OrbitalHub

Roughly 300 light-years from Earth, in a part of the Milky Way that no catalogue has bothered to name, two objects circle each other every 86 minutes and 39 seconds. One is a small star, a red dwarf about 85 times the mass of Jupiter. The other is a brown dwarf about 25 times Jupiter's mass, an object too heavy to be a planet yet too light to fuse hydrogen the way stars do. They orbit so close together that the entire configuration would fit inside the diameter of the Sun. And the smaller one has been quietly leaking material onto the larger one, at a rate slow enough that the larger object will be feeding on its neighbour for billions of years. The system, named ZTF J0440+2325, is described in a paper appearing today in Nature Astronomy by Aaron Householder, Kevin Burdge, and collaborators at MIT and partner institutions, and reported in a parallel MIT News feature by Jennifer Chu. It is the first confirmed case of stable, long-lived mass transfer from a substellar object onto a main-sequence star, and it pulls a tidy assumption about what happens to planets and brown dwarfs that end up too close to a star.

The tidy assumption is the swallow-engulfment model. When a planet, a brown dwarf, or any other substellar object drifts close enough to its host star, the standard picture is that the host tidally strips or disrupts it on a short timescale. The engulfed material gets swallowed and incorporated into the star, the companion disappears, and the binary turns into a single object with a slightly higher mass. This is the fate that astronomers have routinely inferred from observations of close white-dwarf and neutron-star binaries, and it is the fate that theory predicts for the Earth when the Sun leaves the main sequence in five billion years and expands into a red giant. The brown dwarf in ZTF J0440+2325 was supposed to be on the same conveyor belt, dragged inward by tidal friction and consumed on a timescale of thousands to millions of years. Instead, it is doing something much slower and stranger. It is leaking material onto its companion at a rate of about one hundred-thousandth of an Earth mass per year, a number the team works out in the Nature Astronomy paper using the photometric wobble of the binary and the assumption that the donor is overflowing its Roche lobe along the L1 Lagrange point. That is about 40 million dump trucks of material per year by the team's own analogy, a quantity that sounds large but is a tiny fraction of the brown dwarf itself. The star will not run out of food before the Galaxy ages out from under it.

What the team saw first was the triangular waveform. The Zwicky Transient Facility, a wide-field optical time-domain survey that uses a camera on the 48-inch Samuel Oschin Telescope at Palomar Observatory in California, had been picking up a strange light curve from a star in Taurus for several years. Supernova light curves are bell-shaped: gradual rise, peak, gradual fade. Buried in the ZTF archive was a signal that did not look like a bell at all. It looked like a triangle, repeating on a period of about 87 minutes, with no counterpart at other wavelengths and no obvious explanation. Burdge, an assistant professor of physics at MIT who had been searching the ZTF data for periodic variables, noticed it during a search for a different kind of system. He originally suspected a "black widow binary," the colloquial name for a millisecond pulsar that is slowly eroding a low-mass stellar companion with its particle wind and radiation. In a black widow, the optical light curve wobbles because the companion is being whipped around the much heavier neutron star, and the orbit usually shows a clean Doppler signature of that motion. The ZTF light curve did not wobble in the same way, which is what tipped the team off that the two objects were not a David-and-Goliath pair but a more balanced system in which neither dominates. Householder, a graduate student in MIT's Department of Earth, Atmospheric and Planetary Sciences, reanalysed the ZTF photometry and combined it with follow-up time-series spectroscopy. The wobble was small, the spectrum showed hydrogen and helium emission lines characteristic of an M dwarf, and the orbital fit locked the masses at roughly 85 and 25 Jupiter masses. The orbit fit inside the Sun.

