HPS 0410 Einstein for Everyone

Back to main course page

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

Geometric Morals
and a few more

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

Linked documents:
Einstein on Kant
More from Einstein on the Conventionality of Geometry

In earlier chapters, I reviewed some of the ways that special relativity has been judged to have a significance that extends beyond its immediate physical content. This chapter (and new chapters still in the planning stage) take up the same exercise for general relativity.

As before, our goal is to look beyond its physical content to discern how the coming of general relativity affected our conceptions of things in the broader context. In the present chapter, I will pursue those aspects that are connected with geometry. Since the other chapters planned are still unwritten, the chapter will conclude with a synopsis of further morals planned for those chapters.

As before, the goal is to bring some philosophical rigor to the assessment of the various morals claimed. So the reader is urged to review the early sections of Skeptical Morals since they sketch the method to be used. I will also include a statement of my view. Those statements are not to be taken as text-book truths. They are merely illustrations of how one philosopher of science--me--has thought through the issues.

General relativity continued the assault on our older confidence in the certainty of the results of science. So some of the general Skeptical Morals described earlier can equally be traced to general relativity. We need not repeat them here. We need only add the observation that general relativity increased the sense of fragility of older science. It was not just that things behaved a little oddly when they moved very quickly, as special relativity showed. General relativity now revealed a failure of the basic ontology of Newtonian science, that is, of the list of entities fundamental to it. In Newtonian science, gravity is understood to be a force, where forces are one of the most basic existents in the Newtonian scheme. In general relativity, gravity is not longer a force. Rather the effects of gravity are associated with a curvature of spacetime.

Failure of the Kantian Synthetic A Priori

The laws of geometry are not synthetic a priori truths.

In the decade following Einstein's completion of the general theory of relativity, one of the most important ramifications in philosophy was that it forced a reappraisal of the nature of geometrical knowledge. The work of nineteenth century mathematicians had shown that geometries other than Euclid's were logically possible. There was a sentiment growing among mathematicians that we should take these other geometries seriously as physical possibilities for the geometry of our space. For example, we shall see below that, prior to general relativity, Poincaré argued that the choice between Euclidean and other geometries as the physical geometry of space was conventional.

Einstein's discovery of general relativity forced the issue. It was no longer a matter of pure speculation that Euclid's geometry might fail somewhere, somehow, so that we would need a different geometry. Perhaps, it had been conjectured, Euclidean geometry might fail in the very large or in unusual circumstances, if our measuring rods were composed of exotic materials. Rather, Einstein showed us that our best theory of space, time and gravity had made the speculated failure a truth. His theory demanded that Euclidean geometry hold at best only as an approximation.

The immediate casualty was one of the most successful epistemologies of Einstein's era, that of Immanuel Kant. The core novelty of that system was the idea of synthetic a priori truths. A synthetic proposition is one says more of the subject than is given in the subject's definition. An a priori proposition is one that can be known prior to experience. Kant's epistemology was based on the idea that there are truth that are both: synthetic a priori truths..

The laws of geometry were a prominent, early example. Einstein now showed that these laws were not even truths, let alone known a priori. I have described the episode in more detail in earlier chapters, so the details need not be repeated here.

For Einstein's view in his own words, see this.

While Einstein's discovery considerably damaged the Kantian program, it continues to the present day in the revised form of various neo-Kantian traditions.

My View

Kantianism has been an epistemic failure from the start, especially as far as geometry is concerned. According to it, the most mundane facts of experience are not true of the world. Take the facts that the reader is, perhaps, sitting in a room, separated spatially from the rest of the world by walls, a floor and a ceiling; and, presumably, the Earth's South pole is spatially very much more distant than the walls. These spatial facts are geometric and, therefore, not a direct expressions of things in the themselves. Their spatial content is somehow invented by our apparatus of perception. If facts such as these are not really facts at all, then what do we know of the world?

The situation is worse when we ask if we have saved science from Humean skepticism as Kant had intended. The concern was that Euclid had apparently found universal truths of geometry with a security that outstripped our meager epistemic means. Kant recovered the security. We are, he assured us, inventing these general truths through our perceptual apparatus. That is what forces the angles of a triangle to sum to two right angles. Hence we are assured that our experience will always deliver this result. However, in assuring us of that, we have given up the part that matters most. This basic truth of Euclidean geometry is no longer a truth about the things in themselves. The science is not saved. It is demoted to a fabrication, albeit an enduring one.

It is a mystery to me why we needed the advances of the 19th and 20th century to shake confidence in the Kantian picture. It is a greater mystery to me why modern neo-Kantians persist in efforts to save the program.

Perhaps the only salvageable residue is the idea that our conceptual systems provides a medium for conveying facts of the world and that we may have great trouble separating the facts of the world from the medium. That is an interesting if familiar idea. However we must never forget that the medium is not the message of facts of the world. If the regularities of experience we discover are merely regularities in the medium, we have not discovered truths about the world. All we have found are our own fabrications.

Conventionality of Metrical Geometry

The metrical geometry of space is not given factually but is chosen by us as a convention.

While general relativity clearly demonstrated the failure of the Kantian a priori in geometry, there remained a lingering sense that something in the Kantian view was correct. We were somehow supplying something that was being mistaken for bare fact. The thesis of conventionality of geometry identified that something as the metrical geometry of space. It is that part of geometry that specifies distances between points, as will be explained in a greater detail below. The claim is that this geometry is not something that can be known factually, but is something we freely chose from many possible candidates. Where Kant had urged that we imposed Euclidean geometry on experience, the conventionalist now asserted that we freely chose that geometry from among many possibilities.

