Exotic Four-dimensional Manifolds with Planck-scale Obstacles

Another abstract of a fantasy paper.

We perceive our spacetime as four-dimensional, at least macroscopically. This is purely an empirical observation, and our best current theories do not favor four dimensions over any other number, and string theories in particular appears more naturally in ten or more dimensions. However, mathematically non-compact four-dimensional manifolds are known to be special in that they admit uncountably many “exotic” differential structures. These have been investigated in some detail, most notably by Asselmeyer-Maluga and Brans. The standard formulation of the path integral fixes the dimensionality of the spacetime, and often its topology. We show that a more generalized form of the path integral taken over all (non-negative, integer) spacetime dimensions, all topological manifolds, and all inequivalent differential structures on them, can be formulated in a well defined way. We further show that the contribution to the path integral is dominated, in the stationary phase approximation, by non-compact four-dimensional manifolds everywhere punctuated by compact sets non-smoothly-embeddable in a three-ball. These sets presents a natural obstacle to putting a smooth spacetime metric on these exotic manifolds, except in a coarse-grained approximation. These obstacles limit the resolution of any potential measurement, making it tempting to identify them with the Planck scale of our own spacetime.

Elastic Gravity

An abstract of a fantasy quantum gravity paper

The idea that gravity is emergent and not fundamental has been around since Andrei Sakharov. Among the more recent approaches to emergent gravity are the entropic gravity (Verlinde) and gravity from entanglement (Van Raamsdonk). In this paper we construct a model where macroscopic gravity emerges in a quantum system with a large number of highly entangled states. The linearity and unitarity of the quantum world is then translated into the elasticity in the gravitational world: any local attractive effects are counteracted by the larger-scale repulsive effects, producing a globally hyperbolic spacetime admitting a spatial slice with zero mean curvature. We identify this slice with a comoving frame in a Friedman-Robertson-Walker-de Sitter spacetime, and the effective large-scale elastic repulsive force with dark energy. We then find that the matching conditions between the two scales manifest as interaction-free dust, mimicking the appearance of dark matter. We also explore some of the small-scale predictions of this model. One of the more intriguing consequences is the Penrose-like upper limit on the complexity of a coherent state, resulting in the largest possible Schrodinger’s cat mass of about 1 Planck mass. This result offers hints for possible approaches toward modeling the quantum/classical transition and a deterministic view of the Born rule as a manifestation of the hidden residual entanglement of a classical detector and a quantum system under measurement.

Entangled gravity

Image from: https://physicsworld.com/a/quantum-gravity-could-be-probed-by-entangled-masses/

We know that gravity is related to entropy, and entropy is a property of an entangled many-state system. Let’s review what we know about the connection between entropy, entanglement and gravity.

A quantum system is called entangled when it cannot be cleanly separated into subsystems, i.e. where the description of the whole system can only be constructed by combining the descriptions of each subsystem with interactions between them.

Entanglement entropy of a subsystem is determined by the number of entangled states it has with the rest of the system (often the rest of the world). The states it has that have no connection to the outside world don’t count.

The above statement that only the connections with the outside world matter can be expressed in the area law of entanglement entropy , that states that the entanglement entropy “grows at most proportionally with the boundary between the two partitions”.

Black hole horizons are the ultimate way to separate the world into two “partitions”, one inside and the rest outside.

A non-surprising conclusion: the entanglement entropy of a black hole is proportional to the area of its event horizon.

Entropy is information: all our knowledge about a subsystem we are not a part of comes from the subsystem’s entanglement with the world. So entropy can be represented as information: “the information entropy of a system is the amount of “missing” information needed to determine a microstate, given the macrostate.” So one can measure entropy the same way one measures information: by counting the number of bits needed to describe it.

A black hole horizon is a perfect insulator: once something is inside it, all that’s left is the entanglement of that something with the world outside, represented by the number of bits and nothing else. The horizon thus corresponds to the smallest area that can “fit” a certain number of bits. There is a minimum area because of the Planck limit on how small things can be, so the minimum area per bit corresponds to the square of Planck length, about 10^{-35}m\times 10^{-35}m.

Image from http://backreaction.blogspot.com/2017/05/can-we-probe-quantization-of-black-hole.html

So this was the entanglement entropy, that is determined by the number of quantum-mechanically entangled states. But what about the classical entropy, where one does not need to distinguish entangled states from the non-entangled ones? Well, the quantum states start interacting with everything around them (i.e. other quantum states) really quickly, and get entangled with them, so after a short time every single state in a classical system is entangled with every other one. So we don’t need to worry about the entanglement, can just count all the microstates, the way Boltzmann did.

Since any horizon that is a closed surface partitions the world into at least two subsystems, such a horizon automatically has entropy associated with it, describing the entanglement of the severed classical connections between the subsystems. It can be a cosmological horizon in an expanding universe, for example.

The cosmological horizon is quite different from the event horizon of a black hole. First of all, we are “inside” it, and it’s our only link entangled with the “outside” universe. Second, it is different at different points in space and in time. But the salient point is the same: entropy per unit area is constant and corresponds to the maximal possible entanglement with the part of the universe that is classically out of reach. But how can that be? The universe was tiny with a lot of interactions, and now it is huge, so wouldn’t we end up with untold amount of entanglement? Maybe this “untold amount” is what determines the size of the horizon, and what makes the cosmological constant so small, in order to make the horizon big enough to “fit all the entanglement”. This could provide a hint of a connection between entanglement and gravity, namely the former creating the latter, at least where the cosmological constant is concerned.

