Gravitational waves and ultra-condensed matter physics

In 2016, when I saw the first results from the LIGO gravitational wave interferometer my natural caution and skepticism kicked in. They had just observed one signal in an incredibly sensitive measurement. A lot of data analysis was required to extract the signal from the background noise. That signal was then fitted the results of numerical simulations of the solutions to Einstein's gravitational field equations describing the merger of two black holes. Depending on how you count about 15 parameters are required to specify the parameters of the binary system [distance from earth, masses, relative orientations of orbits, .... The detection events involve displacement of the mirrors in the interferometer by about 30 picometres!

What on earth could go wrong?!

After all, this was only two years after the BICEP2 fiasco which claimed to have detected anisotropies in the cosmic microwave background due to gravitational waves associated with cosmic inflation. The observed signal turned out to be just cosmic dust! It led to a book, by the cosmologist Brian Keating, Losing the Nobel Prize: A Story of Cosmology, Ambition, and the Perils of Science’s Highest Honor

Well, I am happy to be wrong, if it is good for science. Now almost one hundred gravitational wave events have been observed and one event GW170817 has been correlated with an x-ray observation.

But detecting some gravitational waves is quite a long way from gravitational wave astronomy, i.e, using gravity wave detectors as a telescope, in the same sense as the regular suite of optical, radio, X-ray, ... detectors. I was also skeptical about that. But it does not seem that gravity wave detectors are providing a new window into the universe.

A few weeks ago I heard a very nice UQ colloquium by Paul Lasky, What's next in gravitational wave astronomy?

Paul gave a nice overview of the state of the field, both past and future. 

A key summary figure is below. It shows different possible futures when two neutron stars merge.

The figure is taken from the helpful review

The evolution of binary neutron star post-merger remnants: a review, Nikhil Sarin and Paul D. Lasky

A few of the things that stood out to me.

1. One stunning piece of physics is that in the black hole mergers that have been observed the combined mass of the resulting black hole is three solar masses less than the total mass of the two separate black holes. The resulting loss of mass energy (E=mc^2) of three solar masses is converted into gravitational wave energy within seconds. During this time the peak radiant power was more than fifty times the power of all the stars in the observable universe combined!

I have fundamental questions about a clear physical description of this energy conversion process. First, defining "energy" in general relativity is a vexed and unresolved question with a long history. Second, is there any sense in which needs to describe this in terms of a quantum field theory: specifically conversion of neutron matter into gravitons?

2. Probing nuclear astrophysics in neutron stars. It may be possible to test the equation of state (relation between pressure and density) of nuclear matter. This determines the Tolman–Oppenheimer–Volkoff limit; the upper bound to the mass of cold, non-rotating neutron stars. According to Sarin and Lasky

The supramassive neutron star observations again provide a tantalising way of developing our understanding of the dynamics of the nascent neutron star and the equation of state of nuclear matter (e.g., [37,121,127–131]). The procedure is straight forward: if we understand the progenitor mass distribution (which we do not), as well as the dominant spin down mechanism (we do not understand that either), and the spin-down rate/braking index (not really), then we can rearrange the set of equations governing the system’s evolution to find that the time of collapse is a function of the unknown maximum neutron star mass, which we can therefore infer. This procedure has been performed a number of times in different works, each arriving at different answers depending on the underlying assumptions at each of the step. The vanilla assumptions of dipole vacuum spin down of hadronic stars does not well fit the data [37,127], leading some authors to infer that quark stars, rather than hadronic stars, best explain the data (e.g., [129,130]), while others infer that gravitational radiation dominates the star’s angular momentum loss rather than magnetic dipole radiation (e.g [121,127]).

As the authors say, this is a "tantalising prospect" but there are many unkowns. I appreciate their honesty. 

3. Probing the phase diagram of Quantum Chromodynamics (QCD)

This is one of my favourite phase diagrams and I used to love to show it to undergraduates.

