Physics

When the inverse square stops working

In physics inverse square forces are among the most reliable things. You can trust that electric and gravitational fields from monopole charges decay like 1/r^2. Sure, dipoles and multipoles may add higher order terms, and extended conductors like wires and planes produce other behaviour. But most of us think we can trust the 1/r^2 behaviour for spherical objects.

I was surprised to learn that this is not at all true recently when a question at the Astronomy Stack Exchange asked about whether gravity changes near the surface of dense objects.

Electromagnetism does not quite obey the inverse square law

The cause was this paper by John Lekner, that showed that there can be attraction between conductive spheres even when they have the same charge! (Popular summary in Nature by Philip Ball) The “trick” here is that when the charged spheres approach each other the charges on the surface redistribute themselves which leads to a polarization. Near the the other sphere charges are pushed away, and if one sphere has a different radius from the other the “image charge” can be opposite, leading to a net attraction.

Charge distribution on two spherical conductors with the same net charge.
Charge distribution on two spherical conductors with the same net charge.

The formulas in the paper are fairly non-intuitive, so I decided to make an approximate numeric model. I put 500 charges on two spheres (radius 1.0 and 2.0) and calculated the mutual electrostatic repulsion/attraction moving them along the surface. Iterate until it stabilizes, then calculate the overall force on one of the spheres.

Force between two equally charged (blue) and two oppositely charged (red) spheres, and force time squared distance.
Force between two equally charged (blue) and two oppositely charged (red) spheres, and force time squared distance.

The result is indeed that the 1/r^2 law fails as the spheres approach each other. The force times squared distance is constant until they get within a few radii, and then the equally charged sphere begins to experience less repulsion and the oppositely charged spheres more attraction than expected. My numerical method is too sloppy to do a really good job of modelling the near-touching phenomena, but it is enough to show that that the inverse square effect is not true for conductors close enough.

Gravity doesn’t obey the inverse square law either

My answer to the question was along the same lines: if two spherical bodies get close to each other they are going to deform, and this will also affect the force between them. In this case it is not charges moving on the surface, but rather gravitational and tidal distortion turning them into ellipsoids. Strictly speaking, they will turn into more general teardrop shapes, but we can use the ellipsoid as an approximation. If they have fixed centres of mass they will be prolate ellipsoids, while if they are orbiting each other they would be general three-axis ellipsoids.

Calculating the gravitational field of an ellipsoid has been done and has a somewhat elegant answer that unfortunately needs to be expressed terms of special functions. The gravitational potential in the system is just the sum of the potentials from both ellipsoids. The equilibrium shapes would correspond to ellipsoids with the same potential along their entire surface; maybe there is an explicit solution, but it does look likely to be an algebraic mess of special functions.

I did a numeric exploration instead. To find the shape I started with spheres and adjusted the semi-major axis (while preserving volume) so the potential along the surface became more equal at the poles. After a few iterations this gives a self-consistent shape. Then I calculated the force (the derivative of the potential) due to this shape on the other mass.

Semi-major axis, force, and force times distance squared for two self-gravitating unit volume ellipsoids at different center-of-mass distances.
Semi-major axis, force, and force times distance squared for two self-gravitating unit volume ellipsoids at different center-of-mass distances.

The result is indeed that the force increases faster than 1/r^2 as the bodies approach each other, since they elongate and eventually merge (a bit before this they will deviate from my ellipsoidal assumption).

This was the Newtonian case. General relativity is even messier. In intense gravitational fields space-time is curved and expanded, making even the meaning of the distance in the inverse square law problematic. For black holes the Paczyński–Wiita potential is an approximation of the potential, and it deviates from the U(r)=-GM/r potential as U_{PW}(r)=-GM/(r-R_S) (where R_S is the Schwarzschild radius). It makes the force increase faster than the classical potential as we approach r=R_S.

Normally we assume that charges and masses stay where they are supposed to be, just as we prefer to reason as if objects are perfectly rigid or well described by point masses. In many situations this stops being true and then the effective forces can shift as the objects and their charges shift around.

Newtonmas fractals: rose of gravity

Continuing my intermittent Newtonmas fractal tradition (2014, 2016, 2018), today I play around with a very suitable fractal based on gravity.

The problem

On Physics StackExchange NiveaNutella asked a simple yet tricky to answer question:

If we have two unmoving equal point masses in the plane (let’s say at (\pm 1,0)) and release particles from different locations they will swing around the masses in some trajectory. If we colour each point by the mass it approaches closest (or even collides with) we get a basin of attraction for each mass. Can one prove the boundary is a straight line?

User Kasper showed that one can reframe the problem in terms of elliptic coordinates and show that this implies a straight boundary, while User Lineage showed it more simply using the second constant of motion. I have the feeling that there ought to be an even simpler argument. Still, Kasper’s solution show that the generic trajectory will quasiperiodically fill a region and tend to come arbitrarily close to one of the masses.

The fractal

In any case, here is a plot of the basins of attraction shaded by the time until getting within a small radius r_{trap} around the masses. Dark regions take long to approach any of the masses, white regions don’t converge within a given cut-off time.

Gravity fractal for N=2.
Gravity fractal for N=2.

The boundary is a straight line, and surrounding the simple regions where orbits fall nearly straight into the nearest mass are the wilder regions where orbits first rock back and forth across the x-axis before settling into ellipses around the masses.

The case for 5 evenly spaced masses for r_{trap}=0.1 and 0.01 (assuming unit masses at unit distance from origin and G=1) is somewhat prettier.

Gravity fractal for N=5, trap radius = 0.1.
Gravity fractal for N=5, trap radius = 0.1.
Gravity fractal for N=5, trap radius = 0.01.
Gravity fractal for N=5, trap radius = 0.01.

As r_{trap}\rightarrow 0 the basins approach ellipses around their central mass, corresponding to orbits that loop around them in elliptic orbits that eventually get close enough to count as a hit. The onion-like shading is due to different number of orbits before this happens. Each basin also has a tail or stem, corresponding to plunging orbits coming in from afar and hitting the mass straight. As the trap condition is made stricter they become thinner and thinner, yet form an ever more intricate chaotic web oughtside the central region. Due to computational limitations (read: only a laptop available) these pictures are of relatively modest integration times.

I cannot claim credit for this fractal, as NiveaNutella already plotted it. But it still fascinates me.

 

 

Wada basins and mille-feuille collision manifolds

These patterns are somewhat reminiscent of the classic Newton’s root-finding iterative formula fractals: several basins of attraction with a fractal border where pairs of basins encounter interleaved tiny parts of basins not member of the pair.

However, this dynamics is continuous rather than discrete. The plane is a 2D section through a 4D phase space, where starting points at zero velocity accelerate so that they bob up and down/ana and kata along the velocity axes. This also leads to a neat property of the basins of attraction: they are each arc-connected sets, since for any two member points that are the start of trajectories they end up in a small ball around the attractor mass. One can hence construct a map from [0,1] to (x,y,\dot{x},\dot{x}) that is a homeomorphism. There are hence just N basins of attraction, plus a set of unstable separatrix points that never approach the masses. Some of these border points are just invariant (like the origin in the case of the evenly distributed masses), others correspond to unstable orbits.

