- 3050 - EINSTEIN’S - Pythagorean Theory of Relativity? Deriving E = mc^2 Using a Teeter-totter. - Every kid knows how a teeter-totter works. You lay a flat board on a fulcrum. If a boy and a girl weigh the same, put the fulcrum in the middle and the teeter-totter is perfectly balanced.- This is the same simple concept we will use to derive E = m * c^2. Energy = mass * speed of light squared.
------------------ 3050 - EINSTEIN’S - Pythagorean Theory of Relativity?
- If Pythagoras had discovered the Theory of Relativity it would have been a lot easier to understand.
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-------------------------- E = mc^2
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- This formula only applies to a mass that is at rest. If a mass is in motion you must consider its Total Energy to include Kinetic Energy and Momentum. Momentum is mass * velocity. So, if the mass is in motion the Energy equation expands to:
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------------------- E^2 = ( mc^2)2 + (mv* c ) ^2
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- This can be recognized as the Pythagorean Theorem for Relativity. Treat Energy as a vector. “E” is the hypotenuse, “(mc^2)” is the opposite side of a right triangle and (mv*c) is the adjacent side of the right triangle. The Theorem says that the hypotenuse squared is equal to the sum of the squares of the other two sides of a right triangle. So:
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---------------- E^2 = ( mc^2)2 + (mv* c ) ^2
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- “(mc^2) is the opposite vertical side. When the velocity is zero the horizontal line collapses into the vertical line. The hypotenuse E becomes the same as the vertical line mc^2 and E = mc^2 when the velocity is zero.
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- Likewise, if the mass is zero the vertical line collapses into the horizontal line and the hypotenuse E = mv^2 * c.
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- The energy of the massless object is a photon. But, its momentum is not zero even though the mass is zero there is always a quantum of energy in each cycle of oscillation.
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- The Energy of photons is Planck’s Constant * Frequency of oscillation. E = h* f. However, if there is just the slightest bit of mass of another particle the hypotenuse will always be greater than the adjacent side. What we can say is the smaller the mass becomes the closer E = mv*c, Energy = the momentum times the speed of light.
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- Reformatting the equation to solve for the velocity of the mass
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------------------------------- v = mv*c / E * c
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- As mv *c / E approaches “one”, (mv*c / E = 0.9999), the velocity approaches the speed of light
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------------------------------ v = c
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- Light can be thought of as particles, photons, with zero mass but still momentum. Their motion can only be one speed, 186,000 miles per second.
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- All light, electromagnetic radiation, travels at this single speed. It has been traveling at this speed ever since the Big Bang. We say the size of the Universe is its age in lightyears. That is how far light travels in one year = 5,880,000 million miles.
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- So the age of the Universe is 13.8 billion years and the distance the light has traveled is 13.8 billion lightyears distance.
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- If you point your telescope due east and see to the edge of the Observable Universe you can only see 13.8 billion lightyears distance. But, if you point your telescope to the west you could see another 13,8 billion lightyears in that direction. In that sense we the observer are in the center of the Universe. Of course , every observer can say that.
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- The Observable Universe is therefore 27.5 billion lightyears in diameter.
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- But, during the 13,8 billion years of time that it takes the light to reach us the Universe has been expanding. So, the Universe we can not see, because the light has not reached us yet , is out 46 billion lightyears in both directions. The diameter of the Universe today is 93 billion lightyears across.
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- Comparing the size of Earth ( 10^7 meters) to the size of the Universe (10^27 meters) is like comparing the size of a virus (10^-6 meters) to the size of the Solar System (10^14 meters).
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- And, we know that the Whole Universe is much, much larger than the 93 billion lightyears. Because the Universe is homogeneous in all directions and the geometry is flat, meaning parallel lines neither diverge or converge then the Universe must be infinitely bigger.
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- The Cosmic Inflation that occurred during the first seconds of the Big Bang expanded the Universe faster than the speed of light.
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- When we conceive the Whole Universe we are no longer in the center. There is no center. Space and time are expanding in all directions, everywhere.
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- Today we measure this expansion of space to be traveling at 47,000 miles per hour per each million lightyears distance. Once we see out a billion light years distance of expanding space we see galaxies receding at 47,000,000 miles per hour.
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- When we get to the edges of the Observable Universe over 46 billion lightyears the galaxies are moving away from us at 2,162,000,000 miles per hour. That is over 3 times faster than the speed of light. The speed of light is 670,633,500 miles per hour. That light will never reach us. Never.
