woensdag 5 juni 2013

Trying to figure out spacetime bending

A few years ago I was obsessed about Einstein's theory of gravity. I wanted to know everything about it, so I browsed a lot on the internet, trying to find information about the topic (I could have gone to the library, however I was still in the frame of mind that everything on the internet is correct). So one day I came across the following website: Bending Spacetime in the Basement. Wow, can one do this at home?!?

First take a good look at the website. He's got a point by stating that he is bending spacetime. It unfortunately has nothing to do with warp-drives and wormholes (Which was what I was looking for), but his experiment is quite cool nonetheless. It somehow proves (I will discuss this later on) that the theory of gravitation found by Newton, not only works for planet and stars, but also for smaller everyday objects. Everything attracts each other.

Impressed by the simplicity of the experiment and the fact that it is about gravitation (I think this is one of the most interesting forces in physics), I showed it to my fellow classmates (I was still studying to become an Aerospace Engineer) and it resulted in heated discussion. Was it really a gravitational pull of the masses that caused the motion in the experiment, or were different forces in play? Maybe it was just tension in the string or a possible magnetic torque. It could even have been a draft (wind) in the basement of the experiment. I always believed (believed in the sense of a good educated guess) it was gravitation at work, but how could I prove this?

My current job is to model the effect of gravitation on the surface of the Earth. The above problem should therefore be a walk in the park for me. I start by assuming that the video coverage is genuine and reports exactly what was going on during the experiment. I also trust our basement spacetime bender and all its statements on his website (again, everything on the internet is true). This enables me to obtain the timing records of the motion, which I can use in my proof.

From the above two stills of the video we can deduce that the full motion - from rest to first contact with the cubic stones - takes about 4:30 min (±15s, he made a time lapse out of it, so the time-resolution of the movie is quite coarse). This gives us an observation with which we can test our models.

So let's make a model that describes this motion. Starting with the law of gravitation of two particles (let's assume that the stones and round-metal objects are point masses. This is what we call modeling, start simple) invented by Newton (He did not know that mass actually bends space and time, but who cares (Einstein did!)):

(Actually this depicts Newton's law of gravitation and his third law of motion) Here, F is the force experienced by the numbered particle, G is the universal gravitational constant, m is the mass of the numbered particle and d is the distance between the two particles. I know equations in a blog will lower its popularity, but hey, it's a physics blog. To clarify, I made a sketch of the situation, and because of symmetry, I only sketched the right part of the rotation device.

Here, M1 is the fixed stone and M2 the round metal object. M2 can only move along the dashed line, s. Half the length of the rotation device is denoted by r and the angle between the initial and current state is depicted by phi (yes, the strange wriggly shape in the bottom corner. It's Greek!). The whole problem can be modeled as a 1D-motion. To find out whether it's only gravitation that moves the rotation device, we calculate to what extend the gravitational pull between the two bodies will affect their movement. The gravitational pull between the two bodies works along the line, d (illustrated in red).

The motion of M2 from its initial resting state can be modeled using a mathematical trick I learned at Aerospace Engineering, namely the state equation and the Euler integrator (I know, this is not the best one, but if you just reduce the timesteps, it will work (hopefully)).
If we can find a relation for double dotted s (as this is the only unknown in this scheme, it really is!), the motion of M2 can be calculated. Double dotted s only depends on the gravitational pull between the masses in the experiment (let's ignore the giant ant, and assume symmetry (which is not entirely correct, but let's start with that and ignore the attraction of the other M1 mass)). So after some proper geometric brain-crunching we obtain a solution for the acceleration felt by M2 (I leave this open for you to do. Hint, it is a tiny-tiny acceleration).

So now it only remains to find values for the geometry of the problem and the masses used: M2 is a lead sinker weight of 169 grams, the rotation bar is 30 cm long, so r = 15 cm, and M1 (the pavement stone, or in Dutch "kinderkopje", which is a bit of an odd name) is 2 kilogram. The difficult bit is to determine the difference between the initial position of the rotation device and the time of impact (in other words what is the angle between the two positions). Or just look at the following figure:

Let's say (wet finger approach (WFA)) this angle is 15 degree ±5 degrees (with this added uncertainty it becomes science ;)). This scientifically approved estimation of the geometry enables us to test the model and find out the duration of the movement. I used the following Matlab script:

clear all;close all;clc;
% The spacetime bending experiment
m1 = 2;
m2 = 0.169;
G = 6.673e-11;
r = 0.15;
% start Euler integration
dt = 1;
tnew = 0;
phinew = pi/2-pi*(15/180);
snew = r*phinew;
sdotnew = 0;
while r*cos(phinew)>0   % M2 hits M1
    % use new values for the calulation
    t = tnew;
    phi = phinew;
    sn_1 = snew;
    sdot_n = sdotnew;
    % Force (gravitation)
    d = sqrt(2.*r^2.*(1-cos(pi/2-phi)));
    alpha = pi/4 + phi/2;
    Fz = G*(m1*m2)./d.^2;
    Falong = Fz.*sin(alpha);
    sddot = Falong./m2;
    % State equation integration
    snew = sn_1 + sdot_n*dt;
    sdotnew = sdot_n + sddot*dt;
    tnew = t + dt;
    phinew = snew/r;
    % plot the results in a figure
    hold on
hold off
axis([0 0.4 0 0.4])

The duration of the motion calculated in the simulation is 12.5 ± 5.7 minutes. Looking back at the 4.5 minutes it took in the video, this suggests that the initial angle was 10 degrees (if it is only gravitational attraction that is playing a role). But the overal exercise tells us, that it could be that gravitational attraction between the two masses is the main contender for the motion. Isn't that cool!

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