Neutron stars are close to the first-order phase transition associated with quark deconfinement.

When the neutron stars merge it may be that the phase boundary is crossed.

Listing mistakes in Condensed Matter Physics: A Very Short Introduction

Someone told me that the day after your book is published you will start finding errors. They were correct.

Here are the first errors I have become aware of.

On Page 2 I erroneously state that diamond "conducts electricity and heat very poorly."

However, the truth about conduction of heat is below, taken from the opening paragraph of this paper.

Diamond has the highest thermal conductivity, L, of any known bulk material. Room-temperature values of L for isotopically enriched diamond exceed 3000 W/m-K, more than an order of magnitude higher than common semiconductors such as silicon and germanium. In diamond, the strong bond stiffness and light atomic mass produce extremely high phonon frequencies and acoustic velocities. In addition, the phonon-phonon umklapp scattering around room temperature is unusually weak.

Figure 2 on page 4 has a typo. Diamond is "hard" not "hand".

On page 82 I erroneously state that for the superfluid transition, the "critical exponent alpha was determined to have a value of -0.0127, that is to five significant figures."  The value actually has three significant figures. 

I thank my engineering friend, Dave Winn, for pointing out the first and third errors.

Update. August 27, 2024.

On page 40, I erroneously state that shear sound waves exist in a liquid. I thank Jean-Noel Fuchs for pointing out this error. Below, I have drafted a corrected paragraph

"In a fluid (gas or liquid) one way to distort a cubic volume of the fluid is to compress the cube into a shape (a rectangular prism) where the lengths of the sides are not identical, but the angles between the sides of the shape are still 90 degrees. Sound waves in air consist of this type of compression: oscillations in the density and pressure of the air occur in the same direction that the sound wave travels. 

      In an isotropic solid there is a second type of distortion: the shape of the cube is changed to that of a rhombohedron, the angles are no longer 90 degrees, but the lengths of the sides remain the same. Associated with these two types of distortions, there are two distinct ways in which sound can travel through a solid. A second type of sound wave corresponds to the second type of distortion, and is called a shear wave. The two types of sound travel at different speeds. An earthquake produces both types of waves: pressure waves and shear waves, the latter travelling slower. Observing and comparing the two types of waves plays an important role in seismology and in the detection of earthquakes."

There are some subtle issues here that go beyond what is appropriate in a VSI. Damped shear waves can exist in a liquid for wavevectors larger than some critical value. I will discuss the issues in a separate blogpost.

Please do write other errors in the comments below. This will help with future revisions.

Amazing things about Chandrasekhar's white dwarf mass limit

This is ultra-condensed matter physics!

In 1931, Subrahmanyan Chandrasekhar published a seminal paper, for which he was awarded the Nobel Prize in 1983. He showed that a white dwarf star must have a mass less than 1.4 solar masses, otherwise it will collapse under gravity. White dwarfs are compact stars for which the nuclear fuel is spent and electron degeneracy pressure prevents gravitational collapse. 

The blog Galileo unbound has a nice post about the history and the essential physics behind the paper.

There are a several things I find quite amazing about Chandrasekhar's derivation  and the expression for the maximum possible mass. 

m_H is the mass of a proton. M_P is the Planck mass. The value of the mass limit is about 1.4 solar masses.

Relativity matters

If the electrons are treated non-relativistically then there is no mass limit. However, when the star becomes dense enough the Fermi velocity of the electrons approaches the speed of light. Then relativistic effects must be included.

A macroscopic quantum effect

Degeneracy pressure is a macroscopic quantum effect. The expression above involves Planck's constant.

Quantum gravity

The mass formula involves the Planck mass, M_P.  On the one hand, this phenomena does not involve quantum gravity because there is no quantisation of the gravitational field. On the other hand, the effect does involve the interplay of gravity and quantum physics.

A "natural" scale

Formula that involve Planck scales usually represent scales of length, energy, mass, time, and temperature that are "unreal", i.e., they are vastly different from terrestial and astrophysical phenomena. For example, the temperature of the Hawking radiation from a black hole of one solar mass is 60 nanoKelvin. 

