Time turners, Energy Conservation and General Relativity

Time turner from the Harry Potter series (and from the Eliezer Yudkowsky’s venerable HPMoR fanfic) is a very useful device if you have some unfinished business in the recent past, like attending an extra class or saving a friend from a certain death. However, General Relativity has a few words to say about them, and they are not very flattering. I will only address one issue here: Energy conservation. TL;DR: if you use a time turner to vanish into the past, those around you will see you blown to tiny bits of Merlin knows what quickly disappearing from view.

Before we get to the time turners, however, let us consider an aside.

Let us start with a common question: if the Sun stop shining this instant, when would we notice? The common answer: it takes light 8.5 minutes to travel the distance of 150,000,000 km between the Sun and the Earth, so that’s how long it will take. This glosses over the issue of what does “this instant” mean exactly at two different points in space, which is not so trivial given the relativity of simultaneity in Special Relativity. It is easily patched up, however, once we fix a global frame of reference. The Cosmic Microwave Background (CMB) is a natural one to use, and both the Earth and the Sun travel with a negligible fraction of the speed of light relative to the CMB. Anyway, the answer is still very close to 8.5 min.

Now another, deceptively similar question: if the Sun disappears this instant, how long before the Earth will stop orbiting the point where it used to be? The common answer: gravity travels with the speed of light, so also 8.5 min. This answer is obvious, simple and wrong. Yes, dead wrong. Why? because static gravity is not like light, it’s more like electric field, only worse.

Let’s first think of how you would make the Sun disappear. Maybe it turned into a black hole? Well, this would not really mean disappearance of gravity, the mass of the black hole will still be that of the Sun, and the Earth will happily (or unhappily, as the case may be) continue orbiting the Sun’s corpse. So, in this case the answer is “it won’t stop orbiting”.

OK, so black hole was a bad example. How about a wormhole instead? You know, the evil Vogon-like aliens need to clear the room for a hyperspace bypass, and they build a wormhole from far away and suck all the matter in the Sun through it out of the way. What would happen then? There are a couple of hints: one is that from outside a wormhole is indistinguishable from a black hole, and the other is the Gauss Law. Both hints lead one to the same answer: just like with turning the Sun into a black hole, there is very little gravitational effect on the surrounding space. The rest of the now ex-Solar system will continue merrily on its way around the point where our Sun used to be.

An aside for those curious about the Gauss Law argument. It’s on a boring side. The law in its integral form states that the flux of the gravitational field inward through any closed surface encompassing the Sun is proportional to the Sun’s mass. To change the field, you need to remove some mass from inside this imaginary surface, by having it physically cross the surface. This last point may not be obvious, but it follows from General Relativity. Specifically, the Einstein’s most misunderstood theory says that the spacetime curvature is determined by the (past and present) distribution of matter in spacetime. There are some exceptions, like the fixed-mass spherical objects, such as black holes and wormholes, which contains no matter, and gravitational radiation, which can carry away energy. But if you take a spherical object like the Sun and try to calculate what happens if you decrease its mass, General Relativity tells you that this mass has gone outward from the Sun in all directions in some form. It is not fussy about the form, as long as just the right amount of mass/energy has gone out.

Let me repeat for those who skipped the above paragraph: if you take the Sun and decrease its mass, the only way it can happen if this mass leaves the Sun outward and disappears into space. This happens all the time, of course, the Sun constantly loses its mass through radiation and solar wind, or in more drastic cases through Supernova explosions. Effects like this propagate no faster than light, of course. So they take forever to propagate all the way to infinity.

Now, back to the time turners. Hermione Granger might be but a small if incredibly studious girl, but she still has mass. If you were to peek at her using a time turner and disappear, her mass, small though it may be, still has to go some place, just like the disappearing Sun’s mass had to go some place. The options are few: she can blow into tiny pieces flying past you, or disappear in a flash of brilliant light (and it takes a lot of light to carry away 50kg, what’s with E=mc^2) . Basically, it will not be a pretty sight. What cannot happen is her simply vanishing, with no ill effects whatsoever. Well, it cannot happen if we are willing to keep Relativity around. Maybe we don’t have to, what’s with a certain deputy mistress turning into a cat and back, probably instantly changing her mass, with no ill effects on her or her surroundings. But if you give up on General Relativity, quite a few things will unravel, like all four Newton’s laws.

