# Locking Clocks in Strong Gravity: Synchronization in General Relativity

(Guest Post with Moira Andrews)

… GR combined with nonlinear synchronization yields the novel phenomenon of a “synchronization cascade”.

Imagine a space ship containing a collection of highly-accurate atomic clocks factory-set to arbitrary precision at the space-ship factory before launch.  The clocks are lined up with precisely-equal spacing along the axis of the space ship, which should allow the astronauts to study events in spacetime to high accuracy as they orbit neutron stars or black holes.  Despite all the precision, spacetime itself will conspire to detune the clocks.  Yet all is not lost.  Using the physics of nonlinear synchronization, the astronauts can bring all the clocks together to a compromise frequency—locking all the clocks to a common rate.  This blog post shows how this can happen.

## Synchronization of Oscillators

The simplest synchronization problem is two “phase oscillators” coupled with a symmetric nonlinearity. The dynamical flow is

where ωk are the individual angular frequencies and g is the coupling constant. When g is greater than the difference Δω, then the two oscillators, despite having different initial frequencies, will find a stable fixed point and lock to a compromise frequency.

Taking this model to N phase oscillators creates the well-known Kuramoto model that is characterized by a relatively sharp mean-field phase transition leading to global synchronization. The model averages N phase oscillators to a mean field where g is the coupling coefficient, K is the mean amplitude, Θ is the mean phase, and ω-bar is the mean frequency. The dynamics are given by

The last equation is the final mean-field equation that synchronizes each individual oscillator to the mean field. For a large number of oscillators that are globally coupled to each other, increasing the coupling has little effect on the oscillators until a critical threshold is crossed, after which all the oscillators synchronize with each other. This is known as the Kuramoto synchronization transition, shown in Fig. 2 for 20 oscillators with uniformly distributed initial frequencies. Note that the critical coupling constant gc is roughly half of the spread of initial frequencies.

The question that this blog seeks to answer is how this synchronization mechanism may be used in a space craft exploring the strong gravity around neutron stars or black holes. The key to answering this question is the metric tensor for this system

where the first term is the time-like term g00 that affects ticking clocks, and the second term is the space-like term that affects the length of the space craft.

## Kuramoto versus the Neutron Star

Consider the space craft holding a steady radius above a neutron star, as in Fig. 3. For simplicity, hold the craft stationary rather than in an orbit to remove the details of rotating frames. Because each clock is at a different gravitational potential, it runs at a different rate because of gravitational time dilation–clocks nearer to the neutron star run slower than clocks farther away. There is also a gravitational length contraction of the space craft, which modifies the clock rates as well.

The analysis starts by incorporating the first-order approximation of time dilation through the  component g00. The component is brought in through the period of oscillations. All frequencies are referenced to the base oscillator that has the angular rate ω0, and the other frequencies are primed. As we consider oscillators higher in the space craft at positions R + h, the 1/(R+h) term in g00 decreases as does the offset between each successive oscillator.

The dynamical equations for a system for only two clocks, coupled through the constant k, are

These are combined to a single equation by considering the phase difference

The two clocks will synchronize to a compromise frequency for the critical coupling coefficient

Now, if there is a string of N clocks, as in Fig. 3, the question is how the frequencies will spread out by gravitational time dilation, and what the entrainment of the frequencies to a common compromise frequency looks like. If the ship is located at some distance from the neutron star, then the gravitational potential at one clock to the next is approximately linear, and coupling them would produce the classic Kuramoto transition.

However, if the ship is much closer to the neutron star, so that the gravitational potential is no longer linear, then there is a “fan-out” of frequencies, with the bottom-most clock ticking much more slowly than the top-most clock. Coupling these clocks produces a modified, or “stretched”, Kuramoto transition as in Fig. 4.

In the two examples in Fig. 4, the bottom-most clock is just above the radius of the neutron star (at R0 = 4RS for a solar-mass neutron star, where RS is the Schwarzschild radius) and at twice that radius (at R0 = 8RS). The length of the ship, along which the clocks are distributed, is RS in this example. This may seem unrealistically large, but we could imagine a regular-sized ship supporting a long stiff cable dangling below it composed of carbon nanotubes that has the clocks distributed evenly on it, with the bottom-most clock at the radius R0. In fact, this might be a reasonable design for exploring spacetime events near a neutron star (although even carbon nanotubes would not be able to withstand the strain).