The simulation work clinched the picture. Once the team had a mass, a radius, and an orbital geometry, the next question was whether the brown dwarf was overflowing its Roche lobe, the teardrop-shaped region of space around a body in a binary inside which material is bound to that body. If the brown dwarf were inside its Roche lobe, no mass transfer would be possible; the two objects would simply orbit forever without exchanging material. If the brown dwarf were overfilling the lobe, material would spill through the L1 point and rain down on the companion's surface. The team ran a set of smoothed-particle hydrodynamics simulations with the measured masses, the measured orbital separation, and a brown-dwarf radius tuned to match the photometric brightness. The simulated test particles do not orbit the brown dwarf indefinitely; they stream across the L1 point and pummel the surface of the red dwarf at high speed. The team describes the geometry as more like an asteroid hitting the Moon than the usual accretion disk around a black hole or a neutron star, where material spirals inward through a flattened disk before settling onto the central object. In ZTF J0440+2325, the brown dwarf is too small for its own gravity to capture the inflowing material into a disk, so the stream lands directly on the surface and heats a roughly circular hot spot at the impact point. That hot spot is what produces the triangular waveform. As the brown dwarf orbits, the hot spot rotates in and out of view, and the system brightens when the hot spot faces Earth and dims when it faces away. The peak of the triangle is the moment of direct face-on view, when the impact fireball is brightest.

The mass-transfer rate is low enough that the system is genuinely stable on astrophysical timescales. Most binary mass transfer is unstable on some level: the donor shrinks as it loses mass, the orbit responds by widening or shrinking in ways that either accelerate or choke off the transfer, and the system either spirals toward a common-envelope merger or settles onto a longer-period detached configuration. What the simulations show for ZTF J0440+2325 is the rarer case in which the mass-loss timescale of the brown dwarf and the angular-momentum loss timescale of the orbit are tuned just right that the rate stays roughly constant for hundreds of thousands of years and may hold for the entire remaining lifetime of the brown dwarf, on the order of billions of years. The brown dwarf, as a substellar object, cannot fuse hydrogen and so does not have a main-sequence lifetime in the usual stellar sense, but it has a thermal-cooling timescale, and the team argues that the binary will outlive that timescale many times over. The brown dwarf will not be eaten quickly. It will be eaten slowly, continuously, and quietly, and the Galaxy will be a different place by the time the meal is finished.

What the team plans to look for next is more of the same. The triangle-shaped light curve is a distinctive signature, and the ZTF archive contains years of survey photometry that have not been systematically searched for it. The paper names a second candidate, ZTF J1444+4820, with an even shorter 67-minute period and a similar wobble signature. If that system confirms, the slow-eating configuration stops being a one-off and becomes a routine endpoint of binary evolution when the masses and separation land in roughly the right place. The triangular waveform, the small wobble, and the absence of a Doppler signature that would betray a heavy compact object are the three ingredients of the search. The same configuration, in principle, applies to small stars and giant planets, and the team plans to look for those as well. The conventional picture of close planetary companions has to share the page with a quieter alternative: a slow, steady meal that outlasts planets.

Abstract

The blogpost explores relativistic effects on proof-of-work based cryptocurrency protocols. Cryptocurrencies are here to stay and it is quite plausible that future human colonists spread across the solar system and beyond will use a decentralized cryptocurrency as opposed to a fiat currency issued by a central authority. The low transaction fees, the ubiquitous access, not being bound by exchange rates or interest rates, not being controlled by financial institutions who are serving foreign interests -- these are some of the advantages cryptocurrencies will enjoy in the thriving exo-economy.

Motivation

At present, on a cryptocurrency network, the information exchanged by the nodes in the network reaches each node almost instantaneously. The speed at which the TCP/IP packets travel on the network and the fact that the Internet spans only the Earth and the LEO makes this possible. However, once the region the network is spread across will reach certain boundaries, the network size would have negative impact on the network: increasing communication failures due to network delays, more frequent and longer blockchain forks as part of the proof-of-work protocol, network segregation into local sub-networks (let us call them topological forks), just to name a few.

Nullius in verba... as they say. Let's go deeper into details and figure out a feasible solution for a truly interplanetary cryptocurrency.

