The result is; distances are contracted in the direction of motion. t = t/(1 v 2 /c 2 ) 0.5 d = d(1- v 2 /c 2 ) 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 v 2 /c 2 ) 0.5 t earth = t ship /(1 (297,000/300,000) 2 ) 0.5 t ship = (9 years)(0.141) t ship = 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 9 th 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.
30 How can the astronaut get to Alpha Centauri and back in less than 9 years. 1. She had to travel faster than light 2. The distance was smaller for her 3. It really took 9 years, but it only seemed like 1.3 years. 40 33% 33% 33% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 21 22 23 24 25 26 27 28 29 30 1 2 3
The astronaut sees length contraction. The distance from Alpha Centauri to Earth is like a meter stick. And it contracted. D ship = D earth (1- (297,000/300,000) 2 ) 0.5 D ship = 9 light years(0.14) D ship = 1.27 light years (round trip)
What about light? For a photon of light that travels at c = c. D light = (9 light years)(1-(c/c) 2 ) 0.5 D light = (9 light years)(0) D light = 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?
30 For a photon of light, what would the distance from Earth to Andromeda be? 1. Zero distance 2. 2.2 million light years 3. Quite a bit less than 2.2 million but bigger than zero. 33% 33% 33% 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 21 22 23 24 25 26 27 28 29 30 1 2 3
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.
In 2-D any motion in x and y can be described by the hypotenuse of a triangle. Δs 2 = Δx 2 + Δy 2 Δy Δx
This is Pythagorean's Theorem In 3 dimensions it is just as easy to figure out the location of the an object when it has moved. This is just Pythagorean s Theorem in 3D Δs 2 = Δx 2 + Δy 2 + Δz 2 The universe we live in is NOT 3D. It is actually 4-dimensional. Time must be included
The Space-Time Metric In 4-D space-time Pythagorean's Theorem becomes: Δs 2 = Δx 2 + Δy 2 + Δz 2 - c 2 Δt 2 Here Δs 2 describes your path through the universe. So, right now, sitting in your seats, Are you moving relative to the others in the classroom?
Are you moving relative to your classmates. 30 1. No, I am not changing my position so I must not be moving. 2. Yes, because time is going by, so I am moving. 3. No, you are actually co-moving through time with everyone. 0 30 33% 33% 33% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3
You are actually co-moving with everyone. Even when you perceive yourself as sitting still, you are moving in 4-D because time is going by. So your after 10 seconds you have moved: Δs 2 = Δx 2 + Δy 2 + Δz 2 - c 2 Δt 2 where Δx 2 = Δy 2 = Δz 2 = 0 But Δs 2 = -(3 x 10 8 m/s) 2 (10 s) 2
Δs 2 = - 9 x 10 18 meters 2 But since everyone in the classroom has the same time, they have all moved an equal distance as you have. You are all co-moving. BUT you are moving in Space-Time. Because time is going by. For someone that is moving past you at close to the speed of light, their clocks will run at a different time. But they will also see you moving relative to them.
Everyone agrees on Δs 2 The person flying past, will see your clock running slow, but they will also see you moving in some direction, maybe in Δx. The difference between what the clocks read and the relative motion, always work in such a way as to make all observers agree on Δs 2 This metric only works when Space-Time is Euclidean. When the total degrees in a triangle equals 180 o.
Non-Euclidean Geometry This right triangle on the surface of a sphere has a total of 270 o. It is not on a flat plane.
When there are no accelerations, spacetime is Euclidean, and the space-time metric from before works great. This is true for special relativity, since there are no accelerations. But it is not true if you try to take gravity into account, because gravity produces accelerations. Einstein to the rescue. General Relativity (1915)
This is why Einstein is considered a genius. F = ma (mass is defined by inertia. The objects resistance to motion.) F = -Gm 1 m 2 /r 2 (mass is defined by the gravitation force it exerts.) Everyone assumes that inertial and gravitational mass are the same. And so did Einstein. But he realized what this really meant.
The Equivalence Principle Suppose that you have a goofy friend that always thinks aliens are about to abduct her. You are at a party and you and your friend both end up crashing. The next morning you both wake up and you are in a small room with no windows. You worry that you might be in a Saw movie, but your friend says that aliens abducted you both, and right now you are out in the middle of interstellar space, thousands of light years from Earth.
30 Can you prove to your friend that you are not in the middle of interstellar space? 1. Drop your cell phone, if it falls like normal, you are probably on Earth. 2. Look for alien finger-prints, if you can t find any, your probably on Earth 3. Argue using Occam s razor. The simplest answer is usually correct. 4. None of the above 40 25% 25% 25% 25% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 21 22 23 24 25 26 27 28 29 30 1 2 3 4
But it turns out that your goofy friend is actually smarter than you think. She immediately says that you might be in a spaceship which is constantly accelerating at 9.8 m/s 2. The exact same acceleration that gravity has on the surface of the Earth. Could you come back with a way to disprove this idea?
30 Idea to disprove goofy friend s acceleration statement. 1. Drop your cell phone. Even if you feel the acceleration on your feet, there is no gravity. The cell phone will still float. 2. No there is no way to disprove this. 50% 50% 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 21 22 23 24 25 26 27 28 29 30 1 2
There is no way to tell the difference.
Dropped ball appears to fall to the floor like on Earth. But actually the floor is moving up to hit the ball. Everything is moving up at 100 m/s One second later the ball is still moving at 100 m/s but you and the spaceship are now going 109.8 m/s
Any experiment that you try gives you the same result as if you were on Earth. BUT, What if we use light? As laser beam cross the room, the room moves up. From inside it looks like the laser spot hits lower on wall.
Inside spaceship the laser beam like this.
Newton s Law says that gravity has no effect on light. Light is massless, so no Force of gravity acting on it. But Einstein takes the Equivalence Principle to the Extreme. He says there is absolutely NO way to tell if you are in a gravity field, or being accelerated. He predicts that light will bend the exact same way in a box on Earth. He predicts that gravity will bend light. At the time no one had any idea that this bending of light occurred.
But there is more. Newton s law of gravity doesn t bend light. So if light bends due to gravity, Newton s Law must be wrong. What is happening? Consider this. A person, Mr Green, moving along side the spaceship, at the velocity that your spaceship had when the light left the laser will say: The laser traveled in a straight line. You are the one that moved up.
Mr Green sees the shorter, straight, green path and Mr. Red sees the longer, curved, red path.
30 How can Mr. Green and Mr. Red see two different paths for the laser beam. 1. The speed of light is different for Mr. Red and Mr. Green 2. The paths look different but they are actually the same length. 3. Mr. Red s clock is running slower than Mr. Green s 0 30 33% 33% 33% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3
In an accelerated frame, time runs slow compared to a non-accelerated frame. The Equivalence Principle tells us that there is no difference between the accelerating ship and the gravity on the Earth. Einstein s conclusion: Near massive objects, time runs slow. It is this slower rate of time, that makes things move in a curved path. There is NO FORCE OF GRAVITY. Mass causes Space-Time to bend. (i.e. Clocks to run slow.)