Space-Time

Discussion

historical context

Newton's laws of motion and universal gravitation, the laws of conservation of energy and momentum, the laws of thermodynamics, and Maxwell's equations for electricity and magnetism were all more or less nearly complete at the end of the Nineteenth Century. They describe a universe consisting of bodies moving with clockwork predicatability on a stage of absolute space and time. They were used to create the machines that launched two waves of industrial revolution — the first one powered by steam and the second one powered by electric current. They can be used to deliver spacecraft to the ends of the solar system with hyper-pinpoint accuracy. They is mathematically consistant in the sense that no one rule would ever violate another. They agree with reality to a high degree of accuracy as tested in experiment after experiment.

At the end of the Nineteenth Century, physics appeared to be at an apex. Several people are reported to have said something like this

There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

Source: Unknown

This has been attributed to William Thomson, Lord Kelvin (1824–1907) in an address to the British Association for the Advancement of Science in 1900, but I haven't been able to find a primary source to back this claim up. A similar comment was made by the German-American scientist Albert Michelson (1852–1931) in 1894.

[I]t seems probable that most of the grand underlying principles have now been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice…. An eminent physicist has remarked that the future truths of physical science are to be looked for in the sixth place of decimals.

Albert Michelson, 1894

And again in 1903.

The more important fundamental laws and facts of physical science have all been discovered, and these are so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote…. [I]nstances might be cited, but these will suffice to justify the statement that "our future discoveries must be looked for in the sixth place of decimals".

Albert Michelson, 1903

It is said that Michelson's "eminent scientist" was Kelvin, but as I will discuss in a future in this book, Kelvin did not seem to believe that physics was nearing its end.

Frayed edges on the tapestry of Classical physics leading to Modern Physics …

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

postulates

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.

time dilation: photons see eternity pass in an instant

t' =  t
√(1 − v2/c2)

relativity of simultaneity

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

ℓ' = ℓ √(1 − v2/c2)

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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