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------------- 2379 - TIME - Deriving Time Dilation from the Pythagorean Theorem
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- Time Dilation is a concept that says, to a stationary observer a moving clock seemingly runs slow. By making the speed of light constant it bound the equations for electricity and magnetism into a single electromagnetic energy. However, at the same time these same equations did some strange things to space and time.
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- For example, if you were in a space ship traveling at 150,000 kilometers per second (50% the speed of light ). And you fired a rocket at 225,000 kilometers per second ( 75% the speed of light ). To a stationary observer how fast would the rocket whiz by? Would it be 375,000 kilometers per second? That would be 125% the speed of light.
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- You can not add velocities that exceed the speed of light. Not allowed. The speed of light is a constant 300,000 kilometers per second. So, if velocity is constant then time and distance have to change in order to make the math work out. Velocity = time / distance must work out to 300,000 kilometers per second or less.
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- Let’s illustrate this concept with a fast train ride. You are in the dining car drinking a beer. You measure the exact time between taking two sips of beer. The time duration is “T” for time clocked on the train. At the same time a stationary observer on the platform watches you whiz by and measures the time you take between drinks. His time is tg, for time measured from the ground. The time he measures is:
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----------------------------- tg = T / square root of ( 1- v^2/c^2)
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----------------------------- v is the velocity of the train.
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----------------------------- c is the velocity of light.
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- This equation represents Time Dilation. As the velocity of the train gets closer and closer to the speed of light, the denominator gets smaller and smaller. Time measured on the ground, tg, gets longer and longer compared time measured on the train.
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- Time dilates.
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----------------------------- tg = T / square root (1 - .1^2) at 10% the speed of light
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----------------------------- tg = 60.3 seconds between sips of beer
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----------------------------- tg = 60 / square root (1 - .5^2) at 50% the speed of light
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----------------------------- tg = 86.6 seconds
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----------------------------- tg = 60 / square root (1 - .9^2) at 90% the speed of light
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----------------------------- tg = 137 seconds
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- Already you are drinking twice as fast as he thinks you are.
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- Note that at slower velocities relative to the speed of light, tg, stationary time is nearly equal to moving time.
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- If the train were traveling a million miles per hour (447,040 meters per second ):
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----------------------- tg = 60 years / square root(1 – (10^6 mph)^2 / (6.7 * 10^8 mph)^2
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----------------------------- tg = 60 / square root ( 1 – 10^12 / 4.4975 * 10^17)
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----------------------------- tg = 60 / square root ( 1 – .00000222)
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----------------------------- tg = 60 / square root (1 - .9999778 )
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----------------------------- tg = 60.0001322 , for all practical purposes you can not detect any change in drinking habits at a million miles per hour. There is only 133 millionths of a second time dilation at that speed.
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- This same formula can be applied to distances, where distances get shorter and shorter as the velocity approaches the speed of light.
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- The formula can also apply to mass. Fast moving mass gets larger and larger compared to rest mass as the velocity approaches the speed of light.
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- This is the same reason no mass can achieve a velocity exceeding the speed of light. As the mass’s velocity approaches the speed of light the mass itself gets infinitely large. There is never enough energy to push an infinitely large mass any faster. Infinite forces and infinite energies just do not exist.
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- To illustrate these concepts further and to actually calculate the formula for time dilation let’s put a clock on the train.
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- We will build our clock using two parallel mirrors facing each other, separated by a distance = L.
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- We place the mirrors such that we can bounce a tennis ball between them. One complete bounce will be counted as one tick on the clock.
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- To calculate the time for one tick we divide the distance by the velocity:
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----------------------------- T = 2 L / c
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----------------------------- “c” is the velocity of the tennis ball.
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- An observer on the ground, observing the train go by at velocity, v, will see the ball bounce in the shape of a triangle as the ball is going up and as it is coming down the train is moving sideways elongating the base of the triangle.
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- “cg” is the velocity of the ball as measured by the observer on the ground. The observer perceives the ball as traveling a longer distance up and down the two sides of the triangle. We will now calculate the perceived time it takes the ball to travel this distance using the Pythagorean Theorem.
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- The height of the triangle is still, L.
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- The sides of the triangle are the distance = velocity / time
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----------------------------- distance = cg * tg / 2
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- The bottom side of the triangle is x = v * tg. We use one half of this distance to get the bottom side of the right triangle.
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- Pythagoras said that, “ the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its sides .” ( See Footnote 1)
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----------------------------- The hypotenuse = cg * tg / 2
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----------------------------- The vertical side = L
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----------------------------- The bottom side = v * tg / 2
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----------------------------- cg^2 * tg^2 / 4 = L^2 + v^2 * tg^2 /4
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----------------------------- tg ^2 ( c^2-v^2) /4 = L^2
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----------------------------- tg^2 = 4 L^2 / cg^2 - v^2
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----------------------------- Previously we determined : T = 2L / c
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----------------------------- L = T * c / 2
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----------------------------- tg^2 = c^2 * T^2 / cg^2 - v^2
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- Now we can calculate the time observed from the ground for the time of the bouncing tennis ball on the train.