Further, since only the entangled matter contributes to the horizons, and the area of the horizon corresponds to the mass of what’s inside, in case of a black hole, or the cosmological constant in the universe, in case of the cosmological horizon.

Image from: https://www.wikihow.com/Make-a-Girl-Attracted-to-You

This suggests that any isolated systems that do not contribute to the entanglement entropy do not contribute to the perceived gravity/spacetime curvature, either. This would be very counter-intuitive! If it does not interact, it does not attract. (Side note: But what about dark matter? Doesn’t it interact only gravitationally? Or maybe it’s an artifact of entanglement, a gravitational relic of interactions between luminous matter?)

So the general theme is: only interacting systems gravitate. In other words, gravitation does not create entanglement, instead it’s entanglement that creates gravitation. Which means that one would not observe gravity from a cat in superposition, but only based on its classical (non-quantum) position/trajectory. So, we’d observe the cat’s gravitational field as that of a live cat, up until we open the box and check, entangling the cat state with the measurement system. But that means that the act of opening the box must create gravitational radiation, “informing” the universe about the cat’s untimely demise.

Image from https://physics.aps.org/articles/v6/115

Schrödinger’s cattraction

So the proverbial cat is in a well isolated box, in a superposition of dead and alive state, and remains that way, as far as you are concerned, until you take a peek.

Only there is a problem. Cat has mass, and mass creates gravity. So, whether you peek in or not, dead cat and live cat, each create their own gravity, right? And, since one version is standing and the other, sadly not anymore, they pull you, and the rest of the world, in slightly different directions. So, theoretically, we could peek into the box without ever having to open it, just by measuring the cat’s attraction to us. Take that, Quantum Mechanics! But that’s easier said than done.

Cats are small, gravity is weak, so the difference in the gravitational attraction is tiny. Fortunately, we are able to measure it easily since time of the Eötvös experiment. Unfortunately, the cat, even a tiny one, is too big to be properly isolated from the rest of the world for any length of time. The smaller an object is, the easier it is to isolate it, and perform quantum tricks with it. Current state of the art is equivalent to about 10,000 hydrogen atoms. Quite an impressive feat, but still a billion billion times less than the number of atoms in even a tiny kitty. The precision of the experiments is ever increasing, however, and some day soon the size gap between measurable quantum mechanics and measurable gravitational attraction will disappear. And we will finally know what happens in the box when no one is looking. Right? Right?!

Well, it’s not really obvious in advance what we will measure. Quantum Mechanics so far aced every experimentally tested prediction it made, but it is rather silent on the topic of gravity. In fact, it actively dislikes gravity, if we were to carry this anthropomorphism to extremes. Why? Because Quantum Mechanics (QM) and Quantum Field Theory (QFT) assume an a known spacetime as the playing field on which the quantum interactions take place. This playing field does not necessarily need to be flat, it can be very much curved, like a saddle or a sphere, it can be expanding or contracting in time, it can have black holes and singularities in it… So it can be more like a golf course than like a football field, and QM does not really mind, though it does get weirder. For example, in a non-flat spacetime the whole concept of a “particle” loses its meaning. In a playing field that is more like Hogwarts with moving staircases the concept of “energy” also loses its meaning. Yet we can still apply QM to it.

We can go even further, and take into account “backreaction”, where quantum fields affect the spacetime itself through their classical energy, if one can be calculated. But this is already a stretch. The stronger quantum fields affect the underlying spacetime, the harder it is to figure out how to mesh the two theories together. That’s why Quantum Gravity, the Holy Grail of fundamental Physics, remains so elusive.

But high energies is not the only situation in which the two theories clash. The quantum Schrodinger’s cat attraction does not require high energies, and the spacetime is very much flat, and yet… Einstein famously never liked quantum entanglement, and his baby, the theory of General Relativity (GR), follows suit. Does having two cat states mean that we have two different spacetimes that merge together into a single one once we take a peek? GR balks at that possibility, because it does not make sense to talk about two different spacetimes in the same space at the same time — whatever that might mean.

Maybe the spacetime is a single one, but the two cat states in a superposition act sort of like two different “lighter” cats that together weigh as much as the “full” un-split cat? Maybe the weight is slowly shifting from a live cat state to the dead cat state as time passes and the odds of the poison-filled needle having pricked the poor kitty get larger and larger? That is a possibility that seems reasonable on the face of it, and it also has interesting consequences for the limits on how big and how physically separated an entangled quantum system can be in principle, before it is ruined by the inescapable gravitational interaction between itself and the world, itself included. So maybe that is what will be found experimentally, once the ingenuity of the experimental physicists and the sensitivity of their apparata get to where something like that can be measured.