The sense of convention here is the same as the one we saw earlier in the thesis of the conventionality of simultaneity. As before the choice is quite akin to the decision in different countries over which side or the road is to be driven on by traffic. Some drive on the right; some on the left. There is no correct fact as to the proper side. Each country makes its own choice on the basis of convenience.

drive on right    or    drive on left
Image: http://fr.wikipedia.org/wiki/Fichier:Beijing_traffic_jam.JPG

Metrical Geometry: the Basic Idea

Metrical geometry is that part of geometry that specifies the distances between points in the space. We can get a sense of how this is a part of the larger geometry by looking at a familiar example.

A common map in old school atlases is a one of the world given in the Mercator projection, such as this one:

Mercator world
From J. G. Bartholomew, Graphical Atlas and Gazetteer of the World, New York: Thomas Nelson & Sons, 1892, p.3.

To begin with, a map like this contains extensive topological information about the relations among the various parts in the space. We see that Canada and the US touch, but that the North American continent is separated from Europe by an ocean. If we could look in more closely, we would see that Austria is fully land-locked within Europe, being surrounded by other countries on all its borders.

More precisely, the topological properties of a space are just those that are unaffected by continuous deformation of the space. Figuratively that corresponds to stretching and shrinking the space like a rubber sheet, but without cutting, tearing or rejoining parts. (Hence topology is sometimes called "rubber sheet geometry.")

Here's the same map after it has undergone such a deformation. This deformed map and the original map contain the same topological information. Canada and the US still touch. North America is still separated from Europe by an ocean. And so on.

deformed Mercator map

What is missing from the topological information is anything about distances. We have all the countries in both maps. But they have different sizes. Information about their sizes--how much distance, say, separates their Easternmost and Westernmost points--changes when we deform the map. That information is not topological.

We need extra information to know how much distance there is been two points in the map. That extra information is "metrical." More precisely, the metrical structure specifies the distances along any curve we may draw in the space.

The metrical structure of the map is supplied by the grid of lines of longitude and latitude. They are emphasized below:

Mercator lines

This grid of line supplied the metrical or distance information only indirectly. But it is there as long as one knows how to read it. To start, consider the equator. We track positions along the equator with the lines of longitude. A nautical mile is defined so that one degree of longitude at the equator corresponds to 60 nautical miles. Hence we have:

ten degrees of longitude at the equation = 600 nautical miles

We will now use this 600 miles as the unit of a measuring rod--red sticks in the figures below--that we will transport over the surface of the earth.

The detail of the map at the right shows the rods laid out along the equator. The meridians (vertical grid lines) are separated by 20 degrees of longitude, so, at the equator, we fit two rods between the successive meridians shown.

As we move North from the equator, however, the meridians approach. They will eventually meet at the North Pole. Once we have moved northward beyond Japan to 60 degrees of latitude, the distance between the adjacent meridians is halved. So now we can only fit one of our 600 nautical mile measuring rods into the space between the meridians.

Why is it halved at 60 degrees of latitude? The general formula is that the distance between the meridians shrinks with the cosine of the latitude; and cos (60) = 1/2.

However the distance between the meridians on the paper of the map remains the same. Hence the map stretches true distance in the East-West direction as we move North and South from the equator.
Mercator detail

What about distances in the North-South direction? An important, even defining, characteristic of the Mercator projection is that the amount it stretches distances is the same in both directions: North-South and East-West. That tells us that we can use the same rods, rotated from the East-West orientation to the North-South orientation, at each point of the map. Here is a map with more 600 nautical mile measuring rods spread over its surface.

Mercator with rods

Here's some map trivia.

This last fact, that the projection stretches equally in both directions, is an artifact of the original purpose of the projection. The Mercator projection was designed to be useful to sailors. It has the important property that the path of boat sailing on a constant bearing comes out as a straight line. The map detail shows a course that is everywhere North-Easterly. If that is to come out as a straight line, the distance on the map sailed in the Easterly direction must correspond to the distance on the map sailed in the Northerly direction. That requirement just amounts to the map stretching distance equally in both directions.

This is a great convenience to navigators at sea plotting their courses on their nautical charts. It is great annoyance to geographers who lament that the Mercator projection grossly exaggerates the size of countries in far Northern and Southern latitudes.
Mercator course

The Mercator projection captures the components of the geometry at issue here. The topological structure of the space comprises all those properties that are unaffected if the map were printed on a rubber sheet and continuously deformed. The metrical structure is the full catalog of distances along curves in the map. That information is supplied by the grid of lines of longitude and latitude, once one knows how to read it.

The Poincaré Disk

The thesis of the conventionality of geometry asserts that the metrical structure of a space may be chosen conventionally.

One of the earliest to advocate this view was Henri Poincaré and he did it at a time prior to general relativity. He argued in his Science and Hypothesis (1905, p. 58) that geometry is really a reflection of the physical properties of the bodies we use as measuring rods, when we survey a space. He concluded (his emphasis):

"The geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions."

He introduced what has become an iconic representation of his claim. He asked us to imagine a disk with a strong temperature gradient over its surface. It is hottest at the middle and cools off to absolute zero at the circumference. We have measuring rods that can be placed over the surface. They instantly adopt the temperature of the part of the disk on which they are placed. They contract in direct proportion to the temperature.

