Outsmarting Einstein From The Trenches

The true story of one of the greatest minds of the last century.

Even if you don’t consider yourself as an huge fan of physics, you’ve certainly heard of Albert Einstein.

He is undoubtedly the most famous physicist who ever existed and he truly deserve all that fame. His contributions to science come from so many different area of physics that it’s almost unbelievable.

Mainly, he is known for postulating the theory of General Relativity, probably the most popular idea from all of physics, describing with precision the true nature of spacetime and the effects gravity has on it.

Safe to say, outsmarting Einstein is the ultimate intellectual flex—a feat so rare that only a handful of people in history can actually claim it.

Karl Schwarzschild is without a doubt one of them.

This is the story that earned him his well deserved spot in the history of physics.

1. The impossible challenge

The year is 1915 and Einstein just shared with the world his famous field equations:

\( R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}= \frac{8\pi G}{c^4} T_{\mu\nu} \tag{1}\)

To this day, this remains one of the most remarkable result of theoretical physics.

On the left side of the equations we have objects describing the geometry of space, its deformation or basically how bent and stretched it is. Mainly the so called Ricci and curvature tensors (the two R’s terms in order).

On the right side instead we have the so called energy-stress tensor (T) which instead describes the motion and structure of mass (multiplied by a series of constants).

This equation is the true heart of General Relativity and where all of its elegance resides.

With just a line of weird symbols, Einstein managed to give meaning to gravity which changed from a weird and mysterious invisible force to a true tangible effect of spacetime deformations.

A good way to visualize it is the following:

A representation of the curvature of space time made by ESA.

A 2d spacetime could be seen as a sheet of some sort like a blanket or similar.

If you put an heavy object like a bowling ball on one side of the tensed sheet, it will create some kind of deep deformation but if you do the same with a snooker ball the deformation will be far smaller and if the sheet you’re holding is not that big, the smaller mass will “fall” in the deformation generated by the bowling ball.

That’s gravity!

Now, as for any equation, we can distinguish between values and the unknown quantity.

You know, if you have to solve:

\(x + 2 = 3\)

You are just asking yourself: “What number added to 2 gives me three?” and that’s clearly 1 so the solution is x = 1.

In Einstein’s case, the unknown quantity was:

\(g_{\mu\nu} \tag{2}\)

Also known as the metric of spacetime.

This object is basically the “map” of spacetime, it tells us the true deformed distance between two objects and how time actually flows due to gravity.

Basically, it’s a pretty big deal.

Now, the Einstein field equations are actually very hard to solve.

Einstein himself could only get “approximations” of solutions and was pretty much convinced that there was no way to get a complete and accurate result for the metric.

It was simply a problem to hard to solve that would have requested years if not decades of work from scientists all over the world.

2. The man who solved the impossible

Karl Schwarzschild (1910 ca.)

Karl Schwarzschild, born in 1873, always showed a strong predisposition towards anything physics or mathematics related.

At seventeen years old he started publishing his first works on the German scientific journal Astronomische Nachrichten regarding the orbits of binary stars systems.

He quickly entered the world of academia starting his PhD in astronomy in 1896 at the Munich University and the next year he was also hired as assistant at the Kuffner Observatory near Vienna.

In 1907 he developed a specific method for astrophotography, which was later labeled as Schwarzschild effect, that quickly became fundamental to classify celestial objects based on their observable quantities. He chose to never file the patent for it.

Also, just because he had some free time to spare, he managed to lay down the basis of quantum mechanics at the same time of Niels Böhr (widely regarded as one of the founding fathers of the same theory).

But, as well as having possibly one of the most complicated surnames in the history of physics (Chandrasekar is definitely a close second), his greatest accomplishment was still yet to come.

In 1914, at the beginning of World War I, Schwarzschild enlisted to fight and was sent to the second lines of the western front, in occupied Belgium, where he directed a meteorological station.

The next year, he was then transferred to the eastern front where he was assigned to artillery to compute difficult ballistic calculations for his regiment.

And it was here — immersed in death, fear, mud and gunpowder — that he managed to solve the impossible.

3. The impossible solution

On November 25, 1915 — Einstein published his field equations, claiming that a solution to describe the curvature of space time was basically impossible to find and it would have requested years and collaboration from all over the world to get something out of it.

December 22, 1915 — not even a month later, Albert Einstein received a letter from the eastern Russian front, written by his friend and collaborator Karl Schwarzschild.

In it, an elegant solution to his equation:

\( ds^2 = -\left(1 - \frac{2GM}{c^2r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \tag{3} \)

In less than a month, while actively fighting on the first lines of the most brutal war ever fought by men, Scwharzschild was able to find a solution to a problem deemed impossible by Albert Einstein himself.

In his letter, he showed the simplest possible solution to the equation, the one describing the outside of a star, where only space time is present and no relevant masses.

To do so, he simply put the energy stress tensor describing mass (the T in equation 1) equal to zero and simplified greatly the equations.

With it, he managed to find the impact a simple and super common object like a star has on the structure of spacetime. It showed one of the most important applications of general relativity: time dilation.

The closer we get to a massive object the slower time passes for us.

Gravity influences time.

Which means that basically time passes slower for your feet with respect than your head because they are usually closer the Earth (of course it’s an insignificant difference that has no real consequence on us, but it’s there nonetheless).

4. Impossibly dangerous consequences

So heavy masses actually bend spacetime, but is there a limit to it?

Is there a place of no return for space and time?

Does such a massive objects exist?

That was the scary question Schwarzschild had to face.

While looking at the worst side of mankind, he faced the abyss for too long until it looked back at him in the form of a Black Hole.

A “simulation” made by astrophysicist Jean Pierre Luminet in 1979. At the time, it was still considered an heavy theoretical object.

A mysterious object, so heavy that not even light can escape from it.

So heavy that spacetime is twisted to the point that your perception of time freezes if you get too close to it.

An anomaly, an aberration, a singularity.

A discovery way ahead of his time.

A prediction so accurate that it seemed almost impossible.

A result worthy of humanity’s highest aspirations, born from the mud and blood of its lowest hour.