Each mass is surrounded by a set of trajectories hitting it exactly, which we can parametrize by the angle they make and the speed they have inwards when they pass some circle around the mass point. They hence form a 3D manifold \theta \times v \times t where t\in (0,\infty) counts the time until collision (i.e. backwards). These collision manifolds must extend through the basin of attraction, approaching the border in ever more convoluted ways as t approaches \infty. Each border point has a neighbourhood where there are infinitely many trajectories directly hitting one of the masses. They form 3D sheets get stacked like an infinitely dense mille-feuille cake in the 4D phase space. And typically these sheets are interleaved with the sheets of the other attractors. The end result is very much like the Lakes of Wada. Proving the boundary actually has the Wada property is tricky, although new methods look promising.

The magnetic pendulum

This fractal is similar to one I made back in 1990 inspired by the dynamics of the magnetic decision-making desk toy, a pendulum oscillating above a number of magnets. Eventually it settles over one. The basic dynamics is fairly similar (see Zhampres’ beautiful images or this great treatment). The difference is that the gravity fractal has no dissipation: in principle orbits can continue forever (but I end when they get close to the masses or after a timeout) and in the magnetic fractal the force dependency was bounded, a K/(r^2 + c) force rather than the G/r^2.

That simulation was part of my epic third year project in the gymnasium. The topic was “Chaos and self-organisation”, and I spent a lot of time reading the dynamical systems literature, running computer simulations, struggling with WordPerfect’s equation editor and producing a manuscript of about 150 pages that required careful photocopying by hand to get the pasted diagrams on separate pieces of paper to show up right. My teacher eventually sat down with me and went through my introduction and had me explain Poincaré sections. Then he promptly passed me. That was likely for the best for both of us.

Appendix: Matlab code

showPlot=0; % plot individual trajectories
randMass = 0; % place masses randomly rather than in circle

RTRAP=0.0001; % size of trap region
tmax=60; % max timesteps to run
S=1000; % resolution

x=linspace(-2,2,S);
y=linspace(-2,2,S);
[X,Y]=meshgrid(x,y);

N=5;
theta=(0:(N-1))*pi*2/N;
PX=cos(theta); PY=sin(theta);
if (randMass==1)
s = rng(3);
PX=randn(N,1); PY=randn(N,1);
end

clf

hit=X*0; 
hitN = X*0; % attractor basin
hitT = X*0; % time until hit
closest = X*0+100; 
closestN=closest; % closest mass to trajectory

tic; % measure time
for a=1:size(X,1)
disp(a)
for b=1:size(X,2)
[t,u,te,ye,ie]=ode45(@(t,y) forceLaw(t,y,N,PX,PY), [0 tmax], [X(a,b) 0 Y(a,b) 0],odeset('Events',@(t,y) finishFun(t,y,N,PX,PY,RTRAP^2)));

if (showPlot==1)
plot(u(:,1),u(:,3),'-b')
hold on
end

if (~isempty(te))
hit(a,b)=1;
hitT(a,b)=te;

mind2=100^2;
for k=1:N
dx=ye(1)-PX(k);
dy=ye(3)-PY(k);
d2=(dx.^2+dy.^2);
if (d2<mind2) mind2=d2; hitN(a,b)=k; end
end

end
for k=1:N
dx=u(:,1)-PX(k);
dy=u(:,3)-PY(k);
d2=min(dx.^2+dy.^2);
closest(a,b)=min(closest(a,b),sqrt(d2));

if (closest(a,b)==sqrt(d2)) closestN(a,b)=k; end
end
end

if (showPlot==1)
drawnow
pause
end
end
elapsedTime = toc

if (showPlot==0)
% Make colorful plot
co=hsv(N);
mag=sqrt(hitT);
mag=1-(mag-min(mag(:)))/(max(mag(:))-min(mag(:)));
im=zeros(S,S,3);
im(:,:,1)=interp1(1:N,co(:,1),closestN).*mag;
im(:,:,2)=interp1(1:N,co(:,2),closestN).*mag;
im(:,:,3)=interp1(1:N,co(:,3),closestN).*mag;
image(im)
end

% Gravity 
function dudt = forceLaw(t,u,N,PX,PY)
%dudt = zeros(4,1);
dudt=u;
dudt(1) = u(2);
dudt(2) = 0;
dudt(3) = u(4);
dudt(4) = 0;

dx=u(1)-PX;
dy=u(3)-PY;
d=(dx.^2+dy.^2).^-1.5;
dudt(2)=dudt(2)-sum(dx.*d);
dudt(4)=dudt(4)-sum(dy.*d);

% for k=1:N
% dx=u(1)-PX(k);
% dy=u(3)-PY(k);
% d=(dx.^2+dy.^2).^-1.5;
% dudt(2)=dudt(2)-dx.*d;
% dudt(4)=dudt(4)-dy.*d;
% end
end

% Are we close enough to one of the masses?
function [value,isterminal,direction] =finishFun(t,u,N,PX,PY,r2)
value=1000;
for k=1:N
dx=u(1)-PX(k);
dy=u(3)-PY(k);
d2=(dx.^2+dy.^2);
value=min(value, d2-r2);
end
isterminal=1;
direction=0;
end

Throwing balls on torus-earth

A question came up on Physics Stack Exchange: how does thrown object trajectories look on a toroidal planet?

Locally we should expect them to be like on Earth: there is constant gravitational acceleration orthogonal to the ground, so they will just look like parabolas.

But if the trajectory is longer the rapid rotation ought to twist it, since there is a fair Coriolis effect. So the differential equation will be \mathbf{x}''=\mathbf{g}+2\mathbf{x}'\times\mathbf{\Omega}. If we just look at the velocity vector we get \mathbf{v}'=\mathbf{g}+2\mathbf{v}\times\mathbf{\Omega}.

That is, the forcefield will twist the velocity around if it is large and orthogonal to the angular velocity vector. If the velocity is parallel it will just be affected by gravity. For a trajectory near the pole it will become twisted and tilted:

Trajectories of a ball thrown from the surface with the angular velocity vector parallel to the gravity vector.
Trajectories of a ball thrown from the surface with the angular velocity vector parallel to the gravity vector.

For a starting point on the equator the twisting gets a bit more complex:

Trajectories of a ball thrown from the surface with the angular velocity vector orthogonal to the gravity vector.
Trajectories of a ball thrown from the surface with the angular velocity vector orthogonal to the gravity vector.

One can also recognise the analogy to an electron in an electromagnetic field: $latex \mathbf{v}’ = (q/m)(\mathbf{E}+\mathbf{v}\times \mathbf{B})$. Without gravity we should hence expect thrown balls to just follow helices around the omega-vector direction just like charged particles follow magnetic field-lines. One can eliminate the electric field from the equation by using a different velocity coordinate $latex \mathbf{v_2}=\mathbf{v}-\mathbf{E}\times\mathbf{B}/B^2$. Hence we can treat ball trajectories like helices plus a drift velocity in the \mathbf{g}\times\mathbf{\Omega} direction. The helix radius will be v/2\Omega.