## Kuramoto versus the Black Hole

Against expectation, exploring spacetime around a black hole is actually easier than around a neutron star, because there is no physical surface at the Schwarzschild radius RS, and gravitational tidal forces can be small for large black holes. In fact, one of the most unintuitive aspects of black holes pertains to a space ship falling into one. A distant observer sees the space ship contracting to zero length and the clocks slowing down and stopping as the space ship approaches the Schwarzschild radius asymptotically, but never crossing it. However, on board the ship, all appears normal as it crosses the Schwarzschild radius. To the astronaut inside, there is is a gravitational potential inside the space ship that causes the clocks at the base to run more slowly than the upper clocks, and length contraction affects the spacing a little, but otherwise there is no singularity as the event horizon is passed. This appears as a classic “paradox” of physics, with two different observers seeing paradoxically different behaviors.

The resolution of this paradox lies in the differential geometry of the two observers. Each approximates spacetime with a Euclidean coordinate system that matches the local coordinates. The distant observer references the warped geometry to this “chart”, which produces the apparent divergence of the Schwarzschild metric at RS. However, the astronaut inside the space ship has her own flat chart to which she references the locally warped space time around the ship. Therefore, it is the differential changes, referenced to the ships coordinate origin, that capture gravitational time dilation and length contraction. Because the synchronization takes place in the local coordinate system of the ship, this is the coordinate system that goes into the dynamical equations for synchronization. Taking this approach, the shifts in the clock rates are given by the derivative of the metric as

where hn is the height of the n-th clock above R0.

Fig. 5 shows the entrainment plot for the black hole. The plot noticeably has a much smoother transition. In this higher mass case, the system does not have as many hard coupling transitions and instead exhibits smooth behavior for global coupling. This is the Kuramoto “cascade”. Contrast the behavior of Fig. 5 (left) to the classic Kuramoto transition of Fig. 2. The increasing frequency separations near the black hole produces a succession of frequency locks as the coupling coefficient increases. For comparison, the case of linear coupling along the cable is shown in Fig. 5 on the right. The cascade is now accompanied with interesting oscillations as one clock entrains with a neighbor, only to be pulled back by interaction with locked subclusters.

Now let us consider what role the spatial component of the metric tensor plays in the synchronization. The spatial component causes the space between the oscillators to decrease closer to the supermassive object. This would cause the oscillators to entrain faster because the bottom oscillators that entrain the slowest would be closer together, but the top oscillators would entrain slower since they are a farther distance apart, as in Fig. 6.

In terms of the local coordinates of the space ship, the locations of each clock are

These values for hn can be put into the equation for ωn above. But it is clear that this produces a second order effect. Even at the event horizon, this effect is only a fraction of the shifts caused by g00 directly on the clocks. This is in contrast to what a distant observer sees–the clock separations decreasing to zero, which would seem to decrease the frequency shifts. But the synchronization coupling is performed in the ship frame, not the distant frame, so the astronaut can safely ignore this contribution.

As a final exploration of the black hole, before we leave it behind, look at the behavior for different values of R0 in Fig. 7. At 4RS, the Kuramoto transition is stretched. At 2RS there is a partial Kuramoto transition for the upper clocks, that then stretch into a cascade of locking events for the lower clocks. At 1RS we see the full cascade as before.

## Note from the Editor:

This blog post by Moira Andrews is based on her final project for Phys 411, upper division undergraduate mechanics, at Purdue University. Students are asked to combine two seemingly-unrelated aspects of modern dynamics and explore the results. Moira thought of synchronizing clocks that are experiencing gravitational time dilation near a massive body. This is a nice example of how GR combined with nonlinear synchronization yields the novel phenomenon of a “synchronization cascade”.

## Bibliography

Cheng, T.-P. (2010). Relativity, Gravitation and Cosmology. Oxford University Press.

Contributors to Wikimedia projects. (2004, July 23). Gravitational time dilation – Wikipedia. Wikipedia, the Free Encyclopedia; Wikimedia Foundation, Inc. https://en.wikipedia.org/wiki/Gravitational_time_dilation

Keeton, C. (2014). Principles of Astrophysics. Springer.

Marmet, P. (n.d.). Natural Length Contraction Due to Gravity. Newton Physics – Links to Papers, Books and Web Sites. Retrieved April 27, 2021, from https://newtonphysics.on.ca/gravity/index.html

Nolte, D. D. (2019). Introduction to Modern Dynamics (2nd ed.). Oxford University Press, USA.