The figure below shows a diameter AA' of the disk. Unit measuring rods are laid along it and they shrink as the rods near the coldest edge. If a rod were to be placed at the edge of the disk, it would shrink to no size, for the temperature there is absolute zero. However the successive placement of the rods will never allow this. Since they shrink towards zero size the closer we come to the circumference, it turns out that infinitely many unit rods will fit along the diameter.

That is, as far as the geometry revealed by the rods is concerned, the diameter is infinitely long. That is, the space is infinite!


Poincare disk

The diameter AA' turns out to be a geodesic of the measured geometry of the surface. That is, it is a line of shortest length. What about other geodesics? They manifest as arc of circles in the geometry of the paper or screen of this page. For example, the line BB' is a straight line of the geometry, that is, it is a geodesic of shortest length. By diverting inwards toward the hot center, the line needs fewer rods to traverse the disk. It is also infinitely long, as is AA'


Poincare disk 2

Finally, the geometry turns out to be the familiar hyperbolic geometry of 5MORE That we have discussed in an earlier chapter. We can see how the alternative to Euclid's fifth postulate, 5MORE, is implemented in the geometry in the above figure. Consider the point P not on the straight line AA'. Line DD' through P is also a straight line of infinite length that does not intersection AA'; that is, DD' is parallel to AA'. There are many more such lines. CC' and EE' through P are also infinitely long straight lines that do not intersect AA' and are thus also parallel to it.

Poincaré's point is this. Here we have a disk and we measure its geometry with measuring rods. One sort of rod that does not respond to temperature will tell us the metrical geometry is Euclidean. Another sort of rod that does respond to the temperature will tell us it is hyperbolic. There is no independent fact of the metrical geometry of the surface. Which metrical geometry we recover depends on our choice of which measuring rods we choose to use. That is, we choose the geometry.

Einstein's "Geometry and Experience"

Poincaré's ingenious analysis of metrical geometry may well have faded from view, were it not for a strong, later endorsement by Einstein. In 1921, Einstein gave a lecture, "Geometry and Experience," before the Prussian Academy of Science. In it he gave his version of Poincaré's argument:

"Why is the equivalence of the practically-rigid body and the body of geometry—which suggests itself so readily—rejected by Poincaré and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behavior, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterizes Poincaré's standpoint. Geometry (G) predicates nothing about the behavior of real things, but only geometry together with the totality (P) of physical laws can do so. Using symbols, we may say that only the sum of (G) + (P) is subject to experimental verification. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions. All that is necessary to avoid contradictions is to choose the remainder of (P) so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometry and the part of natural law, which has been given a conventional status, appear as epistemologically equivalent."

(G) + (P)

This was a powerful endorsement of the conventionality thesis from Einstein just as he was rising to celebrity status, after the success of the 1919 eclipse expeditions that tested his general theory of relativity. Earlier in the lecture, Einstein had stressed the importance of the starting point: that geometry is an empirical science whose laws are to be discovered from measurements. That view, he asserted, was important in his original discovery of general relativity. He explained:

"I attach special importance to the view of geometry, which I have just set forth, because without it I should have been unable to formulate the theory of relativity. Without it the following reflection would have been impossible: in a system of reference rotating relatively to an inertial system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inertial systems on an equal footing, we must abandon Euclidean geometry. Without the above interpretation the decisive step in the transition to generally covariant equations would certainly not have been taken."

Here Einstein is recalling his analysis of rotating disk from his work of 1912. (We discussed it here.) Einstein's rotating disk and Poincaré's disk are not the same, but they are obviously close in their approaches.

Poincare disk      rotating disk

Reichenbach's Universal Forces

The notion of a conventionality of geometry was soon picked up by Einstein's foremost expositor and philosophical commentator, Hans Reichenbach. His 1927 Philosophy of Space and Time gave the conventionality of geometry pride of place in the opening Chapter 1 on space.

His version of the Poincaré disk involved a flat glass surface "G" with a hemispherical hump in the center. Reichenbach's figure is redrawn below:

Recihenbach redrawn

Humans on the glass surface G would explore its surface with measuring rods and soon find that their surface has a hump in it. Reichenbach now imagined a light high above that cast shadows of the rods of G onto a second flat surface E. These shadow rods would return a non-Euclidean spherical geometry in the vicinity of the hump. While all the rods of the surface G would be the same length, their shadows on E would not. So the distance A'B' would equal that of B'C' on the glass hump. But the shadow AB would not be the same length as BC on the surface E. That is, they would differ according to the Euclidean geometry native to E. Reichenbach
Hans Reichenbach

But now imagine, Reichenbach continues, that the measuring rods available to the E-people are forced to conform to the lengths of the shadows, then those rods would return a spherical geometry. What might force them to behave so? One might have temperature differences that enforce the requisite expansions and contractions.

Reichenbach's analysis matches that of Poincaré in general outline. One major difference is that Poincaré considered only geometries of constant curvature. Reichenbach uses a surface which has zero curvature in one place and positive curvature in another. This is an extension that, we might guess, would be unwelcome to Poincaré.

    thermometer Reichenbach did address the obvious weak point in the Poincaré analysis. The use of a temperature based disturbance asks a lot of the reader. We must ignore that the large changes in temperature Poincaré imaged are something we can detect independently of the contraction of rods. As we move closer to absolute zero, different substances liquefy and freeze at different temperatures, which would immediately reveal temperature based disturbances. Even if we ignore those effects, we have the problem that different materials contract and expand differently for the same temperature differences. (Hence a Mercury in glass thermometer can only work because Mercury expands more on heating than does glass.) That would mean that measuring rods made of different substances would return different geometries.