How large is the Coriolis effect? On Earth \Omega=2\pi/86400\approx 0.0000727. On Donut it is 0.000614 and on Hoop 0.000494, several times higher. Still, the correction is not going to be enormous: for a ball moving 10 meters per second the helix radius will be 69 km on Earth (at the pole), 8.1 km on Donut, and 10 km on Hoop. We hence need to throw the ball a suborbital distance before the twists become really visible. At these distances the curvature of the planet and the non-linearity of the gravitational field also begins to bite.

I have not simulated such trajectories since I need a proper mass distribution model of the worlds, and it is messy. However, for an infinitely thin ring one can solve orbits numerically relatively easily (you “just” have to integrate elliptic integrals):

Some orbits around a massive ring.
Some orbits around a massive ring.

Beside the “normal” equatorial orbits and torus-like orbits winding themselves around the ring, there are internal halo-orbits and chaotic tangles.

Blueberry Earth

[Update: I have a paper version of this essay on arXiv:1807.10553, extending and correcting some of the results.]

On Physics Stackexchange billybodega asked the question:

BlueberrySupposing that the entire Earth was instantaneously replaced with an equal volume of closely packed, but uncompressed blueberries, what would happen from the perspective of a person on the surface?

Unfortunately the site tends to frown on fun questions like this, so it was in my opinion prematurely closed while I was working out the answer. So here it is, with some extra extensions:

The density of blueberries has been estimated to 625.56 kg/m3, WillO on Stackexchange estimated it to 13% of Earth’s density (5510*0.13=716.3 kg/m3), so assuming it to be around \rho_{berries}=700 kg/m3 appears to be reasonable. Blueberry pulp has a density similar to water,  980 to 1050 kg per m3 although this is both temperature dependent and depends on how much solids there are. The difference to the whole berries is due to the air between the berries. Note that these are likely the big, thick-skinned “American” blueberries rather than the small wild thin-skinned blueberries (bilberries) I grew up with; the latter would have higher density due to their smaller size and break far more easily.

So instantaneously turning Earth into blueberries will reduce its mass to 0.1274 of what it was. Gravity will become correspondingly weaker, g_{BE}=0.1274 g.

However, blueberries are not particularly sturdy. While there is a literature on blueberry mechanics (of course!), I did not manage to find a great source on their compressive strength. A rough estimate is possible: stacking a sugar cube (1 g) on a berry will not break it, while a milk carton (1 kg) will; 100 g has a decent but not certain chance. So if we assume the blueberry area to be one square centimetre the breaking pressure is on the order of P_{break}=0.1 g / 10^{-4} \approx 10,000 N/m2. This allows us to estimate at what depth the berries will start to break: z=P_{break}/g_{BE}\rho_{berries} = 11.4188 m. So while the surface will be free blueberries they will start pulping within a few meters of the surface.

This pulping has an important effect: the pulp separates from the air, coalescing into a smaller sphere. If we assume pulp to be an incompressible fluid, then a sphere of pulp with the same mass as the initial berries will be \rho_{pulp} r_{pulp}^3 = \rho_{berries}r_{earth}^3, or r_{pulp} = (\rho_{berries}/ \rho_{pulp} )^{1/3}r_{earth}. In this case we end up with a planet with 0.8879 times smaller radius (5,657 km), surrounded by a vast atmosphere.

The freefall timescale for the planet is initially 41 minutes, but relatively shortly the pulping interactions, the air convection etc will slow things down in a complicated way. I expect that the the actual coalescence will take hours, with some late bubbles from the deep interior erupting fairly late.

The gravity on the pulp surface is just 1.5833 m/s2, 16% of normal gravity – almost exactly lunar gravity. This weakens convection currents and the speed with which bubbles move up. The scale height of the atmosphere, assuming the same composition and temperature as on Earth, will be 6.2 times higher. This means that pressure will decline much less with altitude, allowing far thicker clouds and weather systems. As we will see, the atmosphere will puff up more.

The separation has big consequences. Enormous amounts of air will be pushing out from the pulp as bubbles and jets, producing spectacular geysers (especially since the gravity is low). Even more dramatic is the heating: a lot of gravitational energy is released as the mass is compacted. The total gravitational energy of a constant density sphere of radius R is

\int_0^R G [4\pi r^2 \rho] [4 \pi r^3 \rho/3] / r dr  = (16\pi^2 G\rho^2/3) \int_0^R r^4 dr
=(16\pi^2 G/15)\rho^2 R^5

(the first factor in the integral is the mass of a spherical shell of radius r, the second the mass of the stuff inside, and the third the 1/r gravitational potential). If we ignore the mass of the air since it is small and we just want an order of magnitude estimate,  the compression of the berry mass gives energy

E=(16\pi^2 G/15)(\rho_{berries}^2 r_{earth}^5 - \rho_{pulp}^2R_{pulp}^5) \approx 4.3594\times 10^{29} J.

This is the energy output of the sun over half an hour, nothing to sneeze at: blueberry earth will become hot. There is about 573,000 J per kg, enough to heat the blueberries from freezing to boiling.

The result is that blueberry earth will turn into a roaring ocean of boiling jam, with the geysers of released air and steam likely ejecting at least a few berries into orbit (escape velocity is just 4.234 km/s, and berries at the initial surface will be even higher up in the potential). As the planet evolves a thick atmosphere of released steam will add to the already considerable air from the berries. It is not inconceivable that the planet may heat up further due to a water vapour greenhouse effect, turning into a very odd Venusian world.

Meanwhile the jam ocean is very deep, and the pressure at depth will be enough to cause the formation of high pressure ice even if it is warm. If the formation process is slow there will be some separation of water into ice and a concentration of other chemicals in the jam ocean, but I suspect the rapid collapse will instead make some kind of composite pulp ice. Ice VII forms above 9 GPa, so if we just use constant gravity this happens at a depth z_{ice}=P_{VII}/g_{BE}\rho_{pulp}\approx 1,909 km, about two-thirds of the radius. This would make up most of the interior. However, gravity is a bit weaker in the interior, so we need to take that into account. The pressure from all the matter above radius r is P(r) =(3GM^2/8\pi R^4)(1-(r/R)^2), and the ice core will have radius r_{ice}=\sqrt{1-P_{VII}/P(0)}  \approx 3,258 km. This is smaller, about 57% of the radius, and just 20% of the total volume.

The coalescence will also speed up rotation. The original blueberry earth would of course make one rotation every 24 hours, but the smaller result would have a smaller moment of inertia. The angular momentum conservation gives (2/5)MR_1^2(2\pi/T_1) = (2/5)MR_2^2(2\pi/T_2), or T_2 = (R_2/R_1)^2 T_1, in this case 18.9210 hours. This in turn will increase the oblateness a bit, to approximately 0.038 – an 8.8 times increase over Earth.

