**Critical Questions:**

- What can we say about General Relativity without using math?
- What does ‘relativity’ mean?

Before I even say one word about General Relativity, I feel obliged to issue a stern warning: prepare to be frustrated.

You see, most of the rest of the physics you’ll see on this site originated within one or two hundred years of Isaac Newton and the invention of calculus. This means that although there is some quite difficult mathematics behind it, most of it is based on direct observation and can be understood from a conceptual standpoint without worrying too much about the math.

But by the time Einstein came along, the field of mathematics had made some significant progress. All of the physics theories loosely called ‘Modern Physics’ (the Theories of Special and General Relativity and Quantum Mechanics being the main elements) involve lots and lots of extremely difficult math, and General Relativity is no exception. Read almost anything on the subject, and you’ll only need to encounter a few words like ‘semi-Riemannian metric’, ‘Lorentz invariance’, or ‘tensor fields’ before realizing that a true understanding of this stuff requires a small library of books, a PhD, and godlike determination.

So when it comes to modern physics, if you want to avoid the math, you’ll be limiting yourself to hearing only the consequences of the theory rather than getting satisfying arguments for why it must be true.

But with that said, the consequences are so important and yet so eye-gougingly bizarre that they deserve some mention.

To understand Einstein’s Theory of General Relativity and where it came from, we have to go back to an older theory of relativity, which comes from Galileo. Galilean Relativity says that the laws of physics^{1} must hold true whether the observer is stationary or moving at a constant speed.

As proof of this, Galileo used the thought experiment of a ship moving in a steady wind on a perfectly smooth sea. For our purposes, I’d like you to imagine a person standing in a boxcar on a moving train. Imagine that this person tosses an apple straight up into the air and catches it. From his point of view, the apple made a perfectly reasonable motion, just like it would have done if he’d been standing on solid ground.

Now imagine another person standing outside the train and watching it go by. She sees the apple get tossed in the air, but rather than moving straight up and down, from her point of view, the apple moved in a long arc as the train moved forwards. All she has to do to make sense of this is to take into account the fact that the ball was moving sideways as it was being thrown upwards, and then the situation looks just like one person tossing a baseball to another. Everything can be worked out using the same set of laws.

But what would have happened if that train were accelerating forwards? From the passenger’s point of view, if he was facing forwards, he could have tossed the apple up and found it inexplicably hitting him in the face. Trying to apply Newton’s Laws from his point of view wouldn’t work, because the ball would seem to accelerate backwards without a force acting on it.

Of course, from the stationary observer’s point of view, everything would have looked fine, because she could see the train and its passengers accelerating. The apple would have been accelerating too, but the moment it left the man’s hand, it stopped getting pushed forward and so got left behind, following Newton’s First Law.

So we can now differentiate between two different points of view, or ‘reference frames’. An inertial reference frame is one that is either not moving at all, or else it’s moving at a constant speed. We call it ‘inertial’ because the law of inertia (Newton’s First Law) holds. An accelerating point of view is called a non-inertial reference frame, because objects seen from that viewpoint seem to violate the law of inertia.

This theory worked very well for hundreds of years, mostly because scientists rarely dealt with any phenomena that had high enough levels of energy to notice the problems with it. But at the beginning of the twentieth century, Albert Einstein poked two holes into Galilean Relativity that changed pretty much everything.

First, in 1905, he showed that the rate at which an observer is moving really does change the way he sees things. This was the Theory of Special Relativity, as it later came to be called. (Even though it came first, chronologically speaking, I’ve saved it for later in the book because I think it fits better there and because it seemed silly to end a chapter on gravity without talking about General Relativity.)

Next, he took a good hard look at those non-inertial reference frames. He found that by thinking hard about accelerating points of view, he could incorporate gravity into relativity and finally figure out how gravity really worked. More on that in the next section.

But the last thing we need to talk about before we get into General Relativity (GR) is… geometry. That’s right, parallel lines and intersections and all that. That’s because GR changed geometry as we know it.

The formal study of geometry originated in Ancient Greece. People around the world had been working out the math of triangles, squares, and other shapes since before recorded history, but it wasn’t until Euclid wrote his book *Elements* around 300 BC that things really got going. Euclid’s stroke of genius was to come up with just a handful of simple statements that, when used in combination, could be used to derive any geometric principle at all.

These statements, or ‘axioms’, were assumptions that seemed so obvious that nobody had even really thought about them before. The most interesting one was the fifth and last, which involved parallel lines. Although originally it was stated differently, it amounts to the following: imagine a straight line drawn on a flat piece of paper. Then imagine a dot drawn nearby, separate from the line. The fifth axiom says that there is only one possible line you can draw that intersects the dot and never intersects the line.

In Euclidean geometry, we would call those two lines parallel. It’s such a simple statement, and yet it has caused a lot of controversy over the years. The most interesting problem is that a number of mathematicians have come up with sets of axioms that contradict the fifth postulate and yet still can be used to develop a logically consistent, non-Euclidean system of geometry.

But for most of history, physicists were convinced that reality followed the rules of Euclid. In this view, light always travelled in straight lines – when you shone two laser beams parallel to each other, they could go on forever and never meet. The inner angles of triangles always added up to 180º. Everyone was happy.

Until Einstein came along.

**Big Ideas:**

- We’re going to miss out on the derivations of General Relativity because we’re not doing the math.
- Galilean relativity says that the laws of physics work in inertial (non-accelerating) frames of reference.
- Non-Relativistic physics (everything before Special and General Relativity) assumes that the geometry of space is Euclidean, so that parallel lines never meet.

Next: 3.6 – General Relativity

Previous: 3.4 – Orbits

- In this case, the laws Galileo was referring to are best described as Newton’s Laws, even though Newton’s work came after Galileo’s death. For this reason, Galilean Relativity is often called Newtonian Relativity. ↩