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Quiz #8 • On the H-R diagram, a high mass star that is evolving off the main sequence will become redder in color and have and a constant luminosity. Write out the equation for luminosity in terms of surface temperature and radius. Then discuss which parameter is primarily controlling the luminosity as the star evolves.
How about this? • You are on the Earth and a friend flies past the Earth in a spaceship which is traveling 200,000 km/s. You decide to signal your friend by shining a laser beam past the ship. The laser beam is light, so it travels at 300,000 km/s.
How fast does your friend see the laser beam moving? 30 • 100,000 km/s forward • 300,000 km/s forward • 100,000 km/s backward 0 30
Einstein’s two postulates of Special Relativity (1905) • 1) For objects moving with a constant velocity (no accelerations) all motion is relative. • 2) The speed of light in a vacuum is constant for all observers, no matter how they are moving.
Postulate 1, tells us that there is no such thing as an absolute rest frame. There is nowhere in the universe where you can say, that a thing is not moving. It has zero velocity, absolutely. All you can only measure is relative motion. • So, on a plane you do not feel like you are moving. You look out the window and it looks like the ground is scrolling past you in the opposite direction that you are sitting. Which is really moving? It is impossible to say.
Consider a person on a train moving at 100 MPH relative to the ground, and a second person on the ground watching. Mr. Green throws the ball up in the air. 100 MPH
Who will see the ball travel a greater distance? 30 • Mr. Green (on train) • Mr. Red (on ground) • Both measure the same distance traveled. 0 30
Mr. Green sees the ball move on the green path and Mr. Red sees the ball move on the red path
Both Mr. Green and Mr. Red agree on the time that the ball is in the air. • But Mr. Green sees the ball travel a much smaller distance than Mr. Red. • This is because Mr. Green sees the ball moving only in the up-down direction. Mr. Red sees the ball moving up-down and also to the right at 100 MPH. • This means the measured speed of the ball is much larger for Mr. Red than it is for Mr. Green.
Here is how speed is measured. • Velocity = distance/time (example miles/hour) • V = D/t • We can rearrange this equation to read. • D = V*t • Both Red and Green agree on the flight time, t • Red sees a bigger distance traveled, DR > DG • So this means that vR > vG And it is because Mr. Red sees the ball moving right at 100 MPH. • That’s the way our normal world works.
But what happens when the ball is replaced by light and the train is now traveling at nearly the speed of light, c. mirrors V ~ c
Mr. Green sees the light follow the green path and Mr. Red sees the light follow the red path
Clearly, Mr. Red sees the light move a greater distance than Mr. Green. • BUT… Here in lies the problem. • This is light. Both Mr. Green and Mr. Red agree that the light is moving at c = 300,000 km/s, vR = DR/t and VG =DG/t So, DR > DG but vR = vG = c DR/t = DG/t How can this be?
How can this be? 30 • They have to measure a different velocity for light • The distance traveled must be the same • The flight time must be different. 0 30
Time is not the same for both observers! DR/tR = DG/tG Since DR > DG Then tR > tG Time moves at a different rate. When Mr. Green looks at his clock he may see 1 minute go by. But when he looks at Mr. Red’s clock he sees that 2 minutes have gone by. And not just clocks… Mr. Green sees Mr. Red moving around in fast motion.
So if time runs at different speeds depending on whether you are moving or not, why don’t we ever see this effect? • The effect only becomes obvious at speeds close to the speed of light. • If you move, even in a jet at speeds faster than the speed of sound, it is still very slow compared to the speed of light. • Consider being on a train moving 100 MPH. If the mirrors in the light experiment were separated by 1.5 meters, or an up-down distance of 3 meters.
Light travels at 3 x 108 m/s. So it would take: • D = vt or t = D/v = 3m/3 x 108 m/s t = 1 x 10-8 seconds to go up and down. In that amount of time, the train, moving at 100 MPH, has only moved 4 x 10-7 meters. That’s 1000 times smaller than the width of a human hair !!!
So the red triangle would have a base that is the size of an atom. In our normal world, we never see the time dilation effects. 1000 times smaller than the width of a human hair.
The effect is measurable using today’s technology • Atomic clocks have to be corrected according to their latitude on the Earth. • The Global Positioning Satellites (GPS) have to correct for time dilation effects since they are moving rapidly in orbit around the Earth. • In particle accelerators, radio-active isotopes live much longer than in the lab, when they are moving close to c.
When speeds are close to “c”, not only does time run slow, but distances are also compressed. Meter stick When Mr Red looks at both ends of the meter stick at the same time, the light from the far away end had to leave before the light from the close end. Because it has a greater distance to travel.
When speeds are close to “c”, not only does time run slow, but distances are also compressed. Meter stick v ~ c Because of the different travel times, when Mr. Red sees the close end of the meter stick aligned with him, the light coming from the far end had to leave before the stick was lined up. This makes the meter stick appear shorter than a meter to Mr. Red.
The result is; distances are contracted in the direction of motion. • t’ = t/(1 – v2/c2)0.5 • d’ = d(1- v2/c2)0.5 • These are the Lorentz equations.
The Twin-Paradox. • A woman astronaut is going to fly to Alpha Centauri and back. (round trip = 9 light years) • She is going to be traveling at 99% the speed of light, v = 297,000 km/s. The day she is ready to leave she gives birth to identical twins. • One of the twins stays behind on the Earth with the husband. The other twin heads out to Alpha Centauri with the mother. • At this speed, clocks on board the ship run at a much slower rate than the clocks on Earth.
Time on the ship when traveling at 99% the speed of light. • t’ = t/(1 – v2/c2)0.5 • tearth = tship/(1 – (297,000/300,000)2)0.5 • tship = (9 years)(0.141) • tship = 1.27 years
When the baby that went to Alpha Centauri and back returns home, it is 15 months old. Just starting to talk and walk arounds. • The identical twin who stayed on Earth, is now celebrating her 9th birthday. Here sister is almost eight years younger than her. • How can this be? It takes light 4.5 years to reach us from Alpha Centauri. Yet the mother and twin traveled to Alpha Centauri and back in about 1 year and 3 months.
How can the astronaut get to Alpha Centauri and back in less than 9 years. 30 • She had to travel faster than light • The distance was smaller for her • It really took 9 years, but it only seemed like 1.3 years. 0 40
The astronaut sees length contraction. • The distance from Alpha Centauri to Earth is like a meter stick. And it contracted. Dship = Dearth(1- (297,000/300,000)2)0.5 Dship = 9 light years(0.14) Dship = 1.27 light years (round trip)
What about light? • For a photon of light that travels at c = c. • Dlight = (9 light years)(1-(c/c)2)0.5 • Dlight = (9 light years)(0) • Dlight = 0. • We say the distance between Earth and the Andromeda Galaxy is 2.2 million light years. What would a photon say the distance is?
How about time? • t’ = t/(1-1)0.5 • t’ = t/0 Division by zero is undefined. • But as you get extremely close to the speed of light, the time in the outside world approaches infinity. • So it takes an infinite amount of time to go nowhere. • A photon of light would be everywhere in the universe at the same time.
Space-time. • In relativity, distances (space) and time get tangled together. You can’t say where you are without saying how your clock is running. It is part of the coordinate system. Four dimensional Space-time • Sometimes clocks run slow and sometimes distances contract. Sometimes both happen. But there is one thing that is invariant. The Metric distance.
