9.2  Up To Diffeomorphism


The mind of man is more intuitive than logical, and comprehends more than it can coordinate.

                                                                                                Vauvenargues, 1746


Einstein seems to have been strongly wedded to the concept of the continuum described by partial differential equations as the only satisfactory framework for physics. He was certainly not the first to hold this view. For example, in 1860 Riemann wrote


As is well known, physics became a science only after the invention of differential calculus. It was only after realizing that natural phenomena are continuous that attempts to construct abstract models were successful… In the first period, only certain abstract cases were treated: the mass of a body was considered to be concentrated at its center, the planets were mathematical points… so the passage from the infinitely near to the finite was made only in one variable, the time [i.e., by means of total differential equations]. In general, however, this passage has to be done in several variables… Such passages lead to partial differential equations… In all physical theories, partial differential equations constitute the only verifiable basis. These facts, established by induction, must also hold a priori. True basic laws can only hold in the small and must be formulated as partial differential equations.


Compare this with Einstein’s comments (see Section 3.2) over 70 years later about the unsatisfactory dualism inherent in Lorentz’s theory, which expressed the laws of motion of particles in the form of total differential equations while describing the electromagnetic field by means of partial differential equations. Interestingly, Riemann asserted that the continuous nature of physical phenomena was “established by induction”, but immediately went on to say it must also hold a priori, referring somewhat obscurely to the idea that “true basic laws can only hold in the infinitely small”. He may have been trying to convey by these words his rejection of “action at a distance”. Einstein attributed this insight to the special theory of relativity, but of course the Newtonian concept of instantaneous action at a distance had always been viewed skeptically, so it isn’t surprising that Riemann in 1860 – like his contemporary Maxwell – adopted the impossibility of distant action as a fundamental principle. (It’s interesting the consider whether Einstein might have taken this, rather than the invariance of light speed, as one of the founding principles of special relativity, since it immediately leads to the impossibility of rigid bodies, etc.) In his autobiographical notes (1949) Einstein wrote


There is no such thing as simultaneity of distant events; consequently, there is also no such thing as immediate action at a distance in the sense of Newtonian mechanics. Although the introduction of actions at a distance, which propagate at the speed of light, remains feasible according to this theory, it appears unnatural; for in such a theory there could be no reasonable expression for the principle of conservation of energy. It therefore appears unavoidable that physical reality must be described in terms of continuous functions in space.


It’s worth noting that while Riemann and Maxwell had expressed their objections in terms of “action at a (spatial) distance”, Einstein can justly claim that special relativity revealed that the actual concept to be rejected was instantaneous action at a distance. He acknowledge that “distant action” propagating at the speed of light – which is to say, action over null intervals – is remains feasible. In fact, one could argue that such “distant action” was made more feasible by special relativity, especially in the context of Minkowski’s spacetime, in which the null (light-like) intervals have zero absolute magnitude. For any two light-like separated events there exist perfectly valid systems of inertial coordinates in terms of which both the spatial and the temporal measures of distance are arbitrarily small. It doesn’t seem to have troubled Einstein (nor many later scientists) that the existence of non-trivial null intervals potentially undermines the identification of the topology of pseudo-metrical spacetime with that of a true metric space. Thus Einstein could still write that the coordinates of general relativity express the “neighborliness”  of events “whose coordinates differ but little from each other”. As argued in Section 9.1, the assumption that the physically most meaningful topology of a pseudo-metric space is the same as the topology of continuous coordinates assigned to that space, even though there are singularities in the invariant measures based on those coordinates, is questionable. Given Einstein’s aversion to singularities of any kind, including even the coordinate singularity at the Schwarzschild radius, it’s somewhat ironic that he never seems to have worried about the coordinate singularity of every lightlike interval and the non-transitive nature of “null separation” in ordinary Minkowski spacetime.


Apparently unconcerned about the topological implications of Minkowski spacetime, Einstein inferred from the special theory that “physical reality must be described in terms of continuous functions in space”. Of course, years earlier he had already considered some of the possible objections to this point of view. In his 1936 essay on “Physics and Reality” he considered the “already terrifying” prospect of quantum field theory, i.e., the application of the method of quantum mechanics to continuous fields with infinitely many degrees of freedom, and he wrote


To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, on principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space.


In his later search for something beyond general relativity that would encompass quantum phenomena, he maintained that the theory must be invariant under a group that at least contains all continuous transformations (represented by the symmetric tensor), but he hoped to enlarge this group.


