The physics of the impossibly tiny


According to Newtonian gravitation, the attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Einstein refined this somewhat, but as long as there aren’t crazy speeds or non-flattish spacetime involved, Newton’s formulation is accurate. As far as we know.

I read this interesting article, which spun off the thought experiment below. The EM Drive mechanism keeps confounding some Really Smart people who would genuinely like to discredit it. What’s going on here? I’m probably butchering the explanation, but the article posits that very small accelerations don’t behave in a completely smooth manner. In other words, they’re subject to quantum effects.

Here’s a thought experiment. What is the Newtonian gravitational attraction between this hydrogen atom in my little finger, and one in, say, the Andromeda galaxy? (Pedantically, one that was in the Andromeda galaxy 2.5 million years ago, from the Earth’s frame of reference)

I put this into Wolfram Alpha (which helpfully supplied an value for the big-G constant) and came up with an answer of:

1.419 * 10-109 N (newtons)

That’s not a typo. There’s 109 places to the right of the decimal.

I don’t even have a good analogy for how slight that force is. A trillionth trillionth of the seismic energy of a dandelion seed landing, measured from a trillion km away?? Still too much, by a lot. It’s a reasonable question whether the universe even goes to that level of detail. Could such a slight force even be said to exist in a meaningful sense? In other words, could it be measured, even in principle? I’m no expert, but I doubt it. So maybe physics isn’t completely smooth down to that level.

Is there a limit to how small a force can be and still act like a force?

To accurately compute the complete gravitational affect on this hydrogen atom in my little finger, I’d have to take into account the 1080 particles in the observable universe, the vast majority of which make unmeasurably tiny contributions to the overall sum. That’s for a single instant. For a continuous number, I’d need to repeat this enormous calculation something like 1043 times per second. So all you universe simulator writers out there, take note. Some simplification is probably warranted. :)

(A realistic simulator would only need to take into account the mass of the Earth, Moon, Sun, and for a few decimal places more, Jupiter. But I’m talking about laws running the universe, not engineering hacks.)

It seems there are still some quite interesting things left to discover in the universe. I keep going back to the chapter Surprises from the Real World in Lee Smolin’s Trouble With Physics. Exciting times ahead! -m

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