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, to sharpen and hone your math and science

skills.

In fact, Brilliant has a whole interactive course on cosmology and within it, a quiz

specifically titled “The fate of the Universe” that was tailor-made for giving you a deeper

understanding than you can possibly gain from simply watching a video like this one.

Brilliant also has fun daily challenges, which are bite-sized math and science-puzzles - like

this one about what happens to a thermometer if you put it in space, and then rotate it.

Does it still read the same temperature?

Or hotter or colder?

Brilliant is offering 20% off of a premium subscription to the first 200 MinutePhysics

viewers to go to brilliant.org/minutephysics - that lets Brilliant know you came from here,

and gets you full access to all of Brilliant’s courses, puzzles, and daily challenges.

Again, that’s brilliant.org/minutephysics so that you don’t mess up like Einstein.