- 3164 - PYTHAGOREAN - theorem to derive Relativity? Einsteins’s equation can be recognized as the Pythagorean Theorem for Relativity. “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. Here is the derivation:
- ----------------- 3164 - PYTHAGOREAN - theorem to derive Relativity?
- We use the Pythagorean Theorem to calculate the height of a pyramid or a flagpole. The square of the hypotenuse is equal to he sun of the square of the other two sides of any right triangle. Use the shadow of the flagpole.
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- If Pythagoras had discovered the Theory of Relativity it would have been a lot easier to understand. So what does Einstein’s formula have to do with triangles? Energy is equal to the mass times the speed of light squared.
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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. “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. The energy of the massless object is a photon.
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- But, its momentum is not zero even though the mass is zero there is always a quantum of energy in each cycle of oscillation. The Energy of photons is:
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------------------ Planck’s Constant * Frequency of oscillation. E = h* f.
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- 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” 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.
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- 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. 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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- 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. Today we measure this expansion of space to be traveling at 47,000 miles per hour per each million lightyears distance.
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- 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.
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- 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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- That Pythagoras was a pretty smart guy. It works on a lot more than just right triangles.
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- May 19, 2021 PYTHAGOREAN - derive relativity? 1582 3154
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