# Chandrasekhar’s Limit

Arthur Eddington was the complete package—an observationalist with the mathematical and theoretical skills to understand Einstein’s general theory, and the ability to construct the theory of the internal structure of stars.  He was Zeus in Olympus among astrophysicists.  He always had the last word, and he stood with Einstein firmly opposed to the Schwarzschild singularity.  In 1924 he published a theoretical paper in which he derived a new coordinate frame (now known as Eddington-Finkelstein coordinates) in which the singularity at the Schwarzschild radius is removed.  At the time, he took this to mean that the singularity did not exist and that gravitational cut off was not possible [1].  It would seem that the possibility of dark stars (black holes) had been put to rest.  Both Eddington and Einstein said so!  But just as they were writing the obituary of black holes, a strange new form of matter was emerging from astronomical observations that would challenge the views of these giants.

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.

## White Dwarf

Binary star systems have always held a certain fascination for astronomers.  If your field of study is the (mostly) immutable stars, then the stars that do move provide some excitement.  The attraction of binaries is the same thing that makes them important astrophysically—they are dynamic.  While many double stars are observed in the night sky (a few had been noted by Galileo), some of these are just coincidental alignments of near and far stars.  However, William Herschel began cataloging binary stars in 1779 and became convinced in 1802 that at least some of them must be gravitationally bound to each other.  He carefully measured the positions of binary stars over many years and confirmed that these stars showed relative changes in position, proving that they were gravitational bound binary star systems [2].  The first orbit of a binary star was computed in 1827 by Félix Savary for the orbit of Xi Ursae Majoris.  Finding the orbit of a binary star system provides a treasure trove of useful information about the pair of stars.  Not only can the masses of the stars be determined, but their radii and densities also can be estimated.  Furthermore, by combining this information with the distance to the binaries, it was possible to develop a relationship between mass and luminosity for all stars, even single stars.  Therefore, binaries became a form of measuring stick for crucial stellar properties.

One of the binary star systems that Hershel discovered was the pair known as 40 Eridani B/C, which he observed on January 31 in 1783.  Of this pair, 40 Eridani B was very dim compared to its companion.  More than a century later, in 1910 when spectrographs were first being used routinely on large telescopes, the spectrum of 40 Eridani B was found to be of an unusual white spectral class.  In the same year, the low luminosity companion of Sirius, known as Sirius B, which shared the same unusual white spectral class, was evaluated in terms of its size and mass and was found to be exceptionally small and dense [3].  In fact, it was too small and too dense to be believed at first, because the densities were beyond any known or even conceivable matter.  The mass of Sirius B is around the mass of the Sun, but its radius is comparable to the radius of the Earth, making the density of the white star about ten thousand times denser than the core of the Sun.  Eddington at first felt the same way about white dwarfs that he felt about black holes, but he was eventually swayed by the astrophysical evidence.  By 1922 many of these small white stars had been discovered, called white dwarfs, and their incredibly large densities had been firmly established.  In his famous book on stellar structure [4], he noted the strange paradox:  As a star cools, its pressure must decrease, as all gases must do as they cool, and the star would shrink, yet the pressure required to balance the force of gravity to stabilize the star against continued shrinkage must increase as the star gets smaller.  How can pressure decrease and yet increase at the same time?  In 1926, on the eve of the birth of quantum mechanics, Eddington could conceive of no mechanism that could resolve this paradox.  So he noted it as an open problem in his book and sent it to press.

## Subrahmanyan Chandrasekhar

Three years after the publication of Eddington’s book, an eager and excited nineteen-year-old graduate of the University in Madras India boarded a steamer bound for England.  Subrahmanyan Chandrasekhar (1910—1995) had been accepted for graduate studies at Cambridge University.  The voyage in 1930 took eighteen days via the Suez Canal, and he needed something to do to pass the time.  He had with him Eddington’s book, which he carried like a bible, and he also had a copy of a breakthrough article written by R. H. Fowler that applied the new theory of quantum mechanics to the problem of dense matter composed of ions and electrons [5].  Fowler showed how the Pauli exclusion principle for electrons, that obeyed Fermi-Dirac statistics, created an energetic sea of electrons in their lowest energy state, called electron degeneracy.  This degeneracy was a fundamental quantum property of matter, and carried with it an intrinsic pressure unrelated to thermal properties.  Chandrasekhar realized that this was a pressure mechanism that could balance the force of gravity in a cooling star and might resolve Eddington’s paradox of the white dwarfs.  As the steamer moved ever closer to England, Chandrasekhar derived the new balance between gravitational pressure and electron degeneracy pressure and found the radius of the white dwarf as a function of its mass.  The critical step in Chandrasekhar’s theory, conceived alone on the steamer at sea with access to just a handful of books and papers, was the inclusion of special relativity with the quantum physics.  This was necessary, because the densities were so high and the electrons were so energetic, that they attained speeds approaching the speed of light.

