Announcements

1. Announcements • Project abstracts are graded Astronomy 201 Cosmology - Lecture 16

2. Lecture 16Special relativity III: space + time = space-time or what is now, then, and there ? Astronomy 201 Cosmology - Lecture 16

3. Einstein’s new relativity • Galileo: • The laws of mechanics are the same in all inertial frames of reference • time and space are the same in all inertial frames of reference • Einstein: • The laws of physics are the same in all inertial frames of reference • the speed of light in the vacuum is the same in all inertial frames of reference  time spans and distances are relative Astronomy 201 Cosmology - Lecture 16

4. Boost factor • Boost factor: • time dilation: • length contraction: • Examples: • v=0.1c  = 1.005  0.5% boost • v=0.5c   = 1.155  15.5% boost • v=0.9c   = 2.294  129.4% boost • v=0.999c   = 22.37 Astronomy 201 Cosmology - Lecture 16

5. Energy: E = m c2 • Newton: • kinetic energy: Ekin= ½ m v2 • v=0  Ekin= 0 • Einstein: • E =  m0 c2 • v=0   =1  E= m0 c2 “rest energy” • Ekin = (-1) m0 c2 • Example:The rest energy of an average person can cover the world energy need for 45 days !!!! Astronomy 201 Cosmology - Lecture 16

6. This is messy, so let’s clean up a bit • proper time: time measured by a clock at rest with respect to a specific observer clock at the fastest possible rate • proper length: length of an object as measured in its own rest frame  largest possible length • time and length in other inertial frames can be calculated by the so-called Lorentz transformation (i.e. multiplying with or dividing by the boost factor ) Astronomy 201 Cosmology - Lecture 16

7. Minkowski’s spacetime • As we have seen, time intervals, lengths, and simultaneity is relative and depend on the relative velocity of the observer. • velocity connects time and space • Let’s stop separating space and time, let’s rather talk about spacetime • spacetime is 4 dimensional, 3 spatial + 1 time dimensionbut is space and time really the same thing ? Astronomy 201 Cosmology - Lecture 16

8. light: x=ct ct x Minkowski diagram time space Astronomy 201 Cosmology - Lecture 16

9. world line of a particle ct x World lines — slowly moving Astronomy 201 Cosmology - Lecture 16

10. world line of a particle ct x World lines — fast moving Astronomy 201 Cosmology - Lecture 16

11. ct x Faster than speed of light ? Astronomy 201 Cosmology - Lecture 16

12. ct x World lines — accelerated Astronomy 201 Cosmology - Lecture 16

13. ct x World lines — decelerated Astronomy 201 Cosmology - Lecture 16

14. y x geometrical interval (x1,y1) y1 s2= y2 + x2 y (x2,y2) y2 x x1 x2 Astronomy 201 Cosmology - Lecture 16

15. ct x Spacetime interval (x1,t1) ct1 s2= (ct)2 – x2 ct (x2,t2) ct2 x x1 x2 Astronomy 201 Cosmology - Lecture 16

16. Spacetime interval • – sign: difference between space and time • s2 is invariant under Lorentz transformation • for particle moving at speed of light:x = ct s2= 0 light like (null) distance s2= (ct)2 – x2 Astronomy 201 Cosmology - Lecture 16

17. Character of spacetime intervals • s2>0  ct > x • spatial distance can be traveled by speed of light • there exist an inertial frame, in which the two events happen at the same position • but they never happen simultaneously time like distance • s2<0  ct < x • spatial distance cannot be traveled by speed of light • there exist an inertial frame, in which the two events happen simultaneously • but they never happen at the same place space like distance Astronomy 201 Cosmology - Lecture 16

18. elsewhere elsewhere Future, past, and elsewhere Future s2>0 s2<0 s2<0 Past s2>0 Astronomy 201 Cosmology - Lecture 16

19. All observers agree that B is in the past of A and C is in the future C D • Some see A happen first, some see D happen first B Principle of causality • Cause must always precede the effect A must not influence D and vice versa nothing can movefaster than speed of light A Astronomy 201 Cosmology - Lecture 16

20. Announcements • Project abstracts are graded Astronomy 201 Cosmology - Lecture 16