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 to Brilliant.org, this video’s sponsor,
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so that you don’t mess up like Einstein. Video source: https://www.youtube.com/watch?v=0RApKeMGDnE
