Is an interstellar clock orbitally possible?

Question

A Kardashev Type III civilisation has need of a long-scale timepiece, and as such has arranged a series of planets around a black hole such that their orbital periods can be used to tell the time in the same way an analogue clock would be on Earth (don't worry about how they know where 12 o' clock is).

However there is an issue: This clock is meant to last a long time (on the order of tens of billions of years), and the orbits of the planets must remain precisely calibrated to their original orbital periods. Another slight wrinkle is that this race plans to leave the universe (long story. Blame a space wizard) for a little while and so won't be able to actively modify any of the orbits while they're away. A drift of 1 part in 10¹⁴ (the same as the TAI) is permissible.

Given that this civilisation is capable of carting around black holes and planets (potentially to an extragalactic location if needed), is this 'orbital clock' possible? If not, what is the maximum accuracy one could hope to achieve (given that there are multiple planets in this clock)?

Please note: The hard science tag really matters here. Equations to prove yes or no are a must.

Answer 1

Your main problem is that the planets will comprise a chaotic system. This is why the stability of the Solar System is so difficult (possibly impossible) to determine. Our best models are valid for perhaps $\sim10^8$ years - at best (see Laskar et al. (2004)). This is the Lyapunov time, over which orbits are definitely chaotic (the 50 million years quoted there is an extreme lower estimate). Determining that requires calculating the system's Lyapunov exponent, which is not easy and which I will not do here. The two are related, though, by the equation $$|\delta Z(t)|=e^{\lambda t}|\delta Z_0|$$ for Lyapunov exponent $\lambda$, time $t$, and separation $Z(t)$. If we assume that this system has a similar Lyapunov exponent, then over even $\sim10^9$ years, the system is chaotic, and the clock becomes essentially useless.

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Another issue that makes the idea of a central clock - or an extragalactic clock - a bit of a pain in general is time dilation.

Some postulates:

1. Being a Type III civilization, these beings have control of the entire galaxy and likely inhabit it. They therefore need the clock to work in all parts of it.
2. The galaxy has a non-uniform density, and therefore a non-uniform potential. Even treating it as a large disk and neglecting individual effects of bodies, there will be gravitational time dilation.
3. The civilization will survive for a long period of time, and therefore any discrepancies must be tiny; otherwise, the effects will snowball.

I'll take the calculations from John Rennie's answer here. If we assume a central potential of $\Phi(r=0)=6.4\times10^{11}\text{ J kg}^{-1}$, and time dilation of the form $$\Delta t_0=\Delta t_{\infty}\sqrt{1-\frac{2\Delta\Phi}{c^2}},\quad\Delta t_{\infty}\approx\Delta t_{\text{edge}}$$ where $\Delta t_{\text{edge}}$ is a time interval at the edge of the galaxy, then we find that $$\Delta t_{\text{edge}}-\Delta t_0\sim7\times10^{-7}\Delta t_{\text{edge}}$$ That causes a discrepancy much greater than one part in $10^{14}$ - not from inaccuracies in the clock, but just because time will tick differently at different points in the galaxy. This clock cannot be used throughout the galaxy - or even in a small portion of it. It would be much better to use different clocks in different regions. And if you put it outside the galaxy, you can't even use it in the majority of galactic locations.

Yes, you could make corrections depending where you are in the galaxy, but the equation given above - and in the linked answer - is only an approximation. It takes some more complicated computations to correctly figure out gravitational time dilation in the galaxy to enough precision, and frankly, there's a point at which it's just not worth it.

Answer 2

Edit: This answer plays around with one extreme of the question with a large orbital radius and a large orbital velocity that immediately self-destructs. It probably misuses special and general relativity. Also, it requires a civilization with energies at it's disposal beyond the Kardashev scale. Please take it with a grain of salt. It is a thought experiment gong wrong. (Though even negative thought experiment results are still results.)

Let's suppose you give the planet a large orbital radius R. As R approaches infinity the effects of the black hole approach zero. What you're left with is a planet moving through the vacuum of space. Now the question becomes "how consistently can a planet moving linearly through space be used as a clock". Nobody knows how many meteors are moving through the void between galaxies so let's get back to it later. Now the only things that can change the velocity of the planet are radioactive decay from the planet's surface, fluctuations in radiation pressure from the stars coming from the night sky of space, quantum tunneling, and unpredictable variations caused in the gravitational warping of space time. Most of these effects are extraordinarily tiny.

Radioactive decay from the planet's surface can be lowered to below humanity's ability to measure by using non-reactive substances. Quantum mechanics becomes unmeasurable once you hit macroscopic scales, on planet-sized scales it is completely irrelevant. Radiation pressure is stronger than both of these and the random pressure from solid objects hitting our planet even if it's just cosmic dust is greater than radiation pressure because an individual dust particle imparts much more momentum than an individual photon. The momentum of a cosmic dust particle is about 10km/s * 10^-4kg = 10kg m/s. The momentum of the greatest cosmic ray ever measured has an energy of 3 x 10^20 eV. Divide this by the speed of light and we get momentum of 1.6x10^-7kg m/s, a difference of eight orders of magnitude. The effects of cosmic dust are smaller than the effects of cosmic meteors. How many cosmic meteors are out there? Nobody knows.

