9.2 Up To Diffeomorphism

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 x^mn 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?