Space-Time

Discussion

shorthand notation using gamma

γ =  1
 √ 
1 −  v2
  c2

therefore

t' = γt  &  ℓ' = 
γ

lorentz transformation

      t −  vx            
t' =  c2  = γ
t −  vx
  y' = y
 √ 
1 −  v2
c2  
  c2          
     
x' =  x − vt  = γ (x − vt)   z' = z
 √ 
1 −  v2
 
  c2          

velocity addition

u' =  u + v
1 + uv/c2

doppler effect

λ0  =  ƒ  = √ 
1 − v/c
λ ƒ0 1 + v/c

 

v  =  1 − (ƒ/ƒ0)2  =  1 − (λ0/λ)2
c 1 + (ƒ/ƒ0)2 1 + (λ0/λ)2

odds 'n' ends

quotes

Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That's relativity. Albert Einstein.

A man sits with a pretty girl for an hour, it seems like a minute. He sits on a hot stove for a minute, it's longer than any hour. That is relativity. Albert Einstein.

Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion. Albert Einstein.

Henri Poincaré (1854–1912) France
Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence…. The two propositions: "The earth turns round" and "it is more convenient to suppose the earth turns round" have the same meaning; there is nothing more in the one than in the other.
La Science et l'hypothèse.

Bertrand Russell (1872–1970) England
Since everyday life does not confront us with such swiftly moving bodies, Nature, always economical, had educated common sense only up to the level of everyday life.

spacetime interval, sign conventions

The spacetime interval is invariant. Prove it. (Appropriate for this book?)

Time is written first since it's considered the zeroth coordinate in four dimensional spacetime (t = x0, x = x1, y = x2, z = x3)

Δs2 = − (cΔt)2 + Δx2 + Δy2 + Δz2 Δs2 = (cΔt)2 − Δx2 − Δy2 − Δz2
mostly pluses (− + + +)mostly minuses (+ − − −)

displacement

Δr2 = Δx2 + Δy2 + Δz2

(in)equality interval type region(s)
(cΔt)2 > Δr2 timelike past and future
(cΔt)2 = Δr2 lightlike light cone
(cΔt)2 < Δr2 spacelike elsewhen
Spacetime Intervals

light cone

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