## Balloons and general relativity

- by Jaakko Moisio
- March 1, 2020

When I was a teenager, my grandfather gave me the following problem:

Imagine you are in a car. There is a helium balloon floating between the seats, with its string attached to the floor of the car. As the car starts accelerating, from the frame of reference of the driver and passengers, which way does the balloon swing?

I was attending high school at that time, so I had barely started my formal education in physics one or two years earlier. I had read some popular books on modern physics: those books that give a high overview of cutting edge theoretical physics, with lots of nice analogies and colorful interpretation of sometimes bleak mathematical results. In short, I was aware of how awesomely cool the field of physics was, but my ability to do physical reasoning wasn't very mature yet.

“Well”, I said to him, “My first intuition is that it swings backward, but since you're asking, there's probably a catch. Maybe the molecules in the air move to the rear end of the car, *pushing the balloon forward* instead.” Indeed that is the case, as he explained. “But can you prove this? Think about the equivalence principle in general relativity. You know, gravity and acceleration are indistinguishable to an observer.”

My grandfather had a remarkable role in sparking my interest in physics, a field I ended up majoring in the university. He also gave me most of those books I mentioned earlier.

The very elegant solution he gave to the balloon problem was that an observer in an accelerating frame of reference is like an observer in a gravitational field. The buoyancy experienced by the balloon points “up”, but up in the frame of reference of the passengers “up” is tilted forward. Thus the balloon swings toward the front of the car.

In school physics was taught as a collection of distinct phenomena. There is gravity governing the motion of a baseball or a satellite. There is heat governing how solids become liquids, and liquids become gas. There are charged particles affected by electromagnetism. There is a framework where all this happens (Newton's laws of motions for example), but somehow these phenomena are governed by different sets of rules and equations. There is a problem. You look the right equation at the back of the textbook, you plug in the numbers from the problem statement, and you have a solution.

As most of our school knowledge, that's not the whole picture. What is cool about physics is that everything is connected. Theories like general relativity or quantum mechanics are built by postulating a limited set of very general assumptions. In the special relativity you assume two things:

- that the laws of physics are the same for all observers in inertial (non-accelerating) frames of reference
- that everybody observes the same speed of light

Everything else in special relativity inevitably follows: that time progresses slower and lengths become contracted when a body is moving fast, and that \( E = mc^2 \). Add charged particles and you have classical electromagnetism nailed down to every last detail. Add the equivalence of acceleration and gravity, and frame it in mathematics involving pseudo-Riemannian geometry, and you have the theory of general relativity. All those things you learn about general relativity in school and reading popular science publications — curved spacetime and expanding universe — are just inevitable consequences of a few basic principles.

Although I never pursued a career in physics, I'm glad my grandfather and other important people in my life pushed me toward that path. The way to approach solving problems in my physics education — analyzing them and drilling down until getting to the basics — is applicable in many professions. In software development that's often what you do when hunting for bugs in a large (legacy) codebase with many interacting components. Modeling the complicated world with simple enough concepts is at the core of software engineering. I'm sure my fellow students who ended up in finance or management consulting or other fields completely unlike basic research will have similar stories to tell.