Lost in Math: A Book Review

I recently finished reading a thought-provoking book by theoretical physicist Sabine Hossenfelder, Lost in Math: How Beauty Leads Physics Astray. Hossenfelder writes in an engaging and accessible style, and I hope you will enjoy reading this book as much as I did. Do we have a crisis in physics and cosmology? You be the judge. She presents convincing arguments.

The basic premise of Hossenfelder’s book is that when theoretical physicists and cosmologists lack empirical data to validate their theories, they have to rely on other approaches—”beauty”, “symmetry”, “simplicity”, “naturalness“, “elegance”—mathematics. Just because these approaches have been remarkably successful in the past is no guarantee they will lead to further progress.

One structural element that contributes to the book’s appeal is Hossenfelder’s interviews with prominent theoretical physicists and cosmologists: Gian Francesco Giudice, Michael Krämer, Gordon Kane, Keith Olive, Nima Arkani-Hamed, Steven Weinberg, Chad Orzel, Frank Wilczek, Garrett Lisi, Joseph Polchinski, Xiao-Gang Wen, Katie Mack, George Ellis, and Doyne Farmer. And, throughout the book, she quotes many other physicists, past and present, as well. This is a well-researched book by an expert in the field.

I also like her “In Brief” summaries of key points at the end of each chapter. And her occasional self-deprecating, brief, soliloquies, which I find reassuring. This book is never about the care and feeding of the author’s ego, but rather giving voice to largely unspoken fears that theoretical physics is stagnating. And an academic environment hell-bent on preserving the status quo isn’t helping matters, either.

Anthropic Principle

Do we live in a universe fine-tuned for life? If so, is it the only possible universe that would support life? Recent work indicates that there may be more than one set of parameters that could lead to a life-supporting universe.

Beauty is in the Eye of the Beholder

Is our sense of what is “beautiful” a reliable guide to gaining a deeper understanding of nature? Or does it sometimes lead us astray? We know from history that it does.

In the past, symmetries have been very useful. Past and present, they are considered beautiful

When we don’t have data to guide our theory development, aesthetic criteria are used. Caveat emptor.

Experiment and Theory

Traditionally, experiment and observation have driven theory. Now, increasingly, theory drives experiment, and the experiments are getting more difficult, more expensive, and more time consuming to do—if they can be done at all.

Inflation

The rapid expansion of the universe at the time of the Big Bang is known as cosmic inflation, or, simply, inflation. Though there is some evidence to support inflation, that evidence is not yet compelling.

Mathematics

Mathematics creates a logically consistent universe all its own. Some of it can actually be used to describe our physical universe. What math is the right math?

Math is very useful for describing nature, but is math itself “real”, or is it just a useful tool? This is an ancient question.

Memorable Quotations

“I went into physics because I don’t understand human behavior.” (p. 2)

“If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.” (p. 9)

“In our search for new ideas, beauty plays many roles. It’s a guide, a reward, a motivation. It is also a systematic bias.” (p. 10)

On artificial intelligence: “Being unintuitive shouldn’t be held against a theory. Like lack of aesthetic appeal, it is a hurdle to progress. Maybe this one isn’t a hurdle we can overcome. Maybe we’re stuck in the foundations of physics because we’ve reached the limits of what humans can comprehend. Maybe it’s time to pass the torch.” (p. 132)

“The current organization of academia encourages scientists to join already dominant research programs and discourages any critique of one’s own research area.” (p. 170)

Multiverse

The idea that our universe of just one of a great many universes is presently the most controversial idea in physics.

Particles and Interactions

What is truly interesting is not the particles themselves, but the interactions between particles.

Philosophy

Physicists and astrophysicists are sloppy philosophers and could stand to benefit from a better understanding of the philosophical assumptions and implications of their work.

Physics isn’t Math

Sure, physics contains a lot of math, but that math has traditionally been well-grounded in observational science. Is math driving physics more than experiment and observation today?

Quantum Mechanics

Nobody really understands quantum mechanics. Everybody’s amazed but no one is happy. It works splendidly well. The quantum world is weird. Waves and particles don’t really exist, but everything (perhaps even the universe itself) is describable by a probabilistic “wave function” that has properties of both and yet is neither. Then there’s the many-worlds interpretation of quantum mechanics, and quantum entanglement

Science and the Scientific Method

In areas of physics where experiments are too difficult, expensive, or impossible to do, some physicists seem to be abandoning the scientific method as the central pillar of scientific inquiry. Faith in beauty, faith in mathematics, faith in naturalness, faith in symmetry. How is this any different than religion?

