Or How I Stopped Worrying and Learned to Love Infinities…This is also a partial response to a comment by Bruno Suric.
It's a common whine of pseudo-scientists, that mainstream science has lost its way, introducing near-mystical entities like dark matter & dark energy, infinities, etc. to explain what many might want to view as 'common-sense' phenomena.
Of course, they conveniently forget that such 'common-sense' interpretations of the natural world ruled the human race for thousands of years of recorded history. Our modern world of space flight, communications satellites, microelectronics, etc. came about only over the past 400 years or so as we abandoned a view that Nature operates based on our notions and supernaturalism, and developed techniques which allowed mathematics to make predictions that can be tested by experiments which paved the way to precision engineering on large scales down to the atomic level.
While pseudo-scientists like to complain about modern theories with features they don't like, they conveniently forget (or most likely they never actually knew) that many older theories (which they are not complaining about) had annoying infinities and other 'common-sense'-defying characteristics that challenged and perplexed researchers of the day. Some of these issues are still unresolved, but otherwise largely ignored because they occur in regimes not yet accessible by experiment. But even with these problems, the theory is still very useful in areas we can reach with experiment.
Older theories made many 'nonsense' predictions, a number of which were even verified by experiments. Electromagnetism is loaded with them.
- Poisson's Spot or Arago's Spot (Wikipedia): A small bright spot in the center of a shadow, a consequence of the wave theory of light. Poisson predicted the bright spot would exist in the center of a shadow using the wave theory of light and claimed that such a nonsense result was evidence against the wave theory. Then Arago set up and experiment and found it. This is actually a simple experiment for a reasonably equipped optics lab - I have even demonstrated it myself.
- Maxwell's equations predicted atoms composed of negative charges occupying a large space around a central positive charge were unstable would collapse in less than a millionth of a second because the negative charge in orbit around the positive center would radiate away its energy. We know that atoms do not collapse.
- Maxwell's electromagnetism predicted that heat from a fireplace would emit a dangerous amount (essentially an infinite amount) of high-energy radiation, yet that is not observed (Wikipedia: Ultraviolet Catastrophe).
- Electromagnetism has a similar problem to Newtonian gravity at r=0. If we compute the energy of an electron of a finite radius, we get a diverging result as the radius approaches zero. We can choose a finite radius where the internal energy matches the electron mass, and obtain the Classical Electron Radius of 2.82e-15 meters (Wikipedia: Classical Electron Radius). But when we slam electrons together in accelerators, we find the electrons penetrate significantly closer than this. Today, we still only have an 'upper limit' of 1e-22 meters which means that is the LARGEST the electron could be, and it places no constraints on how much smaller it could be (Wikipedia: Electron-Fundamental Properties). The Standard Model assumes the electrons, neutrinos, and quarks are all structureless point-particles. Does the electron have structure? The honest answer is scientists don't know, but are trying to find out.
- Standard Model of particle physics treats all particles as point-masses and charges. While some theories proposed beyond the Standard Model hypothesize a finite size for the electron, none of those theories have yet been verified. But for a few anomalies (neutrino mass, electron magnetic moment, etc.) this works quite well, but it is clearly not complete (Wikipedia: Point Particle, Standard Model)
- Even circuit analysis was not immune from infinities. LC-resonance circuits have an infinity at their resonance frequency (Wikipedia: Electrical Resonance, LC Circuit), which is only resolved with the recognition that real circuits have some resistance which eliminates the infinity (Wikipedia: RLC Circuit). Here's an interesting question which maybe readers can find the answer - has anyone built an LC circuit using superconductors which ostensibly have zero resistance? If so, what happens at the resonance frequency?
- Fluid dynamical equations, such as the Navier-Stokes equations (Wikipedia), treats liquids as a continuum at all scales. In the math, fluids are not ultimately made of atoms and molecules. As a result, mathematics can generate singularities (Backreaction: Singularities in your Kitchen).
- Then there's all the counter-intuitive phenomena in quantum-mechanics, such as tunneling (Wikipedia) which are fundamental in the operation of the integrated circuits in the computer on which you are reading this. Quantum mechanics also generates it's own set of infinities. So far, they are dealt with by a number of techniques that can at best be called 'hacks' (Wikipedia: Renormalization) that allow use to obtain experimentally testable numbers, but they still indicate the theory is not complete.
New Knowledge from Counter-Intuitive Behavior of Physical TheoriesAlmost invariably, the odd solutions to physical theories often generate useful insights. Sometimes these odd solutions are just ignorable, often the researcher has just made an error or oversight in applying the theory. That's why physical science is more of a social endeavor, as researchers provide checks on each others work.
- Einstein-Podolsky-Rosen (EPR) argument (wikipedia). It took many years before we could actually test it and when we did, we found the bizarre 'spooky action-at-a-distance' which Einstein said was proof of the failure of quantum mechanics turned out to be exactly how the universe worked! Even so, some researchers continue to explore if this strange behavior has a more 'logical' origin.
