(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 ω_{k} 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 g_{c} 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 g_{00} 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 g_{00}. The component is brought in through the period of oscillations. All frequencies are referenced to the base oscillator that has the angular rate ω_{0}, 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 g_{00} 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 R_{0} = 4R_{S} for a solar-mass neutron star, where R_{S} is the Schwarzschild radius) and at twice that radius (at R_{0} = 8R_{S}). The length of the ship, along which the clocks are distributed, is R_{S} 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 R_{0}. 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 R_{S}, 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 R_{S}. 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 h_{n} is the height of the n-th clock above R_{0}.

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 h_{n} can be put into the equation for ω_{n} 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 g_{00} 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 R_{0} in Fig. 7. At 4R_{S}, the Kuramoto transition is stretched. At 2R_{S} there is a partial Kuramoto transition for the upper clocks, that then stretch into a cascade of locking events for the lower clocks. At 1R_{S} 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”.

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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.