It would be most beautiful if one were to succeed in expanding the group once more in analogy to the step that led from special relativity to general relativity. More specifically, I have attempted to draw upon the group of complex transformations of the coordinates. All such endeavours were unsuccessful. I also gave up an open or concealed increase in the number of dimensions, an endeavor that … even today has its adherents.


The reference to complex transformations is an interesting fore-runner of more recent efforts, notably Penrose’s twistor program, to exploit the properties of complex functions (cf Section 9.9). The comment about increasing the number of dimensions certainly has relevance to current “string theory” research. Of course, as Einstein observed in an appendix to his Princeton lectures, “In this case one must explain why the continuum is apparently restricted to four dimensions”. He also mentioned the possibility of field equations of higher order, but he thought that such ideas should be pursued “only if there exist empirical reasons to do so”.  On this basis he concluded


We shall limit ourselves to the four-dimensional space and to the group of continuous real transformations of the coordinates.


He went on to describe what he (then) considered to be the “logically most satisfying idea” (involving a non-symmetric tensor), but added a footnote that revealed his lack of conviction, saying he thought the theory had a fair probability of being valid “if the way to an exhaustive description of physical reality on the basis of the continuum turns out to be at all feasible”. A few years later he told Abraham Pais that he “was not sure differential geometry was to be the framework for further progress”, and later still, in 1954, just a year before his death, he wrote to his old friend Besso (quoted in Section 3.8) that he considered it quite possible that physics cannot be based on continuous structures. The dilemma was summed up at the conclusion of his Princeton lectures, where he said


One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers… but this does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory.


The area of current research involving “spin networks” might be regarded as attempts to obtain an algebraic basis for a theory of space and time, but so far these efforts have not achieved much success. The current field of “string theory” has some algebraic aspects, but it seems to entail much the same kind of dualism that Einstein found so objectionable in Lorentz’s theory. Of course, most modern research into fundamental physics is based on quantum field theory, about which Einstein was never enthusiastic – to put it mildly. (Bargmann told Pais that Einstein once “asked him for a private survey of quantum field theory, beginning with second quantization. Bargman did so for about a month. Thereafter Einstein’s interest waned.”)


Of all the various directions that Einstein and others have explored, one of the most intriguing (at least from the standpoint of relativity theory) was the idea of “expanding the group once more in analogy to the step that led from special relativity to general relativity”. However, there are many different ways in which this might conceivably be done. Einstein referred to allowing complex transformations, or non-symmetric, or increasing the number of dimensions, etc., but all these retain the continuum hypothesis. He doesn’t seem to have seriously considered relaxing this assumption, and allowing completely arbitrary transformations (unless this is what he had in mind when he referred to an “algebraic theory”). Ironically in his expositions of general relativity he often proudly explained that it gave an expression of physical laws valid for completely arbitrary transformations of the coordinates, but of course he meant arbitrary only up to diffeomorphism, which in the absolute sense is not very arbitrary at all.


We mentioned in the previous section that diffeomorphically equivalent sets can be assigned the same topology, but from the standpoint of a physical theory it isn't self-evident which diffeomorphism is the right one (assuming there is one) for a particular set of physical entities, such as the events of spacetime.  Suppose we're able to establish a 1-to-1 correspondence between certain physical events and the sets of four real-valued numbers (x0,x1,x2,x3).  (As always, the superscripts are indices, not exponents.)  This is already a very strong supposition, because the real numbers are uncountable, even over a finite range, so we are supposing that physical events are also uncountable.  However, I've intentionally not characterized these physical events as points in a certain contiguous region of a smooth continuous manifold, because the ability to place those events in a one-to-one correspondence with the coordinate sets does not, by itself, imply any particular arrangement of those events.  (We use the word arrangement here to signify the notions of order and nearness associated with a specific topology.)  In particular, it doesn't imply an arrangement similar to that of the coordinate sets interpreted as points in the four-dimensional space denoted by R4


To illustrate why the ability to map events with real coordinates does not, by itself, imply a particular arrangement of those events, consider the coordinates of a single event, normalized to the range 0-1, and expressed in the form of their decimal representations, where xmn denotes the nth most significant digit of the mth coordinate, as shown below


                                    x0 = 0.  x01 x02 x03 x04   x05 x06 x07 x08  ...

                                    x1 = 0.  x11 x12 x13 x14   x15 x16 x17 x18  ...

                                    x2 = 0.  x21 x22 x23 x24   x25 x26 x27 x28  ...

                                    x3 = 0.  x31 x32 x33 x34   x35 x36 x37 x38  ...


We could, as an example, assign each such set of coordinates to a point in an ordinary four-dimensional space with the coordinates (y0,y1,y2,y3) given by the diagonal sets of digits from the corresponding x coordinates, taken in blocks of four, as shown below


                                    y0 = 0.  x01 x12 x23 x34   x05 x16 x27 x38  ...

