The spacetime coincidence

In a footnote of his fundamental article on General Relativity [61] Einstein wrote: ``We assume the possibility of verifying simultaneity for events immediately proximate in space, or, to speak more precisely, for immediate proximity or coincidence in spacetime, without giving a definition of this fundamental concept''.

In order to give a formal interpretation to this sentence, we consider a set $\mathcal{O}$ of ``objects'' and we assume that one can define operationally in a unique objective way an equivalence relation between them called spacetime coincidence. The equivalence classes are the elements of the spacetime $\mathcal{M}$, namely the events.

The elements of $\mathcal{O}$ are characterized by geometric properties and possibily also by dynamical properties, like energy, momentum, mass and so on. The interplay of geometric and dynamical quantities in theories with a modified spacetime has been discussed by several authors [62,63] and finds perhaps its origin in refs. [64,65].

If we disregard provisionally the dynamical properties, it is natural to identify the set $\mathcal{O}$ with the manifold $\mathcal{S}_n$. If it has a structure of fibre bundle as we have assumed in Chapter 1, the fibers are the equivalence classes for the relation of spacetime coincidence and Einstein's assumption is satisfied.

The full structure of principal fibre bundle is not necessary for a spacetime interpretation of the theory. We may more simply assume that the equivalence classes are $(6 + n)$-dimensional differentiable submanifolds of $\mathcal{S}_n$ and that, with a suitable choice of a basis in $\mathcal{T}_n$, the vector fields $A_{\alpha}$ with $\alpha = 4,\ldots, 9+n$ are tangent to them. The Lie bracket of two of these fields is also tangent to the equivalence classes and we obtain the condition

$$F_{\alpha \beta}^i = 0, \qquad \alpha, \beta = 4,\ldots, 9 + n, \qquad i = 0,\ldots, 3.$$ (2.2)

One may ask if the condition (2.2) is sufficient for a spacetime interpretation. It means that the subspaces of $T_s \mathcal{S}_n$ generated by the vectors $A_4(s),\ldots, A_{9+n}(s)$ (we continue to call them ``vertical'' subspaces) form an integrable distribution, namely they satisfy the condition of the Frobenius theorem [25,26] which assures the existence of a foliation of $\mathcal{S}$. This means that for every point $s \in \mathcal{S}$ there are connected $(6 - n)$-dimensional submanifolds, called integral manifolds containing $s$ and tangent to the subspaces that form the distribution. One of these submanifolds contains all the others and is called the leaf containing $s$.

The same problem can also be treated by means of the forms $\omega^{\alpha}$. The vertical subspaces can be defined by means of the differential system

$$\omega^i = 0.$$ (2.3)

These differential forms must vanish when restricted to an integral manifold and their exterior derivatives

$$d \omega^i = - 2^{-1} F_{\alpha \beta}^i \omega^{\alpha} \wedge \omega^{\beta}$$ (2.4)

must have the same property. In this way we obtain again the condition (2.2), which is the integrability condition for the differential system (2.3).

We can consider the leaves as the equivalence classes of a relation of spacetime coincidence and consider them as the elements of a set $\mathcal{M}$. We indicate by $\pi$ the projection of $\mathcal{S}$ on $\mathcal{M}$ and we say that a set $I \subset \mathcal{M}$ is open if its inverse image $\pi^{-1}(I)$ is open. With this definition $\mathcal{M}$ is a topological space, but the Hausdorff separation axiom is not necessarily satisfied.

A minor problem is that, at least if $\mathcal{S}$ is a bundle of frames, every equivalence class (fiber) has two connected components containing left-handed and right handed tetrads. As a consequence it is composed of two leaves and every point of the spacetime is obtained twice. Alternatively, one can say that one obtains a double covering of $\mathcal{M}$.

One can also try to define a structure of differentiable manifold on $\mathcal{M}$. In the proof of Frobenius' theorem one shows that in a neighborhood of every point of $s \in \mathcal{S}_n$ one can find a cubic coordinate system $\xi^{\alpha}$ with $\Vert\xi^{\alpha}\Vert <d$, in such a way that the surfaces (slices) defined by fixing the values of the four coordinates $\xi^0,\ldots, \xi^3$ are integral manifolds and therefore individuate a point of $\mathcal{M}$. From a local point of view we can consider the functions $\xi^0,\ldots, \xi^3$ as local spacetime coordinates and say that, as a consequence of eq. (2.2) the theory has a local spacetime interpretation.

From a global point of view, the functions $\xi^0,\ldots, \xi^3$ provide a local coordinate system of $\mathcal{M}$ only if different values of them correspond to different points of $\mathcal{M}$, namely if different slices belong to different leaves. In this case we say that the cubic coordinate system is regular. If near every point of $\mathcal{S}$ one can find a regular cubic coordinate system, one can consider $\mathcal{M}$ as a differentiable manifold and a global spacetime interpretation is established.

Unfortunately, often this regularity condition is not satisfied. It is useful to illustrate the situation by means of a simple example. We assume that the Poincaré group $\mathcal{P}$ acts on $\mathcal{S}$ on the right transitively, but not freely. The infinitesimal transformations are described by the vector fields $A_{\alpha}$ and the structure coefficients are the structure constants of the Poincaré Lie algebra, so that the condition (2.2) is satisfied. The manifold $\mathcal{S}$ is a homogeneous space and is described by the quotient $\mathcal{H} \backslash \mathcal{P}$ where $\mathcal{H}$ is the stabilizer of a given element $\hat s \in \mathcal{S}$, a closed subgroup of $\mathcal{P}$.

We assume that $\mathcal{H}$ is the Abelian discrete subgroup containing the elements

$$\exp\left( p A_{[01]} + (p a + q b) A_2 \right),$$ (2.5)

where $p$ and $q$ are integers, $a$ and $b$ are real and $a^{-1} b$ is irrational. The numbers of the form $p a + q b$ are dense on the real line and one can find the sequences $\{p_r\}$ and $\{q_r\}$ with the properties

$$\lim_{r \to \infty} p_r = + \infty, \qquad \lim_{r \to \infty} (p_r a + q_r b) = 0.$$ (2.6)

We obtain

$$\lim_{r \to \infty} \hat s \exp\left( p_r A_{[01]}\right) = ... ...nfty} \hat s \exp\left(- (p_r a + q_r b) A_2 \right) = \hat s$$ (2.7)

All the elements of this sequence belong to the same leaf, but not to the same slice, since $p_r a + q_r b \neq 0$. It follows that the regularity condition is not satisfied.

This example shows that there is no hope to assure the existence of a global space-time interpretation by means of local conditions on the structure coefficients. Since we are mainly interested in the study of local field equations, we shall not consider this problem any more.

The equivalence relation considered above can be called primary local spacetime coincidence, because it is defined directly by means of the fundamental vector fields $A_{\alpha}$. In Chapter 7 we define a more general concept of secondary local spacetime coincidence, that also depends on the structure coefficients.

If $n > 0$, one can also define the weaker concept of physical equivalence of extended frames. We say that two extended frames are physically equivalent if they differ by a gauge transformation, namely they belong to the same fiber of $\mathcal{S}_n$ considered as a principal fiber bundle with base $\mathcal{S}$ and structural group $\mathcal{G}$. The whole treatment given above can be repeated in this different context and we find the necessary condition

$$F_{a b}^{\alpha} = 0, \qquad a, b = 10,\ldots, 9 + n, \qquad \alpha = 0,\ldots, 9.$$ (2.8)