Reichenbach made this problem a focus of extended discussion. He distinguished all those forces that act differently on different substances as "differential forces." Those would not be the basis of his analysis. Instead, he would consider "universal forces." They are, by supposition, free of these effects. They act in the same way on all materials; and there is no way to shield materials from them. To make his and Poincaré's example work, we are to assume that the rods used to explore the space are acted upon by universal forces.

Reichenbach's version of the conventionality thesis could now be stated. The measuring rods we use to explore the geometry of a space are real rods whose lengths have been corrected for all differential forces. If the space is warmer in one area, we know how much that leads our steel rods to expand. We correct for it. However there is no way we can make a corresponding factual correction for universal forces. For there is no way to ascertain by measurement just which are acting. We must stipulate that there are none. This need to stipulate away universal forces makes our choice of idealized measuring rods a matter of definition. It is a "coordinative definition" that coordinates the corrected lengths of real rods with the notion of length in the geometry.

Reichenbach's text became a defining document in philosophy of space and time in the mid and later part of the 20th century and the thesis of the conventionality of geometry figured prominently in it.

Einstein Again

That celebration of the thesis, however, overlooked the fact that Einstein's endorsement of it had been equivocal. Immediately after his summary of it in "Geometry and Experience," Einstein moved to what amounted to a retraction:

"Sub specie aeterni Poincaré, in my opinion, is right. The idea of the measuring-rod and the idea of the clock coordinated with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which must not play any independent part in theoretical physics. But it is my conviction that in the present stage of development of theoretical physics these concepts must still be employed as independent concepts; for we are still far from possessing such certain knowledge of the theoretical principles of atomic structure as to be able to construct solid bodies and clocks theoretically from elementary concepts."

Einstein makes clear here that his endorsement of the conventionality thesis depends upon a particular view of measuring rods (and clocks). They are not to be considered as something outside the scope of the physics under discussion. Their constitution and behavior are to be part of the physics investigated. He is imagining here how he expects physics to be if he could complete his program of the unified field theory. Under that program, all of the geometry of space and time, gravity and all matter fields would be combined into a single unified field. With that goal achieved, Einstein expected, there would be considerable latitude in our division of this field into geometrical and matter parts. As a consequence we could choose a different geometry by making the division in different ways.

What Einstein makes clear is that this time has not yet come. We did not know at the time of his writing (or even now) how to merge everything into one unified field. So we must proceed as we did before. We must take the rods and clocks as idealized elements supplied independently of the physics at issue. Then there is no convention. We must settle for whichever geometry the measurements deliver.

For another, lengthier statement by Einstein of his view on the conventionality of geometry, stated in a 1924 book review, see this.

My View

The thesis of the conventionality of metrical geometry achieved widespread attention because of its connection to Einstein, Reichenbach and, indirectly, general relativity. However, in the form of the thesis given by Poincaré and Reichenbach, there is no logical connection to general relativity. Their arguments for the thesis could have been made at any time since Euclid. They were not since, I believe, in the larger perspective, they are unappealing.

I find it hard to see that an interesting sense of convention in geometry has been identified. Absent the complete reconfiguration of physics imagined by Einstein with his unified field theory, the situation with geometry is just one of fairly straightforward discovery. We have many instruments for measuring length, some less reliable, some more so. We routinely correct the measurement for known deficiencies in the instruments. No careful measurement of a long distance with a metal tape can be precise if we do not correct for the small sag in the tape. When we perform these measurements of space by these different instruments, we recover a single metrical geometry of space. The instruments agree on it. That is our geometry.

The arguments above for conventionality of geometry depend on the notion of a universal force, or something like it. They are forces that, supposedly, act on all of us. Yet there is no way to establish independently what their magnitudes are or even that they are present at all. They are entities protected from evidential scrutiny by careful contrivance. In our earlier treatment of verificationism, I concluded that we should regard such constructs with great suspicion. That applies to universal forces.

If we persist in allowing universal forces in our analysis, then we eliminate any interesting distinction between the conventional and the factual. For analogs of universal forces can be invoked to render any measured quantity conventional. All we need is that the quantity is measured by some instrument or instruments. We posit a universal disturbing force that acts equally on all instruments that measure the magnitude. Hence, there is no empirical way to detect the universal disturbing force. Following Reichenbach, it is a matter of convention that we set the force to zero. That, supposedly, means that we are setting the magnitude by convention. If we accept the arguments that favor the conventionality of geometry, we must also conclude that every measured magnitude is chosen conventionally.


Gravity Geometrized

The general theory of relativity has geometrized gravity.

The thesis is stated tersely since any attempt to expand it leads us directly to the principal difficulty with the claim. It has so many meanings that one despairs of finding a single claim that is really intended across the literature. This is a case in which our main task is merely to discern what the thesis claims. Once we have more precise formulations of the many readings, we need worry less about the arguments that support them. Some are so obviously false that they are dismissed automatically. Others are so clearly asserting truths that we need not labor over them, although we will feel that the truths asserted are much less exciting than the grand image conjured by "geometrization."

Let me lay out some readings.

a. Geometrization as reduction to spatial geometry.

This is the most natural reading that comes to beginners who are struggling to learn general relativity. It is not what is intended by the experts who write the popular texts that speak with enthusiasm of how general relativity had geometrized gravity. However it is how they are understood by their eager and forgiving readers, struggling to grasp an abstruse theory.