There are other possibilities, as well. Maybe the result will be akin to the double-slit experiment, where a particle appears to go through both slits at once only when no one is looking, but only through one of the slits when something is trying to catch it in the act. What does it mean for the Schrodinger’s cat-traction? It means that when you measure the gravitational attraction to the cat in the box, it works the same way as with the double slit experiment, or with the classic Schrodinger’s cat experiment: sometimes you measure the dead cat and sometimes you measure the live cat, but nothing in between. Basically, your apparatus becomes entangled with the kitty in question, and, in the many worlds picture, “splits” into two worlds, one where the cat is dead and one where the cat is alive.

If gravity is something like other fundamental forces and can be quantum, this is what we would expect. But this would leave open the question I mentioned earlier: gravity is, classically, a curvature of spacetime, so the whole spacetime would have to “split”, somehow, one spacetime per cat state. There are no good theoretical models of how this might actually work, but it would not be very surprising experimentally.

What other possibilities are there? It is not out of the question that both the “average cat” and the “either/or cat” can be combined in some way. Maybe in some circumstances we would observe the former, and in others we would observe the latter. Or maybe we would observe a mix of the two, or a rapid fluctuation between the two, as if Nature could not decide which way to go.

One could imagine something even weirder: that at small enough scales gravity cannot be measured at all, for whatever reason. It could be a collective phenomenon that gradually kicks in when more and more particles are brought together, creating a curved spacetime. After all, there are many hints that gravity is related to entropy, a measure of disorder in a physical system. Black holes have been shown to possess the largest possible entropy per unit area, and they are purely gravitational objects. So, maybe there is no gravity at all at the single particle scale? That is, single particles still feel gravitational attraction, they just don’t create much of their own, relative to their rest mass, unless they get entangled with other particles. Very unlikely, but not observationally ruled out. Also, very weird, but then it would take something really weird to reconcile the microscopic quantum world with the macroscopic world where gravity is important.

Whatever the experimental results will be, getting some hints about how gravity and quantum behave when they meet would be extremely exciting! Probably as exciting as all the experiments in the early 20th century that led to completely new and unexpected ideas, and ultimately to technological breakthroughs no one could have anticipated.

For a very technical review of the situation, this is a good overview.

Entanglement, Many Worlds and General Relativity

Based on a reply on a forum. Let me try to explain the entanglement part at a layperson level, without all this cumbersome math.

Nothing travels faster than light. More than that, the remote entangled particle’s state does not instantaneously change in response to the local measurement, though it certainly tempting to think in these terms. Einstein, in his famous “EPR” paper and his debates with Bohr understood the former. The latter is still not something obvious, and in some ways mysterious, as you note.

One of the most common entangled states, fancily known as the “singlet” can be expressed as an agreement between two entangled particles, each one is telling the other the same thing:

when someone checks up on you, IF YOU ARE UP, I’M DOWN!

So, at the basic level, this seems like a conspiracy, one of them yelling at the other LOOK, LOOK, I’ve reported that I’m UP, you better report DOWN (or the opposite), and this yelling reaches the other particle instantly, even though there is no unique definition of what counts as “instant” in relativity. One person’s instant is another person’s traveling back in time, at least a little. Yet, frustratingly, there is no way to use this apparent instant shouting to pass any information across. Each side would be seen as UP or DOWN with 50/50 odds.

One popular (among physicists and non-physicists alike) explanation for this is known as Many Worlds. Notice that you yourself consist of quantum particles (or, more accurately, quantum fields), so when you check a particle’s state, you interact with in at the quantum level, becoming entangled with it. This is known as the Wigner’s Friend paradox: by measuring a quantum state you end up splitting into two “you’s”, one seeing “UP” and one seeing “DOWN”. Only the two of you are not able to see each other, because… uh… of another quantum thing called decoherence, where all the shared states between the two disappear very very fast. And shared states are needed for seeing each other. Because “seeing” is another term for interacting quantum mechanically.

So, there are two instances of you, totally oblivious to each other’s existence, each in a world of their own, the only difference, at least initially, being that one measured a particle in the UP state, and the other in the DOWN state. And in each of these worlds if someone checks up on the other entangled particle, they get DOWN for the UP world, and UP for the DOWN world. They automatically end up in the “right” world. Or you do, from their perspective. And voila, there is no faster-than-light traveling, but at the cost of creating an extra world, utterly undetectable but just as real as your own.

if you feel uncomfortable with this “world split”, you are not alone! Physicists and philosophers have been trying to figure out how this might work in detail for the last 50 years or so, ever since one Hugh Everett wrote his unassuming PhD thesis about this topic. While there has been some modest progress in the area, including what is known as “Quantum Darwinism”, a more satisfying explanation is unlikely to emerge until there is some progress in unifying Quantum Mechanics with Einstein’s General Relativity, the theory of gravitation.

Why are the two related, you might ask? Think about it this way. Let’s suppose that, based on your measurement, you decide to shoot some massive object either UP or DOWN, depending on what the outcome of your measurement is. Then in one world this massive object flies up, and in the other it flies down, right? But massive things exert gravitational pull on everything around them. So, some other object, completely uninvolved in this interaction, would feel this pull. But which one? UP or DOWN? Or maybe both? The experiments show, unsurprisingly, that this pull matches the visible direction of travel of this massive body. But, according to Einstein. gravity is the curvature of spacetime, so the two worlds must have different spacetime curvatures. Thus Quantum Mechanical Many Worlds are not just invisible worlds in the same spacetime, they somehow create their own spacetime each! There is nothing in Einstein’s relativity that describes or even allows something like that. Thus a new theory would be required to explain what happens.

