No Boundary Proposal

INTRODUCTION

In seminal works, Penrose and later Hawking, discovered that a universe of any shape which obeys Einstein’s theory of General Relativity must have originated from a point of infinite density and at a finite time in the past [1].  If true, such a beginning would be inexplicable by the methods of science.  That mathematical cosmology has this deficiency and instead so strongly suggests a “super-natural” creation event came as something of a surprise.

It had been hoped that a topology lacking perfectly symmetry could have been the result of an eternal universe with a history before the “big bang”.  But as Hawking wrote, his calculations unambiguously demand a beginning before which nothing existed [2].   This original state is called a “singularity” and is unfortunately scientifically inexplicable.   That this is true is not diminished by our invention of labels for things which are demonstrably unknowable.

Strictly speaking, a point, similar to the concept of infinity, has no physical meaning. Rather they can be endlessly approached but never reached.  A point might better be described as the imagined result of an “operation” in which a tiny bit of space shrinks without limit.   If the rate of shrinking is constant or increasing, after some definite time the space would vanish entirely meaning it wouldn’t exist anymore. As for infinity, nothing in the real world is larger than any finite measure.

And please note it is not the case that as we go backwards in time the universe forever shrinks to a smaller and smaller size.  Rather the universe shrinks exponentially faster as we go back to a point of “infinite” density at a specific moment in the past.   This means the universe did not exist eternally but has a measurable age dating from the moment of creation.

Because science can only attempt to predict what we can observe, it is unable to describe a state of “nothingness” before nature, to include space, time, gravity, laws of quantum mechanics, or indeed any other aspects of reality, existed.  Rather a description of the creation event, ex nihilo, necessarily requires by definition some sort of “super-natural” explanation.

NO BOUNDARY

The Hartle-Hawking “no boundary” proposal thus only attempts to describe how the universe evolved and not how it was created [3].   Because we can’t use natural law to explain how nature behaved before nature existed, physicists run their equations backwards in time from our present reality.  The idea is to handle the singularity as a limiting case as we get closer and closer.

Since the universe is expanding, it must have been smaller in the past.  In the earliest epochs, the universe would have been crammed into such a small space that quantum mechanical effects had to dominate. The extreme densities would also have produced a massive time dilation as described by Einstein’s Theory of General Relativity.  And of course quantum fluctuations in space and time, which are normally invisible in our macroscopic world, would have had a significant influence on its evolution.

Unfortunately, when we try to apply these well tested theories, the mathematics of the singularity blows up. In particular, integrals are summations, and when we use them to calculate sums over many paths, they give nonsensical results when encountering infinite quantities.  The calculations diverge to no specific value at all.  Sometimes two infinite quantities will cancel each other but this is not the case for our universe beginning in a singularity.

To get around this difficulty, Hartle and Hawking tried to find metrics that remain finite as we approach the moment of creation. The model thus pulls a slight-of hand mathematical trick.

To avoid infinite quantities, time is first represented as an imaginary number in the equations.  Imaginary numbers are not a measure of anything in the real world. They are defined using the variable “i” representing the square root of minus one; and thus violate the distributive law of algebra.  Nevertheless they are useful in representing the phase angle of an electromagnetic wave or in describing Schrodinger’s wave function in quantum mechanics.  But they must always be transmogrified into a real number to give us anything we could use to describe something physical.

In any event, the result is that time in the model is initially so diluted as to be nearly non-existent.  At the beginning there was only a point of explosively expanding but nearly empty space.  Initially, matter didn’t exist, as previously believed, in an infinitely dense ball of plasma.  Rather quantum fluctuations only gradually brought matter into existence later by robbing the energy of gravitational fields.

Note that time never really stops but only asymptotically slows down the closer we get to the moment of creation.  Since time is the measure of the rate of change of physical configurations, there could be no change or evolution without it.  And of course, time is meaningless before creation when there was nothing physical which was capable of change.

Also, one cannot say that space is “nothing” because space has its own physical existence and structure.  And indeed even a perfect a vacuum is thought to consist of a “quantum foam”.   And this is experimentally observed in the Casmir Effect and the Lamb Shift.  The point is that quantum fluctuations require a volume of space in which to fluctuate.  Since there was none before creation, fuzzy-minded and ill-informed recourse to quantum mechanics won’t work either.

