Schwarzschild, Master of the Universe

When Einstein introduced his theory of relativity in 1915, there weren’t many people who had the requisite background to understand it.

The geometry used to describe his theory, manifolds and metrics and geodesics and so on, were relatively new.  Riemann didn’t give his foundational lecture “On the foundations of geometry” till 1854.

Most physicists didn’t know or understand this new kind of geometry.1

But there were a few who did. One of them was Karl Schwarzschild. (It’s German, so something like “SHVARTZ-shield.”)

schwartzchild

In Einstein’s original papers on relativity2, Einstein never writes out an explicit solution to his own equations. Instead, Einstein used an approximate solution to derive the light bending effect, among other effects.

But within a few months of Einstein’s publication, Schwarzschild sent Einstein a letter containing the first exact solution to the Einstein equations.3

A bit of history for Schwarzschild is needed to really appreciate this.

Karl Schwarzschild was born in 1873 in Frankfurt as a German Jew.

german

At that time, fortunately, being a Jew wasn’t so much a problem.

He was a bit of a genius, publishing two scientific papers before the age of 16. He got his doctorate in 1896.

Schwarzschild was very successful in his career. He was a professor at a major university, where he got to work with Hilbert and Minkowski. For a while he was the director of the observatory at that university, before he got the most prestigious job for an astronomer in Germany, the directorship of the Astrophysical Observatory in Potsdam.

In 1914, World War I broke out.

At that time, antisemitism was on the rise. Schwarzschild was proud of his country, and wanted to prove that a Jew could be a good German.

So he joined the army.

endbadly

He did well in the war, rising to the rank of lieutenant in the artillery. (Mathematics, it turns out, is useful in aiming those things.)

However, in 1915, he contracted a horrible skin disease called pemphigus.4

That year, as you recall, was when Einstein finally published his theory of relativity.

Schwarzschild, stuck in a hospital near the Russian front, read Einstein’s work, understood it, and produced the first exact solution to the theory.

In the letter Schwarzschild wrote to Einstein, he concluded by saying, “As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas.”5

He died within six months.

rip

I think we can all learn something from Schwarzschild’s view on life.

mathissobeautiful

Now, let’s get back to Schwarzschild’s solution.

Schwarzschild’s solution is supposed to model gravity outside of a star, or a similar large, spherical mass. As this models the gravity around our sun, or around the Earth, this is a particuarly basic and important example of gravity.

In order to find a useful, but still exact, solution to Einstein’s equations, Schwarzschild made a number of simplifying assumptions.6

One of the most basic and important instances of gravity is the gravity around the sun. While there are planets and moons and asteroids and such, the sun is by far the biggest contributor to gravity.7 So, Schwarzschild’s first simplifying assumption was that there was no mass or energy except for the sun. And, in fact, he ignored the sun and just assumed there was no mass anywhere.8

We call this kind of spacetime a vacuum spactime, relating it to the vacuum of space.

vacuumofspace
It’s a picture of the vacuum of space, I promise.

In terms of the Einstein equations, R_{ij} - \frac12 R g_{ij} = T_{ij}, that means that the stress-energy tensor T_{ij} is exactly zero. Using some geometry formulas, you can show that we can simplify the equations to just R_{ij} = 0. Thus, the spacetime has to have no Ricci curvature, i.e., be Ricci flat.

With four spacetime dimensions, Ricci flat does not mean that there is no curvature, but simply that the curvature is particularly simple.9

Next, Schwarzschild assumed the sun (or whatever the mass is) was perfectly spherical, which is true to about one part in 10^5. That’s close enough, and, mathematically, greatly simplifies the analysis.

sphericalsun

This assumption means that if we look at a sphere a certain distance around the center, the metric, which measures gravity, should be the same everywhere on the sphere. In other words, the spacetime is spherically-symmetric in the space directions.

Finally, Schwarzschild assumed that nothing in the spacetime was changing. As we expect the sun to be essentially the same as it is now for the next billion years, this seems reasonable. In other words, he assumed that the spacetime is static.

dayover4billion

So, Schwarzschild was looking for a vacuum, spherically-symmetric, static spacetime.

helloanyone

The reason that Einstein didn’t think it would be possible to find exact solutions was that the Einstein equations are a giant system of 16 nonlinear partial differential equations. Usually, finding an exact solution for one linear partial differential equation is hard…

However, Schwarzschild’s three assumptions are really restrictive. Combined, they reduce 16 difficult differential equations, to a single simple one. Solving that one equation, we can find an exact spacetime solution.

In spherical spacetime coordinates (like we used last time to describe special relativity), Schwarzschild’s metric is ds^2 = -\left(1-\dfrac{2M}{r}\right)dt^2 + \left(1-\dfrac{2M}{r}\right)^{-1} dr^2 + r^2 (d\phi^2 + \sin^2(\phi)d\theta^2).

The letter M is a constant representing the mass of the object in the middle, say the sun.10

Which is weird, since the spacetime is vacuum…

But remember, the Schwarzschild spacetime represents the spacetime outside the star in the middle. How much the spacetime bends far away depends on how much mass there is in the middle, even though there might be no matter at that point of the spacetime.

It turns out this is the only vacuum, spherically-symmetric, static spacetime.11 It’s also true, though harder to prove, that you don’t actually need to assume the spacetime is static.12 In other words, the Schwarzschild spacetime is the only vacuum, spherically-symmetric spacetime.

onlyone

Let’s compare this metric to the similar metric of special relativity, also in spherical spacetime coordinates.