Three Pieces of Rationalore on Determinism

(Epistemic status: nothing really new, but maybe I am missing something important.)

“There’s no free will,” says the philosopher; “To hang is most unjust.”
“There is no free will,” assents the officer; “We hang because we must.”

Ambrose Bierce

Here these pieces are, possibly in the order of increasing controversy:

  1. Consciousness is an algorithm.
  2. Free will is a feeling.
  3. Agency is “God of the gaps”.

The first one is the least controversial. The mind is a process in the brain, and brains are a part of the physical universe, which is described by the laws of physics, and so its parts can be simulated on a computer, whether classical or quantum, depending on the required simulation level. There is no immaterial soul that would make such a simulation impossible.

The second one is uncomfortable, but hard to argue with, especially after Sam Harris was through with it. Every physical system evolve according to the laws of physics, and humans are no exception, though the human actions are better described by the higher abstractions, the emergent laws of biology, sociology, psychology and so on. Still, there is no extra input into these laws other than those of the underlying physics. This leads to the inescapable conclusion that free will is not some extra-physical force in the brain, but a feeling that humans have, one of the many qualia. We may feel like we have freedom of choice and observe others behaving like they do, but that feeling is just an artifact of the algorithm in the brain, possibly evolved to improve the odds of survival.

The last one is generally the hardest to accept, however inescapable it is. If we accept 1 and 2, then agency is “event-causal” and we face the “disappearing agent argument“. Agency as “the ability to make the choice to act”, “intentional actions are initiated by the agent” is one of the convenient abstractions. We distinguish agents from the “objects reacting to natural forces involving only unthinking deterministic processes”, but this only a distinction of necessity. When we do not have a good explanation of something, we instinctively anthropomorphize it. Once we understand the mechanics of its behavior, and it looks less mysterious, its apparent “ability to make a choice” dissolves into the underlying mechanisms devoid of agency. This is generally easier for us to see and accept when only physical processes are involved. Biological processes evoke the feeling of agency, probably because we do not make the connection to the underlying physics as naturally. The same behavioral pattern in a biological system as in a physical system appears to us to have more agency, even though everything is just physics deeper down, and evolves according to the laws of quantum mechanics, only abstracted multiple levels up. The continuous improvement of our understanding of the mechanisms driving various “living” organisms to act the way they do pushes their perceived level of agency out of the parts we understand and into those still looking like black boxes. Until eventually no black-box parts remain, and the whole system looks completely mechanistic. What used to be seen as an intelligence acting to shift the weight of probabilities from all possible futures toward the preferred ones becomes yet another physical process. This final extinguishing of irreducible agency may require super-intelligence.

One consequence of disappearing agency is that a super-intelligence will necessarily see humans (and likely itself) as predictable and repeatable algorithms, up to the inescapable quantum-mechanical uncertainty. The job of the AI Alignment research effort is then to make sure that the hypothetical future super-intelligent AI keeps the otherwise-superfluous labels of ‘agency” and “suffering” attached to human phenomena, and does a better job of it than us humans playing the Sims game. I discussed this in my previous post, AI Alignment: Bubblicious.

What if the humanity fails at creating a human-aligned AI and we all get paper-clipped? Who should we blame? To that the deterministic approach provides a clear if not a satisfying answer: whatever happens will be determined by the laws of physics, including making us “fail”, and including any fears and thoughts of the failure. Does it mean that we should or can give up trying? The answer is the same, it is not something we control, even if we think that we do. Does it mean that this deterministic approach is useless, since it does not make any difference whether we agree with it or not? Not quite. Accepting determinism is useful for constructing a good decision theory and making good decisions, especially in the presence of predictors. Whether you find this worthwhile, you pretend to decide.

AI Alignment: Bubblicious

So we want the super-intelligent AI that is immeasurably smarter than current humans to “robustly advance human interests”, instead of some equivalent of turning the universe into paper clips. Here I will leave aside the question whether “human interests” is a coherent idea, given all the differences between humans of various cultures, backgrounds, mental, physical, emotional and psychological makeup and whether there is a non-empty intersection of multiple sufficiently large groups of humans, let alone all humans. Not even survival of humans in some form is an uncontroversial idea. This is an obvious and well-discussed question, and there is little free energy left there without significant research. Instead I will assume that this question is solved, and there is a set of interests that all or most humans or human groups can agree on. Given that assumption, I will look into why a super-intelligent AI would go along with this interest, or adopt and modify it in a way that today’s humans would find unobjectionable.

To that end, I will start with the concept of “human interest”. Which, in this context, seems similar to other agenty terms like “goals” and “values”. Which, in turn, presuppose that whoever has those interests is an agent. Someone who can make choices and have desires. To put it in perspective, as humans, what kind of less intelligent entities do we pay attention to, in terms of considering their needs and desires, and why?

Let’s first bracket what might be a reasonable range of entities. What might be a comparable gap in intelligence. Probably more agency than a rock. Given a likely huge gap between a self-improving AI and a human, what could be the least intelligent agent we could still care about?

