion is a real difference. For now there is only the original coordinate system. These light cones show that the metric g'(x) disagrees with the original g(x) on how the trajectories of light signals and proper times and distances are spread over the spacetime.

To make the difficulty more concrete, consider the two light signals shown in the figure. If they follow the light cone structure arising from g(x), they intersect at the origin event E of the coordinate system at t = x = 0. There is also a massive body shown in the figure whose timelike worldline takes it through the same event E:

If however they were to follow the light cone structure arising from g'(x), that intersection event would be displaced by one coordinate unit to the right in the +x direction to the point E'. That is to be expected, for we saw above that the dragging effect took the structures at t=0, x=0 and moved them to t=0, x=+1.

Where will the light signals meet? At the event E? Or at the event E'? Nothing in the full specification of the spacetime outside the hole determines it. For g(x) = g'(x) outside the hole, where the two spacetimes they represent agree in all physical properties. This is the troubling form of indeterminism that Einstein felt sufficiently worrisome to render a generally covariant theory g(x) = g'(x).

Einstein's Statement of It

Here is Einstein's most careful and complete version of the hole argument from a paper of 1914:

Einstein's 1914 version of the hole argument in German

Here is my translation:

"§12. Proof of the necessity of a restriction on the choice of coordinates.

     We consider a finite region of the continuum Σ, in which no material process takes place. Physical happenings in Σ are then fully determined if the quantities gμν are given as functions of the xν in relation to the coordinate system K used for description. The totality of these functions will be symbolically denoted by G (x).
      Let a new coordinate system K' be introduced, which coincides with K outside Σ, but deviates from it inside Σ in such a way that the g'μν related to the K' are continuous everywhere like the gμν (together with their derivatives). We denote the totality of the g'μν symbolically with G'(x'). G'(x') and G(x) describe the same gravitational field. In the functions g'μν we replace the coordinates x'ν with the coordinates xν i.e., we form G'(x). Then, likewise, G'(x) describes a gravitational field with respect to K, which however does not correspond with the real (or originally given) gravitational field.
      We now assume that the differential equations of the gravitational field are generally covariant. Then they are satisfied by G'(x') (relative to K') if they are satisfied by G (x) relative to K. Then they are also satisfied by G'(x) relative to K. Then relative to K there exist the solutions G(x) and G'(x), which are different from one another, in spite of the fact that both solutions coincide in the boundary region, i.e., happenings in the gravitational field cannot be uniquely determined by generally covariant differential equations for the gravitational field."

A. Einstein, "Die formale Grundlage der allgemeinen Relativitätstheorie." Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1914): 1030-1085 on p. 1067.

My presentation has simplified Einstein's original argument by considering the "source free" case in which the source term in Einstein's gravitational field equations, the stress-energy tensor Tμν, is zero.  In Einstein's original argument above, he considers the more general case in which the source Tμν is non-zero outside the hole, but is zero within the hole. ("no material processes") In hindsight, the complication adds nothing to the argument, so I have presented a simpler version above. For Einstein in 1913, the difference may have been important. It meant that he could show that the matter distribution outside the hole fails to fix the structure of spacetime within it. This would not have sat well with his early Machian sensibilities. It would directly contradict the version of Mach's principle given in Einstein's 191 paper.

The Point-Coincidence Argument

The hole argument provided only temporary relief for Einstein's enduring doubts about this theory. By November 1915, Einstein had realized that his abandoning of general covariance was mistaken. He made his celebrated return to general covariance and completed the general theory of relativity.

Einstein now had a new problem. In 1915, he announced to his colleagues in physics a generally covariant theory. The previous year he had given apparently compelling reasons against such a theory. Einstein needed to retract the hole argument and to do it in an equally compelling way.

That retraction came in the form of the "point-coincidence" argument, which appeared in his 1916 review article. It was offered as an argument for general covariance and read:

"That this requirement of general co-variance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time.

The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. We allot to the universe four space-time variables x1, x2, x3, x4 in such a way that for every point-event there is a corresponding system of values of the variables x1 ... x4. To two coincident point-events there corresponds one system of values of the variables x1 ... x4, i.e. coincidence is characterized by the identity of the co-ordinates. If, in place of the variables x1 ... x4, we introduce functions of them, x'1, x'2, x'3, x'4, as a new system of co-ordinates, so that the systems of values are made to correspond to one another without ambiguity, the equality of all four co-ordinates in the new system will also serve as an expression for the space-time coincidence of the two point-events. As all our physical experience can be ultimately reduced to such coincidences, there is no immediate reason for preferring certain systems of co-ordinates to others, that is to say, we arrive at the requirement of general co-variance."

