Is gravity a force?

 Believe it or not, this is a question that even physicists don’t seem to agree on. Viktor T Toth, whom I’ve referenced in previous posts and whom I follow on Quora, is almost dismissive of the question. Paraphrasing, he said something like, ‘If you don’t think gravity is a force, just drop a rock onto your foot.’ Ouch! I’d say it’s one of those paradoxes that seem to pop up everywhere in our comprehension of the Universe. 

This question takes me back to my teenage years when I first encountered Newton’s universal theory of gravitation. I struggled to understand how all bodies from the most massive to the smallest could all fall at the same rate in the same gravitational field – it made no sense to me.

 

The equation F = ma is one of the most basic in physics and embodies Newton’s second law in a succinct formula that is universal – it applies everywhere in the Universe. 

 

Leaving gravity aside for the moment, if you apply a uniform force to different masses they will accelerate at different rates. But in gravity, we get a converse relationship. We have different forces dependent on the mass, so we always get the same acceleration. It’s like gravity adjusts its ‘force’ according to the mass it effects. This was the dilemma I couldn’t resolve as a high school physics student.

 

Around the same time, I remember watching a documentary on Einstein (when TV was still in black and white), which was inspiring and thought-provoking in equal measure. One of the more mind-bending ideas I remember was someone explaining how Einstein had changed the concept of gravity as a force to gravity as a curve. This made absolutely no sense to me and it says a lot that I still remember it after more than 5 decades. Of course, I didn’t realise at the time, that these 2 conundrums I was contemplating were related.

 

I’ve recently been reading a book called Emmy Noether’s Wonderful Theorem by Dwight E. Neuenschwander, which is really a university-level text book, not a book for laypeople like me. Emmy Noether famously developed a mathematical formulation to show the relationship between conservation laws and symmetry. Basically, energy is conserved with transformations in time, momentum is conserved with transformations in space and angular momentum is conserved with transformations in rotation. She originally developed this for relativity theory but it equally applies to quantum mechanics. As someone who is not known outside of the physics community, she’s had an enduring and significant input into that field. In her own time, she was not even paid for giving lectures to students, such was the level of prejudice towards women in the sciences in her time (pre WW2).

 

Reading Neuenschwander’s book, I was surprised to learn how much a role the Lagrangian played in her work. Relevant to this topic, Noether was the first to apply the Lagrangian to general relativity, which is actually the easiest way to understand it.

 

To quote from Neuenschwander:

 

The general theory of relativity came along in 1915, and by 1918 the equation of motion of a particle falling in a gravitational field was shown, especially by Emmy Noether and David Hilbert, to be derivable from a variational principle: The world line of a freely falling particle would be that for which the elapsed time between two events was maximised, making the world line a geodesic in spacetime.

 

David Hilbert was Noether’s mentor, and also her greatest champion it has to be said. Hilbert was arguably the greatest living mathematician of his day. The last part of that quote is effectively a description of the Lagrangian, which the author compares to Fermat’s principle. In other words, Fermat’s principle gives the path of least time for light being refracted, and the Lagrangian in a gravitational field gives the geodesic, which is the path of maximum elapsed time for a particle.

 

Notice that in that description of a particle following a geodesic in a gravitational field there is no mention of its mass. This means that its path and its elapsed time is independent of its mass. In fact, we even know it applies to photons, which are, to all intents and purposes, massless.

 

And this revelation finally resolves the conundrum I wrestled with in high school. If you are in free fall, say in a falling lift, or in an orbiting space station; in both cases, you will experience weightlessness.

 

So where is the force? The force is experienced when an object is stopped from following the geodesic, which is the normal everyday experience we all have and don’t even think about. We call it the force of gravity, because gravity is the underlying cause, but the force is actually created by whatever it is that is stopping you from falling, and not the other way round. And, of course, that force is proportional to your mass otherwise you wouldn’t be stationary, relative to whatever’s holding you up. And if you drop a rock on your foot (not recommended) you’ll experience a force that is directly proportional to its mass; no surprise there either.

This is a video, by someone who knows more about this than me, which is even more mind-bending. He argues, quite convincingly, that we are all accelerating by just standing still - on Earth.