The conception is simple. There are numerous cases in science of successful reductions. We once thought that living matter differed in some principled way from dead matter. It was animated by some sort of living spirit without which no mere laboratory experiment could convert dead matter into living matter. Doctor Frankenstein could not reanimate his monster without re-energizing it with the spark of life. Then we learned that all life is just chemical; there is nothing in a living being beyond the chemistry of constituents.

A second example, closer to physics, is light. At the start of the 19th century, light, electricity and magnetism all appeared to be independent things. You could have any one without the others. By mid century, Maxwell was able to show that light really was nothing more than the propagation of a wave-like disturbance in the electromagnetic field. Light was reduced to electromagnetism.


In both cases, the key result is that the chemical properties of a living being are really all there is to its life properties; and the electromagnetic processes in space are all there is to any light present.

The beginner's entirely predictable reading is that general relativity has done the same to gravity. We had thought that gravity was something independent of geometry in this sense. We now learn that gravity really reduces to the geometry of space. The geometry of space is really all there is to any gravity present.

Here the essential point is that geometry is understood in its common meaning: it is the study of the metrical properties of ordinary space. It deals with the distance between points in space, the areas encompassed by lines, and so on. That the geometric facts can also exhaust all gravitational facts is implausible to the beginner and definitely mysterious. Somehow, the beginner is assured, it all works out once we attend to the possibility of a non-Euclidean geometry.

Experts in relativity theory may find it hard to see that this misreading is fostered. They are forgetting the single most misleading image of popular accounts of the theory. It is image of a rubber sheet deformed by mass of a planet. The deformed sheet embodies the non-Euclidean geometry. Somehow a ball rolling around on the sheet illustrates the presence of gravity.

rubber sheet geometry

Beginners are wondering--entirely correctly--if gravity has been geometrized away, what force is pulling the planet down so that it deforms the sheet. The sad truth is that the figure is an irretrievably muddled confusion of ideas.

For a reminder of the part of the physics that this diagram does get right, see The Geometry of Space in the chapter Gravity Near a Massive Body.

b. Geometrization as incorporation into background spacetime structure.

This is the best reading I can find for the thesis of geometrization of gravity.

Prior to general relativity, gravity was conceived as a force deflecting masses away from their natural inertial motion. One would expect that the the greater the mass of the body, the stronger the pull of gravity on it and so the greater the deflection. The curious result was that the greater mass of the body was precisely compensated in Newtonian theory by a correspondingly greater inertia. Hence all bodies in free fall, heavy or light, traced out the same trajectories.

Einstein explained this otherwise curious coincidence. He relocated the trajectories of free fall into the background spacetime structure. Bodies light and heavy followed those trajectories independently of their masses, just as trains, large and small, follow the same track over the mountain.

A Minkowski spacetime was already equipped with preferred trajectories. They are the timelike geodesics that coincide with inertial motions. What Einstein recognized was that free fall trajectories under gravity could be incorporated into this background structure by the simple expedient of introducing curvature into the background. (For more, see Uniqueness of Free Fall in the chapter General Relativity.)

So far, we have a clear and fair accounting of how general relativity altered our theoretical representation of gravity. The difficult question is this: Why should we describe it as geometrization? Recall that the meaning of the term "geometry," at least for our first few millennia, is study of lengths and related metrical properties of of ordinary space. We are now dealing with a spacetime. Why is geometry the right word?

We can see some rationale for it, if we take one step back and recall Minkowski's contribution to special relativity. He found a spacetime formulation of the theory that had striking analogies to the ordinary geometry of space. We saw in an earlier chapter (Spacetime: Minkowski Spacetime Geometry) that they both dealt with a notion of distances in a space. For geometry it was ordinary length. For special relativity it was the interval, which combined measured lengths and times. The analogy was quite close in some aspects. The circles of Euclid are the analogs of the hyperbolas of Minkowski.

circles   hyperbolas

We explain why the airline path from the US to Europe looks curious on an ordinary map. It is a geodesic of the curved geometry of space that is poorly adapted to the Euclidean surface of the map. Correspondingly we explain why a planet in free fall orbits the sun: the orbital trajectory is a geodesic of the spacetime.
    great circle Earth's orbit

The disanalogies remain and are important. Space is the same in all directions, whereas spacetime distinguishes the spatial from the temporal direction. Spacetime must respect the distinction between spatial and temporal. Changes in time are quite distinct from differences of position, even if they all are collected in the one spacetime picture.

This is as close as I can bring the general relativistic account to the spatial geometric account. The two are clearly analogous. But to say that explanations in general relativity are analogous to explanations given in spatial geometry is not the same as saying they are explanations in spatial geometry.

If matters could be left in this state, they would be difficult enough. The complication is that there has a been a slow migration and broadening of the terms "geometric," "geometrize," and so on in the modern literature. Now they have meanings that are so broad that it is hard to know precisely which sense is intended when they are invoked.

One broadening retains the idea that the trajectories of free fall are relocated in the background spacetime structure. However it does it in a way that breaks the analogy between Euclidean geometry and Minkowski spacetime described above. The essential similarity is that both are metrical geometries; that is, they are both concerned essentially with a notion of distance. There are spacetime versions of Newtonian theory that relocate the free fall trajectories into the background spacetime structure in a quite different way. The so-called "affine structure" in a spacetime merely picks out which are the lines designated as straight, without supplying a fully developed notion of times elapsed and distances passed. "Geometrized Newtonian Theory" relocates the free fall trajectories into this structure by curving it. There are other cases of this broadening. One arises in gauge field theories in quantum theory (which I will not try to explain here).

c. Geometric as Coordinate Independent

This is a third way that the term "geometry" is used. It no longer designates physical ideas. Rather it denotes a decision by a theorist to use a particular mathematical formalism. This quite distinct sense of geometrical is commonly tangled up in claims that general relativity has geometrized gravity, for general relativity was the first physical theory to be developed fully in the geometric-as-coordinate-independent manner. The muddling of the two senses is, I suspect, a problem that besets even the experts.