Another effect is the orbit of the Moon. Now the two bodies have about equal mass. Is the Moon bound to blueberry earth? A kilogram of lunar material has potential energy GM_{BE}/r_{moon} \approx 1.6925 \times 10^{5} J, while the kinetic energy is 2.6442\times 10^5 J – more than enough to escape. Had it remained the jam ocean would have made an excellent tidal dissipation mechanism that would have slowed down rotation and moved blueberry earth towards tidal lock with the moon much earlier than the 50 billion years it would otherwise have taken.

So, to sum up, to a person standing on the surface of the Earth when it turns into blueberries, the first effect would be a drastic reduction of gravity. Standing on the blueberries might be possible in theory, except that almost immediately they begin to compress rapidly and air starts erupting everywhere. The effect is basically the worst earthquake ever, and it keeps on going until everything has fallen 714 km. While this is going on everything heats up drastically until the entire environment is boiling jam and steam. The end result is a world that has a steam atmosphere covering an ocean of jam on top of warm blueberry granita.

Admitting blog infidelity with StackExchange

I have neglected Andart II for some time, partly for the good reason of work (The Book is growing!), partly because I got a new addiction: answering (and asking) questions on Physics and Astronomy StackExchange. Very addictive, but also very educational. Here are links to some of the stuff I have been adding, which might be of interest to some readers.

Overcoming inertia

Balls

The tremendous accelerations involved in the kind of spaceflight seen on Star Trek would instantly turn the crew to chunky salsa unless there was some kind of heavy-duty protection. Hence, the inertial damping field.
— Star Trek: The Next Generation Technical Manual, page 24.

For a space opera RPG setting I am considering adding inertia manipulation technology. But can one make a self-consistent inertia dampener without breaking conservation laws? What are the physical consequences? How many cool explosions, superweapons, and other tropes can we squeeze out of it? How to avoid the worst problems brought up by the SF community?

What inertia is

As Newton put it, inertia is the resistance of an object to a change in its state of motion. Newton’s force law F=ma is a consequence of the definition of momentum, p=mv (which in a way is more fundamental since it directly ties in with conservation laws). The mass in the formula is the inertial mass. Mass is a measure of how much there is of matter, and we normally multiply it with a hidden constant of 1 to get the inertial mass – this constant is what we will want to mess with.

There are relativistic versions of the laws of motion that handles momentum and inertia for high velocities, where the kinetic energy becomes so large that it starts to add mass to the whole system. This makes the total inertia go up, as seen by an outside observer, and looks like a nice case for inertia-manipulating tech being vaguely possible.

However, Einstein threw a spanner into this: gravity also acts on mass and conveniently does so exactly as much as inertia: gravitational mass (the masses in F=Gm_1m_2/r^2) and inertial mass appear to be equal. At least in my old school physics textbook (early 1980s!) this was presented as a cool unsolved mystery, but it is a consequence of the equivalence principle in general relativity (1907): all test particles accelerate the same way in a gravitational field, and this is only possible if their gravitational mass and inertial mass are proportional to one another.

So, an inertia manipulation technology will have to imply some form of gravity manipulation technology. Which may be fine from my standpoint, since what space opera is complete without antigravity? (In fact, I already had decided to have Alcubierre warp bubble FTL anyway, so gravity manipulation is in.)

Playing with inertia

OK, let’s leave relativity to the side for the time being and just consider the classical mechanics of inertia manipulation. Let us posit that there is a magical field that allows us to dial up or down the proportionality constant for inertial mass: the momentum of a particle will be p=\mu m v, the force law F=\mu m a and the formula for kinetic energy K=(1/2) \mu m v^2. \mu is the effect of the magic field, running from 0<\mu<\infty, with 1 corresponding to it being absent.

I throw a 1 g ping-pong ball at 1 m/s into my inertics device and turn on the field. What happens? Let us assume the field is \mu=1000. Now the momentum and kinetic energy jumps by a factor of 1000 if the velocity remains unchanged. Were I to catch the ball I would have gained 999 times its original kinetic energy: this looks like an excellent perpetual motion machine. Since we do not want that to be possible (a space empire powered by throwing ping-pong balls sounds silly) we must demand that energy is conserved.

Velocity shifting to preserve kinetic energy

Radiation shieldingOne way of doing energy conservation is for the velocity to go down for my heavy ping-pong ball. This means that the new velocity will be v/\sqrt{\mu}. Inertia-increasing fields slow down objects, while inertia-decreasing fields speed them up.

Forcefields/armour

One could have a force-field made of super-high inertia that would slow down incoming projectiles. At first this seems pointless, since once they get through to the other side they speed up and will do the same damage. But we could of course put in a bunch of armour in this field, and have it resist the projectile. The kinetic energy will be the same but it will be a lower velocity collision which means that the strength of the armour has a better chance of stopping it (in fact, as we will see below, we can use superdense armour here too). Consider the difference between being shot with a rifle bullet or being slowly but strongly stabbed by it: in the later case the force can be distributed by a good armour to a vast surface. Definitely a good thing for a space opera.

Spacecraft

A spacecraft that wants to get somewhere fast could just project a low \mu field around itself and boost its speed by a huge 1/\sqrt{\mu} factor. Sounds very useful. But now an impacting meteorite will both have an high relative speed, and when it enters the field get that boosted by the same factor again: impacts will happen at velocities increased by a factor of 1/\mu as measured by the ship. So boosting your speed with a factor of a 1000 will give you dust hitting you at speeds a million times higher. Since typical interplanetary dust already moves a few km/s, we are talking about hyperrelativistic impactors. The armour above sounds like a good thing to have…

Note that any inertia-reducing technology is going to improve rockets even if there is no reactionless drive or other shenanigans: you just reduce the inertia of the reaction mass. The rocket equation no longer bites: sure, your ship is mostly massive reaction mass in storage, but to accelerate the ship you just take a measure of that mass, restore its inertia, expel it, and enjoy the huge acceleration as the big engine pushes the overall very low-inertia ship. There is just a snag in this particular case: when restoring the inertia you somehow need to give the mass enough kinetic energy to be at rest in relation to the ship…

Cannons

This kind of inertics does not make for a great cannon. I can certainly make my projectile speed up a lot in the bore by lowering its inertia, but as soon as it leaves it will slow down. If we assume a given amount of force F accelerating it along the length L bore, it will pick up FL Joules of kinetic energy from the work the cannon does – independent of mass or inertia! The difference may be power: if you can only supply a certain energy per second like in a coilgun, having a slower projectile in the bore is better.

Physics

Note that entering and leaving an inertics field will induce stresses. A metal rod entering an inertia-increasing field will have the part in the field moving more slowly, pushing back against the not slowed part (yet another plus for the armour!). When leaving the field the lighter part outside will pull away strongly.

Another effect of shifting velocities is that gases behave differently. At first it looks like changing speeds would change temperature (since we tend to think of the temperature of a gas as how fast the molecules are bouncing around), but actually the kinetic temperature of a gas depends on (you guessed it) the average kinetic energy. So that doesn’t change at all. However, the speed of sound should scale as \propto 1/\sqrt{\mu}: it becomes far higher in the inertia-dampening field, producing helium-voice like effects. Air molecules inside an inertia-decreasing field would tend to leave more quickly than outside air would enter, producing a pressure difference.