Probably something like a bacterium. From the human point of view, it is alive, and has certain elements of agency, like the need to feed, which is satisfies by, say, moving up a sugar gradient toward richer food sources so it can grow. It also divides once it is mature enough, or reached a certain size. It can die eventually, after multiple generations, and so on.

The above is a very simplified black-box description of bacteria, but still enough to make at least some humans care to preserve it as a life form, instead of coldly getting rid of it and reusing the material for something else. Where does this compassion for life come from? In the following we contend that it comes from the lack of knowledge about the inner workings of the “agent” and consequently lack of ability to reproduce it when desired.

We give a simple example to demonstrate how lack of knowledge makes something look “alive” or “agenty” to us and elicits emotional reactions such as empathy and compassion. Enter


Let’s take a… pot of boiling water. If you don’t have an immediate intuitive picture of it in mind, here is a sample video. Those bubbles look almost alive, don’t they? They are born, they travel along a gradient of water pressure to get larger, while changing shape rather chaotically, they split apart once they grow big enough, they merge sometimes, and they die eventually when reaching the surface. Just like a bacteria.

So, a black-box description of bubbles is almost indistinguishable from a black-box description of something that is conventionally considered alive. Yet few people feel compelled to protect bubbles, say, by adding more water and keeping the stove on, and have no qualms whatsoever to turn off the boiler and letting the bubbles “die”. How come?

There are some related immediate intuitive explanations for it:

  • We know “how the bubbles work” — it’s just water vapor after all! The physics is known, and the water boiling process can be numerically simulated from the relevant physics equations.
  • We know how to make bubbles at will — just pour some water into a pot and turn the stove on.
  • We don’t empathize with bubbles as potentially experience suffering, something we may when observing, say, a bacteria writhe and try to escape when encountering an irritant.
  • We see all bubbles as basically identical, with no individuality, so a disappearing bubble does not feel like a loss of something unique.

Thus something whose inner workings we understand down to the physical level and can reproduce at will without loss of anything “important” no longer feels like an agent. This may seem rather controversial. Say, you poke a worm and it wriggles and squirms, and we immediately anthropomorphize this observation and compare it to human suffering in similar circumstances. Were we to understand the biology and the physics of the worm, we may have concluded that the reactions are more like that of a wiggling bubble than that of a poked human, assuming the brain structure producing the quale “suffering” does not have an analog in the worm’s cerebral ganglion. Alternatively, we might conclude that worms do have a similar structure, producing suffering when interacted with a certain way, and end up potentially extending human morals to cover worms, or maybe also bacteria. Or even bubbles.

To a super-intelligence humans will likely be as transparent as, well, water vapor bubbles are to humans. Provided the AI alignment work is successful, and provided that empathy and compassion are present in the AI in some form, the aligned AI might consider avoiding causing human suffering in some way. However, given that humans have no agency, as far as AI is concerned, it would have no reason to respect, let alone advance human interests, in large part because they do not exist for a system devoid of agency. It will have multiple options that similar to those humans have with respect to bubbles. We mention some of those:

  • Instantiate humanity from the first principles, equivalent to putting a kettle on the stove and turning the heat on to make bubbles.
  • Run a simulation of humanity on what passes for a computer in a super-intelligent AI. Possibly using reversible quantum computation to play around with the possibilities.
  • Formulate the relevant equations and initial conditions, and avoid expending resources to either instantiate or simulate humanity unless useful for the AI’s own purposes.

Also see my old post on LW on the perception of agency by humans.

So my thesis for the AI Alignment is “As long as humanity can be simulated or instantiated when desired, there is no point in keeping real humans around.”


If the Sun suddenly disappeared…

A summary of a post-talk chat

That’s what every site on the internet gets wrong as far as gravity is concerned. What would happen if “the sun suddenly disappeared”? The usual logic is that the planets would fly away on straight line, since the gravitational attraction of the Sun to the Earth would disappear as well (though people differ whether it would be instantaneous or after 8 min, when we see the Sun go dark).

That is not how General Relativity works. This is not how physics works. In physics a lot of things are conserved. Locally or globally. So, let’s try to actually think through how the sun might disappear without violating the laws of physics. Because if we break the laws, why cherry-pick, just break them all and then any answer is as good as any other.

If an electron suddenly disappeared…

Well, not an electron. The nucleus. Let’s think about a similar situation with electricity. Say, we have an atom. A similar question would be “What would happen if the the nucleus suddenly disappeared?” Again, what is implied, is “… without breaking the laws of physics.” We know that electric charge is conserved. The only way to get rid of positive charge is to either add negative charge, or to physically take the charged object and move it some place else. There is no such thing as “suddenly disappearing”in the Maxwell Equations, the laws of electromagnetism.

But! We can be clever! What if we create a wormhole to the charged object and sneakily grab it and drag it some place else that way. Wouldn’t it be as good as it disappearing?

