Cosmic joke

General Relativity and Quantum Mechanics are famously incompatible theories—no one yet knows how to combine them in a self-consistent way.

And yet, there is something that GR and QM have in common.  They both give us elegant, simple equations that defy solution, even with the largest computers available.

The fundamental GR equation looks like this:

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}

Hidden in those Greek subscripts are 64 interdependent partial differential equations in 64 unknowns.

The fundamental equation of QM is the Schrodinger Equation, even simpler:

c4cp02258k-t1_hi-res

But looks are deceiving!  This is not an equation about how particles move but about how entire systems of particles evolve, as a single connected system. The equation has the property that it gives a beautifully precise solution for one electron in the isolated Hydrogen atom.  But adding a second electron makes the equation a million times harder to solve, and each subsequent electron you add requires a million times as much computing power as the previous one.

The most complicated classical equation that has ever been solved exactly is a computer simulation of 10 billion galaxies.

Bolshoi-Cosmology-Simulation.pngThe most complicated quantum equation that has ever been solved exactly is a computer simulation of 2 electrons (in this case, the Helium atom).

Helium-Schrodinger-solution

It’s as if God had said to physicists, “You want an equation that tells you how the world works—OK, try this one!”

What does it mean to have a theory that you believe describes the world very accurately, but you can never compare it to experiment because you can’t solve the damn equations?  It means that quantum physicists are always coming up with new and ever more clever approximations, then comparing with experiment to adjust the solutions and make a better fit.

It also means that quantum physicists can’t (or don’t) make bold predictions, and are subject to experimental surprises.  High-temperature superconductors.  Cold fusion.  Quantum biology?  Extra-sensory perception?

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2 thoughts on “Cosmic joke

  1. The GR equation is the fundamental equation of the Big Bang. Since the time of Einstein, it is customary to apply the solution in which the entire universe is smooth and uniform everywhere. But this is only because only in very simple geometry do we know how to solve the equation. In fact, the world we see through the telescope looks lumpy at every scale.

    For 20 years now, we have known that the Big Bang solution doesn’t match what we see through the telescope in another way. The expansion is getting faster and faster, so we add Dark Energy to the recipe. But then the clumps won’t condense into galaxies, so we add Dark Matter to close the gap. What we don’t know is whether we might obviate the need for DM and DE if we were able to solve the equations of GR for a lumpy universe.

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