Momentum conservation is a headache

Atlas 6Changing the velocity so that energy is conserved unfortunately has a drawback: momentum is not conserved! I throw a heavy object at my inertics machine at velocity v, momentum mv and energy (1/2)mv^2, it reduces is inertia and increases the speed to v/\sqrt{\mu}, keeps the kinetic energy at (1/2)mv^2, and the momentum is now mv/\sqrt{\mu}.

What if we assume the momentum change comes from the field or machine? When I hit the mass M machine with an object it experiences a force enough to change its velocity by w=mv(1-1/\sqrt{\mu})/M. When set to increase inertia it is pushed back a bit, potentially moving up to speed (m/M)v. When set to decrease inertia it is pushed forward, starting to move towards the direction the object impacted from. In fact, it can get arbitrarily large velocities by reducing \mu close to 0.

This sounds odd. Demanding momentum and energy conservation requires mv = mv/\sqrt{\mu} + Mw (giving the above formula) and mv^2 = \mu m(v/\sqrt{\mu})^2 + Mw^2, which insists that w=0. Clearly we cannot have both.

I don’t know about you, but I’d rather keep energy conserved. It is more obvious when you cheat about energy conservation.

Still, as Einstein pointed out using 4-vectors, momentum and energy conservation are deeply entangled – one reason inertics isn’t terribly likely in the real world is that they cannot be separated. We could of course try to conserve 4-momentum ((E/c,\gamma \mu m v_x, \gamma \mu m v_y, \gamma \mu m v_z)), which would look like changing both energy and normal momentum at the same time.

Energy gain/loss to preserve momentum

Buffer stopsWhat about just retaining the normal momentum rather than the kinetic energy? The new velocity would be v/\mu, the new kinetic energy would be K_1=(1/2) \mu m (v/\mu)^2 = (1/2) mv^2 / \mu = K_0/\mu. Just like in the kinetic energy preserving case the object slows down (or speeds up), but more strongly. And there is an energy debt of K_0 (1-1/\mu) that needs to be fixed.

One way of resolving energy conservation is to demand that the change in energy is supplied by the inertia-manipulation device. My ping-pong ball does not change momentum, but requires 0.999 Joule to gain the new kinetic energy. The device has to provide that. When the ball leaves the field there will be a surge of energy the device needs to absorb back. Some nice potential here for things blowing up in dramatic ways, a requirement for any self-respecting space opera.

Spacecraft

If I want to accelerate my spaceship in this setting, I would point my momentum vector towards the target, reduce my inertia a lot, and then have to provide a lot of kinetic energy from my inertics devices and power supply (actually, store a lot – the energy is a surplus). At first this sounds like it is just as bad as normal rocketry, but in fact it is awesome: I can convert my electricity directly into velocity without having to lug around a lot of reaction mass! I will even get it back when slowing down, a bit like electric brake regeneration systems.  The rocket equation does not apply beyond getting some initial momentum. In fact, the less velocity I have from the start, the better.

At least in this scheme inertia-reduced reaction mass can be restored to full inertia within the conceptual framework of energy addition/subtraction.

One drawback is that now when I run into interplanetary dust it will drain my batteries as the inertics system needs to give it a lot of kinetic energy (which will then go on harming me!)

Another big problem (pointed out by Erik Max Francis) is that turning energy into kinetic energy gives an energy requirement $latex dK/dt=mva$, which depends on an absolute speed. This requires a privileged reference frame, throwing out relativity theory. Oops (but not unexpected).

Forcefields/armour

Energy addition/depletion makes traditional force-fields somewhat plausible: a projectile hits the field, and we use the inertics to reduce its kinetic energy to something manageable. A rifle bullet has a few thousand Joules of energy, and if you can drain that it will now harmlessly bounce off your normal armour. Presumably shields will be depleted when the ship cannot dissipate or store the incoming kinetic energy fast enough, causing the inertics to overload and then leaving the ship unshielded.

Cannons

This kind of inertics allows us to accelerate projectiles using the inertics technology, essentially feeding them as much kinetic energy as we want. If you first make your projectile super-heavy, accelerate it strongly, and then normalise the inertia it will now speed away with a huge velocity.

Physics

A metal rod entering this kind of field will experience the same type of force as in the kinetic energy respecting model, but here the field generator will also be working on providing energy balance: in a sense it will be acting as a generator/motor. Unfortunately it does not look like it could give a net energy gain by having matter flow through.

Note that this kind of device cannot be simply turned off like the previous one: there has to be an energy accounting as everything returns to \mu=1. The really tricky case is if you are in energy-debt: you have an object of lowered inertia in the field, and cut the power. Now the object needs to get a bunch of kinetic energy from somewhere. Sudden absorption of nearby kinetic energy, freezing stuff nearby? That would break thermodynamics (I could set up a perpetual motion heat engine this way). Leaving the inertia-changed object with the changed inertia? That would mean there could be objects and particles with any effective mass – space might eventually be littered with atoms with altered inertia, becoming part of normal chemistry and physics. No such atoms have ever been found, but maybe that is because alien predecessor civilisations were careful with inertial pollution.

Other approaches

Gravity manipulation

Levitating morris dancersAnother approach is to say that we are manipulating spacetime so that inertial forces are cancelled by a suitable gravity force (or, for purists, that the acceleration due to something gets cancelled by a counter-acceleration due to spacetime curvature that makes the object retain the same relative momentum).

The classic is the “gravitic drive” idea, where the spacecraft generates a gravity field somehow and then free-falls towards the destination. The acceleration can be arbitrarily large but the crew will just experience freefall. Same thing for accelerating projectiles or making force-fields: they just accelerate/decelerate projectiles a lot. Since momentum is conserved there will be recoil.

The force-fields will however be wimpy: essentially it needs to be equivalent to an acceleration bringing the projectile to a stop over a short distance. Given that normal interplanetary velocities are in tens of kilometres per second (escape velocity of Earth, more or less) the gravity field needs to be many, many Gs to work. Consider slowing down a 20 km/s railgun bullet to a stop over a distance of 10 meters: it needs to happen over a millisecond and requires a 20 million m/s^2 deceleration (2.03 megaG).

If we go with energy and momentum conservation we may still need to posit that the inertics/antigravity draws power corresponding to the work it does . Make a wheel turn because of an attracting and repulsing field, and the generator has to pay the work (plus experience a torque). Make a spacecraft go from point A to B, and it needs to pay the potential energy difference, momentum change, and at least temporarily the gain in kinetic energy. And if you demand momentum conservation for a gravitic drive, then you have the drive pulling back with the same “force” as the spacecraft experiences. Note that energy and momentum in general relativity are only locally conserved; at least this kind of drive can handwave some excuse for breaking local momentum conservation by positing that the momentum now resides in an extended gravity field (and maybe gravitational waves).

Unlike the previous kinds of inertics this doesn’t change the properties of matter, so the effects on objects discussed below do not apply.

