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5.1.2.       General Relativity & Astrophysics   <CJ chap 28.8 & not in text  >

5.1.2.1. Discussion

5.1.2.1.1.     Special relativity addresses observers moving with relative constant velocity only

5.1.2.1.2.     General relativity deals with cases where one observer is accelerated relative to the other

5.1.2.1.3.     Rotating platform:  Einstein argued that a rotating platform gives a non-Euclidian (curved) geometry

5.1.2.1.3.1. As one moves outward, the Lorentz contraction shortens circumferences to ever smaller values

5.1.2.1.3.2. Also as one moves outward, clocks slow down because of time dilation

5.1.2.1.3.3. Far from the center, where v is almost equal to c, the circumference  is near 0 & time stands still

5.1.2.1.3.4. So space and time in accelerated frames is unquestionably curved (not ‘flat’)

5.1.2.1.4.     Elevator experiment: Einstein compared an accelerated elevator to the same one in gravity with a=g

5.1.2.1.4.1. No experiment with regular matter would distinguish g from a as all mass has the same g

5.1.2.1.4.2. Yet light is not bent by gravity (as per Newtons equation) but light ‘appears’ bent with acceleration

5.1.2.1.4.3. Einstein argued that by symmetry, light should be bent the same amount by g as by a

5.1.2.1.4.4. This violates the Newton formula for gravity as light has a mass of zero

5.1.2.1.4.5. His prediction that light from a distant star is bent by the sun was verified

5.1.2.1.5.     Gravity (and acceleration) is thus seen as a warped space time where masses follow geodesics

5.1.2.1.6.     The integration of Einstein’s theory is still not reconciled with modern theories of other forces

5.1.2.2. Mathematical

5.1.2.2.1.     The circumferences is shortened by the Lorentz contraction: C = C₀ (1-v²/c²)^1/2 

5.1.2.2.1.1. and one can compute at what point the circumference begins to get smaller and at v=c is zero

5.1.2.2.2.     At larger distances from the center, time dilation effects slow time by t = t₀ /(1-v²/c²)^1/2 

5.1.2.2.2.1. where t is the observed length and  t₀ is the length in its own rest frame

5.1.2.2.3.     In both equations, v = r w where w is the angular velocity of the platform

5.1.2.3. Advanced

5.1.2.3.1.     The mathematical theory of curved spaces is called Riemannian geometry or differential geometry

5.1.2.3.2.     The fundamental concept is the metric g_mn  which is used to define scalar products thus length & angle

5.1.2.3.3.     Particles (as well as light) follow the shortest distances (called geodesics) in such curved spaces

5.1.2.3.4.     Einstein’s theory thus relates g_mn  for the space to T^mn  which is the energy momentum tensor density