A. Einstein, "The Foundation of the General Theory of Relativity," 1916 as translated in H. A. Lorentz et al. The Principle of Relativity. New York: Dover on p. 117.

Here Einstein does not say explicitly that this argument is to supercede the hole argument. However it is clear that it does. For the two spacetimes associated with g(x) and g'(x) in the hole argument agree on all coincidences. For example, the meeting of the two light signals coincides with their meeting the worldline of the massive point particle. That is true in the spacetime associated with g(x); and it is equally true in the spacetime associated with g'(x). The only difference between the two spacetimes lies in how the tables of numbers forming g(x) and g'(x) are spread over the one coordinate system x. Those differences are invisible to the point-coincidences of physical systems like light signals and massive particles. For when g(x) is dragged and spread anew over x to form g'(x), all these material coincidences are carried along with the new spreading. Nothing in them is changed.

This resolution of the hole argument could be put in positive terms concerning the ontology of coordinate systems. In his generally covariant theory, the coordinate systems have no independent physical reality. Here is how he put it in a letter to his friend and confidant, Michele Besso in early 1916:

"There is no physical content in two different solutions G(x) and G'(x) existing with respect to the same coordinate system K. To imagine two solutions simultaneously in the same manifold has no meaning and the system K has no physical reality."

Einstein to Michele Besso, January 3, 1916 (Papers, Vol. 8A, Doc. 178.)

The same point was made more figuratively by Einstein much later in a 1952 appendix to his popular text on relativity:

"On the basis of the general theory of relativity, on the other hand, space as opposed to "what fills space", which is dependent on the co-ordinates, has no separate existence. Thus a pure gravitational field might have been described in terms of the gik (as functions of the coordinates), by solution of the gravitational equations. If we imagine the gravitational field, i.e. the functions gik, to be removed, there does not remain a space of the type (1) [Minkowski spacetime], but absolutely nothing and also no topological space.

There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field."

Albert Einstein, Relativity: the Special and the General Theory. Trans. R. W. Lawson. 15th rev. ed. London: Methuen, 1954 (1977) on p. 155.

What Einstein says here gives a vivid picture of the precise step in the hole argument where it fails. Figuratively speaking, the hole argument asks us to start with a metric g(x) in a coordinate system x, remove g(x) from x and replace it with g'(x), in the same coordinate system x.


The failure comes with the intermediate step. We are supposed to make physical sense of the coordinate system x without the metric g. Then we are apply the new metric g' to this very same coordinate system. Einstein repudiates the idea of the coordinate system x existing independently of g. As he says, taking away g does not leave us with a bare coordinate system x upon which we can carry out further operations. It leaves us with nothing.


The Modern Hole Argument

The account above describes the hole argument and Einstein's resolution of it, at it appeared in 1914-1916. The modern literature in philosophy of space and time also draws on the hole argument and offers a solution. The issues in both literatures are essentially the same. This sameness, however, is obscured by the more sophisticated mathematical clothing used in the modern literature.

For an introduction to the modern formulation and a survey of responses, see my "The Hole Argument," Stanford Encyclopedia of Philosophy.

Readers who know the modern version of the hole argument can see the similarity of the two literatures once they know how to translate Einstein's considerations into the modern context. The modern context represents the events of spacetime as a mathematical object known as a manifold. Call it "M." It is a set of events along with a specification of the particular way those events can be collected into neighborhoods. The metric is then introduced as a codification of intervals between nearby events.

The structure corresponding to this spacetime manifold is the coordinate system Einstein uses in formulating the hole argument. For in both Einstein's original formulation and in the modern formulation, the essential manipulation is to spread a metric g in two different ways over a coordinate system x or a spacetime manifold M.

Einstein concludes through the point-coincidence argument that the coordinate system x has no existence independently from the metric tensor g associated with it. In the modern literature, the initial reading of the import of the hole argument was similar. One  might imagine that the spacetime manifold of events has an existence independent of what is in spacetime. Since independent existence is a defining characteristic of substances, this amounts to a "substantival" view of the manifold. The the indeterminism of the hole argument is escaped by denying the independent existence of the spacetime manifold. That is, we deny "spacetime substantivalism" or, more exactly, " spacetime manifold substantivalism."

Copyright John D. Norton. November 12, 2019.