One problem is edge tidal effects. Somewhere there is going to be a transition zone where there is a field gradient: an object passing through is going to experience some extreme shear forces and likely spaghettify. Conversely, this makes for a nifty weapon ripping apart targets.

One problem with gravity manipulation is that it normally has to occur through gravity, which is both very weak and only has positive charges. Electromagnetic technology works so well because we can play positive and negative charges against each other, getting strong effects without using (very) enormous numbers of electrons. Gravity (and gravitomagnetic effects) normally only occurs due to large mass-energy densities and momenta. So for this to work there better be antigravitons, negative mass, or some other way of making gravity behave differently from vanilla relativity. Inertics can typically handwave something about the Higgs field at least.

Forcefield manipulation

This leaves out the gravity part and just posits that you can place force vectors wherever you want. A bit like Iain M. Banks’ effector beams. No real constraints because it is entirely made-up physics; it is not clear it respects any particular conservation laws.

Other physical effects

Here are some of the nontrivial effects of changing inertia of matter (I will leave out gravity manipulation, which has more obvious effects).

Electromagnetism: beware the blue carrot

It is worth noting that this thought experiment does not affect light and other electromagnetic fields: photons are massless. The overall effect is that they will tend to push around charged objects in the field more or less. A low-inertia electron subjected to a given electric field will accelerate more, a high-inertia electron less. This in turn changes the natural frequencies of many systems: a radio antenna will change tuning depending on the inertia change. A receiver inside the inertics field will experience outside signals as being stronger (if the field decreases inertia) or weaker (if it increases it).

Reducing inertia also increases the Bohr magneton, e\hbar/2 \mu m_e. This means that paramagnetic materials become more strongly affected by magnetic fields, and that ferromagnets are boosted. Conversely, higher inertia reduces magnetic effects.

Changing inertia would likely change atomic spectra (see below) and hence optical properties of many compounds. Many pigments gain their colour from absorption due to conjugated systems (think of carotene or heme) that act as antennas: inertia manipulation will change the absorbed frequencies. Carotene with increased inertia will presumably shift its absorption spectra towards lower frequencies, becoming redder, while lowered inertia causes a green or blue shift. An interesting effect is that the rhodopsin in the eye will also be affected and colour vision will experience the same shift (objects will appear to change colour in regions with a different \mu from the place where the observer is, but not inside their field). Strong enough fields will cause shifts so that absorption and transmission outside the visual range will matter, e.g. infrared or UV becomes visible.

However, the above claim that photons should not be affected by inertia manipulation may not have to hold true. Photons carry momentum, p=\hbar k where k is the wave vector. So we could assume a factor of 1/\sqrt{\mu} or 1/\mu gets in there and the field red/blueshifts photons. This would complicate things a lot, so I will leave analysis to the interested reader. But it would likely make inertics fields visible due to refractive effects.

Chemistry: toxic energy levels, plus a shrink-ray

Projectile warningOne area inertics would mess up is chemistry. Chemistry is basically all about the behaviour of the valence electrons of atoms. Their behaviour depends on their distribution between the atomic orbitals, which in turn depends on the Schrödinger equation for the atomic potential. And this equation has a dependency on the mass of the electron and nucleus.

If we look at hydrogen-like atoms, the main effect is that the energy levels become

E_n = - \mu (M Z^2 e^4/8 \epsilon_0^2 h^2 n^2),

where M=m_e m_p/(m_e+m_p) is the reduced mass. In short, the inertial manipulation field scales the energy levels up and down proportionally. One effect is that it becomes much easier to ionise low-inertia materials, and that materials that are normally held together by ionization bonds (say NaCl salt) may spontaneously decay when in high-inertia fields.

The Bohr radius scales as a_0 \propto 1/\mu: low-inertia atoms become larger. This really messes with materials. Placed in a low-inertia field atoms expand, making objects such as metals inflate. In a high inertia-field, electrons keep closer to the nuclei and objects shrink.

As distances change, the effects of electromagnetic forces also change: internal molecular electric forces, van der Waals forces and things like that change in strength, which will no doubt have effects on biology. Not to mention melting points: reducing the inertia will make many materials melt at far lower temperatures due to larger inter-atomic and inter-molecular distances, increasing it can make room-temperature liquids freeze because they are now more closely packed.

This size change also affects the electron-electron interactions, which among other things shield the nucleus and reduce the effective nuclear charge. The changed energy levels do not strongly affect the structure of the lightest atoms, so they will likely form the same kind of chemical bonds and have the same chemistry. However, heavier atoms such as copper, chromium and palladium already have ordering rules that are slightly off because of the quirks of the energy levels. As the field deviates from 1 we should expect lighter and lighter atoms to get alternative filling patterns and this means they will get different chemistry. Given that copper and chromium are essential for some enzymes, this does not bode well – if copper no longer works in cytochrome oxidase, the respiratory chain will lethally crash.

If we allow permanently inertia-altered particles chemistry can get extremely weird. An inertia-changed electron would orbit in a different way than a normal one, giving the atom it resided in entirely different chemical properties. Each changed electron could have its own individual inertia. Presumably such particles would randomise chemistry where they resided, causing all sorts of odd reactions and compounds not normally seen. The overall effect would likely be pretty toxic, since it would on average tend to catalyze metastable high-energy, low-entropy structures in biochemistry to fall down to lower energy, higher entropy states.

Lowering inertia in many ways looks like heating up things: particles move faster, chemicals diffuse more, and things melt. Given that much of biochemistry is tremendously temperature dependent, this suggests that even slight changes of \mu to 0.99 or 1.01 would be enough to create many of the bad effects of high fever or hypothermia, and a bit more would be directly lethal as proteins denaturate.

Fluids: I need a lie down

Inside a lowered inertia field matter responds more strongly to forces, and this means that fluids flow faster for the same pressure difference. Buoyancy cases stronger convection. For a given velocity, the inertial forces  are reduced compared to the viscosity, lowering the Reynolds number and making flows more laminar. Conversely, enhanced inertia fluids are hard to get to move but at a given speed they will be more turbulent.

This will really mess up the sense of balance and likely blood flow.

Gravity: equivalent exchange

I have ignored the equivalence of inertial and gravitational mass. One way for me to get away with it is to claim that they are still equivalent, since everything occurs within some local region where my inertics field is acting: all objects get their inertial mass multiplied by \mu and this also changes their gravitational mass. The equivalence principle still holds.

What if there is no equivalence principle? I could make 1 kg object and a 1 gram object fall at different accelerations. If I had a massless spring between them it would be extended, and I would gain energy. Beside the work done by gravity to bring down the objects (which I could collect and use to put them back where they started) I would now have extra energy – aha, another perpetual motion machine! So we better stick to the equivalence principle.

Given that boosting inertia makes matter both tend to shrink to denser states and have more gravitational force, an important worldbuilding issue is how far I will let this process go. Using it to help fission or fusion seems fine. Allowing it to squeeze matter into degenerate states or neutronium might be more world-changing. And easy making of black holes is likely incompatible with the survival of civilisation.