Here's how the notion looks in the simple case of picking out the inertial trajectories in a Minkowski spacetime.

Coordinate dependent approach.

We label events in spacetime with four coordinates: three numbers (x,y,z) for spatial position and one number t for temporal position. To have things work out well, we choose the four coordinates carefully, so that differences of the spatial coordinates correspond to measured spatial lengths in an inertial frame; and differences of the time coordinate correspond to differences of proper time for clocks of the frame.

An inertial trajectory is a timelike curve along the spatial coordinates, x, y, z, are each proportional to the time coordinate t.

This condition is coordinate dependent in the sense that we have to pick the right coordinate system in which to implement it. Use another coordinate system that does not have the close connection between the coordinates and measured spaces and times and the condition fails.

Coordinate independent approach.

We saw earlier that we can pick out the inertial trajectories in a Minkowski spacetime by what we would now call a "geometric" rule:

The inertial trajectories are those timelike curves along which the maximum proper time elapses.

This condition is coordinate independent in the sense that it does not matter which coordinate system we may happen to work in. The proper time elapsed along a timelike curve is the same in all of them.
    geodesic coordinate    geodesic geometric

Generally speaking, more mathematically oriented physicists have come to prefer formalisms that work equally well, no matter which coordinate system is employed, no matter how messy the coordinate system might be. Their fear is that sometimes an odd selection of coordinates can deliver effects that look physical but are merely an artifact of badly chosen coordinates. The structures that work equally well in all coordinate systems are called "geometric objects." A physical law expressed in terms of geometric objects will be written the same in all coordinate systems.

A geometrical treatment of some piece of physical theory is simply one that employs these sophisticated mathematical methods. Since they are merely a mode of description, employing them makes no difference to the physical content of the theory. The use of geometrical methods, in this sense, is merely a warning to the reader that certain advanced techniques will be used. Announcing a geometrization of gravity in this sense makes no physical assertion at all.

Einstein's View

Translations of Einstein quotes in this section are drawn from Dennis Lehmkuhl, "Why Einstein never really cared for geometrization" (Reference link coming soon; in the meantime, contact Dennis.) This work should be consulted for a full account of Einstein's perspective on geometrization in general relativity. One might imagine that Einstein was in the forefront of those urging that general relativity has geometrized gravity. This is not the case at all. Einstein was solidly opposed to the claim and publicly argued against it.

Einstein's clearest statement come in his review of Émile Meyerson 1925 La Déduction Relativiste. In a commentary published in 1928, he first characterized one of Meyerson's theses as a reduction claim:

"Meyerson sees another essential correspondence between Descartes’ theory of physical events and the theory of relativity, namely the reduction of all concepts of the theory to spatial, or rather geometrical, concepts..."

Einstein is sharp and unequivocal in his repudiation:

"...I have an entirely different opinion on the matter. I cannot, namely, admit that the assertion that the theory of relativity traces physics back to geometry has a clear meaning...."

He explains a little later, mentioning the "metric tensor," which is the mathematical quantity that assigns spacetime intervals to displacements along curves in a spacetime:

"...The fact that the metric tensor is denoted as “geometrical” is simply connected to the fact that this formal structure first appeared in the area of study denoted as “geometry”. However, this is by no means a justification for denoting as “geometry” every area of study in which this formal structure plays a role, not even if for the sake of illustration one makes use of notions which one knows from geometry. Using a similar reasoning Maxwell and Hertz could have denoted the electromagnetic equations of the vacuum as “geometrical” because the geometrical concept of a vector occurs in these equations."

In effect Einstein is repudiating the strength of the analogy sketched above between Euclidean geometry and the spacetime structure of general relativity. That we first learned of structures like those used in general relativity from ordinary geometry does not license us to call them all geometrical.

My View

My view is easy to state: I think Einstein is right. And he is right for the reasons he gave.

To review the three senses of "geometrize" given above, (a) proves to be factually incorrect and not really intended by any relativists. It survives mostly as an artifact of oversimplified and hence failed popularizations. (c) is physically empty and thus tells us nothing about the world. Something like (b) is correct and important, but geometrization is a poor description of its content.


Further Morals

There are many, further morals that should be represented in this treatment of the philosophical significance of general relativity. Here I will sketch a few more. These remarks are intended only as a brief inventory. A fuller discussion is needed. I hope that, one day, I will be able to write more chapters that develop them more thoroughly.

Ontology of Space and Time

These theses pertain to the place of space, time and various spacetime structures in our catalog of the basic entities that comprise the world (that is, our "ontology").

Antirealism/Anti-absolutism Concerning Spacetime

Einstein's own conception of the foundations of general relativity involved various antirealist or anti-absolutist theses. The included:

• General relativity has generalized the relativity of motion from inertial motion to accelerated motion.

• General relativity eliminated an absoluteness of inertial frames of reference that was objectionable in the same way as had been Newton's Absolute Space and Time.