Indeed, Schrodinger’s equation consists entirely of terms for the wave functions’ rate of change with distance and in time.  Without any volume of space or something physical which is able to change as measured by some sort of clock, quantum fluctuations simply couldn’t exist either.   Quantum effects may seem strange and violate common sensibilities but they are not magical and instead seem to obey well defined laws, when they have even a little bit of nature to work with.

Another difficulty is that the model has to describe the universe before the “Planck time” or about 10^(-43) second after creation. But according to Heisenberg’s Uncertainty Principle, it is theoretically impossible to know about anything earlier than the Planck time.  So in that realm we are free to speculate.  Note that this is even before inflation which lasted from about 10^(-36) to 10^(-32) seconds after creation.  During the inflationary epoch we have reason to believe the universe exploded from being smaller than a proton to about the size of a grain of sand.  This is because we can tweak parameters in the inflation field equation to explain features of our current universe such as its remarkable homogeneity on scales of 250 million light years.  That we are able to create verifiable theories from this earliest of times, even if not earlier, is nothing short of amazing.

To summarize, this model, or indeed any model, is unable to explain how a “point” of space, even empty space but complete with quantum foam and embedded laws of gravity and quantum mechanics, sprang into existence out of absolutely nothing.  And it does not “explain” the “Big Bang” either as that is not simply an ongoing expansion but rather originated before that at what we can only call a “super-natural” act of creation.  And wishful thinking to the contrary based on sensational newspaper headlines is no substitute for logic or science.

The model does however have the advantage of avoiding a mathematical infinity and is thus able to calculate integrals that would otherwise be intractable. Unfortunately, when we employ this model to calculate properties of the universe we now observe, we get nonsensical results [4-5].  Perhaps some modification could rescue the situation but so far nothing has been successful.  So there is still some work to do.

CONCLUSIONS

That this model was celebrated by notable anti-religious personalities who misrepresented the science as proving the nonexistence of God, is shameful.  Apparently atheists understand neither religious reasoning, nor the logic on which it is founded, nor the science with which it is compatible. 

The key question is what brought expanding space into existence and gave it any volume bigger than zero regardless of that initial finite shape.  Note that at the origin, all points have the same shape, which is none.  Going backwards in time from current reality and asymptotically approaching nothing as a limiting case is not the same thing as going forwards and making the leap from nothing to something. 

Misunderstanding this essential distinction, atheists gleefully celebrate the supremacy of science when in fact the model continues to imply the existence of a “super-natural” creator.  Rather to be an atheist, one has to deny the validity of our current scientific understandings and have a blind, almost irrational, faith they will eventually be replaced with something more compatible with personal preferences.

Since science and mathematics and logic have demonstrated that absolute certainty is impossible, we cannot say that will not happen; just that, lacking any reasonable evidence or argument, it is unlikely. 

REFERENCES

1. Penrose was awarded the Nobel Prize for his “Singularity Theorem” as described in
    “The Singularity Theorem (Nobel Prize in Physics 2020)” by Emanuel Malek, in Einstein Online Band 12 (2020). 
    The result was confirmed later by Hawking with even more rigor in
   "Properties of expanding universes" by Stephen Hawking in Cambridge Digital Library. 

2. Despite various misrepresentations in the popular press, Hawking maintained that his calculations demonstrated the universe 
    must have had a beginning as in his lecture “The beginning of time”, (1996).
    https://www.hawking.org.uk/in-words/lectures/the-beginning-of-time

3. Although proposed by Hawking in 1981 at the Pontifical Academy of Sciences held at the Vatican in Rome, Italy, the complete theory was published in
  “Wave function of the Universe” by J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).

4. The mathematical absurdity of this model was criticized by Susskind in
   “Susskind's Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions” by Don N. Page, JCAP 0701:004,2007.

5. The model was found to be incompatible with our currently observed universe in 
    “No Rescue for the No Boundary Proposal” by Job Feldbrugge, Jean-Luc Lehners, and Neil Turok, Phys. Rev. D 97, 023509 (2018).