## What is relativity, anyway?

#### September 2020

Relativity is one of my favorite topics in physics. Its origins were almost pure logic puzzle science, which in my opinion is one of the most fun and satisfying types of science. Most people have a vague sense of what relativity is – the speed of light is finite, and something about time travel? – but they don’t really know what it is, which is too bad because the principles are very simple, and the results are pretty awesome.

There are two halves of relativity as Einstein developed it: special relativity and general relativity.

Special relativity is, at heart, two ideas:

• The laws of physics are the same to everyone everywhere, as long as they are using a non-accelerating (“inertial”) frame of reference.
• The speed of light in a vacuum is the same regardless of frame of reference.

Before Einstein, these two things were widely considered to be paradoxical: how could it be that the speed of light is constant for everyone – its velocity is not added to your velocity, if you are moving – but also physics doesn’t depend on your perspective? What Einstein did was take both of these things seriously and realized that when you put them together, you get a space-time connection and lot of other really powerful stuff.

For example, time dilation immediately falls out. Imagine that you are standing on the platform of a train station while a train flies past, and your friend is onboard the train. If your friend puts a mirror on the ceiling of the train car and, lying on the floor, points a flashlight at it, he will see the light reflected back at him having gone up in a straight line and returned in a straight line to the floor. However, from your perspective on the platform, while the light is traveling up and down, it is also traveling sideways. This means the path it travels is longer than the path it appears to travel to your friend. Since the speed of light is constant, and the distance between the floor and the ceiling don’t change, from your perspective, time must be passing more slowly for your friend than for you. And indeed this is an experimentally tested real effect! (For example, you need to take it into account for GPS to work reliably.) It is important to note that your friend, on the train, never feels anything different: from their perspective, time is passing just fine at the same rate it always does. The effect is entirely about the difference in your perspectives.

The idea that the laws of physics are the same to everyone is sometimes called “general covariance” and is really important for the tractability of a description of physics overall. There isn’t necessarily a guarantee that it would be true in our universe, but it turns out that it is.

General relativity adds one more idea: when an observer is using an accelerating (“non-inertial”) frame of reference, the laws of physics might appear to be different, and that difference shows up as a gravitational field.

For example, if you are in a car accelerating on the highway and you throw a ball into the air, it will veer towards the back of the car. You know that this is because the car is accelerating - but it is also completely equivalent to model this as a gravitational field coming from the back of a non-accelerating car! In fact, the equivalence principle says that if you were in a windowless box that was either being accelerated by a rope or is in the presence of a gravitational field, there is no measurement you could make that would tell them apart.

The language of physics is math, and so while it’s easy to get the big ideas that underlie relativity, getting the full depth of the implications requires going to the equations. The math of special relativity is mostly pretty easy – you can get surprisingly far with high school level math – but general relativity is a completely different beast, and one of the reasons Einstein’s original GR papers are so challenging is because the math really required to do it well (e.g. differential geometry) didn’t exist at the time. Einstein had the realization that the universe has no sense of coordinates, just a way of measuring distances between events, which he described as “no prior geometry,” but it took a while for the full mathematical description of what this actually meant to be worked out.

Anyway, relativity is incredibly trippy and fascinating, so if you’ve liked this sneak peek I highly recommend checking out Einstein’s short 1916 book “Relativity: The Special and the General Theory” which is written for laypeople and uses very little math while still walking you through the core of it all with Einstein’s unique incisive clarity. General relativity is certainly one of the most beautiful things in all of science (possibly the most; quantum field theory is very cool too, though it lacks the intrinsic elegance of GR), and I wish more people had the chance to understand and appreciate it.