Space-Time

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Discussion

Relativity is almost entirely the work of Albert Einstein (1879-1955) Germany.

Problems trying to resolve the conflict between the two major realms of Classical physics: Newtonian mechanics and Maxwell's theory of electrodynamics. This proved to be impossible using the traditional concepts of space and time. Einstein developed a new view of time first and then space.

  1. The laws of physics must be the same in all inertial reference frames. This statement is known as the principle of relativity.
  2. The speed of light in a vacuum has the same value in all inertial reference frames regardless of the velocity of the observer or the velocity of the source.

relativity of simultaneity

time dilation

t' =  t
√ ⎛
⎝		 1 −  v2 ⎞
⎠
c2

length contraction: a merry go round is smaller that 2πr at its circumference.

ℓ' = ℓ  √ ⎛
⎝		 1 −  v2 ⎞
⎠
c2

shorthand notation using gamma

γ =  1
√ ⎛
⎝		 1 −  v2 ⎞
⎠
  c2

therefore

t' = γt & ℓ' =  ℓ
γ

lorentz transformation

x' =  x − vt  = γ(x − vt)
√ ⎛
⎝		 1 −  v2 ⎞
⎠
  c2        
y' =      y      = y      
z' =      z      = z      
      t −  v  x        
t' =  c2  = γ ⎛
⎝		  t −  v  x ⎞
⎠
√ ⎛
⎝		 1 −  v2 ⎞
⎠		 c2
  c2        

velocity addition

u' =  u − v
1 −  uv
  c2

doppler effect

ƒ  = √ ⎛
⎝		 1 ± v/c ⎞
⎠		 ⇒ v  =  (ƒ/ƒ0)2 ∓ 1
ƒ0 1 ∓ v/c c (ƒ/ƒ0)2 ± 1

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.

Jules 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.


[magnify]

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Gravity Probe B is an earth-orbiting relativity experiment developed by NASA and Stanford University to test Albert Einstein's general theory of relativity. The media have reported that Gravity Probe B has gone in search of "the missing inch". What the hell are they talking about? Do the following calculations for a satellite in a circular, low earth orbit (LEO) at an altitude of 642 km. (Ignore the rules of significant digits when calculating distances. Record these answers with as much precision as your calculator will allow.)
    1. Using classical euclidean geometry, determine the circumference of this orbit.
    2. Using classical Newtonian mechanics, determine the speed of a satellite in such an orbit.
    3. Using special relativity, determine the amount by which the circumference of the orbit has been shortened due to the satellite's motion. State your answer in …
      1. centimeters
      2. English inches

    Solution …

    1. Circumference …
       
      C = 2πr
      C = 2π(6,378,100 m + 642.000 m)
      C = 44108589.174932 m
       
    2. The centripetal force is supplied by gravity for a satellite in circular orbit. Set the centripetal force formula equal to Newton's law of universal gravitation.
               
      Fc = Fg ⇒ mv2  =  Gm1m2
      r r2
               
      Solve for speed (v), plug in numbers, and calculate an answer.
                       
      v =  √ ⎛
      ⎝      		 Gm ⎞
      ⎠      		  = √ ⎛
      ⎝      		 (6.67 × 10−11 Nm2/kg2)(5.98 × 1024 kg) ⎞
      ⎠
      r (6,378,100 m + 642,000 m)
      v =  7538 m/s
                       
    3. Apply the length contraction formula to the circumference of the orbit moving at the speed of the satellite.
       
      ℓ =  ℓ0 √ ⎛
      ⎝      		 1 −  v2 ⎞
      ⎠      		  
      c2
      ℓ =  44108589.174932 m √ ⎛
      ⎝      		 1 −  (7538 m/s)2 ⎞
      ⎠
      (3.00 × 108 m/s)2
      ℓ =  44108589.161009 m
       
      Subtract the contracted length from the rest length.
       
      Δℓ = ℓ0 − ℓ
      Δℓ = 44108589.174932 m − 44108589.172716 m
      Δℓ = 0.013923 m
       
      1. Convert to centimeters with a simple shift of the decimal point.
         
        Δℓ = 1.4 cm
         
      2. Convert to English inches using the definition of an inch. (An inch is 0.0254 m by definition.)
         
        Δℓ = 0.013923 m ÷ 0.0254 m/in. = 0.55 inches
         
        OK, so it's closer to half an inch than an inch, but its of the right order. The more important thing to note is that Gravity Probe B is not looking for space-time distortion caused by motion, it's looking for space-time distortion caused by gravity. It's not trying to confirm the predictions of special relativity, it's trying to confirm the predictions of general relativity. The two effects are of about the same size, however, and together they make an inch.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. Imagine that technology has advanced to the point where humans can travel through space at speeds approaching the speed of light.
    1. At what fraction of the speed of light would a second of a space traveler's existence equal a day of existence on earth?
    2. How much would a space traveler age on a journey to the Large Magellanic Cloud (a dwarf galaxy 169,000 light years from earth) were they to travel at this speed?
    3. How much would the earth age on this journey?

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