# Toward Quantum Gravity? Discrete Space, Dark Energy And The Real Line

Unfortunately for previous cosmological theories, the universe appears reasonably convincingly to be expanding at an accelerating rate. This, all physicists know, is no small problem. It is so severe that physicists now invoke "Dark Energy" to explain this accelerating expansion.

Here is the central problem: Throw a ball into the air. If you do so at less than the escape velocity from the earth's gravitational field, it will fly upward, exchange kinetic and gravitational potential energy until the two are equal at the top of its trajectory, then the ball will fall back to earth. If the initial velocity of the ball is just equal to the earth's gravitational field, the ball will "coast to infinity" ever more slowly, reaching infinity at 0 velocity, in the absence of other matter in the universe to provide additional gravitational fields. If the initial velocity of the ball is still higher, it will fly from the earth and coast to infinity at an ever *decreasing*, but finite velocity.

The Big Bang, well, Banged. To put the issue over-simply, vast energy arising from nothing became a quark gluon soup, became matter and fields containing energy, hence, by E=MC^{2}, matter, as the universe expanded. The mutual gravitational attraction of all the mass and energy in the universe is supposed to forever slow the expansion of the universe. Given General Relativity, the universe should either expand then collapse in the Big Crunch, like the ball falling again to earth, it should expand forever to infinity reaching infinity at a zero rate of expansion, a critical state of affairs, or it should expand forever, but at an ever slowing but finite rate.

The first and last cases correspond by General Relativity to a positively curved spacetime, and a negatively curved space time. The critical case corresponds to a flat spacetime. Cosmologists thought the universe was critical.

What the universe should not be doing is expanding at an accelerating rate.

But it seems to be.

Einstein made a "mistake."

He thought the universe was of constant size, and his equations did not predict this. He "fudged" and added the simplest correction term to his equations, the "Cosmological Constant." Using it, Einstein could achieve a universe of constant size. Hubble soon discovered that galaxies recede from one another in proportion to their distance apart and concluded the universe was expanding. Einstein admitted it was the worst mistake of his scientific life.

But the Cosmological Constant has remerged as a start to account for Dark Energy. With it, one can achieve an accelerating expansion of the universe. The Cosmological Constant is as if there were a "repulsive force" in space, pushing any two regions of space further apart, and the more so as their distance apart increases.

Einstein realized that his Cosmological Constant implied an increasing energy in the expanding universe, such that the vacuum energy per unit volume remained constant. This flies in the face of the Big Bang belief that all the energy of the universe came to exist in that Bang. Well, OK, so it does.

The problem is, there is, in a deep sense, no physics behind the Cosmological Constant, it is just a constant added the only place Einstein could in his equations and keep the rest of the physics of General Relativity unmodified.

Thus, the deep question is: What conceivable physical process(es) could constitute the Cosmological constant.

I am going to propose a related set of concepts that might do the job:

- Space is NOT CONTINUOUS. Space, on the smallest length scale, the Planck length of 10
^{34}centimeters, comes in discrete units. These units cannot interpenetrate, otherwise, once again, space would be continuous. An approach to quantum gravity, Loop Quantum Gravity, kindly taught to me by Lee Smolin, a founder of that field, proposes that on the Planck scale, space is comprised of tetrahedra. - These units of space also cannot expand. But this implies that space itself cannot expand! The only way "space" can expand is if more units of space, the tetrahedra, come into existence. In Loop Quantum Gravity this is allowed: tetrahedra can build new tetrahedra on any of their four faces in what are called Pachner moves. So for space to "expand" it must build itself by "cloning" itself.
- Tetrahedra or units of space have a constant "rate" or probability per unit time, of cloning themselves. This postulate implies that space builds itself and expands EVER FASTER. Indeed, like a bacterial colony freshly plated, space expands exponentially. We are stepping towards Dark Energy and a Cosmological Constant which yields an exponentially expanding space. Before I go further, it is clear that the more space now exists, the more tetrahedra there are, so any two regions of space are being "pushed apart" by the cloning of tetrahedra between them.
- Every time a tetrahedron of space is created, new energy enters that tetrahedron from nowhere, so the vacuum energy is constant as space expands at an accelerating rate. This accords with Einstein's realization that the Cosmological Constant could imply can constant energy per unit space as space expanded: Dark Energy we now say. I note that it makes us uncomfortable in imagine energy arising from "nothing", but we accept this magic at the Big Bang It is no more magical here. Some energy per unit space is needed for point 5) next. Also physicists say that once there is matter, energy, expansion, and gravitation, energy is not conserved.
- Newborn tetrahedra are quantum objects. More, by Pachner moves, quantum geometry can be in a superposition of more than one "graph structure" connecting the nodes of the tetrahedra in a diversity of ways. But because they contain energy, they contain mass. At some scale and volume of cloning tetrahedra, considered as a "system" in a still larger spatial volume considered as the "environment" of the "system", the mass in the space in the "system" is sufficient that some number of quantum tetrahedra DECOHERE into CLASSICAL SPACE.
- Now two alternative postulates: 6A, even classical space can give birth to new tetrahedra. 6B, decoherence can be reversed occasionally, and newly quantum tetrahedra can clone themselves and give birth to new tetrahedra. The alternative postulates matter of course, but I will not favor either.

