The Cavendish experiment

Disappointed with this blog? Yeah, sorry again. Too much time between posts and I'm not getting very deep in any subject I deal with. That's because it requires time, and that's a thing I didn't nor I have at the moment.

So, let's talk about a funny experiment finally I've found in the internets. It's called Cavendish experiment and you can get good historical explanations of it here and here.

The experiment seems easy to follow. In Newtonian terms, two masses attract each other.

Proportionally to the masses, inverse proportionality to distance squared. But proportional doesn't mean equal. There's a constant in somewhere you have to add to make things right. Unit conversion stuff. Kinky stuff. If you want to measure how much does it weight Earth (is there anything British-er than asking yourself that kind of questions? ) you need to get that constant first.

But, how can you measure the gravitational constant without knowing Earth mass? And harder still, you are living on Earth. In case you figure out a way of measuring without taking into account Earth gravitational pull, doesn't it still f*ck with your measurements?

Before explaining something you probably already know I need to make clear why am I interested in Cavendish experiment:

1. It's an "easy" experiment which allows us to understand gravity in human size scales.
2. What is its relationship with space-time? Before you say human scales are Newtonian, which is almost truth for every almost example you can find, just keep on reading.

Relationship with general relativity.

So we have started with Cavendish experiment and suddenly I start to derive Newtonian limits from general relativity? Hell, no. Alexander Tolish has an amazing paper on-line which saves me from trouble (pages 14-17). Summarizing:

1. Because things go very slow (compared to speed of light, we are not talking here about a Kubrik movie) trajectories in space-time are reduced from full relativistic worldlines to trajectories in the second law of Newton motion. This is possible because we can say: "ok, these proper time terms in geodesic equation are so small we are not even able to measure its influence, so we don't compute them".
2. Because masses are static (again, slow) and there's no energy flux, nor fluctuations, energy-stress tensor can be reduced to its first element: mass energy distribution. And from that, you can derive a fancy vectorial expression which converts this density into a Laplacian for gravitational potential. Therefore, after simplifications, our famous expression F is proportional to masses and inverse to distance squared.

I guess I'll go deeper later (as usual) in the demonstration of why we can forget about proper time terms in geodesic equation (if I want changes from Newtonian theory, maybe these terms are an interesting Achilles heel to work with) but for the moment, let's focus. Cavendish. Yeah. That was the theme this post was about.

The experiment.

So... Have you ever tried to do something with hardware? It's f*cking painful, dude. Nothing comes out as expected. Lots of things can go wrong and that's when you realize about them. If you don't and you are expecting a result, you can come with the idea your initial hypothesis is wrong.

Saying that, the experiment is simple. You skip Earth influence in measure by attracting masses horizontally. You just have to use a torsion balance big enough to be able to measure something and small enough to fit in one place to avoid wind or disturbances. 

Cavendish torsion balance. It was not really his idea, not really to obtain G either as you can find if you go deeper in the history, but an impressive hardware for an XIII century dude. The drawing, is from wikipedia.

The small masses are hanging from a wire which you can measure its torsional spring constant. The big masses are static and attract (very slowly, remember gravity is small even for big things) the small masses forcing the wire to twist. Once the wire stops twisting (after a long, long, long time) you know the force of attraction between the two pair of masses by measuring how much force the torsional spring is retaining. This guy explains it better than me. Sounds about right, yes? Well, that's what I thought, until I saw a couple of videos about modern torsion balances for repeating the experiment.

A couple of dudes in a lab.
An old italian professor giving a piece of his mind.
CGI stuff
CGI stuff again

• First question, how do you measure how much angle the balance rotates. Easy answer: with a laser. Or maybe like Cavendish did and use some vernier scales or rotational variants (and maybe a magnifying lens). You measure a distance and knowing the distance, you can get the angle by trigonometry. The angle leads to force. Force leads to G. G leads to Earth mass.
• Second question: why does the balance oscillates? You can see this very clearly in the Italian experiment (I know, it's weird and adorable). Well, that is something no one seems to explain clearly when you look info about the experiment. They plant the equations and that's it. The fact is you are working with a spring (a torsional spring) and springs tend to offer dynamic responses. 

 An example of response for a second order system under the influence of a step input. It's from an amplifier. Funny thing an amplifier have the same type of response than the Cavendish experiment. No black magic here, it's just a matter of dynamic systems, poles, zeros and frequency behavior. Beautiful, I know. Again from wikipedia. Love those guys.

When balance is released, masses' self gravity  acts (pretty cool, right?) and produces an attraction force. This attraction force increases as distances reduces, which should lead to a collision. But there's a spring against that force and when you twist too much the wire, torsional force overpass gravity. And the masses start to rotate in the opposite direction. But, as they rotate in this other direction, torque reduces because it's a spring! You are releasing it. And when you release a spring, it produces no force. And then gravity between masses starts playing again.

This cycle repeats until equilibrium because each time wire is released and attracted it tends to its minimum energy state. That happens when wire torque and gravity equals each other.

More questions arise:

• How accurate is this experiment?
• Can I build it on my own?
• I'm really into interferometer stuff... Can I see the same with lasers?

Well, I need to inspect a lot more about this but I can extract a couple of little conclusions/future works which I find interesting.

1. It's cool to see gravity working because you want to. Not gravity in the Earth, but the gravity between two masses you put together to interact. This lead me to imagine every object surrounded by its own gravitational field. Pretty cool.
2. In order to change how space tells matter how to move as we see it in our daily experience, it seems a good idea to modify speed in that matter.
3. In order to change how matter tells space how to curve, one of the things I should study is energy flux influence or what happens when we don't have just a static energy-matter distribution. More than that, it seems there are a lot of terms depending on proper time, so maybe speed in this flux is somehow relevant.
4. How does frequency translates into the space-time picture?

See you as soon as I can!