Einstein’s most revolutionary idea is that gravity is geometry. The theorems of geometry are not eternal truths, but can be changed by mass and energy.

This is my first longer explanation of ideas that are well established in science, but not well understood by the public. To keep this from becoming too long and overwhelming, I have split the six parts of this explanation into three blog posts. (I&II, III&IV, V&VI)

The two parts for today are about black holes and the experimental evidence of general relativity.

---

Prerequisites: My post Gravity is Geometry, Parts III & IV and the Pythagorean Theorem. I am serious about explaining the key ideas of general relativity without using calculus or anything more difficult. If you find something hard to understand, please ask me for more clarification.

Originally Written: October 2016.

Confidence Level: Established science since the early 1900s.

---

---

V. Black Holes

One of the more interesting questions that you can ask about general relativity is: What are the most ridiculous geometries that could happen in our universe?

Non-Euclidean geometry can describe many more shapes than Euclidean geometry can. Similarly, curved spacetime has a lot more weird possibilities than flat spacetime does. In flat space, if you travel in a straight line, you will never return to where you started. On a curved surface, for example on the surface of a doughnut, you definitely can. A similar idea in spacetime is a closed time-like curve. In some geometries, you can travel forward in time and eventually end up back at the same location and time that you started. Non-Euclidean geometries can also describe a surface with a tunnel, called a wormhole, connecting two points that otherwise would be widely separated.

There is no evidence that our universe can contain either closed time-like curves or wormholes. Wormholes require negative mass in order for them to stay open. We cannot produce negative mass. We also know of no mechanism that could create them. Closed time-like curves require assumptions about the universe that we also do not observe.

Don’t get too upset that our universe doesn’t appear to be as cool as general relativity could allow it to be. There are some truly strange geometries in our universe as well. The strangest geometry that we can observe in our universe is a black hole.

Suppose that you have a spherical object with mass $M$. What gravity does this produce? Since gravity is geometry, this question should be phrased as: What is the geometry around the mass?

Newton described the gravitational field outside of a spherical mass as: $$ \vec{g} = – G \frac{M}{r^2}, \ \mbox{pointing towards the mass} $$

$r$ is the distance from the measuring location to the mass and $G$ is a number we’ve measured.This equation forms the foundation of our understanding of the motion of the planets in our solar system and the stars in our galaxy.

Notice that this says nothing about geometry or about spacetime. There is no time in this equation. The Newtonian gravitational field measures the mass of and distance to the object instantaneously.

Schwarzschild was the first to solve Einstein’s Equations ($ G_{\mu \nu} = 4 \pi \, T_{\mu \nu} $, see Part IV for a reminder) to determine the Pythagorean Theorem outside of a spherical mass. The result is: $$ ds^2 = – \left(1 – \frac{2 G M}{c^2 r}\right) c^2 dt^2 + \frac{1}{\left(1 – \frac{2 G M}{c^2 r}\right)} \ dr^2 + r^2 d\theta^2 + r^2 \cos^2\theta \ d\phi^2 $$ There are a lot of interesting things to learn from this equation.

First, notice that the coefficients do not depend on time. The geometry that you observe will change if you move in the $r$ or $\theta$ direction, but not if you move forward in time. The geometry, though strange, is not changing.

Second, consider what happens if you set $ dt = dr = 0 $. If you do this, the equation looks identical to the Pythagorean theorem on the surface of a sphere (see Part II for a reminder). If you do not change the radius or the time, the Schwarzschild Pythagorean Theorem is the same as that of the surface of the sphere. This geometry is thus spherically symmetric, just like the mass which created it.

Third, if you look at the geodesics described by this geometry for large $r$, you will get something equivalent to Newton’s equation for the gravitational field. I won’t go through the details of how you do this, but it is important to point out that general relativity looks like Newtonian gravity if all of the masses are far apart and moving slowly compared to the speed of light. Otherwise, Newton would not have been able to use his theory to understand the solar system.