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.  He discovered that electron degeneracy pressure could balance the force of gravity if the mass of the white dwarf were smaller than about 1.4 times the mass of the Sun.  But if the dwarf was more massive than this, then even the electron degeneracy pressure would be insufficient to fight gravity, and the star would continue to collapse.  To what?  Schwarzschild’s singularity was one possibility.  Chandrasekhar wrote up two papers on his calculations, and when he arrived in England, he showed them to Fowler, who was to be his advisor at Cambridge.  Fowler was genuinely enthusiastic about  the first paper, on the derivation of the relativistic electron degeneracy pressure, and it was submitted for publication.  The second paper, on the maximum sustainable mass for a white dwarf, which reared the ugly head of Schwarzschild’s singularity, made Fowler uncomfortable, and he sat on the paper, unwilling to give his approval for publication in the leading British astrophysical journal.  Chandrasekhar grew annoyed, and in frustration sent it, without Fowler’s approval, to an American journal, where “The Maximum Mass of Ideal White Dwarfs” was published in 1931 [6].  This paper, written in eighteen days on a steamer at sea, established what became known as the Chandrasekhar limit, for which Chandrasekhar would win the 1983 Nobel Prize in Physics, but not before he was forced to fight major battles for its acceptance.

## Chandrasekhar versus Eddington

Initially there was almost no response to Chandrasekhar’s paper.  Frankly, few astronomers had the theoretical training needed to understand the physics.  Eddington was one exception, which was why he held such stature in the community.  The big question therefore was:  Was Chandrasekhar’s theory correct?  During the three years to obtain his PhD, Chandrasekhar met frequently with Eddington, who was also at Cambridge, and with colleagues outside the university, and they all encouraged Chandrasekhar to tackle the more difficult problem to combine internal stellar structure with his theory.  This could not be done with pen and paper, but required numerical calculation.  Eddington was in possession of an early electromagnetic calculator, and he loaned it to Chandrasekhar to do the calculations.  After many months of tedious work, Chandrasekhar was finally ready to confirm his theory at the 1934 meeting of the British Astrophysical Society.

The young Chandrasekhar stood up and gave his results in an impeccable presentation before an auditorium crowded with his peers.  But as he left the stage, he was shocked when Eddington himself rose to give the next presentation.  Eddington proceeded to criticize and reject Chandrasekhar’s careful work, proposing instead a garbled mash-up of quantum theory and relativity that would eliminate Chandrasekhar’s limit and hence prevent collapse to the Schwarzschild singularity.  Chandrasekhar sat mortified in the audience.  After the session, many of his friends and colleagues came up to him to give their condolences—if Eddington, the leader of the field and one of the few astronomers who understood Einstein’s theories, said that Chandrasekhar was wrong, then that was that.  Badly wounded, Chandrasekhar was faced with a dire choice.  Should he fight against the reputation of Eddington, fight for the truth of his theory?  But he was at the beginning of his career and could ill afford to pit himself against the giant.  So he turned his back on the problem of stellar death, and applied his talents to the problem of stellar evolution.

Chandrasekhar went on to have an illustrious career, spent mostly at the University of Chicago (far from Cambridge), and he did eventually return to his limit as it became clear that Eddington was wrong.  In fact, many at the time already suspected Eddington was wrong and were seeking for the answer to the next question: If white dwarfs cannot support themselves under gravity and must collapse, what do they collapse to?  In Pasadena at the California Institute of Technology, an astrophysicist named Fritz Zwicky thought he knew the answer.

## Fritz Zwicky’s Neutron Star

Zwicky and Baade submitted an abstract to the American Physical Society Meeting in 1933, which Kip Thorne has called “…one of the most prescient documents in the history of physics and astronomy” [7].  In the abstract, Zwicky and Baade introduced, for the first time, the existence of supernovae as a separate class of nova and estimated the total energy output of these cataclysmic events, including the possibility that they are the source of some cosmic rays.  They introduced the idea of a neutron star, a star composed purely of neutrons, only a year after Chadwick discovered the neutron’s existence, and they strongly suggested that a supernova is produced by the transformation of a star into a neutron star.  A neutron star would have a mass similar to that of the Sun, but would have a radius of only tens of kilometers.  If the mass density of white dwarfs was hard to swallow, the density of a neutron star was billion times greater!  It would take nearly thirty years before each of the assertions made in this short abstract were proven true, but Zwicky certainly had a clear view, tempered by Baade, of where the field of astrophysics was headed.  But no one listened to Zwicky.  He was too aggressive and backed up his wild assertions with too little substance.  Therefore, neutron stars simmered on the back burner until more substantial physicists could address their properties more seriously.

