Sean Carroll

The classic example of something like this is coordinates, just in, again, ordinary three-dimensional space, right?

So by coordinates, you might have heard, well, I can have Cartesian coordinates.

That's X, Y, and Z with perpendicular axes.

But I can also have spherical coordinates or elliptical coordinates or weird kinds of coordinate systems.

In general relativity, Einstein's theory of curved spacetime, you need to have crazy coordinate systems because spacetime is not flat.

And ordinary Cartesian coordinates, which become Minkowski coordinates in spacetime rather than space, those simple rectilinear coordinates are just not available anymore.

You have to use some curvilinear coordinates.

So people, mathematicians especially, did understand from a relatively early time that you shouldn't confuse the real thing for your convenient description of the real thing.

You shouldn't confuse the thing that actually exists for your choice of labels, which are what coordinates are.

And this is easy to say, and you're all nodding along, yes, yes, yes, it doesn't really matter whether I express a distance in meters or feet, right?

Or whatever, more complicated choices of coordinates, what matters is the physical thing called the length or the distance, okay?

But in fact, in practice, it is super hard for people to truly internalize this lesson.

And here's where I will say that Einstein kind of failed at this over and over again.

You might know that Einstein died, I don't know exactly what year, but it was the 1950s.

He went to his grave not really understanding that there were these things called black holes.

and being very confused about whether or not there were these things called gravitational waves.

You know, these days, black holes and gravitational waves are taken as some of the most important and impressive implications of Einstein's theory of general relativity.

But between 1915 when he wrote down the theory and the 1950s when he passed away, people were very confused about how to extract what was real in general relativity from

from what was simply an artifact of using one coordinate system over another.

So for gravitational waves, there are ways to change coordinate systems so it looks like the wave isn't there.