[ Still, destroying planets with small black holes is harder than it looks. The traditional “everything gets sucked down into the singularity” scenario is surprisingly slow. If you model it using spherical Bondi accretion you need an Earth-mass black hole to make the sun implode within a year or so, and a 3\cdot 10^{19} kg asteroid mass black hole to implode the Earth. And the extreme luminosity slows things a lot more. A better way may be to use an evaporating black hole to irradiate the solar system instead, or blow up something sending big fragments. ]

Another fun use of inertics is of course to mess up stars directly. This does not work with the energy addition/depletion model, but the velocity change model would allow creating a region of increased inertia where density ramps up: plasma enters the volume and may start descending below the spot. Conversely, reducing inertia may open a channel where it is easier for plasma from the interior to ascend (especially since it would be lighter). Even if one cannot turn this into a black hole or trigger surface fusion, it might enable directed flares as the plasma drags electromagnetic field lines with it.

The probe was invisible on the monitor, but its effects were obvious: titanic volumes of solar plasma were sucked together into a strangely geometric sunspot. Suddenly there was a tiny glint in the middle and a shock-wave: the telemetry screens went blank.

“Seems your doomsday weapon has failed, professor. Mad science clearly has no good concept of proper workmanship.”

“Stay your tongue. This is mad engineering: the energy ran out exactly when I had planned. Just watch.”

Without the probe sucking it together the dense plasma was now wildly expanding. As it expanded it cooled. Beyond a certain point it became too cold to remain plasma: there was a bright flash as the protons and electrons recombined and the vortex became transparent. Suddenly neutral the matter no longer constrained the tortured magnetic field lines and they snapped together at the speed of light. The monitor crashed.

“I really hope there is no civilization in this solar system sensitive to massive electromagnetic pulses” the professor gloated in the dark.

Conclusions

Model Pros Cons
Preserve kinetic energy Nice armour. Fast spacecraft with no energy needs (but weird momentum changes). Interplanetary dust is a problem. Inertics cannons inefficient. Toxic effects on biochemistry.
Preserve momentum Nice classical forcefield. Fast spacecraft with energy demands. Inertics cannons work. Potential for cool explosions due to overloads. Interplanetary dust drains batteries. Extremely weird issues of energy-debts: either breaking thermodynamics or getting altered inertia materials. Toxic effects on biochemistry. Breaks relativity.
Gravity manipulation No toxic chemistry effects. Fast spacecraft with energy demands. Inertics cannons work. Forcefields wimpy. Gravitic drives are iffy due to momentum conservation (and are WMDs). Gravity is more obviously hard to manipulate than inertia. Tidal edge forces.

In both cases where actual inertia is changed inertics fields appear pretty lethal. A brief brush with a weak field will likely just be incapacitating, but prolonged exposure is definitely going to kill. And extreme fields are going to do very nasty stuff to most normal materials – making them expand or contract, melt, change chemical structure and whatnot. Hence spacecraft, cannons and other devices using inertics need to be designed to handle these effects. One might imagine placing the crew compartment in a counter-inertics field keeping \mu=1 while the bulk of the spacecraft is surrounded by other fields. A failure of this counter-inertics field does not just instantly turn the crew into tuna paste, but into blue toxic tuna paste.

Gravity manipulation is cleaner, but this is not necessarily a plus from the cool fiction perspective: sometimes bad side effects are exactly what world-building needs. I love the idea of inertics with potential as an anti-personnel or assassination weapon through its biochemical effects, or “forcefields” being super-dense metal with amplified inertia protecting against high-velocity or beam impact.

The atomic rocket page makes a big deal out of how reactionless propulsion makes space opera destroying weapons of mass destruction (if every tramp freighter can be turned into a relativistic missile, how long is the Imperial Capital going to last?) This is a smaller problem here: being hit by a inertia-reduced freighter hurts less, even when it is very fast (think of being hit by a fast ping-pong ball). Gravity propulsion still enables some nasty relativistic weaponry, and if you spend time adding kinetic energy to your inertia-reduced missile it can become pretty nasty. But even if the reactionless aspect does not trivially produce WMDs inertia manipulation will produce a fair number of other risky possibilities. However, given that even a normal space freighter is a hypervelocity missile, the problem lies more in how to conceptualise a civilisation that regularly handles high-energy objects in the vicinity of centres of civilisation.

Not discussed here are issues of how big the fields can be made. Could we reduce the inertia of an asteroid or planet, sending it careening around? That has some big effects on the setting. Similarly, how small can we make the inertics: do they require a starship to power them, or could we have them in epaulettes? Can they be counteracted by another field?

Inertia-changing devices are really tricky to get to work consistently; most space opera SF using them just conveniently ignores the mess – just like how FTL gives rise to time travel or that talking droids ought to transform the global economy totally.

But it is fun to think through the awkward aspects, since some of them make the world-building more exciting. Plus, I would rather discover them before my players, so I can make official handwaves of why they don’t matter if they are brought up.

How much for that neutron in the window?

Zach Weinersmith asked:

That is a great question. I once came up with the answer “50 tons of neutrons are needed” to a serious problem (you don’t want to know). How cheaply could you get that?

Figuring out roughly how many neutrons there are per kilogram of pure elements is pretty easy. Get their standard atomic weights, A, and subtract the atomic number Z since that is the number of protons: N=A-Z. Now we know how many neutrons there are per atom on average (standard atomic weights include the different isotope weights, weighted by their abundance).

[ Since nucleons (protons and neutrons) are about 1830 times heavier than electrons, we can ignore the electrons for an error on order of 0.05%. There is also a binding energy error, since some of the total atomic mass is because of binding energy between nucleons, which is 0.94% or less. These errors are nothing compared to price uncertainties.]

We know that one nucleon weighs about u=1.660539040\cdot 10^{-27} kg, so the number of nucleons per kilogram is N_{\mathrm{nucl}} \approx 1/(Au) and the number of neutrons per kilo is N_n \approx N_{\mathrm{nucl}}(N/A). This ranges from 7.5\cdot 10^{25} for helium down to 1.2\cdot 10^{24} for Oganesson. Hydrogen just has 4.7\cdot 10^{24} neutrons per kilogram, despite having 5.97\cdot 10^{26} nucleons per kilogram – there isn’t that much deuterium and tritium around to contribute neutrons.

Now, the price of elements is badly defined. I can get a kilogram of coal much cheaper than a kilogram of diamond, and ultra-pure elements are very expensive even if the everyday element is cheap. Plus, prices vary. And it is hard to buy plutonium on the open market. Ignoring all that and taking the numbers from Wikipedia (and ignoring the that some values look odd, and some are for compounds, and that the prices are unadjusted for inflation, and that they are lacking for many elements…) we can actually calculate the number of neutrons per dollar:

Neutrons per dollar if one buys one kilogram of the element.
Neutrons per dollar if one buys one kilogram of the element.

And the winner is… aluminium! You can get 8.8\cdot 10^{24} neutrons per dollar from aluminium.

