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Musing

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willorn

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I can already tell this post will have a lot less structure than usual.

I've been thinking about special relativity quite a bit more than usual these past few days, in particular, the twins paradox. We didn't discuss it, but it seems to me that the actual aging is not the paradox involved, but the question of which twin aged how much is the paradox, since the earth twin would believe the other twin to be 40 years older and the space twin would think himself only 4 years older. Secondly, we discussed that special relativity applied to objects either in constant motion or at rest. In other words, objects in an inertial frame of reference.

That being said, the brother traveling in the spaceship must have experienced some sort of acceleration throughout his journey, when he left earth for example, and most likely when he turned around and when returned to earth. Therefore, I do not even think that the laws of special relativity apply to this situation. The question then for me is in that situation what would happen?

I imagine that the twin on earth has aged physically by forty years and that the twin who traveled has aged physically by just four years, and that no paradox exists at all.

Something else I have been thinking about: E=MC^2

I never truly understood the principle, so I looked online for the experiment used to determine this formula, and then attempted to derive it myself. I found that a useful experiment to reference (although theoretical) is this: a box is stationary in a vaccuum. A photon moves through the box from left to right. Since a photon technically has momentum, the box must then move left in order to conserve momentum of the system. When the photon reaches the right side of the box, the impact causes the box to stop moving.

However, since no external forces acted on the box, its center of mass must be in the same position as before (new concept for me!) but the box has moved left. Therefore, Einstein determined the photon must have a mass equivalent in order to satisfy the laws of physics.

I dreged up an equatin devised by Einstien to get started. I wonder if he came up with this expression before or after he determined that E=mC^2, because that would make this post seem rather silly. Since, a photon is massless, I was able to draw a simpler conclusion from his equation.The momentum rho is the momentum of both the box and the photon, by conservation of momentum.

gif.latex?E^2=\rho ^2C^2+m^2v^2 \Rightarrow \rho=\frac{E}{C} \Rightarrow mv=\frac{E}{C}

Running low on ideas, I nosed around some more, and found that I should start thinking about the time it takes the photon to move from side to side. That train of thought led me to the following. The key is that velocity is change in displacement over time and that the time the photon required to cross the box is the length of the box side over the photon's velocity.

gif.latex?m(\frac{\Delta x}{\Delta t})=\frac{E}{C} \Rightarrow\Delta t=\frac{L}{C} \Rightarrow m\Delta x=\frac{EL}{C^2}

Thanks to what I learned this year in class, I know the center of mass of a system can be expressed the sums of products of mass and displacement of all individual parts over the sum of all individual masses.

I determined that if the center of mass did not move, then the position of the center of mass must have been in the same position as the box after the system resolves itself.

gif.latex?\overline{x} = \frac{Mx_{1}+mx_{2}}{M+m} \Rightarrow \frac{Mx_{1}+mx_{2}}{M+m}=\frac{M(x_{1}-x_{after})+mx_{2}}{M+m}

We can substitute X2 (the displacement of the photon) to be L the length of the box because it traveled the full length of the box.

gif.latex?M(x_{1}-x_{after})+mL = Mx_{1} +mx_{2}\Rightarrow -Mx_{after} = mL

Reviving the previous equation created and substituting it for m(delta x):

(I can do this because although the expression reads differently, the displacement after represents the displacement of the photon after colliding with the box's side, and the Mass is of the same object in both cases)

gif.latex?mL = \frac {EL}{C^2} \Rightarrow E = mC^2

I find that deriving an equation always helps me to conceptualize it, and I hope this derivation helps you too! In my probing I also discovered that all mass has a measurable frequency, although it has little or no effect on people. More on that later...

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Way to dive in to a new and challenging topic... I can't wait to hear your analysis of the frequency of mass and the deBroglie Wavelength!

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