Also don’t forget the reverse process: Hermione appearing out of thin air just before walking into her extra class. The above process has to happen in reverse: an amount of matter equivalent to her mass has to travel inwards out of nowhere and coalesce into a person. Where did this matter come from? How did it form before collapsing into a person? How did it know that it would need to time its arrival into a certain point perfectly with whatever time turner will have been set to? That’s some hard-core magic right there. Also, suck it, the Second law of Thermodynamics.

So, let me summarize: mass cannot just disappear, it has to spread out. mass cannot just appear, it has to coalesce. Thus time turners cannot be used inconspicuously, everyone around would be well aware of one’s use. Actually, it probably cannot be used at all without breaking General Relativity and/or Thermodynamics.



Why instrumentalism?

Reposting from Less Wrong:

What started my deconversion from realism is watching smart people argue about interpretations of QM, Boltzmann brains and other untestable ontologies. After a while these debates started to seem silly to me, so I had to figure out why. Additionally, I wanted to distill the minimum ontology, something which needn’t be a subject of pointless argument, but only of experimental checking. Eventually I decided that external reality is just an assumption, like any other. This seems to work for me, and saves me a lot of worrying about untestables. Most physicists follow this pragmatic approach, except for a few tenured dudes who can afford to speculate on any topic they like. Max Tegmark and Don Page are more or less famous examples. But few physicists worry about formalizing their ontology of pragmatism. They follow the standard meaning of the terms exist, real, true, etc., and when these terms lead to untestable speculations, their pragmatism takes over and they lose interest, except maybe for some idle chat over a beer. A fine example of compartmentalization. I’ve been trying to decompartmentalize and see where the pragmatic approach leads, and my interpretation of the instrumentalism is the current outcome. It lets me to spot early many statements implications of which a pragmatist would eventually ignore, which is quite satisfying. I am not saying that I have finally worked out the One True Ontology, or that I have resolved every issue to my satisfaction, but it’s the best I’ve been able to cobble together. And I am not willing to trade it back for a highly compartmentalized version of realism.

Open Problems in Instrumentalism

This deserves a detailed write-up, but the following list should be a start:

  • The issue of outputs: if all there is is inputs and models, what are actions? In other words, how does one improve the chosen model’s accuracy?
  • The place of the mind: TheOtherDave: all models exist in the mind, yet the mind itself is a model, does this mean that the mind contains models, yet the models also contain the mind? Which seems like a contradiction.
  • The source of motivation and preferences: if everything is a model, what would be the reason to prefer one model over another and try to actualize it, and not just perform passion-free measurements?


The hierarchical structure of models

My instrumentalist approach, repeated to people ad nauseum, is that the only assumptions one needs is external inputs and the models we build based on them in order to predict the future values of these inputs. To the next standard and obvious question “But where are those inputs coming from, if not from external reality?”, my answer is “This question is meaningless as stated.” It’s no better than the “question of origin” — “where did everything come from?” An instrumentalist is concerned with making accurate predictions (measuring future inputs), which require modeling these inputs based on the past inputs. I have been talking a lot about it on Less Wrong again lately, starting around http://lesswrong.com/lw/h3p/welcome_to_less_wrong_5th_thread_march_2013/8pvb and going deeper and deeper for several days.

Now, what I really wanted to write about is how the models stack up. And how the inputs stack up, as well. I also wanted to trace how models go all the way down to subconsciousness, intuition and insticnt, and how animals, plants and other lifeforms can be described in terms of modeling inputs. However, there is at least one unresolved issue I’m still trying to get a grip on, and it requires working through the mix of hierarchical inputs and feedback loops. Maybe another day.

Counterfactual Definitions in Physics

paper-machine mentioned on #lesswrong the other day that  “the electric field is defined counterfactually, i.e., “this at this point is related to the force that a test point particle would experience if it were there”. However, there is no such thing as a test particle, as each charge creates its own field, distorting the one that has to be measured. Moreover, there is no such thing as a point charge, because the electric field, inversely proportional to the square of the distance, grows without bounds near the point particle.