But the cosmic dust and cosmic rays don't really matter because you can minimize their effects by speeding the planet up to near-lightspeed. The faster you get the planet going the less small changes in momentum are going to affect it's velocity, thanks to relativity. (Though you do eventually end up producing a nuclear explosion with energies far beyond known physics. At this point you have a bomb, not a clock, and what you're measuring is the shockwave of weirdness.) So if you get a planet moving at 99.999...% the speed of light you can make the clock arbitrarily precise, except for variations in it's course caused by unpredictable gravitational fluctuations warping spacetime. These come from dark matter and dark energy, stuff we also don't understand.

So the answer to your question is that by bending the rules of this situation beyond the breaking point we can make the clock arbitrarily precise (at least up to the Plank limit beyond which time and distance cease to mean anything). The precise accuracy of the clock for a given set of circumstances is far beyond known physics.

But what if we didn't stretch the rules to the breaking point and instead used a simple situation that we understand very well? What if we took an Earth-like planet and placed it around a sun-like star with similar Earth-like conditions? Then the orbit would vary about as much as Earth's does.

Edit: Neighboring Galaxies

Joe asked if the orbital radius would be large enough to be affected by neighboring galaxies. I didn't think of that in my original answer. The answer is yes, so it looks like we'll need a Kardashev Type IV civilization to push around some galaxies to give this clock some space.

First. Let's calculate the orbital radius. We'll start with GMm/R^2=F=mv^2/R. (I'm not sure if this equation doesn't work for scales this large. In fact, I'm not even really sure what'll happen to orbits at relativistic speeds.) This equation reduces to GM/v^2=R. The mass of the black hole in the center of our galaxy is 8.2×10^36 kg. Let's plug this into the equation and use the speed of light as v. This gets us R=5x10^21m. The third-closest galaxy to the Milky Way is 1.6x10^21m so this orbital radius on the order of the distance between galaxies in our neighborhood.

However, the distance between galaxies varies and we're not in the most empty part of the universe. Cosmic voids contain few to no galaxies and have a diameter or 10 to 100 megaparsecs. R = 5x10^21 meters = 0.2 Megaparsecs (Mpc). So you could fix the lightspeed orbital clock in one of these with a 5 Mpc radius to spare. This means the neighboring galaxies may be 20 times farther away thereby exerting at least a 400th of the gravitational force of the black hole at the center of this clock. That is significant.

Technically what we need to worry about here isn't gravity from neighboring galaxies but rather gravity from dark matter which there is more of and that will produce a stronger force. We don't know how the gravitational pull of dark matter fluctuates because we can only measure it's long-term effects on large scales that places dark matter beyond the realm of current science, so let's just focus on the matter galaxies for the moment.

They will certainly throw the clock off significantly if the civilization building this clock but will they do so unpredictibly over tens of billions of years? The universe is only 14 billion years old. I doubt we can predict galactic movements that far out, but you'll have to ask a Kardashev Type physicist to be sure.

Answer 3

The OP requested a hard science answer for a question that essentially doesn't need or effectively never needed a hard science answer.

Why? Because the black hole clock as proposed is an absurdity. Also, it is completely unnecessary. This answer shall not attempt demonstrate why a back hole clock is an absurdity. Many of the other answers demonstrate the problems involved. There is a currently existing technological that will perform the necessary timekeeping for a Kardashev III civilization.

The ideal timekeeper for a galactic civilization is a pulsar clock.

A pulsar clock is a clock which depends on counting radio pulses emitted by pulsars.

In fact, the pulsar clock already exists and is operating in Europe.

The first pulsar clock in the world was installed in St Catherine's Church, Gdańsk, Poland, in 2011.1 It was the first clock to count the time using a signal source outside the Earth. The pulsar clock consists of a radiotelescope with 16 antennas, which receive signals from six designated pulsars. Digital processing of the pulsar signals is done by an FPGA device.

A K3 civilization should be to adapt the same principles and perform timekeeping to orders of magnitude of accuracy better than currently achieved. Exactly what any self-respecting galactic civilization needs to achieve.

Once the rotational frequency of any given of pulsars is known, it will be possible to determine the time anywhere in the galaxy.

In the nearly inertial frame of the Solar-system barycenter, the rotational period of a pulsar is nearly constant, so the time-dependent phase (t) of a pulsar can be approximated by a Taylor expansion

  (t)=0+f(t−t0)+21f(t−t0)2+ 

where 0 and t0 are arbitrary reference phases and times for each pulsar. The important thing about pulsar timing, though, is that the observed rotational phase difference between each of the TOAs must contain an integer number of rotations. Since each TOA corresponds to a different time t, the parameters that we are fitting for, such as f and f, must result in a phase change between each pair of TOAs i and j that is an integer number of turns, or ij=n turns (1 turn = 2 radians). Since all measurements are made with regard to the integrated pulse phase rather than the instantaneous pulse period, the precision with which astronomers can make long-term timing measurements can be quite extraordinary.

Extraordinary long-term timing measurements, isn't that exactly what a K3 civilization will want for its time keeping? No need to shunt black holes and their planets around. The proposed interstellar clock using black holes is a non-issue. The real answer is they wouldn't bother to even think of doing it -- except as an amusing hypothetical exercise -- because pulsar clocks do work, will work, and likely to do so into the far future.

Sources:

The Wikipedia entry for pulsar clock

Pulsar Timing at http://www.cv.nrao.edu/course/astr534/PulsarTiming.html