# Relativity: The Special and the General Theory (Albert Einstein, 1916)

This book was written by Einstein just one year after having published his paper about the general theory of relativity (1915) and 11 years after having published his theory of special relativity (1905). The goal of the book is to explain in simple terms his theories for people who don't have a strong background in maths/physics, but still some basic notions[1]. It's a relatively small book (180 pages). I read the French translation which was published in 1956.

[1] The fourth cover of my French edition recommends a level of *baccalauréat*, which is the French exam to validate high school (I assume with a scientific specialty).

I struggled to understand many of the concepts in the book. Particularly, I failed to get a holistic view of the general theory of relativity, which is significantly more advanced, conceptually and mathematically, than the special one. I guess I shouldn't shy too much from my apparent lack of intelligence, as my understanding (from Wikipedia) is that the general theory of relativity wasn't well understood by the scientific community itself before decades after this book was published.

Still, I found that there was value in the book because of the following: understanding a gross outline of what those theories are and how they work; going through a lot of funny thoughts experiments, which Einstein seemed to love; peeking into Einstein's intellectual process in finding the theories.

## Outline of the theories

### Special relativity

The principle of relativity stipulates that the laws of Nature should work exactly the same in different frames of reference. For example it shouldn't matter whether you're at a fixed point on the ground or traveling on a train, from your point of view, everything you observe should behave according to the same laws.

One of those laws is that light travels at constant speed in a void.

This immediately presents a paradox, because if you observe a ray of light traveling at speed `c`

from the ground, then someone traveling in a train in the same direction than this ray of light is expected to measure its speed as `c`

*minus the speed of the train*, which would mean that either the principle of relativity doesn't hold, either the constancy of the speed of light isn't a fundamental law of Nature.

Since none of those options are satisfactory, Einstein prefers altering the notion of speed itself: the special theory of relativity stipulates that *distances* and *time* aren't the same in the different frames of reference. There even is an exact formula to convert them from one frame to another: the Lorentz transformation.

When you put the speed of light in those transformations, you realize it stays constant, which means that this is a model in which both the principle of relativity and the constancy of speed of light hold, so it's a good model, I guess.

### General relativity

The problem with everything above is that it only works with *inertial* frames of reference, which means frames of reference that move at constant speed in the same direction; they don't accelerate and they don't rotate on themselves. This is what is "special" in the special theory of relativity: it only solves the problem for a very specific kind of frames of reference.

The general theory of relativity, as you can guess, tries to make the principle of relativity work for all kinds of frames of reference, no matter whether they accelerate, rotate, jump around, dance the mambo, or anything else. This is where shit gets serious.

If the train you're in starts braking (it isn't an inertial frame of reference anymore), then from your point of view there is a force pulling you to the front of the train (one of the so-called "inertial forces" when you deal with non-inertial frames of reference in Newtonian mechanics). According to the principle of relativity, the laws of Nature should be the same in all frames of reference, which means that this force should be a fundamental manifestation of Nature. Well, Einstein decides it's ✨ gravity ✨

Now, this is where my understand starts being blurry: I'm not sure we are supposed to interpret this as if actual real gravity appeared out of nowhere in the train to pull you forward. Rather, it should mean that from your point of view in the train, there is no observable difference between assuming you are pulled by an inertial force or pulled by gravity. This is the equivalence principle.

Things get interesting when you mix the equivalence principle with light. Imagine you are in a spaceship that is accelerating in space. Since it is accelerating, you are pushed to the bottom of the ship, which, according to the principle of equivalence, is exactly the same as living in a gravity of some amount. Now, imagine that a straight ray of light is passing through a window of the ship (we assume no refraction). Viewed from space, the ray is straight, but viewed from the interior of the ship, it is curved (since the ship is accelerating). Well, this is where the principle of equivalence starts getting real: it is observed that when a ray of light passes by a massive object generating actual gravity, it is curved, in the exact same way than in the accelerating spaceship.

We can also apply the principle of equivalence to time. Imagine a carousel on which you put 2 clocks: one at the center, and one at the edge. From the frame of reference of the carousel, the clock at the center is not experiencing any particular force. However, the clock at the edge is experiencing some force that is trying to pull it away from the carousel (the centrifugal force). According to the equivalence principle, this means that the clock at the edge is subjected to some form of gravity. Furthermore, because of the principles of special relativity, since the clock at the edge is moving but not the clock at the center, the two clocks aren't experiencing the same *time*. This means that there is a link between gravity and time. Again, the principle of equivalence really works, since massive bodies, which generate a lot of gravity, alter the flow of time, exactly in the way that would be expected on this carousel.

Beyond those fine examples, the book goes on to explain how the entire general theory of relativity holds together (the way the Lorentz transformations make the special relativity hold together), at which point I failed to understand most of what I was reading.

## Experimental verification

From what I've read, the theory has been confirmed again and again by a multitude of experiments throughout the XXth century, but here I'm listing the two early confirmations that are given in the book:

- It can be measured that Mercury exact orbit around the Sun doesn't obey Newtonian mechanics, but obeys relativist mechanics.
- It can be measured that the Sun deflects the light from distant stars, deflection which is explained by relativity.

Einstein also describes a third consequence of the theory: gravitational redshift (which I didn't take the time to try to understand quite well), but he explains in the book that astronomers were still working hard to try to actually observe it (it was first observed in 1925, nine years later).

## Einstein intellectual process

Since the book is written in bottom-up way, and early after the publication of the theories, it sorts of opens a window into Einstein's mind on how he came up with those theories. Einstein usually starts by introducing a clever thought experiment that showcases some paradoxical stuff about Nature, and then, using some form of intuitive judgment of how it could make sense, he works his way up to a theoretical idea.

I found it refreshing how much of a *physics* book it is. When I studied physics in *prépa*, it was all formulas and diagrams and it was very easy to forget that you're supposed to be reasoning about the real world and not some arbitrary abstraction. Einstein, to the contrary, never let physics go out of sight.

What is surprising, however, is that since the manifestations of the theory only happen in extremes conditions (speeds close to the speed of light, extremely massive objects), he had very little actual physics to depend on, but still succeeded in developing the theory to a high level of sophistication, adding details and texture to it, always depending on his thought experiments. This reminded me of the way Waymo or Tesla are training their autonomous cars in software simulations more than on real roads. The man didn't have access to actual physics, so he fabricated the theory by simulating the Universe in his damn mind.