I remember that when I first learn this definition in high school, it bothered me for a short while, and then I got used to it, only noticing this counter-factuality very occasionally. So I will try to trace how it came about.

First, clearly the electric field is measurable, but not very directly. The relevant device measures either the displacement of a charged object, or the force required to keep it in place, depending on the design. In the former case the displacement is due to the force the field exerts on the charge. So, in both cases what is really measure is the force, and the field is inferred from it by calibration.

The jump from the Coulomb force between two charged objects to the mediating electric field generated by one of them and sensed by the other is the leap of faith Michael Faraday made without ever directly measuring the “undisturbed” field. I should really look through his biography and see what thought process he followed.

An insight like that is rare, and I am wondering about other examples where a counterfactual definition produces the most useful and accurate model of some physical phenomenon. The concept of force is probably one of those. Aristotle described “unnatural” motion as “forced”, and it would have worked well, if not for the invisible force of the Earth’s gravity, and probably air resistance. Otherwise Galilean relativity (unforced motion is along a straight line with constant velocity) would have been very intuitive. As it was, however, the world had to wait for Galileo and Newton to rather counterfactually define force as the rate of change of momentum.

I will try to find some more interesting examples of counterfactual definitions next.


Litany of Instrumentarski

Or should it be Pragmatarski? Neither portmanteau sounds good. Anyway, basically a copy of my recent Less Wrong post:

Eliezer Yudkowski created the following affirmation, which he calls the Litany of Tarski:

If the box contains a diamond,

I desire to believe that the box contains a diamond;

If the box does not contain a diamond,

I desire to believe that the box does not contain a diamond;

Let me not become attached to beliefs I may not want.

This is all fine and dandy, but how would an instrumentalist like myself interpret “If the box contains a diamond”? It clearly talks about the territory in the overrated map/territory meme. So I would like to restate it in a more instrumental way, which does not presume existence of territory:

If believing that there is a diamond in the box lets me find the diamond in the box,

I desire to believe that there is a diamond in the box;

If believing that there is a diamond in the box leaves me with an empty box,

I desire to believe that there is no diamond in the box;

Let me not become attached to inaccurate beliefs.

Here I replaced the statement about the territory (“box contains…”) with the statement about models (“believing that there is…”) and inputs: “lets me find the diamond…”

For some reason this caused confusion, people decided that I find diamonds useful, so I modified it a bit to be clear that even if don’t care about diamonds, I do care about good accounting:

If I will find a diamond in the box,

I desire to believe that I will find a diamond in the box;

If I will find no diamond in the box,

I desire to believe that I will find no diamond in the box;

Let me not become attached to inaccurate beliefs.

This more explicitly states that I value accurate models, i.e. those best predicting future inputs. However, Vaniver said that s/he prefers the unmodified version as a stronger one. I am not sure about the reasoning, but I find either one more palatable than the Eliezer’s original. Well, the name still sucks.


The Map/Territory Meme is Overrated

Going through my old posts on Less Wrong, I have found the first one that describes my version of instrumentalism:

The map/territory meme works better in some cases, worse in others. Note that it is only a model, not necessarily “the way things are”. A simpler and just as powerful approach is that repeatably testable models are all there is. It is tempting to assign repeatability of testing to the invisible unachievable “objective reality” behind the veil, something science strives to uncover, but this notion is unnecessary. If this makes you uncomfortable, do the standard exercise and reformulate the question such that it has an answer: “Why do I think that there is objective reality?”. (The answer turns out to be similar to that of “Why do I think I have free will?”)

Again, the minimal approach to the scientific method is: build models and test them. Adopt those with more experimentally confirmed predictive power. And by all means, use Bayes as an aid in constructing less-stupid models.

Typical reaction  (just tested on #lesswrong, again):

But what are you testing your models *against*?

Me: who cares, you can pretend that they come from some thing called territory, which becomes one of your models

And on it goes, about the whether the spaceship disappearing behind the cosmological horizon ceases to exist and so on.