                                    y1 = 0.  x02 x13 x24 x31   x06 x17 x28 x35  ...

                                    y2 = 0.  x03 x14 x21 x32   x07 x18 x25 x35  ...

                                    y3 = 0.  x04 x11 x22 x33   x08 x15 x26 x37  ...


We could also transpose each consecutive pair of blocks, or scramble the digits in any number of other ways, provided only that we ensure a 1-to-1 mapping.  We could even imagine that the y space has (say) eight dimensions instead of four, and we could construct those eight coordinates from the odd and even numbered digits of the four x coordinates.  It's easy to imagine numerous 1-to-1 mappings between a set of abstract events and sets of coordinates such that the actual arrangement of the events (if indeed they possess one) bears no direct resemblance to the arrangement of the coordinate sets in their natural space.


So, returning to our task, we've assigned coordinates to a set of events, and we now wish to assert some relationship between those events that remains invariant under a particular kind of transformation of the coordinates.  Specifically, we limit ourselves to coordinate mappings that can be reached from our original x mapping by means of a simple linear transformation applied on the natural space of x.  In other words, we wish to consider transformations from x to X given by a set of four continuous functions  f i with continuous partial first derivatives.  Thus we have


                                                X0  =  f 0 (x0 , x1 , x2 , x3)

                                                X1  =  f 1 (x0 , x1 , x2 , x3)

                                                X2  =  f 2 (x0 , x1 , x2 , x3)

                                                X3  =  f 3 (x0 , x1 , x2 , x3)


Further, we require this transformation to posses a differentiable inverse, i.e., there exist differentiable functions Fi such that


                                                x0  =  F0 (X0 , X1 , X2 , X3)

                                                x1  =  F1 (X0 , X1 , X2 , X3)

                                                x2  =  F2 (X0 , X1 , X2 , X3)

                                                x3  =  F3 (X0 , X1 , X2 , X3)


A mapping of this kind is called a diffeomorphism, and two sets are said to be equivalent up to diffeomorphism if there is such a mapping from one to the other.  Any physical theory, such as general relativity, formulated in terms of tensor fields in spacetime automatically possess the freedom to choose the coordinate system from among a complete class of diffeomorphically equivalent systems.  From one point of view this can be seen as a tremendous generality and freedom from dependence on arbitrary coordinate systems.  However, as noted above, there are infinitely many systems of coordinates that are not diffeomorphically equivalent, so the limitation to equivalent systems up to diffeomorphism can also be seen as quite restrictive. 


For example, no such functions can possibly reproduce the digit-scrambling transformations discussed previously, such as the mapping from x to y, because those mappings are everywhere discontinuous.  Thus we cannot get from x coordinates to y coordinates (or vice versa) by means of continuous transformations.  By restricting ourselves to differentiable transformations we're implicitly focusing our attention on one particular equivalence class of coordinate systems, with no a priori guarantee that this class of systems includes the most natural parameterization of physical events.  In fact, we don't even know if physical events possess a natural parameterization, or if they do, whether it is unique.


Recall that the special theory of relativity assumes the existence and identifiability of a preferred equivalence class of coordinate systems called the inertial systems.  The laws of physics, according to special relativity, should be the same when expressed with respect to any inertial system of coordinates, but not necessarily with respect to non-inertial systems of reference.  It was dissatisfaction with having given a preferred role to a particular class of coordinate systems that led Einstein to generalize the "gage freedom" of general relativity, by formulating physical laws in pure tensor form (general covariance) so that they apply to any system of coordinates from a much larger equivalence class, namely, those that are equivalent to an inertial coordinate system up to diffeomorphism.  This entails accelerated coordinate systems (over suitably restricted regions) that are outside the class of inertial systems.  Impressive though this achievement is, we should not forget that general relativity is still restricted to a preferred class of coordinate systems, which comprise only an infinitesimal fraction of all conceivable mappings of physical events, because it still excludes non-diffeomorphic transformations.


It's interesting to consider how we arrive at (and agree upon) our preferred equivalence class of coordinate systems.  Even from the standpoint of special relativity the identification of an inertial coordinate system is far from trivial (even though it's often taken for granted).  When we proceed to the general theory we have a great deal more freedom, but we're still confined to a single topology, a single pattern of coherence.  How is this coherence apprehended by our senses?  Is it conceivable that a different set of senses might have led us to apprehend a different coherent structure in the physical world?  More to the point, would it be possible to formulate physical laws in such a way that they remain applicable under completely arbitrary transformations?


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