[math-fun] Science fiction (some of which is permanently gonna be fiction)
There's a lot of garbage science fiction out there that has nothing to do with science, especially in hollywood... but in this post I want to demolish two sci-fi ideas that occur in works of Larry Niven (et alia) which maybe did sound achievable to the naive. But a third such idea (super strong materials) will also be considered and argued to maybe have hope. 1. Sci-fi antimatter. You just keep a chunk of antimatter magnetically suspended in vacuum or something, where it's an excellent energy source for your super-rocket, ray-gun, also an excellent mega-bomb when you turn off the magnetic levitation. Stores the maximum possible amount of energy for that amount of mass. Actually: let's say a single normal-matter gas atom contaminates the vacuum. Hits the antimatter. Boom. Way more energy is released than you need to eject some antimatter atoms at high speed. They in turn hit the normal matter container. Boom boom boom. The resulting normal atom ejecta fly out to hit more antimatter. Etc. This is an exponentially amplifying process. THERE IS NO PRACTICAL WAY TO STORE A CHUNK OF ANTIMATTER. Get over it. 2. "Bussard Ramjet." A magic interstellar travel ship which somehow attracts interstellar hydrogen (with a "magnetic field"?) into its inscoop, pipes it thru a fusion reactor, and spits the hot helium out the back. "Keeps accelerating forever" (preferably at 1G) need not carry fuel, and if you wait long enough you'll achieve speeds arbitrarily near lightspeed enabling you to visit the whole universe in your lifetime thanks to time dilation. Actually: Even if this could be made to work, the impinging hydrogen would slow the ship down, it's a jet not a rocket, and the ultimate speed would be upper-bounded by the ejection velocity of the helium, no matter how long you waited. That is way below lightspeed since the mass loss (hydrogen-->helium) is very small fractionally. Meanwhile, if you just carried a tank of hydrogen fuel massing e times your unfueled mass (genuine rocket), you could achieve slightly greater speed without all the technical difficulty... and if it were deuterium you'd get a good deal faster speed. THIS WHOLE "BUSSARD RAMJET" IDEA IS JUST IRREDEEMABLY STUPID. 3. Super-strong materials. There's lots of them in sci-fi novels. So now let me actually do a little math (order of magnitude estimates); I'm going to attempt to derive upper bounds on the yield strength of any conceivable material. First try: relativity. If you pull on your material with force F enough to stretch it by a factor of (say) 2, then that stores elastic energy or order F*StretchDistance. This energy is bounded by m*c^2. Hence F<m*c^2/length is a rough upper bound on tensile strength of any mass=m object. Expressed in the usual units (pascals) of force per unit cross-sectional area, Strength < MassDensity * c^2. Now most sci-fi applications for superstrong spaceship hulls (armor), superstrong cables, etc envision materials with densities similar to normal materials, let's say 7.85 grams/cc which is the density of steel. Sci-fi super-steel would have a strength upper limit of Strength < 7*10^20 pascals as opposed to the actual strength of available ultra-high-strength steels 5*10^9 Pascals and maybe super-fibers could exceed that by factor 10-100 in our wildest dreams. So... not bad: sci-fi might hope if it could meet the relativity bound, to get strengths exceeding today's strongest stuff by a factor of 10^10. That's pretty good... (but never will be strong enough to protect spacecraft from collisions at 0.01c)... Second try: quantum mechanics. Material is made out of atomic (or subatomic, whatever) particles. Particles are trapped in potential-energy wells. Say the energy of your mass=m particle at position x is E(x) which has a minimum at x=0 locally like (m/2)*w^2*x^2 where w is the angular frequency of the harmonic oscillator. Considering the ground state of harmonic oscillator http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator the particle then is localized with std.deviation=DeltaL=(hbar/(m*w))^(1/2) and energy-gap to the next quantum level DeltaE=hbar*w [I may ignore small constant factors like 2]. If you pull on it with force of order F=DeltaE/DeltaL=(hbar*m)^(1/2)*w^(3/2) you are going to be able to excite the thing into the next state. This corresponds to a tensile strength (force per unit area) of order at most Strength < hbar^(-1/2) * w^(5/2) * m^(3/2) achieved by a material with Density of order Density = m^(5/2) * hbar^(-3/2) * w^(3/2). So with relativity, Strength < c^2 * Density. With quantum mechanics, Strength < (hbar * w / m) * Density. Roughly. The QM bound actually might be able to reach the same order as the relativity bound if we were dealing with particles bound by energies comparable to their mass-energy -- which is the case for quarks (the mass of the proton derives mostly from binding energies, not quark masses). For atoms, though, the fact electrons do the binding and they have way smaller masses than the nuclei, immediately kills you by a factor of at least 2000 versus the relativity bound. Actually only the OUTER electrons do the binding and the nuclei of the useful atoms are around 20000 times heavier than an electron, killing us by a factor of around 20000. This if we had a material whose bonds could only be broken by gamma rays in the 100 KeV range (electron positron annihilation radiation) would be tight, but all the materials we know of have bonds breakable by w in the UV light range of frequencies (<10eV), killing you by a further factor of 10000. This is not merely an empirical observation but actually theoretically inherent (actually "10000" is about 137^2 where the fine structure constant is 1/137) because electron-binding is via photons and the electron-photon nondimensional coupling constant in QED is 1/137. So our QM bound in view of that predicts the highest strength materials one could make with chemistry ought to have strengths 2*10^8 smaller than the relativity bound... which is only at most about 50X stronger than today's strongest stuff. I.e. the fundamental order of magnitude arguments get fairly close to the truth. However, a hypothetical material made of quarks bound into fibers (material would have extremely high density within the fibers, but low density if fibers surrounded by a lot of empty space) could, far as I can see from this crude analysis, genuinely achieve vastly greater strengths than today's materials with comparable densities. To rule that out you'd have to argue every such construct would be inherently unstable or something, which is an argument probably out of reach of today's physicists. Nuclear matter has a lot of surface tension which is why nuclei all are ball-shaped (isoperimetric theorem at work). Cylindrical fibers have too much surface. However, if you made one, then trying to cut it would cause even more surface -- so you might be able to hope for a quark fiber stable against all small-enough perturbations if you could somehow create it in the first place. I'd be worried though, that if anybody DID cut one of your fibers, then that'd cause big energy release, whacking nearby fiber hard enough to destabilize it... a chain reaction... and your chunk of material would turn into a nuclear bomb. CONCLUSION: this 3rd sci-fi fantasy of super-strong materials conceivably is achievable -- anyhow cannot be ruled out using simple arguments that I can see -- but don't try this at home!
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Warren Smith