Well, no, actually it won’t. What would happen is that, even though the nucleus (or whatever charged object we are considering) is gone, its electric field remains in place, with the electric field lines threading the wormhole all the way back to the stolen charged object. To the world outside, it would still appear as if the charge is in the same place it was. The most useless theft ever! Or maybe the most ingenious, no one ever noticed anything, and the thief made it out with a charge! But, but, but! After the wormhole is closed, what’s left where the charge was? And the answer to that is… Who said one can close it, just like that? Think about those electric field lines getting closer and closer as one tries to pinch off the wormhole throat. Closer means the electric field gets stronger! Which takes energy to do. And to eventually pinch it off completely one would need infinite energy. Cheaper to “just” create another charged object in the place of the stolen one, then you can finally close that wormhole. At least the energy cost of creating it is “only” mc². Still, that’s a lot of work for our charge thief. So maybe best leave that wormhole throat in place. It would not be a perfect crime, there is a smoking gun right there, but for most intents and purposes, and if no one is looking too closely, the charge is still there.

Wait! Our thief can be even cleverer! To pinch the throat without messing with the electric field threading it, why not push an opposite charge through the throat, to cancel that pesky leftover electric field? if only! The fields would point in the same direction. The charge one would have to push through has to be the same as the stolen one, both sign and quantity. Oops!

Now, back to the matters both light and grave…

So, back to the Sun! How can we make it disappear without breaking the laws of physics? All that mass and energy has to go somewhere. Maybe we could blow it up somehow, maybe turn it into a supernova? Well, that will not end well for the Earth at all. Maybe we can do that other trick with the wormhole and suck the sun out that way? Let’s give it a shot, but we already have a hint of what will happen from thinking through the case of electric charge.

If we think of gravitational field lines the same way we think of the electric field lines (and, surprisingly, even though Gravity is radically different from Electromagnetism in so many ways, as long as we are far away from really strong gravity, or from really fast accelerating objects, one can think of gravitational field as similar to the electric field), then the same logic as above also works. Only with a bit of a twist.

Let’s create a wormhole into the center of the Sun and start sucking the content away. Surprisingly, but not all that much anymore, the gravity exerted by the Sun onto the planets will not change one bit. Because those gravitational field lines can only be affected by moving or re-centering some mass and sucking stuff through a wormhole doesn’t do any of that. All we will notice is that the Sun is getting smaller with time. And probably colder, but cooling will take a long time. Instead it would look like the Sun is collapsing, like someone removed whatever was keeping it large. Which is exactly what we did!

At this point, it’s worth recalling what happens to a star when it gets smaller and smaller. Eventually it will be so small, it turns into a black hole. Wait, what? How does that even make sense? We sucked out almost all the matter from the Sun through our wormhole, by the time what’s left occupies the visible volume small enough to correspond to a black hole with one solar mass (a mile in diameter or so), there is almost nothing left! How can such a small amount of matter create a black hole as massive as the Sun?

Well. We have already agreed that the planets don’t notice any changes in gravity, haven’t we? So, for all practical (and theoretical) purposes, the solar leftovers are still having the same effect on the world as the whole Sun. How can it be? Remember, if we trace the gravitational field lines all the way in, we will notice that it threads the wormhole and extends all the way through to the matter we sucked out, and presumably arranged somewhere on the other side.

Once the remains of the Sun fit inside the Schwarzschild radius (the fancy term for “as small as the black hole of the same mass”), there will not be any more light coming out. Because, you know, nothing can escape the black hole. Event horizon and all/

So, here is the resulting picture: the Earth and all other objects in the solar system will keep orbiting the dark emptiness in the sky where the Sun used to be. Quickly turning into frozen hulks, except for the gas planets, which have other ways to keep themselves, well, gaseous.

Actually I lied a tiny bit…

Not a lot though. An astute reader would point out by now that stable wormholes need something special to keep them stable and not let them turn into a black hole. Something that has negative mass, the so-called “exotic matter”. This material would counteract the gravity of the solar matter being sucked out, so the gravitational field lines will fade away a bit as they pass through the exotic matter. This does not affect the final conclusion much. The only change would be that if the wormhole is traversable, in which case the even horizon would not form, and we will notice the last remains of the Sun disappearing through the wormhole throat.


Hitching a ride on a black hole, part IV: Going on an 10gnt

You might be wondering why I picked 10g as the star drive’s acceleration and whether it is gives you much vs a leisurely 1g or less. Let us deal with this question.

First, assuming perfect mass-to-energy conversion, how long can we keep accelerating before all the fuel is used up? (We will not be considering ramjet-style refueling by scooping interstellar medium.) An honest calculation would require some calculus, but we can skip that by noting that when rate of change as a fraction of magnitude is a constant, the magnitude decreases exponentially. A well-known example is radioactive decay.

So, suppose we start with a rocket ship with mass M. If every T second we eject m kg of fuel with velocity v, then the thrust pushing the rocket forward is \frac{mv}{T} and its acceleration is a=\frac{mv}{MT}. After T seconds the rocket’s mass is M-m, so to keep the acceleration constant we need to reduce thrust correspondingly, and continue to do so as the rocket gets lighter and lighter. As a result, the rocket’s mass decreases slower and slower with time, getting to about 37% of its original weight after t=\frac va seconds (and to 37% of that after another t seconds). For a light-speed exhaust this works out to be only 35 days at 10g acceleration.