- Negative root prediction of positron. Negative roots in Dirac equation (Wikipedia) suggested existence of a positive counterpart of the electron, initially misidentified with the proton, which was identified a few years later (Wikipedia: positron).
The other reason is these theories work quite well everywhere else, while exhibiting strange behavior under very limited conditions.
Cranks want to replace these theories with bizarre things that generate little or no testable predictions. When we investigate the alternatives presented by the cranks, we encounter:
- no math to provide predictive capability; or
- presenting same math as the mainstream theory, but surrounding it with different terminology; or
- replacing a single theory that makes many different predictions with numerous ad hoc explanations. The big problem with these alternatives is they can only arise as the original theory makes successful predictions after which the pseudo-scientists adopt the results as yet another ad hoc explanation.
- Admit the answer is unknown, then actually doing the REAL work needed to solve it. Continue to use the theory where it has good predictive power and work to test it closer to the limits where the strange behavior occurs.
- Pseudo-scientist 'solution' is to claim problem doesn't exist or is made up; ignore the theory which has it, often without replacement. Of course, such individuals will miss out of the discoveries and inventions possible with even a incomplete theory.
What Infinities Tell UsScience popularizers often overuse some of the odder predictions of various theories such as speculation about the meanings of infinities the theory might have. This is in part to emphasize that there are still discoveries to be made - that science is not complete.
But anyone claiming to do serious SCIENTIFIC critiques would rightfully have their judgement questioned if they base their arguments on the statements made by popularizers. It's legitimate to argue about best analogies, over sensationalizing, or even outright inaccuracies for statements made by popularizers who are aiming to encourage excitement and enthusiasm about the science. Once one gets into describing what happens when we encounter infinities from a scientific theory, popularizers should probably have a red warning banner that reads "SPECULATION" or "HYPOTHESIS".
In reality, history suggests that infinities are signposts:
"Beyond Here May Be NEW PHYSICS".
It's an area on the frontier, seeking solutions, and therefore the most exciting. Sometimes it's just telling us that the approximations break down, atomic scale processes become important in a fluid, resistance becomes important in a circuit, etc. Sometimes there really are new physics. The Pioneer anomaly is an observation that was real and could have been a hint of new physics, but detailed examinations of laboratory phenomena, such as radiation recoil, revealed that over the 40+ years of the mission, even these tiny effects could accumulate sufficiently to be detectable (Wikipedia: Pioneer Anomaly).
So what's up with black holes?Many cranks try to make a big deal about general relativity and black holes, wanting to rant about singularities and such. These singularities in General Relavity are the modern version of these historical scientific problems, and as such get a lot of attention in popularizations.
Is there a singularity at 'center' of black hole?
- Many physicists suspect (myself included) suspect you can never actually cross the event horizon, so what lies at the center while perhaps interesting, is largely irrelevant.
- However, some solutions suggest you can actually cross the horizon and reach the center, but it would be a one-way trip.
Was there anything prior to the 'singularity' at the beginning of the Big Bang?
The real answer at present is we don't really know.
But GR has successfully passed virtually every other test we've thrown at it (Wikipedia: Tests of General Relativity). The bar for replacing relativity is pretty high.
Either way, trying to find a conclusive answer to the question can provide new insights, but in the process generates a horde of speculations. Most of this interaction takes place in the literature and places like the Cornell Preprint server. All the PROFESSIONAL literature I've read on this controversial topic is pretty civil. Some of it receives responses, but things are usually quiet unless some new experimental result suggests a connection, or a university press office desperately needs something to publish.
Not so with the cranks, who take being dismissed, corrected, or ignored as personal attacks and try to use this as a way to get attention, but that's another post...
Many of the cranks who scream about the nonsensical theories seem utterly clueless about the history of the topic, ignoring how many of those 'problem theories' are not only still in use today, but in many cases driving technologies that they use.
But the cranks will find some excuse to ignore these inconvenient facts...
"But those infinities are not as bad as this one in relativity..."Baloney.
Many of these historical problems were sources of serious controversy of their day, just as the modern infinities. Some, such as the infinite self-energy electromagnetic mass issue are still only solved by methods that aren't much different than hacks. Such workarounds suggest that our theory is not truly complete.
But does that mean the theory is useless?
No, it is quite usable in the broader realms, but the problems provide guides for future experiments which may outline the requirements of a more complete theory.
Do mainstream scientists know the solution to the problem? No.
But the cranks know even less.
- Wikipedia: Mechanical Singularity, Gimbal Lock, Gravitational Singularity, Prandtl–Glauert singularity, Penrose–Hawking singularity theorems, Firewall Physics
- Einstein Online: Spacetime Singularities
- Ask a Mathematician/Ask a Physicist: What are Singularities? Do they exist in nature?