If scientists can evaluate a theory using other criteria than that theory’s ability to describe observation, how is that science?

Stagnation

Some areas of physics haven’t seen any new data for decades. In such an environment, theories can and do run amok.

Standard Model (of particle physics)

Ugly, contrived, ad hoc, baroque, overly flexible, unfinished, too many unexplained parameters. These are some of the words used to describe the standard model of particle physics. And, yet, the standard model describes the elementary particles we see in nature and their interactions with extraordinary exactitude.

String Theory

String theory dates back at least to the 1970s, and its origins go back to the 1940s. To date, there is still no experimental evidence to support it. String theory is not able to predict basic features of the standard model. That’s a problem.

Triple Threat: Crises in Physics, Astrophysics, and Cosmology?

Physics: Sure, the Large Hadron Collider (LHC) at CERN gave us the Higgs boson, but little else. No new physics. No supersymmetry particles. Embarrassments like the diphoton anomaly. Do we need a bigger collider? Perhaps. Do we need new ideas? Likely.

Astrophysics: We’ve spent decades trying to understand what dark matter is, to no avail. No dark matter particles have been found.

Cosmology: We have no testable idea as to what dark energy is. Plenty of theories, though.


See Hossenfelder’s recent comments on the LHC and dark matter in her op-ed, “The Uncertain Future of Particle Physics” in the January 23, 2019 issue of The New York Times.


The book concludes with three appendices:

  • Appendix A: The Standard Model Particles
  • Appendix B: The Trouble with Naturalness
  • Appendix C: What You Can Do To Help

Hossenfelder gives some excellent practical advice in Appendix C. This appendix is divided into three sections of action items:

  • As a scientist
  • As a higher ed administrator, science policy maker, journal editor, or representative of a funding body
  • As a science writer or member of the public

I’m really glad she wrote this book. As an insider, it takes courage to criticize the status quo.

References
Hossenfelder, S., Lost in Math: How Beauty Leads Physics Astray, Basic Books, New York (2018).
Hossenfelder, Sabine. “The Uncertain Future of Particle Physics.” The New York Times 23 Jan 2019. https://www.nytimes.com/2019/01/23/opinion/particle-physics-large-hadron-collider.html.

What is a Vacuum?

A vacuum is not nothing.    It is only a region of three-dimensional space that is entirely devoid of matter, entirely devoid of particles.

The best laboratory vacuum contains about 25 particles (molecules, atoms) per cubic centimeter (cm3).

The atmosphere on the surface of the Moon (if you can call it that) contains a lot more particles than the best laboratory vacuum: about 40,000 particles per cm3.  This extremely tenuous lunar atmosphere is mostly made up of the “noble” gases argon, helium, and neon.

The vacuum of interplanetary space contains about 11 particles per cm3.

The vacuum of interstellar space contains about 1 particle per cm3.

The vacuum of intergalactic space contains about 10-6 particles per cm3.  That’s just 10 particles per cubic meter of space.

But what if we could remove all of the particles in a parcel of space?  And somehow shield that empty parcel of space from any external electromagnetic fields?  What would we have then?

It appears that even completely empty space has some inherent energy associated with it.  The vacuum is constantly “seething” with electromagnetic waves of all possible wavelengths, popping into and out of existence on unimaginably short time scales—allowed by Heisenberg’s energy-time uncertainly principle.  These “quantum flourishes” may be a intrinsic property of space—as is dark energy.  Dark matter, on the other hand, is some weird form of matter that exists within space, exerting gravitational influence but not interacting with normal matter or electromagnetic waves in any other way.

Is there any evidence of this vacuum energy, or is it all theoretical?  There are at least three phenomena that point to the intrinsic energy of empty space.  (1) The Casimir effect; (2) Spontaneous emission; and (3) The Lamb shift.

The Casimir effect
Take two uncharged conductive plates and put them very close to each other, just a few nanometers apart.  Only the shortest wavelengths will be able to exist between the plates, but all wavelengths will exist on the other side of the two plates.  Under normal circumstances, this will cause a net force or pressure that pushes the two plates towards one another.

