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Understanding Spacetime: Special Relativity in Astronomy 201

This lecture delves into the principles of Einstein's special relativity, exploring space, time, and their interrelation as spacetime. Key concepts include the boost factor, time dilation, and length contraction, providing mathematical examples of relativistic effects at varying speeds. We will also examine energy equivalence through the equation E=mc² and explore Minkowski's spacetime with graphical representations. Lastly, this lecture discusses the implications of relativistic physics on causality and the nature of simultaneity for different observers within the framework of cosmology.

elijah-contreras
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Understanding Spacetime: Special Relativity in Astronomy 201

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  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

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