Now we can start looking at the truly weird consequences of this equation.

What happens if $ 2 G M = c^2 r $? The coefficient of $ dt^2 $ becomes zero and the coefficient of $ dr^2 $ becomes infinite. The Pythagorean Theorem at this point reduces to: $$ ds^2 = 0 \ dt^2 + \infty \ dr^2 + r^2 d\theta^2 + r^2 \cos^2 \theta \ d\phi^2 $$ Moving forward in time has no effect on the spacetime interval. Changing the radius changes the spacetime interval by an infinite amount. The location where this happens is called the Schwarzschild radius: $$ R_S = \frac{2 G M}{c^2} $$

Now let’s look at even smaller radii.

Recall from Part III that the difference between time and space in the Pythagorean Theorem is that the spatial coordinates have positive coefficients and time has a negative coefficient. For $r > R_S$, this same pattern holds. However, for $r < R_S$, the coefficient in front of $dt^2$ becomes positive and the coefficient in front of $dr^2$ becomes negative. It is as though time and radius have switched roles inside of the Schwarzschild radius. The inexorable passage of time is replaced by an inexorable motion towards $r = 0$.

What happens when $r = 0$? Once again, we get singular coefficients. Now, the coefficient of $dt$ is infinite and the coefficients of the spatial coordinates are zero. $$ ds^2 = \infty \ dt^2 + 0 \ dr^2 + 0 \ d\theta^2 + 0 \ d\phi^2 $$ Here, changing any of the spatial coordinates has no impact on the spacetime interval and changing time changes the spacetime interval by an infinite amount.

If having $\infty$ in the Pythagorean Theorem seems like a problem to you, you’re right. Something extremely weird happens at both of these singularities.

The $\infty$ at the Schwarzschild radius can be removed by writing the geometry in terms of different coordinates. Even in these coordinates, something strange happens here. Typically, if you stay at one spatial location and move forward in time, the spacetime interval is time-like. At the Schwarzschild radius, moving forward in time is a light-like spacetime interval, with $ds^2 = 0$. Inside the Schwarzschild radius, regardless of the geometry, staying at a fixed location and moving forward in time is a space-like interval, with $ds^2 < 0$.

The $\infty$ at $r=0$ occurs regardless of what coordinates you use to describe this geometry. The mass there is infinitely dense.

Does this geometry even make sense?

Recall that Schwarzschild’s solution only applied outside of the spherical mass. Inside the mass, the geometry would be different. We only have to worry about the Schwarzschild geometry at some location if this location is outside of the mass itself. If we plug in numbers for the sun, for example, we find that its Schwarzschild radius is about 5 kilometers. The sun’s radius is much larger than this, so we don’t have to worry about this weird geometry. The strange geometry at and inside of the Schwarzschild radius is only relevant for the densest objects in the universe.

It is possible for something to get that dense. When the largest stars run out of fuel and explode, they can create enough pressure at the center to force more mass than the sun into a radius of less than a few kilometers. Once this happens, a black hole forms. Anything inside the Schwarzschild radius will continue to move towards $r = 0$ as surely as it would move forward in time if it remained outside of the black hole. The Schwarzschild radius of such an object is called its event horizon. Escaping from the event horizon of a black hole is just as difficult as traveling backward in time.

---

VI. Experimental Evidence of General Relativity

Why do we believe that general relativity is a good model of reality?

‘Einstein said so’ is not a good enough reason to believe anything. Instead, the evidence for general relativity should be based in observations.

Gravitational Lensing

In the Newtonian model, gravity is a force which acts on all objects that have mass. Light doesn’t have mass and always moves at the same speed. It cannot experience an acceleration, so it must always travel in a straight line through space.

In Einstein’s model, gravity is the result of curved spacetime. Straight lines no longer exist. Instead, objects move along geodesics unless acted on by an outside force. Light doesn’t have any forces acting on it, so it should also move along (curved) geodesics. General relativity predicts that the path light takes is curved by gravity. The bending of the path light takes due to gravity is called gravitational lensing.