Cryptocurrency 101

You can think of a cryptocurrency as a digital money ecosystem. Plain and simple. A collection of technologies are part of this ecosystem, all of them the result of years of research in the cryptography and distributed systems fields: a decentralized network of computers, a public transaction ledger -- also known as a blockchain, a protocol that consists of a set of rules for transaction validation, a decentralized consensus mechanism. [ANTO]

A decentralized network of computers ensures the resilience of the network. We can think of both computing power and data storage capabilities. As the computing power is distributed across the network, any disruption can be successfully handled by the network. Transaction data resides on all nodes of the network. This implies that even physical damage done to network nodes will not take out the network.

The blockchain is a public distributed ledger which stores all transactions handled by the network. As the name suggests, the blockchain is a list of data blocks. Each of these blocks contains a set of transactions, as many as can fit in the block, given the maximum size of the block (a characteristic of the network). Transactions contain sender and receiver info, and the amount/asset that is changing ownership, and they are broadcasted and added to blocks by network nodes. Blocks are linked to the previous block in the chain by a cryptographic hash (the hash of the previous block becomes part of the current block). This backward link leads all the way to the first block in the chain, the Genesis block. Each cryptocurrency blockchain has one. The cryptographic hashes have an important role in protecting the blockchain from tampering attempts.

The transaction validation rules enforced by the nodes in the network ensure that the content of each block in the blockchain is valid. By far, the most frequent form of fraud is double spending. The validation process makes sure that the inputs on the transactions exist and that they have not been already spent. The transactions marked as invalid are rejected by the network and do not make it on the blockchain.

The consensus mechanism is designed so that all the nodes in the network can agree on the set of transactions to be included in the current block. It shifts the authority and the credibility required by a central clearing house to a network of nodes. Important to mention here that, inherently, the nodes do not trust each other, and they do not have to because the trust is enforced by the consensus mechanism itself.

The Proof-of-Work

We mentioned above that one of the key innovations that make cryptocurrencies possible is the decentralized consensus mechanism. Currently, the de facto consensus mechanism is the proof-of-work.

The proof-of-work consensus mechanism was proposed by Satoshi Nakamoto in 2008 [NAKA]. In general, a proof-of-work is a piece of data that requires costly and time consuming computational effort, but it is very easy to verify. A very good analogy is one of your semester projects in college... it takes you a whole semester to finalize it, but it takes very little time and effort for your supervisor to evaluate and grade it. Similarly, proof-of-work is used by cryptocurrencies for block generation. For the network to accept the block, nodes must complete a proof-of-work which also guarantees the integrity of the transactions included in the block. The difficulty of the proof-of-work is adjusted by the network so that a new block can be generated every fixed time interval. This fixed time interval is characteristic to every cryptocurrency. Most notably, for Bitcoin this time interval is set to 10 minutes.

Satoshi Nakamoto's solution, emobodied by the proof-of-work algorithm above, achieves consensus on the network without a central trusted authority (hence the name decentralized consensus mechanism).

Enter the Ansible

For the astute reader, it is quite obvious at this point that communication delays between the network nodes have a direct effect on the protocol described above. An important assumption made when the protocol was designed is that these delays are small, and this is why they can be handled by the network when the nodes choose to follow and validate the blocks on the longest blockchain fork. Also, this is the reason why all the nodes in the network can have a say in the network consensus. They all can find a solution for the current block, and they all can be rewarded when they find the solution. The communication delays witnessed by the peripheral nodes will cripple their ability to find solutions and these nodes will not be incentivized to remain in the network. Hence, the segregation of the network effect mentioned earlier.

Unfortunately, given the current technological level of our civilization, we do not have at our disposal a technology that allows us to communicate fast and reliable over the large expanse of space. It would take a hell of a wait time to proces a payment made by a mining corporation in the TRAPPIST-1 system, 12 parsecs within the constellation of Aquarius, to a planetary engineering corporation located in the Sol system... twice 39 years plus the block confirmation time.