To remind you, the metric for special relativity is ds^2 = -dt^2 + dr^2 + r^2 (d\phi^2 + \sin^2(\phi)d\theta^2).

For large r, i.e., far away from the sun, 1-2M/r is close to 1, and so the two metrics are very similar. This makes sense; far away from the sun, there shouldn’t be much gravity. Since special relativity is the case where there is no gravity, that means the two metrics should be about the same.

On the other hand, when r is smaller, the two metrics differ quite a lot. Near the star, gravity is compressing the spacetime.

obesegravity

The Schwarzschild spacetime does an excellent job at modeling our solar system.

For instance, in the early 1900’s, it was noticed that Mercury’s orbit was not quite right.

Planetary orbits are essentially ellipses. Given Newtonian gravity, and ignoring the other planets, the orbits are exactly ellipses.

elliptical

However, each planet’s mass slightly alters the orbits of the other plants. The main effect of the other planets is that the orbital ellipse precesses, or rotates, around the sun.

precessed

With some calculations using Newtonian gravity, you can calculate this effect. It’s small, but noticable. And (once we discovered all the planets) the observations agreed with the calculations.

Except for Mercury.

Mercury’s orbit precessed just a bit more than calculated. For a while, it was thought there was another planet even closer to the sun that was effecting Mercury’s orbit, but this hypothetical “Vulcan” was never observed.13

However, using the Schwarzschild spacetime, if you calculate an orbit, you find that relativity causes the orbital ellipse to precess on its own. And the amount of precession is just enough to explain Mercury’s precession.14

canttouchthis

However, while this spacetime works well, mathematically there’s something problematic with it.

If r is either 0 or 2M, you divide by zero!

Of course, this spacetime was not originally meant to be a model for radii that small. For instance, the radius of the Sun is 696 thousand kilometers, while the radius 2M is (in the correct, geometrized units) a mere 3 km.

Since the sun isn’t a vacuum[citation needed], the Schwarzschild spacetime isn’t valid inside the sun. It only models the spacetime outside the sun.

We could just ignore that part of the spacetime, and say it’s “obviously” not valid.

But what if there was something so compact, so dense, that all of its mass fit within a radius of 2M? What kind of object would that be?

mystery

That, my friends, is a black hole.

gaspblackholeshearts

Of course, for many years, no one seriously worried about what happened at this radius, 2M, sometimes called the Schwarzschild radius.

But we now understand that this radius is the boundary of a black hole, a region where gravity is so intense, and spacetime bent so much, that not even light, the fastest of all things, can escape.

woah

But that’ll have to wait till next time.

nooooooo


  1. This is also why it took Einstein so many years to figure out the right equations. To quote the Wikipedia page History of general relativity, this led to the following amusing story. (It should be true, even if it isn’t.) As related by Ludwik Silberstein, during one of Eddington’s lectures he asked “Professor Eddington, you must be one of three persons in the world who understands general relativity.” Eddington paused, unable to answer. Silberstein continued “Don’t be modest, Eddington!” Finally, Eddington replied “On the contrary, I’m trying to think who the third person is.” 
  2. I’m going to stop constantly saying “general” relativity, unless I need to differentiate. 
  3. Well, other than the spacetime of special relativity, of course. 
  4. Here’s a link to the Wikipedia page, but, as a warning, it’s pretty gross. Schwarzschild was not a lucky man… 
  5.  Letter from K Schwarzschild to A Einstein dated 22 December 1915, in “The Collected Papers of Albert Einstein”, vol.8a, doc.#169 
  6. Though no spherical cows. 
  7. The sun contains somewhere between 99.8 and 99.9 percent of the mass in our solar system! 
  8. Don’t worry. While Schwarzschild’s first spacetime was empty, he soon discovered a solution that also included a simple matter model for the star. 
  9. Interestingly, if spacetime were three-dimensional, Ricci curvature is all the curvature there is, so Ricci flat would mean actually flat. So a vacuum spacetime would (locally) have to look like (three-dimensional) special relativity! 
  10. The mass is measured in geometric units. So, that means the speed of light is 1 (unitless), like we did for special relativity. But we also set the gravitational constant G to 1. In these units, the mass of the sun is 1.5 kilometers. (Yeah, kilometers is a unit of mass in geometric units…) 
  11. Of course, since changing coordinates doesn’t actually change the spacetime, you might come up with a seemingly different vacuum, spherically-symmetric, static spacetime. But, if you do, there is going to be some change of coordinates that shows that it’s really the same as the Schwarzschild spacetime. 
  12. This is called Birkhoff’s theorem. 
  13. This was actually a clever thing to do. In fact, a similar mathematical effort due to inconsistencies in the orbit of Uranus led to the discovery of Neptune. 
  14. Einstein actually calculated this effect using an approximate solution, but you can also use the Schwarzschild spacetime. You can read a bit more about this effect on Wikipedia
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4 thoughts on “Schwarzschild, Master of the Universe”

    1. I saw the cool sphere thing on top of his tomb with stars on it. Plus the 2 year old child he left behind. (Super sad…) Were you referring to one of those things?

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      1. If I had to guess, it’d just represent the “celestial sphere.” He was, after all, the head of an observatory. And black holes weren’t really a thing yet in 1950, when the last name was put on that stone.

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