Arche in physics (Hartle-Hawking boundary condition)

I will now stop talking about the mysticism of the 4-dimensional arche as depth and introduce how the arche (the beginning of the universe) is considered in modern cosmology.

As is well known, the mainstream of modern cosmology is based on the Big Bang Theory. According to the big bang theory, the universe suddenly underwent a rapid expansion called inflation from a minimal point called the singularity about 13.7 billion years ago. It continues to expand even now at the rate of 73 km (45.3 mi) per second per 3.3 million light-years away from the Earth (called the Hubble constant).

However, a singularity, which is considered the starting point of the universe, is a great annoyance to physicists. The singularity is inevitably derived from Einstein’s cosmological equations, where the energy density and temperature become infinite. Therefore, no matter what equations are formulated, they become incalculable, and all physical laws break down. This is an untenable situation for physicists who want to imagine somehow how the universe began with the power of reason.

Groundbreaking physicist Stephen Hawking thought out a way to avoid this troublesome singularity. In 1983, Hawking and James Hartle published a novel hypothesis called Hartle-Hawking boundary condition. The theory postulates that an imaginary time universe (also known as the quantum gravity period) existed before the beginning of the real-time universe and mathematically shows what the universe was like before the Big Bang. When imaginary-time (it) is applied to the equation of four-dimensional invariant distance, the distinction between time and space becomes invalid, and the singularity itself disappears.

This situation can be explained intuitively, as shown in the diagrams below. Think of the expanding universe as shown in Figure 1. The expansion with time t is shown as a cone in this figure. The tip of the cone is pointy, and this tip is considered a singularity.

Now, if we set the time term in the 4D invariant distance equation to imaginary timeit(multiplying the time t term by imaginary number i), we get as follows;

Let s be the four-dimensional distance, an invariant of the Lorentz transformation.

　s^2= (x^2+y^2+z^2)+(ict)^2

Now substituting the imaginary-time it for the real-time t, we get

s^2=(x^2+y^2+z^2)+(ic(it))^2

 If we put s as 1 and organize, the equation becomes

x^2+y^2+z^2+(ct)^2 =1

This equation is no different from a 3-dimensional sphere with a distance of 1 from the origin in 4-dimensional Euclidean space. In other words, if we substitute imaginary-time it into the equation for spacetime, then 4D spacetime becomes indistinguishable from 4D space!

This transformation from real-time t to imaginary-time it (called Wick rotation in physics) can correspond to reversing the metric in the four-dimensional direction. A rough geometric image from the equation corresponds to rounding a cone-shaped space-time into a spherical shape (see Fig. 2).

The point is that the introduction of imaginary-time it transforms a four-dimensional cone into a four-dimensional sphere. Such a spherical space-time existed at the beginning of the real-time universe, which is the essence of the boundary condition.

If space-time had a spherical shape, it means that the beginning of the universe had a spherical tip at the apex of a cone, just like when you change the lead of a pointed pencil into the round tip of a ballpoint pen (see Figure 3).

From this perspective, Hawking argues that there was an endless and seamless world at the beginning of the universe that seemed to have no beginning and no end. If this is the case, then the spherical world as Arche becomes an image of the universe in which the place where the Big Bang occurred and the place where the Big Crunch occurred repeat each other repeatedly. In the imaginary time universe, every location is both the beginning and the end of the universe.

This is very similar to the Buddhist image of the universe as “beginningless and endless.” Everywhere in time and space, no matter when or where will be a place where the world of beginningless-and-endless eternity is expanding. This is the emergence of “Kuon (the remote past) gannjo (beginning)—the Mystic Law, uncreated and eternal, of the Buddha of beginningless time*.”

“Kuon is something that is not worked for, that is not improved upon, but that exists just as it always has*.”

“Kuon is now, now is Kuon.”

  (Nichiren 1222-1282, a Japanese Buddhist priest and philosopher)

* Quotes from Dictionary of Buddhism, Nichiren Buddhism Library