• General relativity has implemented Mach's principle. In Einstein's original statement of it, the principle asserted that the inertia of a body results from an interaction between the body and all other bodies. A later version asserts that the matter distribution in spacetime determines the spacetime's metrical geometry completely.

A related thesis that Einstein himself did not stress is that general relativity vindicates a relationist view of spacetime. That view asserts that all that is real in spacetime are relations between events, where some physical occurrence must happen at each related event.

In retrospect it is unclear that any of these theses are vindicated by general relativity. Each remains a matter of debate, as is even the question of how properly to formulate them. Einstein's own presentations have complicated the assessment. His habit was to give expositions of the foundations of his theory that recapitulated the development of his own ideas in the process of discovery of the theory. The danger with such accounts is that we commit a version of the genetic fallacy. That A led Einstein to discover B does not entail that A is foundational or even compatible with B. He also changed the meaning of the principles he judged foundational to general relativity over time.

For a technical survey of some of these issues see my "General Covariance and the Foundations of General Relativity: Eight Decades of Dispute," Reports on Progress in Physics, 56 , pp.791-858

Realism About Spacetime

If one conceives of general relativity as a theory concerning the geometry of spacetime, then a strict realist reading requires us to elevate spacetime itself to a fundamental status in our ontology, that is, in our list of the basic existents of the world. This is sometimes described as a "substantival" view, deriving from the notion in metaphysics of a substance as something that exists independently of other things.

The major obstacle to substantival views about spacetime comes from the "hole argument." The argument was introduced by Einstein during his discovery of general relativity for a different purpose. It was later redirected as an argument against a substantival view of spacetime.

For a survey see my "The Hole Argument," Stanford Encyclopedia of Philosophy. For a development with the least technicalities possible, see Section 4 "The Real" in my “Philosophy in Einstein’s Science," Alternatives to Materialist Philosophies of Science, Philip MacEwen, ed., The Mellen Press, forthcoming.

Theory and Evidence

The general theory of relativity prompted a reappraisal of how evidence bears on theory. There are two sides to the reappraisal. One overturns an older view and the other promotes replacement views.


Since the time of Bacon, philosophers and philosophically-minded scientists have advocated an inductive method for science. The idea was that scientists would gather facts and combine them to form the generalizations that are the laws of science. In the seventeenth century, Bacon urged scientists to assemble tables of absence and presence of some feature of the world and from them to discern the underlying form or law of the feature. In the nineteenth century, Mill urged scientists to see which instances agreed and which differed in some property and from them to infer to the property's cause. The key element is that this was both an inductive logic and a method of discovery.

Through the nineteenth century physical theories became successively more complicated and abstract. However one could still maintain the idea that they were derived somewhat mechanically from experience. Einstein's two principles of special relativity, for example, could be seen as some sort of simple generalization of experience. The principle of relativity, for example, might still be a simple generalization of the failure of ether drift experiments. All ether drift experiments have failed so far. Hence we infer that all ether drift experiments will fail.

With general relativity, however, the gap between experience and the core elements of the theory had become too great. To arrive at general relativity required creative leaps that seemed quite remote from the sorts of mechanical generalizations envisaged by Bacon and Mill. General relativity seemed to provide the refutation of both notions: that physical theories could be arrived at by a process of generalization and that there was an identifiable method governing the process.

Or so Einstein asserts in his 1933 lecture "On the Methods of Theoretical Physics":

"The natural philosophers of those days were, on the contrary, most of them possessed with the idea that the fundamental concepts and postulates of physics were not in the logical sense free inventions of the human mind but could be deduced from experience by "abstraction"--that is to say, by logical means. A clear recognition of the erroneousness of this notion really only came with the general theory of relativity, which showed that one could take account of a wider range of empirical facts, and that, too, in a more satisfactory and complete manner, on a foundation quite different from the Newtonian."

Einstein here alludes to a version of what we would now call the "underdetermination thesis." Both Newtonian theory and general relativity fit empirically very well with the ordinary gravitational phenomena such as were available to Newton. Yet they proceed with very different concepts. Newton represents gravity as a force. Einstein represents it as an aspect of spacetime structure. Thus, Einstein is suggesting, the evidence available to Newton could not pick between the two and thus could not determine the theory. The underdetermination thesis asserts that this situation is widespread to varying degrees according to the version of the thesis.

Einstein concludes the lecture with:

"Physics constitutes a logical system of thought which is in a state of evolution, whose basis cannot be distilled, as it were, from experience by an inductive method, but can only be arrived at by free invention. The justification (truth content) of the system rests in the verification of the derived propositions by sense experiences, whereby the relations of the latter to the former can only be comprehended intuitively."

The idea that physical theories must be arrived at by "free invention" is a distinctive and striking claim by Einstein.

Severe Tests

If inductivism is to be discarded, we need a new account of how evidence bears on theory. Einstein's last remark already showed how he conceived the relation: "The justification (truth content) of the system rests in the verification of the derived propositions..." This is a version of the nineteenth century notion of hypothetico-deductive confirmation: a theory is confirmed when we find that it makes true predictions.

Einstein, however, added an extra twist in his practice that he did not mention above. He sought predictions that would be unlikely to be the case if the theory were false. That is, he chose predictions designed to expose his theory to the greatest chance of failure when the prediction was tested.

Einstein's strategy impressed a young Karl Popper immensely. Popper was uncomfortable with psychoanalysis and other fringe theories, since no matter what happened, the theorists seemed to have an explanation for it. The theories could not fail. It was quite otherwise with general relativity. Popper recalled his experience in 1919 reflecting on the eclipse test of general relativity. (In a 1953 lecture, subsequently published as Science: Conjectures and Refutations.)

