Saturday, December 20, 2008

Earth Stands Still

A hat-tip to the wondrous Kea, who will soon be working at Oxford. In the new version of DAY THE EARTH STOOD STILL, Klaatu is deeply concerned about he mess humans have made on the planet. Again he comes across the scientist's blackboard filled with our primitive math. He begins by crossing out the "lambda" term in Einstein's field equation, which is a useless contrivance. Einstein himself called this "cosmic constant" his greatest blunder. Without it, Einstein might have predicted expansion of the Universe.

Now the equation looks like:

Ruv-½guvR=-κTuv

Klaatu then writes in the solutions of his more advanced species. Today, as was done 2 years ago, we can guess at what he wrote:




Today's physicists, trained for years in our primitive science, may raise petty objections when someone alters their equations. On the question of units, the Friedmann equations are valid in units of mass density or energy density.

The stress-energy tensor T_uv can have units of mass density ρ or energy density ρ(c^2). Removing for a moment the c^2 from Friedmann we would have:
8πGρ/3 = ⅓κρ (c^2)
4πGρ/3 = ⅙κρ (c^2)

To normalise the left-hand and right-hand sides, some physicists chose κ=8πG/(c^2). This led to decades of misconception that Relativity requires a fixed c. Some "geometrized" unit systems give κ=8πG/(c^4), which is too convoluted to describe.

If T_uv has units of energy density, we must use the same for Friedmann:
8πGρ(c^2)/3 = ⅓κρ(c^2)
4πGρ(c^2)/3 = ⅙κρ(c^2)
κ=8πG
This is very trivial. The (c^2) on both sides simply cancels out. Einstein called constant κ "related to the gravitational constant" without mentioning c.

Now the Einstein equation becomes Ruv-½guvR=8πGTuv. The Bianchi identities become:
▽u(Ruv-½guvR)=0
8πG▽u(Tuv)=0
The world is much simpler without that pesky (c^2) factor.

Finally, the Einstein-Hilbert action becomes:
S=∫(16πGR+Lm)d^4(x)
Thus we can do everything General Relativity can without a fixed c. Some problems, like the deflection of bodies by the Sun, work even better with a varying c.

In reality, a human professor would throw Klaatu out for messing with his blackboard. A Universe that can be described in a few equations just might be beyond human understanding. Humans tend to complicate their Universe with epicycles, luminiferous ether, or cosmic constants. Are humans ready for new physics? Perhaps we should ask Klaatu.

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1 Comments:

Blogger qraal said...

Hi Louise

Just saw "Day the Earth Stood Still" on Boxing Day here in Oz. Pretty cool. Lots of astrobiology and SETI subtleties that the usual punters might've missed, and the honorable (but misguided) military was a reminder of a current foolish war. Much to my delight Cleese's character actually portrayed the gravitas needed for the job of "Earth's finest". Being a Ben Bova fan, the reference to "the Precipice" had interesting resonances for me. A reference to VSL would've been icing on the cake ;-)

2:22 PM  

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