## Derivation of the Relativistic Chandrasekhar Limit

White dwarfs are created from the balance between gravitational compression and the degeneracy pressure of electrons caused by the Pauli exclusion principle. When a star collapses gravitationally, the matter becomes so dense that the electrons begin to fill up quantum states until all the lowest-energy states are filled and no more electrons can be added. This results in a balance that stabilizes the gravitational collapse, and the result is a white dwarf with a mass density a million times larger than the Sun.

If the electrons remained non-relativistic, then there would be no upper limit for the size of a star that would form a white dwarf. However, because electrons become relativistic at high enough compaction, if the initial star is too massive, the electron degeneracy pressure becomes limited relativistically and cannot keep the matter from compacting more, and even the white dwarf will collapse (to a neutron star or a black hole). The largest mass that can be supported by a white dwarf is known as the Chandrasekhar limit.

A simplified derivation of the Chandrasekhar limit begins by defining the total energy as the kinetic energy of the degenerate Fermi electron gas plus the gravitational potential energy

The kinetic energy of the degenerate Fermi gas has the relativistic expression

where the Fermi k-vector can be expressed as a function of the radius of the white dwarf and the total number of electrons in the star, as

If the star is composed of pure hydrogen, then the mass of the star is expressed in terms of the total number of electrons and the mass of the proton

The total energy of the white dwarf is minimized by taking its derivative with respect to the radius of the star

When the derivative is set to zero, the term in brackets becomes

This is solved for the radius for which the electron degeneracy pressure stabilizes the gravitational pressure

This is the relativistic radius-mass expression for the size of the stabilized white dwarf as a function of the mass (or total number of electrons). One of the astonishing results of this calculation is the merging of astronomically large numbers (the mass of stars) with both relativity and quantum physics. The radius of the white dwarf is actually expressed as a multiple of the Compton wavelength of the electron!

The expression in the square root becomes smaller as the size of the star increases, and there is an upper bound to the mass of the star beyond which the argument in the square root goes negative. This upper bound is the Chandrasekhar limit defined when the argument equals zero

This gives the final expression for the Chandrasekhar limit (expressed in terms of the Planck mass)

This expression is only approximate, but it does contain the essential physics and magnitude. This limit is on the order of a solar mass. A more realistic numerical calculation yields a limiting mass of about 1.4 times the mass of the Sun. For white dwarfs larger than this value, the electron degeneracy is insufficient to support the gravitational pressure, and the star will collapse to a neutron star or a black hole.

By David D. Nolte, Jan. 7, 2019

[1] The fact that Eddington coordinates removed the singularity at the Schwarzschild radius was first pointed out by Lemaitre in 1933.  A local observer passing through the Schwarzschild radius would experience no divergence in local properties, even though a distant observer would see that in-falling observer becoming length contracted and time dilated. This point of view of an in-falling observer was explained in 1958 by Finkelstein, who also pointed out that the Schwarzschild radius is an event horizon.

[2] William Herschel (1803), Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing, Philosophical Transactions of the Royal Society of London 93, pp. 339–382 (Motion of binary stars)

[3] Boss, L. (1910). Preliminary General Catalogue of 6188 stars for the epoch 1900. Carnegie Institution of Washington. (Mass and radius of Sirius B)

[4] Eddington, A. S. (1927). Stars and Atoms. Clarendon Press. LCCN 27015694.

[5] Fowler, R. H. (1926). “On dense matter”. Monthly Notices of the Royal Astronomical Society 87: 114. Bibcode:1926MNRAS..87..114F. (Quantum mechanics of degenerate matter).

[6] Chandrasekhar, S. (1931). “The Maximum Mass of Ideal White Dwarfs”. The Astrophysical Journal 74: 81. Bibcode:1931ApJ….74…81C. doi:10.1086/143324. (Mass limit of white dwarfs).

[7] Kip Thorne (1994) Black Holes & Time Warps: Einstein’s Outrageous Legacy (Norton). pg. 174

[8] Oppenheimer was aware of Zwicky’s proposal because he had a joint appointment between Berkeley and Cal Tech.

[9] See Chapter 7, “The Lens of Gravity” in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018).