Gravitational lensing was the first experimental test of general relativity. In 1919, shortly after Einstein published general relativity, there was a solar eclipse in the South Atlantic Ocean. During the solar eclipse, stars would be visible whose light passed close to the sun on its way to earth. If general relativity were correct, then the path that light takes would be bent by the sun’s gravity and the stars would appear in slightly different locations than when they are visible at night. Eddington led an expedition from England to go observe this solar eclipse and measure if gravitational lensing had occurred. He not only observed that the path light takes is bent when it passes near the sun, but also that it is bent by the amount predicted by general relativity.

Gravitational lensing has been observed many times since then. Some of the most dramatic examples of gravitational lensing occur when the path that light takes is substantially bent by galaxies or clusters of galaxies. This magnifies whatever is behind them and can even produce multiple images of the same, more distant galaxy.

Indirect Observations of Black Holes

Black holes cannot exist if you assume that Newtonian gravity is correct. Most astronomical objects (planets, stars, galaxies, nebulae, etc.) exist primarily as lumps of matter. The warped geometry around them is less apparent to observers than the matter itself. Blacks holes exist primarily as extremely warped geometry. You cannot detect any matter, or anything else, inside. All you can see is the effect of the warped geometry on other nearby objects. Any observations which indicate the existence of black holes provide evidence that general relativity is correct.

There are two places that we have seen indirect observations of black holes.

Our galaxy has a black hole whose mass is a few million times the mass of the sun at its center. We can measure its mass by looking at the orbits of stars going around it and get an upper bound on its size by noticing that it is too small to appear on our best cameras. Video of the orbits of stars going around our galaxy’s central black hole can be seen here. The only compact object that we know of that could be that heavy is a black hole. Anything else would collapse under the weight of its own gravity. Astronomers suspect that there is a supermassive black hole like this one at the center of every galaxy.

Although we cannot directly measure light from a black hole, we can measure light from matter falling into the black hole. If there is anything currently falling into the black hole, that matter will move extremely quickly, heat up, and emit xrays. We can observe this process within our solar system. Black holes with mass a few times the mass of the sun sometimes have another star nearby. As the black hole and the star orbit each other, the black hole pulls some matter off of the star. We have also seen matter falling into supermassive black holes in other galaxies. In the early universe, supermassive black holes could consume a few solar masses of matter a year. The mass falling into them creates extremely bright objects called quasars, an abbreviation of ‘quasi-stellar.’ These objects are bright enough to be confused with stars in our galaxy, but are actually the material falling onto a supermassive black hole in a galaxy billions of light years away.

Gravitational Waves

Thus far, we have spoken mostly about static geometries, i.e. geometries which do not change in time. The coefficients of the terms in the Pythagorean Theorem could depend on time as well as on location. This is unavoidable if the masses which cause the warped geometry are moving. The locations of the masses is changing in time, so the geometry must change in time as well to continue to satisfy Einstein’s equations.

When the geometry changes in response to a moving mass, it cannot change everywhere simultaneously. As we learned in Part III, nothing can travel faster than the speed of light. Even changes in geometry cannot travel faster than the speed of light. Instead, moving masses cause changes to the laws of geometry close to them and the changes then propagate outward at the speed of light. These propagating changes to the laws of geometry are called gravitational waves.

Gravitational waves are extremely hard to measure. To measure them, you have to detect changes to the Pythagorean Theorem. We (scientists) have built two gravitational wave detectors, called LIGO (Laser Interferometer Gravitational-wave Observatory), one in Livingston, Louisiana and one in Hanford, Washington.

• 

• 

Each detector consists of two long vacuum chambers oriented at a right angles with respect to one another. Each vacuum chamber is 4 km long. Lasers shine through the tubes and reflect off of mirrors at each end. The lasers are used to measure the difference between the lengths of the two tubes. As a gravitational wave goes by, the Pythagorean Theorem changes slightly, changing what the lasers measure the difference in length to be. LIGO has ridiculously high precision. It can observe if the length of the tube changes by a hundredth of the diameter of a proton.