Taking the above non-mathematized schema of a physical a theory of Dark Energy as the cloning of space by space, the first thing to do is estimate the rate of cloning of space needed to account for the observed accelerating expansion of the universe. This is a relatively straightforward job.

The Real Line is a Real Mess. Best if space is discrete on some small scale.

There is a separate reason to consider space on some smallest scale to be discrete: The real line is a real mess. Recall that the real line, a human invention, is a continuous line, for example from 0.0 to 1.0 on some scale like a meter stick. The real line has both an infinity of rational numbers, where rational numbers can be expressed as the ratio of two whole numbers such as 34/6501. But as Pythagoras proved, there are numbers that cannot be expressed as such ratios. These are the irrational numbers and are expressed as INFINITELY LONG sequence of the integers, 0,1,2,3,4,5,6,7,8,9 after the "decimal point". How many irrational numbers are there? Well, between any two rational numbers, no matter how close, there are an infinite number of irrational numbers. The rational numbers are of the same order of infinity as the integers. The irrationals are of second order infinity.

Here is the problem. Mathematics is largely founded on set theory. But suppose I want to form two sets of irrational numbers, each with specific irrational numbers in it. How can I do so? Each irrational number is an infinite series of the first 10 integers above. How can I pick a "specific" irrational number?

One answer would seem easy: have a computer algorithm to compute each such number, like Pi, or the square root of 2, both of which are irrational, and then just form the sets.

Unfortunately, Alan Turing of the Turing Machine that is the basis of all digital classical and quantum computers today, proved that almost no irrational numbers were "effectively computable" using a Universal Turing Machine. So we cannot use this idea for our two sets.

The mathematicians, wishing to preserve set theory, postulated an Axiom of Choice asserting, well what the heck, that one can just go ahead and form the two sets.

Axioms are fine, but Turing tells us there is no effectively computable way to achieve the formation of these two sets for arbitrary choices of irrational numbers.

Why should we care? Because on the one hand, classical physics is based on the continuous line, rational and irrational numbers included. But then, classical physics works at length scales far above the Planck length scale of 10 to the - 34 centimeters.

We don't need a continuum at that scale for classical physics.

On the other hand, if we remember that it is we who invented the idea of the real line and have learned to love it, this does not mean that the real line is the right description of reality. And it is filled with problems. As I said, the real line is a mess.

Then if we give up the real line, space must have a smallest length scale. The Planck length scale is the obvious scale. Then we get to Loop Quantum Gravity's tetrahedra, or some other minimal discrete units of space, avoid the continuous line, and with the postulates that these minimal units of space cannot interpenetrate, cannot expand, and can clone themselves we have the start of an exponentially expanding space. If we add energy per new unit of space and decoherence of quantum units of space, we get classical space and a constant vacuum energy. And if on 6A or 6B space tetrahedra can continue to clone themselves at some rate per unit volume of space we have Dark Energy.

There is as yet no matter or other energy in this space, of course, to add its mutual gravitational attraction to that of vacuum energy, as described by General Relativity, but given CLASSICAL SPACE achieved by the decoherence of quantum regions of space considered as "systems" into larger regions of quantum space considered as "environments, General Relativity seems safe.

This could conceivably constitute a theory, or part of a theory, of quantum gravity. The ideas could just possibly unite General Relativity and Quantum Mechanics. It seems worth thinking about.

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