The Mystical Formula and The Mystery of Khronos

Abstract

In 1908, Minkowski put forward the idea that invariance under what we call today the Lorentz group, GL(1, 3, R), would be more meaningful in a fourdimensional space-time continuum. This suggestion implies that space and time are intertwined entities so that, kinematic and dynamical quantities can be expressed as vectors, or more generally by tensors, in the four-dimensional space-time. Minkowski also showed how causality should be structured in the four-dimensional vector space. The mathematical formulation proposed by Minkowski made its generalization to curved spaces quite natural, leaving the doors to the General Theory of Relativity and many other developments ajar.Nevertheless, it is remarkable that this deceptively simple formulation eluded many researchers of space and time, and goes against our every day experience and perception, according to which space and time are distinct entities. In this contribution, we discuss these contradictory views, analyze how they are seen in contemporary physics and comment on the challenges that space-time explorers face.

Cosmological thinking: cultural heritage and challenge

“…Despite these considerations, it would not be too exaggerated to state that contemporary cosmology is an indissoluble part of the post-modern thinking. Indeed, science, and cosmology in particular, do permeate our culture, and one has already seen some of these manifestations in contemporary arts 2 science fiction, cinema 3 , music [22] and so on 4 . Literature and most particularly, contemporary literature, is not indifferent to the appeal of cosmology.…”

“…Thus, theoretical cosmology has emerged from the understanding that the general theory of relativity was the suitable framework to address the cosmological problem, given the that this theory concerns the global nature of the four dimensional space-time (see Ref. [4] for a discussion of this historical development). In the context of the general theory of relativity, space-time acquires a plasticity that is shaped by matter configurations and can therefore behave in quite unusual and unfamiliar ways (see e.g.…”

“…Figure2depicts her view of the unifying model of dark energy and dark matter, the generalized Chaplygin gas[21] 3. The scene of a boy in one of Woody Allen's films -most likely the director himself -who refuses to eat because the Universe is expanding is an example that strikes the memory of this author 4. One could also recall that "Big Bang", "Neutrinos", "Gamma-Ray Bursts", "Spiders from Mars", "Dark Matter" and the like are the names of some rock bands; "Supermassive Black Holes", the motto of a rock song.…”

The Spirit of Unification: The Wei of Physics

“…1908: Minkowski shows that SR is more meaningful with a further unification of space and time so to treat physical phenomena in a space-time continuum (see Refs. [1,2] for thorough discussions).…”

“…that breaks down, through the Higgs mechanism, to Maxwell's theory U EM (1). The fundamental spontaneous symmetry breaking mechanism that endows the Higgs field with a non-vanishing vacuum expectation value (experimentally, < 0|H|0 >= 246 GeV ) is on its own a very interesting example of the unity of Nature, as it is the very process underlying phase transitions in condensed matter physics, the Higgs field vacuum expectation value playing the role of an order parameter, the Higgs field effective potential being equivalent to Helmholtz's free energy.…”