Most of the objects that we experience are neither massive enough nor moving quickly enough to produce substantial gravitational waves. Only the most dramatic events in the universe produce substantial gravitational waves: collisions / mergers of black holes. We also might be able to detect slightly less dramatic events like supernova (the explosions at the end of the most massive stars’ life) or collisions between neutron stars (objects about the mass of the sun which consist entirely of a single atomic nucleus).^[1]Since I wrote this, we have observed collisions involving neutron stars too.

LIGO began operating with its current capacity in September 2015. LIGO detected its first gravitational waves on September 14ᵗʰ, 2015. The results were published in February 2016 and can be found here. Since I don’t expect you to be able to read current physics papers, I will guide you through the important parts of the paper.

First, take a look at Figure 1 of the paper, part of which is reproduced here. The top two figures show the data from the two detectors. The left plot is from the detector in Washington. The right plot is from the detector in Louisiana. The wave arrived in Washington 7 milliseconds after it arrived in Louisiana, so the data from Louisiana is shifted by that amount to make the waveforms line up. There is clearly a wave in both pictures that is visible for about 0.2 seconds. The waves visible in Washington and Louisiana are almost identical, despite being thousands of kilometers apart. The bottom two figures show theoretical models of what a gravitational wave would look like. The data match the predictions made using general relativity !

Ernest Rutherford once said: “If your experiment requires statistics, then you should devise a better experiment.” The data shown here are a excellent example of a scientific discovery whose results are clear enough to be visible by glancing at the data, without any statistical analysis.

Scientists at LIGO were able to use the data to extract information about the black holes involved in the collision. The two original black holes had masses 36 times the mass of the sun and 29 times the mass of the sun. The final black hole had mass 62 times the mass of the sun. These numbers don’t look like they add up: the final black hole’s mass is less than the sum of the initial black holes’ mass. The missing mass was turned into energy and radiated away as gravitational waves. Three times the mass of the sun was turned into energy and radiated as gravitational waves in 0.2 seconds. This black hole merger was 50 times more luminous than the light from the entire visible universe for the 0.2 seconds that it was happening. The black hole merger took place 1.4 billion light years away from Earth.

This observation was the first direct measurement of gravitational waves, fulfilling a prediction that Einstein made using general relativity about a hundred years ago. This is the first time anyone has ever observed the interaction between two black holes. Previously, we have only been able to observe the interaction between a black hole and a visible object like a star. This is also the first time we have observed an intermediate-mass black hole. Up to now, only two kinds of black holes have been seen: ones $< 10$ times the mass of the sun produced by the explosion of a massive star and ones at the centers of galaxies which are $> 100,000$ times the mass of the sun. Presumably, intermediate mass black holes had to have existed, and they probably still exist, but they have never been seen close to a visible object. Even after this first detection, there is still a wide range of intermediate mass black holes which have yet to be observed. This combination of first times easily makes this event one of the most important scientific discoveries that will happen during our lifetimes.

LIGO didn’t shut off once it made its first observation. LIGO is not just a test of general relativity. It is also an observatory. It continues to observe the gravitational waves from black hole mergers or other extremely violent events. Several detectors which are similar to LIGO are being built or upgraded in Germany, Italy, Japan, and India. When more of the detectors are on, we will be able to better determine what direction the gravitational waves came from.

A second black hole merger was measured on December 25ᵗʰ, 2016. This merger was about the same distance away, but the black holes involved were smaller. The two initial black holes had masses of 14 times the mass of the sun and 8 times the mass of the sun. The final black hole was 21 times the mass of the sun, with one solar mass of energy radiated as gravitational waves.

By observing many black hole mergers in the years to come, LIGO gives us a new tool to do astronomy with. Before, we could only look at light to see what we can see in the cosmos. Now, we are also starting to use gravitational waves to observe astronomical events that we could never observe using light.