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- Nothing can exceed the speed of light. So, how can a galaxy be moving that fast? It must be impossible, and , violating the Theory of Relativity. True, no “ thing” can travel faster than the speed of light. But, nothing says that space can not expand faster than the speed of light. The more space there is the faster things are expanding away from each other.
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- We have not violated the Pythagorean Theorem: E^2 = (mc^2)^2 + (mv*c)^2
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- You can derive Einstein’s E = mc^2 Using a Teeter-totter. Every kid knows how a teeter-totter works. You lay a flat board on a fulcrum. If a boy and a girl weigh the same, put the fulcrum in the middle and the teeter-totter is perfectly balanced.
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- This is the same simple concept we will use to derive --------------- E = m * c^2.
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------------------------ Energy = mass * speed of light squared.
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- If one kid is heavier than the other simply move the board off center and put the heavier kid on the shorter side. The teeter-totter is still in balance.
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- If the kids both weigh the same but the boy had a heavy ball in his lap, then, he would have the short side to keep things balanced. If he threw the heavy ball to the girl on the left side she would go down unless we moved the fulcrum to the right to give her the short side.
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- This is the same simple concept we will use to derive E = m * c^2.
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- Our teeter-totter will need to have a weightless board and frictionless fulcrum. We will start with equal masses on each side of the teeter-totter then shoot a burst of light from one mass to the other.
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- The burst of light leaving one mass makes it lighter. The other mass absorbs the light and becomes heavier. The fulcrum is moved toward the second mass in order to restore the balance between the two masses.
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- Ok, let’s start this thought experiment. Are you thinking? You may need to sketch this on a piece of paper as we go through it. The two masses are balanced and the fulcrum is in the middle. We will call the fulcrum at zero distance. Then, the two masses are located at + x1 distance and – x2 distance.
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- The mass on the right of the teeter-totter gives a burst of radiation toward the mass on the left. The entire teeter-totter recoils in the opposite direction due to action equals reaction and shifts to the right.
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- When the radiation hits the left mass, the teeter-totter comes to rest, halting the movement to the right. The mass at the left has absorbed the radiation and becomes heavier. However, the fulcrum has shifted and the teeter-totter remains in balance. Both masses have shifted to the right by a distance Dx.
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- If the teeter-totter is in balance then the center of mass for the system has not moved. The center of mass is still at zero.
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- Before we started, the teeter-totter was in balance due to this equation:
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----------------------------- Mx1 + mx2 = 0
- If the radiation has mass then the mass at the right lost some amount of mass, Dm.
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- And, the mass at the left must have gained the same amount of mass, Dm.
- If the teeter-totter is still in balance then the new equation must still equal zero:
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--------------------- (m + Dm)( -x + Dx) + (m – Dm)(x + Dx) = 0
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-------------------- -2*Dm * x + 2* m* Dx = 0
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-------------------- Dm = m * Dx / x
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-------------------- Dx is the distance traveled.
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------------------- Dx = velocity * time (from distance = rate * time)
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- The velocity is the velocity of the radiation, and that is the constant speed of light = c.
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- The time is the time of the teeter-totter was moving to the right.
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- The distance across the teeter-totter is 2x, and the time it takes light to cross that distance is:
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------------------------ Time = 2x / c
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- The change in distance that shifted to the right:
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-------------------------- Dx = velocity * time = velocity * 2x/c
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- The change in mass due to the absorbed radiation, repeating the equation above:
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------------------------- Dm = m * Dx / x
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------------------------ Dm = m * velocity * 2 * x / x * c = 2m* velocity / c
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- The momentum of the entire teeter-totter is 2mv. The is from the law of conservation of momentum. The entire mass of the teeter-totter is 2m. The entire system must have the same momentum as when it moved to the right.
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---------------------------- Momentum = P = 2 * m * velocity
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--------------------------- Dm = P / c
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- The energy in a pulse of light is equal to momentum * speed of light:
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----------------------------------- E = P * c , and P = E / c
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---------------------------------- Dm = E / c^2
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- The change in mass, Dm, is the energy given off by the mass.
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- Therefore: E = m * c^2. Albert Einstein was right.
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- Energy is equal to mass times speed of light squared.
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- Energy is equal to mass times 90,000,000,000,000,000 meters^2 / seconds^2.
- That is a large number. No wonder the atomic bomb makes such a big hole in the ground.
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- February 16, 2021. EINSTEIN’S - Pythagorean Theory 2265 1582 3050
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--------------------- Tuesday, February 16, 2021 -------------------------
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