Fortunately, the Sci-Fi literature already offers a solution for our problem. For those of you, Sci-Fi nerds like myself, that have already read Rocannon's World [LEGU] and Ender's Game [CARD], the Ansible device must sound very familiar. The ansible is a fictional device capable of faster-than-light communication. To word this differently, an operator of such device can send and receive messages to and from another device over any distance with no delay.

Hence, even if only in the realm of science fiction, we will be able to devise a solution for the problem that the future cryptocurrency enthusiasts, living in the outer space colonies, will eventually face.

The Special Relativity Theorem

Before elaborating more on a required upgrade of the network protocol, we have to discuss the special relativity theorem and its implications on how time and distance are perceived in reference systems.

Albert Einstein was awarded the Nobel Prize in Physics in 1921. He received it for his contributions to the understanding of the photoelectric effect, after publishing a paper on it in 1905. At that point his contributions to the understanding of gravity through his theory of relativity were well known, but the new perspective on gravity offered by Einstein's theory was so controversial that the Nobel Prize Committee members chose to protect their reputation. They decided that it was appropriate to award Einstein the Nobel Prize for "his services to theoretical Physics, and especially for his discovery of the law of the photoelectric effect."

During his research Einstein attempted to reconcile the principle of relativity with the principle of the constancy of the velocity of light. This attempt led Einstein to the discovery of the special relativity theorem. Einstein's Gedankenexperiment (thought experiment) with a test subject travelling by train is very well known in the scientific community. If our human subject is walking towards the front of the train with velocity w with respect to the train, and the train is moving with velocity v with respect to the embankment, then an observer on the embankment will measure as W = v + w the velocity of our subject on the train. If instead of our traveller, we consider a beam of light propagating with velocity c, the velocity measured by the observer on the embarkment would be v + c. However, this violates the principle that the velocity of light is constant in any inertial reference system and equal with c. Einstein found a solution for this problem and thus resolved the incompatibility.

One direct consequence of the special relativity theorem is the Lorentz transformation.

Before the relativity theorem, time in physics had an absolute significance, independent of the state of motion of the reference system. However, every inertial reference system has its own particular time. We always have to be told which inertial reference system the statement of time refers to. If you look at your watch, the time you read is the time as measured in the Earth inertial reference system. By the way, the Earth circles the Sun on an almost circular orbit, which means that any body maintaining a constant position in the Earth reference system should experience some centripetal force. However, this centripetal component is negligible compared to the gravitational pull of the Earth. Hence, engineers use this aproximation when calculating satellite orbits.

Back to our original trail of thoughts... the axiom that lays the foundation for the Lorentz transformation states that every ray of light possesses the velocity of transmission c relative to any inertial reference system. That is the velocity of transmission in vacuo (in a vacuum).

If following Einstein's thought experiments [EINS], the above-mentioned axiom leads to a set of four equations that explain the relation between dimensional and temporal coordinates in two inertial reference systems:

  x' = (x - vt)/sqr(1 - v²/c²)
  y' = y
  z' = z
  t' = (t - x*v/c²)/sqr(1 - v²/c²)

where:

 x, y, z, t coordinates in inertial reference system K,
 x', y', z' , t' coordinates in inertial reference system K',
 v the relative velocity between inertial reference systems;

also, the expression 1//sqr(1 - v²/c²) is known as the Lorentz factor.

The Lorentz transformation equations are a more general case of the Galilean transformation:

  x' = x - vt
  y' = y
  z' = z
  t' = t

which is the basis of the assumptions of the classical mechanics as to the absolute character of both dimensional and temporal coordinates. The above are the result of replacing velocity c with ∞ in the Lorentz transformation equations.

The Lorentz transformation has a few corollaries which have an impact on time dilation, length contraction, relativistic mass, relativistic momentum, and relativistic kinetic energy. The one that concerns us is the time dilation corollary:

 Δt = γΔt'
 
where γ is defined by

 α = 1/γ = sqr(1 - v²/c²)

Assuming a clock at rest in inertial reference system K, and moving with the velocity v in the inertial reference system K', the time Δt' between two ticks as measured in the frame K' is longer than the time Δt between the same ticks as measured in the rest frame of the clock, K.

