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In 1915, Albert Einstein published a very important equation - no, not that one - the
one he published didn’t just relate mass and energy, but mass, energy and gravity - this
equation replaced the older “Newton’s law of Gravitation,” which you may be familiar
with, and it remains to this day our best description of how gravity works.
Just like how F=ma is a mathematical description of how the acceleration of an object depends
on the forces applied to it, the Einstein Equation of general relativity relates the
motion of mass and energy (the “T” on the right) to the curvature of spacetime (the
“R’s” on the left).
And Einstein didn’t just pull this equation out of thin air - it was the natural consequence
of a long and careful consideration of key principles of physics combined with the advanced
mathematics of curved surfaces, and of course, agreement with the experimental observations
of the day.
The equation, however, is deceptively simple.
This one single line is in fact an incredibly fancy shorthand for what’s actually a system
of ten second order partial differential equations relating mass and energy to the curvature
of spacetime, AND the the curvature R’s themselves are a shorthand for more, um, complex,
expressions.
But the point is this: after figuring out that these equations matched up with Newton’s
law of gravitation for weak gravitational fields and speeds much slower than light speed,
AND after showing that the equations correctly predicted a previously “unexplained-by-Newton’s-law”
anomaly in the orbit of Mercury, Einstein tried to figure out what the equations had
to say about the universe as a whole.
Of course, all the matter and energy in the universe is too complicated to put into the
equations and have any hope of solving them, but if you zoom out enough, you can approximate
the universe as having a roughly constant density everywhere, and in every direction.
And Einstein was able to solve the equations for a very simplified universe with constant
density everywhere - the ten complicated equations reduced to just two simple ones: this one
says the curvature of space in the universe is proportional to the density, so more stuff
in the universe means more curvature of space; and this one says that the density has to
be zero.
Which would mean there can’t be anything in the universe…
Needless to say, this was a problem.
And it turns out that there are two solutions to the problem - the one Einstein took, and
the one he didn’t.
Einstein’s solution was this: he knew (since he had dived deep into the math) that it was
possible to slightly change his equations; you can add a single very simple term without
violating any key principles of physics.
There wasn’t much other motivation for adding this term, but it doesn’t change anything
about how well the equations match up with Newton’s law when gravity is weak, or how
well they predict the orbit of Mercury, or whatever , so maybe it was ok?
AND, crucially for Einstein, the new term changes the equation for the density of the
universe: instead of saying “density equals zero,” it now says “density is proportional
to the new term”.
So if the new term was non-zero, that meant the universe could have stuff in it!
Voila - solution number one - Einstein’s solution.
The other solution to how the universe can have stuff in it was this: don’t assume
(as Einstein had) that the universe is static and unchanging.
The general understanding at the time was that the universe didn’t expand or contract,
and Einstein had also made a small but unfortunate technical error in his calculations which
appeared to prohibit the possibility of a changing universe, so it’s not surprising
that Einstein didn’t see this solution.
But it was there: if you don’t make the mathematical assumption that the universe
is static, and you don't make the technical error Einstein did, you can find a different
valid solution to Einstein’s equations.
Which physicist Alexander Friedmann did.
Actually he used the version of the equations with the new term, knowing he could always
set that term to zero if it wasn’t real.
But the key part is he didn’t assume the universe was static.
Friedmann found that the ten equations again reduced to two: the first equation now describes
how the change in density of the universe relates to its change in size: specifically,
it says that if the universe gets bigger, then it gets less dense, which makes sense
- stuff’s literally spreading out.
The second equation says that the deceleration of the universe is proportional to its density
minus Einstein’s constant; that is, the stuff in the universe attracts itself gravitationally
so the universe would have a tendency to pull inwards on itself, slowing any expansion and
possibly even contracting.
Unless Einstein’s constant were real and had a value big enough to balance or overpower
the gravitational attraction . So that's the solution Einstein didn't see.
Later, once astronomers took sufficiently detailed measurements, it turned out that
the universe WAS indeed expanding: distant galaxies are moving away from us, and from
each other - the universe is not static.
And the measurements indicated that the universe was expanding at a constant rate, at least
within experimental error bars.
So Einstein’s equations didn’t appear to have any need for the extra term he had
added.
Einstein was reported by physicist George Gamow to have called it “his biggest blunder”
- and while there’s no known documentation that he ever actually said or wrote those
words specifically, there’s plenty of record of him expressing disdain in other ways: “away
with the cosmological term,” “I always had a bad conscience,” “I found it very
ugly,” “such a constant appears…unjustified.”
And, during Einstein’s lifetime, that was certainly true - the term did appear unjustified.
However, remember how Friedmann’s equations predicted that the universe should be attracting
itself gravitationally and so the expansion should be slowing down, unless Einstein’s
constant is real?
Well, in 1998 , decades after Einstein’s death, astronomers made the surprising discovery
that the universe’s rate of expansion isn’t constant, and it ISN’T slowing down - it’s
getting faster.
And so in a great, ironic twist, Einstein’s constant does ultimately have a role in describing
the universe… though it turns out to be a very different universe from what he had
imagined.
If you don’t want to make silly math mistakes like Einstein, then you should probably head
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