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- Replace the tennis ball with a light beam reflecting back and forth between the mirrors. One complete bounce represents one tick of the clock. Since the speed of light is constant it measures the same regardless if you are on the ground or on the train. Since velocity = distance / time and velocity stays the same, then distance and time must change in order to keep the ratio constant.
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----------------------------- cg = c
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----------------------------- tg^2 = T^2 / (1 - v^2/c^2)
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----------------------------- tg = T / square root (1 - v^2 / c^2)
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- The formula for Time Dilation tells us the time measured in a stationary position is equal to the time measured in motion divided by the square root of one plus the velocity squared / speed of light squared.
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- The time, or distance, measured by two observers is not absolute. It depends on their frame of reference.
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- Two bike riders riding along. One rider tosses a water bottle to the other at an easy speed so the second rider can catch it. An observer on the ground watching this sees the bottle travel farther and at a faster speed than the riders see it.
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- Time dilation a real thing and it can be measured or proven in an experiment.
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- Richard Keating of the U.S. Naval Observatory did an experiment in 1971 by sending four cesium-beam atomic clocks twice around the world in a passenger aircraft. First flying from west to east with the rotation of the Earth, then from east to west against the rotation of the Earth.
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- When the clocks were returned home to the lab they were out of step to similar clocks that had remained in the lab by exactly the amount calculated for time dilation.
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- The moving clocks had been running faster than the stationary clocks. The westbound clocks gained 273 billionths of a second.
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------------------- The westbound atomic clocks gained 273 billionths of a second.
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------------------- tg = T / square root (1 - v^2 / c^2)
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------------------- T - tg = 273*10^-9
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------------------- The plane traveling at 667 miles per hour ( 2.98 meters / second)
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------------------- square root(1 - v^2 / c^2) = square root (1 - .0000000000009892388) takes a bigger computer than I have to calculate.
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- The passengers on board must age less compared to their relatives on the ground. ( This calculation is complicated by the fact that the airplanes were traveling relative to the Earth’s rotation and the fact that gravity is slightly weaker at the higher altitudes.)
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- Eastbound the slowing of the airborne clocks was sufficient to override the quickening due to weaker gravity. The eastbound clocks ran slow by 59 billionths of a second, so passengers were younger by that amount compared to those that stayed at home. The tampering with time in relativity is a real and lasting thing.
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- A few millionths of a second gained by east bound travel will not make you look much younger. Even astronauts traveling at 10% the speed of light will only lag by one day in 200 days compared with the clocks back on Earth.
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- In 1977 at the CERN atomic cyclotron in Switzerland accelerated a Muon, a heavy electron that spontaneously breaks up after 2 millionths of a second, to 99.94% the speed of light. The high-speed travel prolonged the moon’s life nearly 30 fold.
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- These same muons come to us from outer space crashing into our atmosphere as Cosmic Rays ( See Reference # 26 “Gamma Rays and Cosmic Rays” ). Because they are traveling at such high speeds their longer lives allow them to penetrate the Earth’s atmosphere and smash into the surface. These penetrating muons change our weather at low altitudes and cause genetic mutations in living animals. If it was not for time dilation maybe life would not have evolved here on Earth.
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- The constant speed of light is as close to a sure bet as science has found. It is confirmed to be constant in one part in 100,000,000,000,000,000,000. One part in 10^20.
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- If you take these scientific facts to an extreme in space travel you could start at age 20 accelerating constantly at the surface acceleration of gravity, 9.8 meters / sec^2. Travel to the Andromeda Galaxy which is 2 million lightyears away. Decelerate for a landing on your 50th birthday. Return to Earth in time for you 80th birthday, but when you arrive the Earth would have aged 4,000 years into the future.
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- The distance traveled to Andromeda and back is 4 million lightyears. There are 9.4605 * 10^15 meters per lightyear. Distance is 37 * 10^21 meters. 60 years is 1.896 * 10^9 seconds. If the space ship averages 99.99% the speed of light for the entire trip.
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--------------------------- tg = 60 years / square root (1 - .9999^2) = 60 / square root (1-.9998001) = 60 / square root ( .00019999) = 60 / .01414 = 4,242 years pass back on Earth.
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- Things would be different.
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- But, the speed of light would still be constant.
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------------------ Footnote:
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- Pythagoras was a Greek philosopher born on a Aegean island in 560 B.C. At the age of 31 he moved to southern Italy and started a cult. In fact it was Pythagoras that coined the word, “philosopher”.
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- Members of the cult were fond of scientific discover. Pythagoras was studying sound and he documented the math that defines the tone of a string dependent on its length and its tension.
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- He discovered irrational numbers such as the square root of two. “ No conceivable fraction, however complicated, will give the product of two when multiplied by itself.” The cult was sworn to keep this fact a secret.
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- He came up with his theorem that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its sides. For astronomy Pythagoras identified the morning star and the evening star as the same star called Aphrodite I ( It later became known as Venus).
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- He taught that the Earth was spherical and that the Sun, Moon, and planets did not partake in the same motion as the stars. Each had its own path and was at different distances from Earth. For 2,100 years spheres stuck as the determinant for planetary motions. Then, Kepler came along at corrected things.
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- May 25, 2019 37 397
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