This should give you pause. It sure made me check this simple calculation, just in case. With our star drive we would have to basically annihilate 60% of the star in just one month. For scale, the Sun converts not even 10% of its mass into energy by burning Hydrogen then Helium for over 10 billion years! And we plan to burn that much in one week! Or at least one week by the ship’s clock, as we would be close to the speed of light after only a few weeks, when time dilation firmly sets in. Still, this is very much comparable with supernova explosions, only going on for weeks non-stop.

Now would be a good time for environmental considerations. If the exhaust consists mostly of electromagnetic radiation, then whatever is in the path of the exhaust would not fare well, not within a few light years, anyway. Says Wikipedia:

Gamma rays induce a chemical reaction in the upper atmosphere, converting molecular nitrogen into nitrogen oxides, depleting the ozone layer enough to expose the surface to harmful solar and cosmic radiation

And that’s a few dozen light years away. Using near-light speed massive particles, like Hydrogen plasma jets is just as bad. CEPA, the Cosmic Environmental Protection Agency, might be a bit put out with us for destroying all life on a planet or ten.

Fortunately, there is a way out. Kind of. If the exhaust mostly consisted of neutrinos, the impact on the surrounding Cosmos would be minimal. Neutrinos go through everything almost completely unaffected, so even at a few million km the radiation exposure from them is quite small.

Interestingly enough, most of the energy released in a core-collapse supernova, the kind where a large star runs out of gas and collapses on itself until everything in it turns into neutrons, is in the form on neutrinos. However, this is just way too weak by our standards, as most of the energy is not released, but remains as its neutron core or the resulting black hole. Yes, that’s right, one of the most dreadful star explosions we observe, visible a few galaxies over, just doesn’t produce enough energy to power our star drive.

There are good reasons that it is very hard to turn normal matter into neutrinos completely, most of them are related to conservation laws. Specifically, we currently do not know of any way to turn “quarks”, the stuff of which atomic nuclei are made into “leptons”, which is what neutrinos are. It is entirely possible that as-yet-unknown laws of physics come into play at very high energies, but, if so, this energy would have to be higher than anything a mere supernova can produce, since we don’t see supernovae disappear in a puff of neutrinos.

One bit of good news is that this is not an issue for black/white holes, as quark number is not conserved (or at least not visible) after the black hole is formed. Only mass, angular momentum and electric charge are preserved by the collapse. So there is a hope that if we give up on “real” stars, like white dwarfs and neutron stars, and build our star drive out of the black/white hole combo instead, then there is no restriction on the type of material emitted.

Anyway, back to our calculations. Suppose we have run at full power for some time, and burned the fraction X of our star drive. How fast are we going now? Well, almost at the speed of light if X is large enough. What is more interesting is not the exact speed, but the time dilation/space contraction factor. Because this factor is what tells you how fast you get to where you are going. Like, if the factor is 10, then you get across 10 light years distance in only in one year local time.

So the space contraction factor for a 100%-efficient rocket with light-speed exhaust is \gamma=\frac{X^2+1}{2X}. While the expression itself is simple, I have not found an easy way to derive it, without cranking through the relativistic rocket calculations, so please forgive me for just dropping it here without a proper motivation. And if you know of a way, please comment.

Let’s plug in some numbers! As we have seen earlier, after one month of travel at 10g (or after almost a full year at just 1g) we are left with 40% of the fuel we had at the start. Plugging in X=0.4 in the time-dilation formula above, we get \gamma=1.45, not a very impressive number. It only reduces our subjective travel time by barely 30%. If we give it another month, we are down to X=0.16 and \gamma=3.2. Only when we burn away 95% of the star we get 10 times for our time dilation effect. How long will it take at 10g? About 3 months ship time.

So, we accelerate until our time dilation factor is 10, then what? Turn off the star drive and cruise? But then we are left weightless for most of the flight, until we start decelerating, which kind of defeats the purpose of our ingenious setup to provide a steady 1g for the crew. Also, how long would we have to cruise weightless? With the time dilation factor of 10 it would take full 3 years to get to, say, Vega, if you count acceleration and deceleration. Including 2.5 years with the drive off. And after decelerating we are left with 5% of 5% or just one quarter of one percent of the original star. Somewhat wasteful, is it not? We used up a whole star in a most efficient way imaginable to get not across the Universe, not across the Galaxy, but across less than one one-thousandth of the Milky Way. And it took us three years to do it.

To summarize the disappointing answer to our original question, 10g acceleration gets us the time dilation/space contraction factor of 10 if we accelerate for 3 months and burn away 95% of the fuel. But wait! Not all is lost quite yet. This result is for a conventional rocket, not a black-hole joy ride, where the effects of General Relativity are also important. We will see if it makes any difference next time.

Hitching a ride on a black hole, part III: Edge of your seat

Last time we designed our literal-star-drive-based space ship:

star surfing

and figured out that to provide 10g of acceleration in the frame of the star while surfing its gravitational wake in free-fall, we would have to trail the star drive by about a million kilometers (or miles, the calculations were not very precise). We also noticed that the tidal gravity from the star limits the size of the crew module to tens of kilometers, or the size of a really small moon, like Phobos. Anything larger is going to be stretched beyond breaking point and ripped into pieces. This size is certainly nothing to sneeze at, but it has certain limitations. For example, if you want to travel comfortably without leaving your home planet and instead use it at the crew module, it is probably too large to remain in one piece. So even the largest possible crew module does not provide enough natural gravity to feel comfortable.

