1. Goal: To get to know the ins and outs of relativity (relatively speaking) Objectives: 1)Black holes vs space-time 2)General Relativity 3)Special Relativity 4)Velocities of relativistic objects when you are relativistic

2. Black Hole A black hole is an object that is either so massive or so dense that the escape velocity on its surface is greater than the speed of light. As Einstein discovered nothing can travel faster than the speed of light. Therefore NOTHING, not even light can escape from a black hole!

3. No escape! The radius at which the escape velocity is exactly the speed of light is called the Schwarzschild radius. The Schwarzschild radius is an event horizon. An event horizon is a surface where if something were to pass through it, it is gone (event horizon = goodbye forever).

4. But there’s more! Mass warps space. Time is relative to space. Therefore masses warp time also! Tobject = Tuniversal / (1 – r s / r) 1/2 Where r s is the Schwarzschild radius (the radius of the event horizon of a black hole) r s = 1.5 km * Mass of object / Mass of our sun

5. Black hole astrophysics What would happen if we swapped our sun for a black hole of exact equal mass? A) The earth would be sucked into the black hole B) Time on the earth would slow down C) The earth would be slingshot out of the solar system D) Nothing would happen to the orbit of the earth or the clocks on earth.

6. Black hole astrophysics What would happen if we swapped our sun for a black hole of exact equal mass? D) Nothing would happen to the orbit of the earth or the clocks on earth. Black holes are not vacuum cleaners. They obey gravity just like everything else. In fact it is harder to run into a black hole because it is so frikkn small (diameter of 3 km for one the mass of out sun).

7. Is that all? Nope (but that is all for black holes for now, sorry). Special Relativity Clocks progress at a rate RELATIVE to their position in space. Velocity slows the progress of an object’s clock so that: Tobject = Tuniverse / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2

8. Example 1 Tobject = Tuniverse / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 If v = 0.99c then what is the value for gamma?

9. Example 1 Tobject = Tuniverse / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 If v = 0.99c then what is the value for gamma? Gamma = 1 / (1 – (0.99c) 2 /c 2 ) 1/2 = 7.09 Note that gamma has no units – it is just a factor.

10. Example 2 A spacecraft flies to the Alpha Centauri star system at a velocity of 0.9999 c. Since Alpha Centauri is 4.3 lightyears away find: A) The time that observers on the earth think it took the spacecraft to get there. B) The amount of time that passes by for the crew of the ship. Remember the speed of light is 1 lightyear/year

11. Example A spacecraft flies to the Alpha Centauri star system at a velocity of 0.9999 c. Since Alpha Centauri is 4.3 lightyears away find: A) The time that observers on the earth think it took the spacecraft to get there. D = VT, T = D/V = 4.2 lightyears / (0.9999 lightyears/year) = 4.2 years B) The amount of time that passes by for the crew of the ship. Tship = Tuniverse / gamma Gamma = 1/(1-(0.9999 c / c) 2 ) 1/2 Gamma = 70.7 So, T = 4.2 years / 70.7 = 0.059 years = 21.7 days

12. Lorenz contraction Also, the sizes of moving objects are also RELATIVE to their velocities in space. Linmotion = Lrest / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 So (in the direction they are moving) their length appears to shrink. However their other dimensions stay the same. A sphere for example would appear as a saucer…

13. Lorenz example Linmotion(observed) = Lrest / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 A spacecraft which is 700 m long is traveling directly towards you at a velocity of 0.999 c. What is the observed length of the spacecraft by an observer on earth? What is the length of the spacecraft as observed by the crew of the craft (hint what is the velocity of the spacecraft from the perspective of the crew)?

14. Lorenz example Linmotion = Lrest / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 A spacecraft which is 700 m long is traveling directly towards you at a velocity of 0.999 c. What is the observed length of the spacecraft by an observer on earth? Lobserved = Lrest / gamma = 700 m / gamma Gamma = 1 / (1 – v 2 /c 2 ) 1/2 = 1 / (1 – (0.999c) 2 /c 2 ) 1/2 = 22.4 So, L = 700 m / 22.4 = 31.3 m What is the length of the spacecraft as observed by the crew of the craft (hint what is the velocity of the spacecraft from the perspective of the crew)? 700 m – it appears at rest to any actually on the spaceship

15. But what happens… If you are traveling a fraction of the speed of light and something flies by you? First, a conceptual question, suppose light goes by you when you are traveling 90% of the speed of light. What velocity does the light appear to be traveling?

16. But what happens… If you are traveling a fraction of the speed of light and something flies by you? First, a conceptual question, suppose light goes by you when you are traveling 90% of the speed of light. What velocity does the light appear to be traveling? - the speed of light! Light always appears to go the speed of light in a vacuum! This is why lengths contract and time slows down. If earth was watching though, they would see light move past you at 0.1 c faster than you…

17. Okay now for the equation: Warning: the book is REALLY confusing about this… If you are moving quickly and shot something out at what appears to you as a velcoity of Vob, then the actual velocity of the object is: Vactual = (Vyours + Vob) 1 + Vyours * Vob / c 2 Vyours is your velocity compared to the rest frame of the universe Vactual is the object’s (the one you shot out) velocity compared to the rest frame of the universe

18. Final sample of day You are moving at 75% of the speed of light and you shot out a probe which moves with a velocity compared to you of 90% of the speed of light. What speed is the probe actually moving at (compared to rest frame of universe)?

19. Final sample of day You are moving at 75% of the speed of light and a spacecraft flies by you at 90% of the speed of light. What speed does it appear to be moving at? Vactual = (Vyours + Vob) 1 + Vyours * Vob / c 2 Vactual = (0.75c + 0.90 c) / (1 + 0.75 * 0.9) Vactual = 1.65 c / 1.675 = 0.985 c (or 98.5% the speed of light)

20. Some other famous stuff You have probably heard that E = mc 2 Too bad it is not completely correct… This is only the rest energy of matter. Yes, this means that matter is a form of energy!

21. However The total energy is E = Gamma mc 2 And the kinetic energy is: KE = (Gamma – 1) mc 2 And momentum is: p = gamma * mv So, most of relativity is multiplying or dividing by gamma!

22. Conclusion Relativity is strange but cool, and not as much math as you might think. You basically just have to know how to find gamma, and apply that to everything.