For physics, geometry has an added value. Geometric relationships have an absolute character, that is, they do not depend on the point of view of whoever considers them. An example is the statement that the sum of the angles of a plane triangle is 180 degrees. This is clearly an intrinsic property of the triangle and does not depend at all on the position of the triangle with respect to the observer.
Presumably, the relationships between physical objects must have the same nature and that, therefore, the use of a geometric language for their description seems pertinent. However, by involving the time variable, we move away from the geometric paradigm, since it seems to be restricted to the spatial relationships between objects. These relationships have, in a sense, a timeless character. The formulation of the laws of motion takes as its starting point the notion of force. This is a concept related to the physiological sensation we experience when trying to move objects in space. In Newtonian mechanics, force is the cause that gives rise to the acceleration of a body. Knowing the forces, we are able to project the movement into the future and predict how it will unfold. Although the resulting trajectory has a geometric character, the forces themselves have a different status in theory and are the reason why the object follows this trajectory.
Thus, for example, Newton introduced the idea of a universal gravitational force that depended on the distance at which the considered objects were. The fundamental problem of celestial dynamics consisted, therefore, in the determination of the movement from the changing forces that were operating on the objects as they modified their relative distances. On the way to the ‘geometrization’ of physics, it was crucial to understand that the same trajectory can be traversed in many ways.
The speed, for example, can be small initially and gradually increase over time or vice versa. The geometrization of movement requires a kind of fusion between space and time. If we only consider its spatial part, we do not capture the full essence of it. It was not until the advent of Einstein’s theory of special relativity that this fusion acquired the legitimacy of the great ideas in physics. Einstein showed that space and time are closely related, as much as the width and length of a table can be. It is a question of ‘perspective’. Thus, for example, to determine the time interval between two events for an observer in motion, it is necessary to know both the spatial and temporal separation between these events for an observer at rest. Determining ‘temporal width’ requires knowledge of ‘spatial length’ and ‘temporal width’, both observed from another point of view (namely, that of the observer at rest). This is in every way analogous to the purely spatial case where the determination of the spatial width of a table, from a given perspective, would depend on the knowledge of the spatial length and width observed under a reference angle.
As has been said, the Gordian knot is not, however, in the geometric vision of movement but in that of the concept of force . Although the notion of force, as we have seen, was originally motivated by a purely physiological sensation, it was soon discovered that it was necessary to consider those of a more abstract nature, such as those that act at a distance. The force of gravitation proposed by Newton belonged to this category. It was Einstein who, again, had a pioneering role in this ‘geometrizing trend’ by getting the force of gravity to be replaced by the geometry of space-time itself. The motion of a particle under gravitational action was completely determined by the condition that the path followed by it was a geodesic. The shape of space-time is given by the distribution of matter in it, thus implementing the notion that masses are responsible for the creation of the ‘gravitational field’.
Perhaps surprisingly, the geometric paradigm has required the quantum revolution for its development. This has given rise to the possibility of defining “internal spaces” associated with the different types of particles. In the case of the quantum theory of electricity and magnetism, it is possible to conceive of this space as a small circumference associated with each point in space-time. To give an account of the history of a particle with an electric charge requires selecting a specific place in each of these circumferences, giving rise to a kind of ‘generalized trajectory’ where a line in space-time is replaced by a specific configuration of the called ‘particle wave function’. These more abstract trajectories are determined by the “deformation” that electric charges distributed through space-time generate in internal space.
The generalization of these ideas also allows the geometrization of the weak and strong nuclear interactions, giving rise to the so-called standard model of particle physics. Curiously, the first interaction arranged in purely geometric terms has remained outside the quantum-mechanical framework, and today, this is considered one of the central problems of physics: the quantization of gravity.
We could say that the historical development of physics has led us to be close to being able to ‘see is to believe’, although this forces us to delve into the bowels of matter in order to discover the geometric characteristics of its internal spaces.
Salvador Sánchez is a doctor in Physics and a member of the Faculty of the Degree in Physics at UNIR.