Relativistic Effects on Proof-of-Work

To better understand the relativistic effects on how relative time is perceived in two inertial reference systems (or Galilean reference systems [EINS]) we have to mention the twin paradox. The twin paradox is a thought experiment that involves two identical twins. One of the twins makes a journey into space onboard a relativistic spaceship, and upon his return to Earth discovers that his twin has aged much more than he did. Depending on how fast the spaceship moves through space and/or how long the journey was, our traveller could return to Earth and realize that several generations have passed meanwhile.

As a direct application of the twin paradox, we have Mazer Rackham, International Fleet Admiral and Ender Wiggin's mentor, as portrayed by Orson Scott Card [CARD]. The fighter pilot who destroyed the Formic Fleet Flagship, killed the Hive Queen, and ended the Second Formic Invasion, Mazer Rackham is sent on a journey on a relativistic spaceship and returns to Earth 100 years later in order to assist the International Fleet. Upon his return he has barely aged a few years.

Similarly, if a subnetwork is moving relative to the rest of the network, the nodes in the subnetwork would experience time dilation as predicted by the special relativity theorem. Hence, they are at a disadvantage compared to the rest of the nodes because they have less time available to find a solution for the proof-of-work problem. In order to make thing square for all the nodes in the network, the difficulty of the problem should be adjusted in each subnetwork using the Lorentz factor: in order to compensate for time dilation, the relativistic nodes should have to solve a simpler problem. Also, their block confirmation time should decrease as well.

One element is missing still... how do ansibles synchronize? What value has the Lorentz factor for each one of them? One simple solution for this problem is having one of the ansibles (let us say the Earth-bound one) broadcasting a beacon every fixed number of seconds. The ansibles interfacing to subnetworks would pick up the broadcast, and by measuring the time interval between the beacons, infer the Lorentz factor. Once the Lorentz factor is determined, the nodes mining on the subnetworks will have their proof-of-work difficulty and the block confirmation time adjusted accordingly.

A more Down-to-Earth Solution for the Problem

We will fork the storyline (pun intended) at Enter the Ansible paragraph, and suggest a more down-to-earth solution for the problem. The peripheral nodes are not to participate in the consensus. Their function would be just to relay messages (a.k.a. payment information) across the network. In our multi-planetary scenario, the nodes working on the proof-of-work would reside on/around the Earth, and the network nodes on Mars or any other region in the Sol system would have to wait 2x (network packets travel time) + (block confirmation time) in order to confirm a payment. However, such a solution would have an Achilles heel very hard to defend... if starting its mining engine, any peripheral node could take over the local network and force a topological fork of the network. For a node, the incentive of staying honest on the current network must be stronger than any reward obtained by cheating.

Concluding remarks

While the blogpost contains some elements of fiction, it accurately describes the cryptocurrency ecosystem, the special relativity theorem, the Lorentz transformation and its corollaries. We would like to thank the readers that went through this exercise with us. Undeniably, cryptocurrencies are disrupting the global economy and they are here to stay. Undeniably, we -- as a species -- will explore and migrate farther than the LEO. History has taught us that decentralization is the key to survival and prosperity. Let us together make the future happen.

References

[ANTO] Antonopoulos, Andreas M., Mastering Bitcoin, Second Edition, O'Reilly Media, Inc., June 2017;
[CARD] Card, Orson Scott, Ender's Game, A Tor Teen Book, 2014;
[EINS] Einstein, Albert, Relativity, The Special and the General Theory, Princeton University Press, 2015;
[LEGU] Le Guin, Ursula K., Rocannon's World, Harper & Row, 1977;
[NAKA] Nakamoto, Satoshi, Bitcoin - A Peer-to-Peer Electronic Cash System, www.bitcoin.org, 2008;