General · Language · Problem

Singularities RIP?

John Healy

[Draft] [RIP series]

Credit: Pixabay/CC0 Public Domain

So, this recent SciTechDaily article (below) about the “Singularity Problem” caught my attention today. Because it’s a reminder of what the much fictionalized / mythologized notion is really about. The need for a deeper understanding of nature. The representational limits (approximate character) of mathematical models. A theory’s domain of applicability. [1]

As noted by Wiki:

• Mathematical singularity, a point at which a given mathematical object is not defined or not “well-behaved”, for example infinite or not differentiable.

• Gravitational singularity, in general relativity, a point in which gravity is so intense that spacetime itself becomes ill defined.

• Initial singularity, a hypothesized singularity of infinite density before quantum fluctuations caused the Big Bang and subsequent inflation that created the Universe.

• Penrose–Hawking singularity theorems, in general relativity theory, theorems about how gravitation produces singularities such as in black holes.

• SciTechDaily > “A Quantum Physics Approach to a Singularity Problem” by SCIENCEPOD (February 16, 2022) – A new paper in Nuclear Physics B [26 July 2021] … suggests that extending the treatment of singularities in classical physics into quantum physics could help to solve this disparity between branches of physics. [2]

One of the major issues in general relativity that separates it from other descriptions of the universe, like quantum physics, is the existence of singularities. Singularities are points that when mathematically described give an infinite value and suggest areas of the universe where the laws of physics would cease to exist — i.e. points at the beginning of the universe and at the center of black holes.

“No description of nature is perfect and complete. Every theory has its domain of applicability, beyond which it breaks down and its predictions no longer make sense,” Casadio [Dipartimento di Fisica e Astronomia, Università di Bologna, Italy] says.

Casadio suggests that this can be envisaged as a piece of paper with a small hole in it: “You can move the tip of your pen on the paper, which represents the movement of a particle, but if you reach the hole your pen suddenly stops drawing and the particles suddenly disappear.”

The authors used the tools of QFT [quantum field theory] to construct a mathematical object that can signal the presence of singularities in experimentally measurable quantities. This object, which they have named the “functional winding number” is non-zero in the presence of singularities and vanishes in their absence.

This approach has revealed that certain singularities predicted theoretically do not affect quantities that can in principle be measured experimentally, and therefore remain harmless mathematical constructs.

Notes

[1] And I recall that Feynman said something about when the math “spits out” infinities, nature is telling us something important, that something more interesting is going on.

As noted by other physicists as well:

“That word — infinite — is a hard pill to swallow. When infinites appear in the mathematics, it’s a signpost that we’re doing something wrong, that our machinery isn’t quite up to the task. We’re missing something. … we know our theoretical models (i.e., general relativity) are incomplete. There isn’t really a singularity at the center of a black hole. But we simply don’t understand strong gravity at small scales. That’s the domain of a full-on quantum theory of gravity, which we haven’t cracked despite decades of trying. Hard.” — Paul Sutter, On the Existence of Black Holes, September 11, 2017.

And as similarly noted in my January 11, 2021 comment, this quote from Casey Cep’s article “Science’s Demons, from Descartes to Darwin and Beyond” in New Yorker (January 8, 2021):

[Historian of science Jimena Canales’ book] “Bedeviled: A Shadow History of Demons in Science” (Princeton University Press) … has gathered together in one book demons with very different origins and responsibilities – among them the scientist James Clerk Maxwell’s demon, the physicist David Bohm’s demon, the philosopher John Searle’s demon, and the naturalist Charles Darwin’s demon. … They are not supernatural creatures; rather, they are particular kinds of thought experiments, placeholders of sorts for laws or theories or concepts not yet understood.

… but, in general, scientific demons simply represent gaps in our existing knowledge, … because they often represent theories that violate laws of time or space, machines that exceed human capacities, and technologies that threaten human existence, the use of demons feels appropriate …

[2] After the end of the article, there’s a “We recommend” section which lists related articles, e.g.:

• “Even if a Black Hole Can Be Described With a Mathematical Model, It Doesn’t Mean It Exists in Reality” (2021)

• “A quantum approach to a singularity problem” (2021)