In contrast, the limiting mass is on the same scale as that of our own sun!

It agrees with astronomical observations

Determinations of the masses of hundreds of white dwarfs show most have a mass of about 0.5 solar masses and the highest observed value is 1.3 solar masses.

Hollywood and a Physical Review paper

 I am not sure I have seen this before. If you watched the movie Oppenheimer, you may have noticed that a one point a student excitedly showed Oppenheimer the latest issue of Physical Review and the following image flashed across the movie screen.

On Continued Gravitational Contraction

J. R. Oppenheimer and H. Snyder

A beautiful blog post just appeared on 3 Quarks Daily,

 

September 1, 1939: A Tale Of Two Papers

by Ashutosh Jogalekar

The post describes the scientific and historical significance of the paper, including how it attracted no interest for twenty years, being eclipsed by a paper in the same issue of Physical Review.

The Mechanism of Nuclear Fission

Niels Bohr and John Archibald Wheeler

Have you ever seen a Hollywood movie that explicitly showed the page of a scientific journal article.

Condensed Matter Physics: A Very Short Introduction (hard copies) now available on Amazon USA

My book has finally been released by Amazon in the USA. I don't like Amazon but it is cheap and you can avoid shipping charges.

In Australia Amazon has listed under "Engineering and Transportation" and is currently out of stock. 

Majorana: mysterious disappearance of a particle and of credibility

The theoretical physicist Ettore Majorana mysteriously disappeared in 1938. Unfortunately, Majorana particles are also going to be associated in history with some mysterious disappearances: their own existence, the prospect of a topological quantum computer, some prominent scientists' reputations, and the credibility of Physical Review journals.

Nine years ago I expressed skepticism that there would ever be a quantum computer based on Majorana fermions. I wish I was wrong. It certainly would be cool. Things are even worse than I thought. The issues (scientific, ethical, technological, hype, ...) were recently highlighted in a strange incident involving a paper from Microsoft that was published by PRB.

I found the commentary of Vincent Mourik on the whole incident enlightening and disturbing. His description is "Here's the full background of my involvement with the recent Microsoft Quantum paper. APS pulled off an arcane unofficial off-the-record peer review already one year ago when it was presented at PRX. And then published it anyway at PRB..."

I do have concerns about the unusual practice used by PRX and the precedent of publishing a paper with incomplete details. However, my more significant concern is that, based on Vincent's report, this paper should never have been published in any self-respecting scientific journal. I fear that in order to "compete" with the luxury journals Physical Review has descended to their low scientific and ethical standards.

I thank Doug Natelson and a commenter on his blog for bringing this sorry saga to my attention.

What is a complex system?

What do we mean when we say a particular system is "complex"? We main have some intuition that it means there are many degrees of freedom and/or that it is hard to understand. "Complexity" is sometimes used as a buzzword, just like "emergence." There are many research institutes that claim to be studying "complex systems" and there is something called "complexity theory". Complexity seems to mean different things to different people.

I am particularly interested in understanding the relationship between emergence and complexity. To do this we first need to be more precise about what we mean by both terms. A concrete question is the following. Consider a system that exhibits emergent properties. Often that will be associated with a hierarchy of scales. For example: atoms, molecules, proteins, DNA, genes, cells, organs, people. The corresponding hierarchy of research fields is physics, chemistry, biochemistry, genetics, cell biology, physiology, psychology. Within physics a hierarchy is quarks and leptons, nuclei and electrons, atoms, molecules, liquid, and fluid. 