In second place, nitrogen (7.1\cdot 10^{24}) and in third, hydrogen (6.8\cdot 10^{24})! Hydrogen may be very neutron-poor, but since it is rather cheap and you get lots of nucleons per kilo, this balances the lack.

Given that these prices are dodgy, I would expect an uncertainty on the order of a magnitude (at least). So the true winner, given the cheapest actual source of the element, might be hard to find without excruciating price comparisons. But we can be fairly certain it is going to be something with an atomic number less than 25. Uranium is unlikely to be a cheap neutron source in this sense (and just look at poor plutonium!)

So, given that aluminium is 51.8% neutrons by weight I need 96.5 tons. The current aluminium price is $1,650.00 per ton, so I would have to pay $159,225 for the neutrons in my doomsday weapon – I mean, totally innocuous thought experiment!

Scientific progress goes zig-zag

I recently nerded out about high-energy proton interaction with matter, enjoying reading up on the Bethe equation at the Particle Data Group review and elsewhere. That got me to look around at the PDL website, which is full of awesome stuff – everything from math and physics reviews to data for the most obscure “particles” ever, plus tests of how conserved the conservation laws are.

That binge led me to this interesting set of historical graphs of the estimates of various physical constants in the PDG publications over time:

Historical graph of physical constant estimates from K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014) and 2015 update.
Historical graph of physical constant estimates from K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014) and 2015 update.

The first thing that strikes the viewer is that they have moved a fair bit, including often being far outside the original error bars. 6 of them have escaped them. That doesn’t look very good for science!

Fortunately, it turns out that these error bars are not 95% confidence intervals (the most common form in many branches of science) but 68.3% confidence intervals (one standard deviation, if things are normal). That means having half of them out of range is entirely reasonable! On the other hand, most researchers don’t understand error bars (original paper), and we should be able to do much better.

The PDG state:

Sometimes large changes occur. These usually reflect the introduction of significant new data or the discarding of older data. Older data are discarded in favor of newer data when it is felt that the newer data have smaller systematic errors, or have more checks on systematic errors, or have made corrections unknown at the time of the older experiments, or simply have much smaller errors. Sometimes, the scale factor becomes large near the time at which a large jump takes place, reflecting the uncertainty introduced by the new and inconsistent data. By and large, however, a full scan of our history plots shows a dull progression toward greater precision at central values quite consistent with the first data points shown.

Overall, kudos to PDG for showing the history and making it clearer what is going on! But I do not agree it is a dull progression.

Zigzag to truth

The locus classicus for histories of physical constants being not quite a monotonic march towards truth is Max Henrion and Baruch Fischhoff. Assessing uncertainty in physical constants. American Journal of Physics 54, 791 (1986); doi: 10.1119/1.14447. They discuss the problem of people being overconfident and badly calibrated, and then show the zigzagging approach to current values:

Plot of light-speed measurements 1875-1958. Error bars are standard error. From Max Henrion and Baruch Fischhoff. Assessing uncertainty in physical constants. American Journal of Physics 54, 791 (1986); doi: 10.1119/1.14447.
Plot of light-speed measurements 1875-1958. Error bars are standard error. From Max Henrion and Baruch Fischhoff. Assessing uncertainty in physical constants. American Journal of Physics 54, 791 (1986); doi: 10.1119/1.14447.

Note that the shifts were far larger than the estimated error bars. The dip in the 1930s and 40s even made some physicists propose that c could be changing over time. Overall Henrion and Fischhoff find that physicists have been rather overconfident in their tight error bounds on their measurements. The approach towards current estimates is anything but dull, and hides many amusing historical anecdotes.

Stories like this might have been helpful; it is notable that the PDG histories on the right, for newer constants, seem to stay closer to the present value than the longer ones to the left. Maybe this is just because they have not had the time to veer off yet, but one can be hopeful.

Still, even if people are improving this might not mean the conclusions stay stable or approach truth monotonically. A related issue is “negative learning”, where more data and improved models make the consensus view of a topic move in the wrong direction: Oppenheimer, M., O’Neill, B. C., & Webster, M. (2008). Negative learning. Climatic Change, 89(1-2), 155-172. Here the problem is not just that people are overconfident in how certain they can be about their conclusions, but also that there is a bit of group-think, plus that the models change in structure and are affected in different ways by the same data. They point out how estimates of ozone depletion oscillated, or the consensus on the stability of climate has shifted from oscillatory (before 1968) towards instability (68-82), towards stability (82-96), and now towards instability again (96-06). These problems are not due to mere irrationality, but the fact that as we learn more and build better models these incomplete but better models may still deviate strongly from the ground truth because they miss some key component.

Noli fumare

This is related to what Nick Bostrom calls the “data fumes” problem. Early data will be fragmentary and explanations uncertain – but the data points and their patterns are very salient, just as the early models, since there is nothing else. So we begin to anchor on them. Then new data arrives and the models improve… and the old patterns are revealed as statistical noise, or bugs in the simulation or plotting routine. But since we anchored on them, we are unlikely to update as strongly towards the new most likely estimates. Worse, accommodating a new model takes mental work; our status quo bias will be pushing against the update. Even if we do accommodate the new state, things will likely change more – we may well end up either with a view anchored on early noise, or assume that the final state is far more uncertain than it actually is (since we weigh the early jumps strongly because of their saliency).

This is of course why most people prefer to believe a charismatic diet cultleader expert rather than trying to dig through 70 years of messy, conflicting dietary epidemiology.

Here is a simple example where an agent is trying to do a maximum likelihood estimation of a Gaussian distribution with mean 1 and variance 1, but is hamstrung by giving double weight to the first 9 data points:

Simple data fume model, showing the slow and biased convergence when the first 9 data points are over-weighted. The blue area is a 95% confidence interval for the mean of the generating Gaussian distribution.
Simple data fume model, showing the slow and biased convergence when the first 9 data points are over-weighted. The blue area is a 95% confidence interval for the mean of the generating Gaussian distribution.

It is not hard to complicate the model with anchoring/recency/status quo bias (estimates get biased towards previous estimates), or that early data points are more polluted by differently distributed noise. Asymmetric error checking (you will look for bugs if results deviate from expectation and hence often find such bugs, but not look for bugs making your results closer to expectation) is another obvious factor for how data fumes can get integrated in models.

The problem with data fumes is that it is not easy to tell when you have stabilized enough to start trusting the data. It is even messier when the inputs are results generated by your own models or code. I like to approach it by using multiple models to guesstimate model error: for example, one mathematical model on paper and one Monte Carlo simulation – if they don’t agree, then I should disregard either answer and keep on improving.

Even when everything seems to be fine there may be a big crucial consideration one has missed. The Turing-Good estimator gives another way of estimating the risk of that: if you have acquired N data points and seen K big surprises (remember that the first data point counts as one!), then the probability of a new surprise for your next data point is \approx K/N. So if you expect M data points in total, when K(M-N)/N \ll 1 you can start to trust the estimates… assuming surprises are uncorrelated etc. Which you will not be certain about. The progression towards greater precision may be anything but dull.