"Now the impressive thing about this case is the risk involved in a prediction of this kind. If observation shows that the predicted effect is definitely absent, then the theory is simply refuted. The theory is incompatible with certain possible results of observation--in fact with results which everybody before Einstein would have expected.[1] This is quite different from the situation I have previously described, when it turned out that the theories in question were compatible with the most divergent human behaviour, so that it was practically impossible to describe any human behaviour that might not be claimed to be a verification of these theories. These considerations led me in the winter of 1919-20 to conclusions which I may now reformulate as follows."

What followed was an account of Popper's notion that science proceeds through a cycle of bold conjectures and refutation. It is powered by the theorist seeking to expose the theory to severe test.

No Adjustable Parameters

Closely connected with this last idea was a single example that, oddly enough, involved no prediction at all. It was Einstein's 1915 discovery that his general theory of relativity accommodated the anomalous motion of Mercury exactly. It was not a prediction since the anomalous motion was already known and Einstein had even wondered years before if his developing theory might eventually explain it.

What made the success so powerful was the fact that there were no parameters Einstein could adjust in his theory to fit it to the anomalous motion of Mercury. Either the theory fitted or it didn't. When it did, Einstein gave philosophy of science one of the most powerful examples of how a single datum can lend strong evidence to a theory.

Mathematical Platonism as a Method of Discovery

Einstein himself drew an even stronger lesson from general relativity. Having renounced all hope of an inductive method for discovering scientific theories, he announced in his 1933 Herbert Spenser lecture that there is another method. True laws are expressed in mathematically simple terms, so we can find them merely by sifting through the simple mathematical expressions. He wrote what is surely one of the most astonishing manifestoes of a great scientist:

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed."

For my account of how Einstein derived his Platonism from his experience with general relativity, see "'Nature in the Realization of the Simplest Conceivable Mathematical Ideas¹: Einstein and the Canon of Mathematical Simplicity," Studies in the History and Philosophy of Modern Physics, 31 (2000), pp.135-170.

Cosmic Morals

God and the Big Bang

The King James version of the bible begins in Genesis with the following memorable account of the origin of the world:

In the beginning God created the heaven and the earth.
And the earth was without form and void, and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
And God said, “Let there be light”; and there was light.
And God saw the light, that it was good; and God divided the light from the darkness.

Should we see the big bang as a vindication of this account of the Genesis account of creation? The big bang is hot and bright. The moment of creation happens when God says "Let there be light."

Should we see in the big bang an account of miraculous creation? For big bang cosmology initiates the universe in a singularity, which is a breakdown of physical laws. Are such breakdowns miracles?

These are a vexed questions as is the relationship between science and religion more generally. We find theistic scientists arguing passionately for relations of compatibility and support between science and theism. And we find atheists urging the reverse with equal fervor. For a convenient survey from the atheist perspective, See Ch.4 of Colin Howson, Objecting to God.

My view is that theists cannot use big bang cosmology as support for the Genesis account. The difficulty is obvious. The moment of light is preceded by "darkness upon the face of the deep." Read literally, that tells us that the antecedent stage was a watery one. Big bang cosmology affords no water prior to the big bang. This is a fatal mismatch. If one wants to read the talk of the waters metaphorically, then the possible readings of the few word in Genesis become so great in number that the discoveries of big bang cosmology no longer provide a test of the Genesis account.

More generally, it is tempting to imagine that God is present at the big bang as an agency bringing about the world. Whatever cogency may be in the idea of God's creative agency, there seems to be no additional support for it in big bang cosmology, as opposed to a cosmology with an infinite past in time. For there is no moment of time of the big bang. A God who is metaphorically there is outside time and space, just as a God who creates a world with an infinite time does so from outside space and time.

Finally, the identification of the big bang with a miracle is a suspiciously selective choice that appears driven as much by prior belief as indepdent reasons. There are more singularities in our spacetime theories.What of singularities in black holes? Are we to believe that God places miracles at the end of gravitational collapse? If they are ordinary occurrences with no divine connection, then why are we singling out the big bang singularity as extraordinary? Worse, some cosmological models have the universe ending in a big crunch singularity. Is that awful end the triumph of the devil in some sort of anti-miracle?

Completing An Infinity

A standard notion in the theory of computation is that it is impossible to complete a computation that requires infinitely many steps. It is easy to think of this as a logical requirement. It is not. It is a physical requirement. That has been made clear by the identification of spacetimes in which the infinite lifetime of one entity can be fully contained within the past light cone of another.

In such a spacetime, the first entity is an idealized calculator who is programmed to compute for an infinity of the calculator's proper time. For example, the calculator may check Goldbach's conjecture that every even number is the sum of two primes. The calculator simply goes through the infinite list 2, 4, 6, 8, ... checking each even number. If a counterexample is found, the calculator sends out a signal that will be received by the second entity, the observer. The calculuator will never know at any finite time in its life whether a counterexample is found. However the observer will know if one is found, since the observer can survey the entire, infinite life of the calculator. The calculator-observer combination can now compute things that are normally regarded as uncomputable.

The key element is that we need access to a particular sort of spacetime, which shows that it is a matter of physical contingency that we have no such combinations.

For an account of these spacetimes, see my paper with John Earman, "Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes," Philosophy of Science, 60 (1993), pp. 22-42.

Copyright John D. Norton. February 23, 2013.