Of course, gravity does not have to be natural. One can always produce enough centrifugal forces by spinning the module fast enough. Many sci-fi stories rely on that. A ring of 1 km in radius only needs to make a full turn once a minute to give the illusion of 1g gravity on the inside of its surface. “Only”? Seems a bit dizzying, to see the world around you complete a full rotation in barely one minute. Just over 3 min for a 10 km ring. Still not a lot of fun. Imagine all those distant stars zipping around you every minute. So, just for fun, let’s consider other alternatives. After all we’ve been playing with gravity for some time in this series, why stop now?

But first… Let us return to the issue of stability. If you ever surfed, skied or even walked (and I hope you have done at least that last activity, otherwise you are probably just a brain in a jar), you realize that it requires some effort to remain on the sweet spot of the wave. If you goof, you are either too far down or too far up and probably off your board. And that’s assuming you don’t fall over sideways. This is because the equilibrium you achieve is unstable. It’s more like balancing a long stick vertically on your finger than holding it hanging down. You can do it, but you have to constantly compensate for the damn thing trying to fall. It is the same with our star-drive setup. If the crew module lags a bit behind the sweet spot, it would tend to keep lagging farther and farther, until it is left behind in the darkness, forever lost. If the crew module slides a bit forward, the star’s gravity will pull it harder and it will plunge toward it and disintegrate in the short order, which is also a suboptimal outcome. So we have to maintain constant vigilance and compensate for random deviations from the optimal surfing location. How would we do that? One obvious solution is to have small rocket engines — maneuvering thrusters — to provide course corrections. This means having to carry extra fuel, maintain the engines, etc. While inescapable in other circumstances, using on-board engines for course correction seems a bit silly when we are literally blowing up a star for fuel just ahead of us.

Note that in the diagram above I carefully drew the exhaust skirting the crew module. Mainly because I did not want the crew members to be incinerated by a stream of radiation strong enough to accelerate a star-massed object at freaking 10 gravities! We will get to the question of how terrifyingly awesome this exhaust must be later in this post series. For now let us note that if the star drive goes out of alignment only a little bit and the propellant hits the crew module, the crew does not stand a chance in hell, because hell is what it would feel like for the scant few seconds until they get evaporated. However, in homeopathic doses this same exhaust might be helpful, rather than harmful. (Who said homeopathy doesn’t work?)

If only a small amount of exhaust hits our crew module, it nudges us backward. This can be good if we stray too close to the star. A useful side effect is that it provides some acceleration, which to the people inside would feel like gravity. Isn’t it nice to kill two birds with one stone? Umm, this metaphor might be a bit cruel and dated. Throwing stones at birds might have been considered good clean fun a couple of centuries ago, but not by the enlightened standards of the present day. Anyway, I digress. We want to use a small amount of starlight to stabilize the crew module and to simulate gravity. Again, this idea is not new, using solar sail for interplanetary travel has been discussed for some 100 years. A German-Russian-Soviet scientist and engineer Friedrich Zander was apparently one of the first to do a serious calculation of the solar sail propulsion, during his downtime between designing one of the first liquid-fuel rockets (well, second, after Goddard in the US, but before von Braun in Germany) and figuring out the details of a gravitational slingshot, a now-standard technique for interplanetary travel.

There is at least a couple of ways we can design our solar sail: we can put it in front of the crew module, like a rocket:

Star-surfing with reflector

or behind it, like a parachute:

Star-surfing with parachute

There are advantages and disadvantages in both cases, as evident by the fact that both approaches are in use. A parachute-style sail provides stability for free, while a crew module equipped with a reflector would have to be actively stabilized, the way planes and rockets are. On the other hand, a heavy module precariously suspended on something like ropes… What if one of them breaks? And how will they fair while being bombarded by the exhaust rays? In either case, as you can see, the direction toward the star becomes “down”. The preferred location of the crew module would be somewhat ahead of the free-fall point. How far ahead? Not very much. If the star drive provides 10g acceleration, we only need to get to the 11g zone to get 1g apparent gravity. And since the gravitational force is proportional to the inverse square of the distance, we only need to be 5% closer to the star to get that extra 1g, or around 50,000 km closer.

Interestingly, the solar sail provides us something like a neutral equilibrium: if we get closer to the star, its gravity gets stronger, but so does the exhaust density, which also falls off as inverse square of the distance, or close to it. So we still need to do some course correction, but not nearly as much. There are also some other small effects which affect the crew module, like a slight blueshift of the exhaust during the time it took for it to reach the sails, the gravitational effects of the exhaust left behind, and a few others. However, we cannot properly address them without doing a fully general-relativistic analysis, and they are too small to matter, anyway.

So, as promised, we have improved our star ride to make it more stable, easier to control and more comfortable by diverting a tiny fraction of the ejecta toward the solar sail. We have not yet calculated how big or strong the sail would have to be. Hopefully next time.  We will also look into how far 10g acceleration gets us and how fast. In the meantime, please feel free to comment or ask questions!