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08Relativity

# 08Relativity - Special Theory of Relativity Up to ~1895...

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P460 - Relativity 1 Special Theory of Relativity Up to ~1895, used simple Galilean Transformations x’ = x - vt t’ = t But observed that the speed of light, c, is always measured to travel at the same speed even if seen from different, moving frames c = 3 x 10 8 m/s is finite and is the fastest speed at which information/energy/particles can travel Einstein postulated that the laws of physics are the same in all inertial frames. With c=constant he “derived” Lorentz Transformations u light sees v l =c moving frame also sees v l = c NOT v l = c-u

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P460 - Relativity 2 “Derive” Lorentz Transform Bounce light off a mirror. Observe in 2 frames: A velocity=0 with respect to light source A’ velocity = v observe speed of light = c in both frames barb2right c = distance/time = 2L / t A c 2 = 4(L 2 + (x’/2) 2 ) / t’ 2 A’ Assume linear transform (guess) x’ = G(x + vt) let x = 0 t’ = G(t + Bx) so x’=Gvt t’=Gt some algebra (ct’) 2 = 4L 2 + (Gvt) 2 and L = ct /2 and t’=Gt gives (Gt) 2 = t 2 (1 + (Gv/c) 2 ) or G 2 = 1/(1 - v 2 /c 2 ) mirror L A A’ x’
P460 - Relativity 3 Lorentz Transformations Define β = u/c and γ = 1/ sqrt(1 - β 2 ) x’ = γ (x + ut) u = velocity of transform y’ = y between frames is in z’ = z x-direction. If do x’ barb2right x then t’ = γ (t + β x/c) “+” barb2right “-”. Use common sense can differentiate these to get velocity transforms v x ’ = (v x - u) / (1 -u v x / c 2 ) v y ’ = v y / γ / (1 - u v x / c 2 ) v z ’ = v z / γ / (1 - u v x / c 2 ) usually for v < 0.1c non-relativistic (non-Newtonian) expressions are OK. Note that 3D space point is now 4D space-time point (x,y,z,t)

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P460 - Relativity 4 Time Dilation Saw that t’ = γ t. The “clock” runs slower for an observer not in the “rest” frame muons in atmosphere. Lifetime = τ = 2.2 x 10 -6 sec c τ = 0.66 km decay path = βγτ c β γ average in lab
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08Relativity - Special Theory of Relativity Up to ~1895...

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