Spontaneous emission
An example of spontaneous emission is an electron transitioning from an excited state to the ground state, emitting a photon.  What causes this transition to occur when it does?

The Lamb shift
The Lamb shift is a tiny shift in the energy levels of electrons in hydrogen and other atoms that can’t be explained without considering the interaction of the atom with “empty” space.

References
Reucroft, S. and Swain, J., “What is the Casimir effect?”, Scientific American, https://www.scientificamerican.com/article/what-is-the-casimir-effec/.  Accessed 20 Feb 2018.

Koks,D. and Gibbs, P., “What is the Casimir effect?”, http://math.ucr.edu/home/baez/physics/Quantum/casimir.html.  Accessed 20 Feb 2018.

 

Do Dark Matter and Dark Energy Exist?

Numerous searches for the particle or particles responsible for dark matter have so far come up empty.  What if dark matter doesn’t really exist?  Could there be alternative explanation for the phenomena attributed to dark matter?

In the November 10, 2017 issue of the Astrophysical Journal, Swiss astronomer André Maeder presents an intriguing hypothesis that non-baryonic dark matter need not exist, nor dark energy either.  In “Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?” he suggests that scale invariance of empty space (i.e. very low density) over time could be causing the phenomena we attribute to dark matter and dark energy.

What is scale invariance?  In the cosmological context, it means that empty space and its properties do not change following an expansion or contraction.  Scales of length, time, mass, energy, and so on are defined by the presence of matter.  In the presence of matter, space is not scale invariant.  But take the matter away, and empty space may have some non-intuitive properties.  The expanding universe may require adding a small acceleration term that opposes the force of gravity.  In the earlier denser universe, this acceleration term was tiny in comparison to the rate at which the expansion was slowing down, but in the later emptier universe, the acceleration term dominates.  Sound like dark energy, doesn’t it?  But maybe it is an inherent property of empty space itself.

The existence of dark matter is primarily suggested by two  observed dynamical anomalies:

  1. Flat outer rotation curve of spiral galaxies (including the Milky Way)
  2. Motions of galaxies within galaxy clusters

Many spiral galaxies have a well-known property that  beyond a certain distance from their centers, their rotation rate (the orbital velocity of stars at that distance) stays nearly constant rather than decreasing as one would expect from Keplerian motion / Newtonian dynamics (think planets orbiting the Sun in our own solar system— the farther the planet is from the Sun, the slower it orbits).  Only there seems to be evidence that the rotation curves of galaxies when they are young (as seen in the high-redshift universe) do have a Keplerian gradient, but in the present-day universe the rotation curve is flat.  So, it appears, flat rotation curves could be an age effect.  In other words, in the outer regions of spiral galaxies, stars may be orbiting at the same velocity as they did in the past when they were closer to the galactic center.  Maeder writes:

…the relatively flat rotation curves of spiral galaxies is an age effect from the mechanical laws, which account for the scale invariant properties of the empty space at large scales.  These laws predict that the circular velocities remain the same, while a very low expansion rate not far from the Hubble rate progressively extends the outer layers, increasing the radius of the Galaxy and decreasing its surface density like 1/t.

We need to study the rotation curves (as a function of galactocentric radius all the way out to the outermost reaches of the galaxy) of many more galaxies at different redshifts (and thus ages) to help us test the validity of the scale invariant vs. dark matter hypotheses.  Maeder suggests a thorough rotation study of two massive and fast-rotating galaxies, UGC 2953 (a.k.a. IC 356; 50-68 Mly) and UGC 2487 (a.k.a. NGC 1167; 219-225 Mly), would be quite interesting.

The observed motions of galaxies within many galaxy clusters seems to indicate there is a substantial amount of unseen mass within these clusters, through application of the virial theorem.  However, the motions within some galaxy clusters such as Coma (336 Mly) and Abell 2029 (1.1 Gly) may be explainable without the need to resort to “exotic” dark matter.

Then there’s the AVR (Age-Velocity Dispersion Relation) problem which, incidentally, has nothing to do with dark matter.  But it may offer evidence for the scale invariant hypothesis.  It is convenient to specify the motion of a star in a spiral galaxy such as the Milky Way in a galactocentric coordinate system.

