Flow of time and flow of space

Time flows. That much is not surprising. Either time flows through us, or we flow through time—the effect feels the same, as long as the direction stays fixed. We call it the arrow of time. 

The arrow of time mystery

 Here, theory and experience part ways. Physicists and philosophers keep trying to explain—or to explain away—the arrow of time. Is it objective? Or merely subjective? And is its mystery somehow intertwined with that of consciousness? 

 Recently, I attended a seminar on the arrow of time, where one approach drew on the relational theory of Yu. S. Vladimirov. What I noticed was this: while the speaker aimed for an “objective explanation,” he could not avoid repeatedly saying “we.” I resisted the impulse to ask, “Who are these ‘we’ within the framework you propose?” I felt entitled to ask, having already confronted that very question myself—in a paper on the quantum theory of measurement. 

 So we have the arrow of time, we have irreversibility, and we have an unease about time loops—except, perhaps, in our rare dreams.

Here is the new version of my notes, with added Ch. 6.1 "Time-like streams", and 6.2. "Space-like streams," with new Figures 10,11,12.

 

 Link to pdf file.

In the next post we take a look at what kind of life can reside on the two  infinity circles?

Shape of infinity

 Our model toy universe is a torus It is a homogeneous space for the group SO₀(2,2). It does not carry any natural metric. But it carries, as we will see later, a natural conformal structure. Thus there are no ``geodesics'' -- ``shortest'' lines connecting points, but there are natural isotropic lines, or ``null'' lines - they represent ``light rays''. If we select a point p, there are two light rays emanating from p, one ``to the right'', and one ``to the left''. 

Fig. 1 Two Minkowski spacetimes with their boundaries at infinity.

They form two circles. So we have, automatically two circular light rays intersecting at p, and also in the opposite point (-p). These two light rays form p^⟘ .

Shape of infinity

These are red and blue circles on Fig. 1. These two circles of light form the ``infinity'' boundary for the complement, which splits into two disjoint open regions -- red and blue. Each region carries a natural structure of an affine space. It can be identify with Minkowski space-time, endowed with Minkowski space-time quadratic form (or ``metric''). But this Minkowski metric depends on the choice of p. Thus the torus as a whole does not have a natural metric. The stability group of $p$ consists of time and space translations, Lorentz boosts, and dilations. It We can arbitrarily select an ``origin'' in one of these two Minkowski spaces. This further breaks the symmetry to just Lorentz boosts and dilations. Every light ray through a point in one Minkowski space intersects with two light rays at infinity, somewhere in the middle between (p) and (-p). But no such light ray passes through the points (p) or (-p). This opens the following question: which paths in Minkowski space goes to (p) and (-p)?

We will address this question in the forthcoming post.

The whole document containing the details is shown below. You can also download the pdf here. The old notes have been essentially expanded and updated.