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The result is; distances are contracted in the direction of motion.

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- tâ€™ = t/(1 â€“ v2/c2)0.5
- dâ€™ = d(1- v2/c2)0.5
- These are the Lorentz equations.

- 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.

- tâ€™ = t/(1 â€“ v2/c2)0.5
- tearth = tship/(1 â€“ (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.

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- 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.

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- 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)

- 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?

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- Zero distance
- 2.2 million light years
- Quite a bit less than 2.2 million but bigger than zero.

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- tâ€™ = t/(1-1)0.5
- tâ€™ = 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.

- In relativity, distances (space) and time get tangled together. You canâ€™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.

Î”s2 = Î”x2 + Î”y2

Î”y

Î”x

- In 3 dimensions it is just as easy to figure out the location of the an object when it has moved. This is just Pythagoreanâ€™s Theorem in 3D
Î”s2 = Î”x2 + Î”y2 + Î”z2

The universe we live in is NOT 3D. It is actually 4-dimensional. Time must be included

- In 4-D space-time Pythagorean's Theorem becomes:
Î”s2 = Î”x2 + Î”y2 + Î”z2 - c2Î”t2

Here Î”s2describes your path through the universe. So, right now, sitting in your seats, Are you moving relative to the others in the classroom?

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- No, I am not changing my position so I must not be moving.
- Yes, because time is going by, so I am moving.
- No, you are actually co-moving through time with everyone.

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- 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:
Î”s2 = Î”x2 + Î”y2 + Î”z2 - c2Î”t2

where Î”x2 = Î”y2 = Î”z2 = 0

But Î”s2 = -(3 x 108 m/s)2(10 s)2

- Î”s2 = - 9 x 1018 meters2
- 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.

- The person flying past, will see your clock running slow, but they will also see you moving in some direction, maybe in Î”x.
- The difference between what the clocks read and the relative motion, always work in such a way as to make all observers agree on Î”s2
- This metric only works when Space-Time is Euclidean. When the total degrees in a triangle equals 180o.

This right triangle on the surface of a sphere has a total of 270o. It is not on a flat plane.

- When there are no accelerations, space-time 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)

- F = ma (mass is defined by inertia. The
objects resistance to motion.)

F = -Gm1m2/r2 (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.

- 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.

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- Drop your cell phone, if it falls like normal, you are probably on Earth.
- Look for alien finger-prints, if you canâ€™t find any, your probably on Earth
- Argue using Occamâ€™s razor. The simplest answer is usually correct.
- None of the above

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- 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/s2. The exact same acceleration that gravity has on the surface of the Earth.
- Could you come back with a way to disprove this idea?

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- Drop your cell phone. Even if you feel the acceleration on your feet, there is no gravity. The cell phone will still float.
- No there is no way to disprove this.

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One second later the ball is still moving at 100 m/s but you and the spaceship are now going 109.8 m/s

Everything is moving up at 100 m/s

- 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.

- 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.

- 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.

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- The speed of light is different for Mr. Red and Mr. Green
- The paths look different but they are actually the same length.
- Mr. Redâ€™s clock is running slower than Mr. Greenâ€™s

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- 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.
- Einsteinâ€™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.)