In More is Different, Anderson states that as one goes up the hierarchy the system scale and complexity increases. This makes sense when complexity is defined in terms of the number of degrees of freedom in the system (e.g., the size of the Hilbert space needed to describe the complete state of the system). On the other hand, the system state and its dynamics become simpler as one goes up the hierarchy.  The state of the liquid can be described completely in terms of the density, temperature, and the equation of state. The dynamics of the fluid can be described by the Navier-Stokes equation. Although that is hard to solve in the regime of turbulence, the system is still arguably a lot simpler than quantum chromodynamics (QCD)! Thus, we need to be clearer about what we mean by complexity.

To address these issues I found the following article very helpful and stimulating.

What is a complex system? by James Ladyman, James Lambert, and Karoline Wiesner 

It was published in 2013 in a philosophy journal, has been cited more than 800 times, and is co-authored by two philosophers of science and a physicist.

[I just discovered that Ladyman and Wiesner published a book with the same title in 2020. It is an expansion of the 2013 article.].

In 1999 the journal Science had a special issue that focussed on complex systems, with an Introduction entitled, Beyond Reductionism. Eight survey articles covered complexity in physics, chemistry, biology, earth science, and economics.

Ladyman et al., begin by pointing out how each of the authors of these articles chooses different properties to define what complexity is associated. These characteristics include non-linearity, feedback, spontaneous order, robustness and lack of central control, emergence, hierarchical organisation, and numerosity.

The problem is that these characteristics are not equivalent. If we do choose a specific definition for a complex system, the difficult problem then remains of determining whether each of the characteristics above is necessary, sufficient, both, or neither for the system to be complex (as defined). This is similar to what happens with attempts to define emergence.

Information content is sometimes used to quantify complexity. Shannon entropy and Kolmogorov complexity (Sections 3.1, 3.2) are discussed. The latter is also known as algorithmic complexity. This is the length of the shortest computer program (algorithm) that can be written to produce the entity as output. A problem with both these measures are they are non-computable.

Deterministic complexity is different from statistical complexity (Section 3.3). A deterministic measure treats a completely random sequence of 0s and 1s as having maximal complexity. A statistical measure treats a completely random sequence as having minimal complexity. Both Shannon and algorithmic complexity are deterministic.

Section 4 makes some important and helpful distinctions about different measures of complexity.

3 targets of measures: methods used, data obtained, system itself

3 types of measures: difficulty of description, difficulty of creation, or degree of organisation

They then review three distinct measures that have been proposed logical depth (Charles Bennett), thermodynamic depth (Seth Lloyd and Heinz Pagels), and effective complexity (Murray Gell-Mann).

Logical depth and effective complexity are complementary quantities. The Mandelbrot set is example of a system (set of data) that exhibits a complex structure that has a high information content. It is difficult to describe. It has a large logical depth.

Created by Wolfgang Beyer with the program Ultra Fractal 3. 

On the other hand, the effective complexity of the set is quite small since it can be generated using the simple equation

z_n+1 = c + z_n^2

c is a complex number and the Mandelbrot set is the values of c for which the iterative map is bounded.

Ladyman et al, prefer the definition of a complex system below, but do acknowledge its limitations.

(Physical account) A complex system is an ensemble of many elements which are interacting in a disordered way, resulting in robust organisation and memory. 

(Data-driven account) A system is complex if it can generate data series with high  statistical complexity. 

What is statistical complexity? It relates to degrees of pattern and some they refer to as causal state reconstruction. It is applied to data sets, not systems or methods. Central to their definition is the idea of the epsilon-machine, something introduced in a long and very mathematical article from 2001, Computational Mechanics: Pattern and Prediction, Structure and Simplicity, by Shalizi and Crutchfield.

The article concludes with a philosophical question. Do patterns really exist? This relates to debates about scientific realism versus instrumentalism. The authors advocate something known as "rainforest realism", that has been advanced by Daniel Dennett, Don Ross, and David Wallace. A pattern is real if one can construct an epsilon-machine that can simulate the phenomena and predict its behaviour.

I don't have a full appreciation or understanding of where the article ends up. Nevertheless, the journey there is helpful as it clarifies some of the subtleties and complexities (!) of trying to be more precise about what we mean by a "complex system".