U = component of velocity towards the galaxy center

V = component of velocity in the direction of galactic rotation

W = component of velocity orthogonal to the galactic plane

Maeder writes:

The AVR problem is that of explaining why the velocity dispersion, in particular for the W-component, considerably increases with the age of the stars considered … Continuous processes, such as spiral waves, collisions with giant molecular clouds, etc… are active in the disk plane and may effectively influence the stellar velocity distributions.  However…vertical heating (the increase of the dispersion σW) is unexpected, since the stars spend most of their lifetime out of the galactic plane.

There may be more to “empty” space than meets the eye…

References
Maeder, A., 2017, ApJ, 849, 158
arXiv:1710.11425

Cosmologically Distant Objects Appear Magnified

George F. R. Ellis writes in section 2.3.3 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

…there is a minimum apparent size for objects of fixed physical size at some redshift zc = z depending on the density parameter and the cosmological constant.  The past light cone of the observer attains a maximum area at z; the entire universe acts as a gravitational lens for further off objects, magnifying their apparent size so that very distant objects can appear to have the same angular size as nearby ones.  For the Einstein-de Sitter universe, the minimum angular diameter is at z = 1.25; in low density universes, it occurs at higher redshifts.

Electromagnetic radiation such as visible light that we observe from a source that is in motion relative to us (the observer) experiences a change in wavelength that is given by

This is called redshift and is positive for a source that is moving away from us and negative for a source that is moving towards us.  The higher the relative speed toward or away from us, the greater the magnitude of the redshift.  Superimposed upon the kinematic velocities of individual galaxies relative to our Milky Way galaxy, since 1929 we have known that there is a cosmological redshift (called the Hubble flow) that is always positive and increasing in magnitude with increasing distance between any two galaxies.  In the nearby universe, the redshift (or blueshift) from kinematic velocities (often referred to as “peculiar velocities”) swamp the contribution from the Hubble flow, so some galaxies are actually approaching each other.  A good example of this is M31 and the Milky Way galaxy.  For more distant galaxies, however, the cosmological redshift swamps any contribution from the kinematic velocities.  Thus, redshift becomes a useful proxy for distance at cosmological distances.

From our everyday experience, we know that the further away an object is, the smaller is its angular size.  However, there comes a point where the angular size of an object reaches a minimum, and at even greater distances, its angular size increases!  As George Ellis states above, the entire universe acts as a gravitational lens to magnify distant objects.

Michael Richmond presents an equation for angular size as a function of redshift (based on some classical assumptions about the structure of the universe).  In his equation, the angular size of an object also depends upon the value we choose for H0, the Hubble constant, the matter density parameter, ΩM, and, of course, the physical size of the object of interest.

Let’s work through an example using this equation.  The visible part of the Andromeda Galaxy is estimated to be about 220,000 light years across.  In megaparsecs, that is 0.0675.  This is the value we will use for S.

For the Hubble constant, H0, let use a recent result: 71.9 km/s/Mpc.

And, for the matter density parameter, ΩM, let’s use 1.0.  This indicates that we live in a universe that has just enough matter for the universe to eventually recollapse, were it not for dark energy.  Though Richmond’s equation above only applies to a matter-dominated universe where the dark energy density parameter ΩΛ is zero, as George Ellis indicates above, a minimum angular diameter is still reached in a universe with dark-energy (i.e. low density universe), only this occurs at a higher redshift than that presented here.

I have not been able to find or derive a more general equation for angular size as a function of redshift that will work for a dark-energy-dominated universe (perhaps a knowledgeable reader will post a comment here providing some insight into this issue), but it will be a useful exercise to continue with the calculation assuming the matter-dominated Einstein-de Sitter universe.

Casting Michael Richmond’s equation into the following SAS program, I was able to find that the Andromeda galaxy would reach a minimum angular size of 11.3 arcseconds at z = 1.25, as shown below.

In principle, measuring the angular size of a “standard” object at various redshifts could allow us to determine what kind of universe we live in.  But there’s a problem.  As we look further out into space we are also looking further back in time, so there is no guarantee that a “standard” object in today’s universe (say, a spiral galaxy such as M31) would have looked the same or even existed billions of years ago.

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Richmond, Michael, Two classic cosmological tests
[http://spiff.rit.edu/classes/phys443/lectures/classic/classic.html]