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5.1.1.       Special Relativity   <CJ chap 28  >

5.1.1.1. Discussion

5.1.1.1.1.     Constancy of c, the velocity of light, to all observers presents a conflict between Newton & Maxwell

5.1.1.1.1.1. Maxwell EM equations predict that c = (e₀m₀)^-1/2  = 3E8 m/s in vacuum to all frames & observers

5.1.1.1.1.1.1.     Michelson & Morley repeatedly proved this was true using the earths motion: Explain

5.1.1.1.1.1.2.     Attempts to explain c=const. with ‘ether’ theories etc were flawed.

5.1.1.1.1.2. Newtonian space time is related by x’=x-Vt & t’=t   which implies v’ = v – V ie velocities add

5.1.1.1.1.2.1.     This is confirmed by our intuition and everyday experience – Examples of cars:

5.1.1.1.2.     Einstein assumed three postulates and allowed for a more general relationship for x & t

5.1.1.1.2.1. Assumption 1: The laws of physics are identical in inertially related (constant v) frames

5.1.1.1.2.2. Assumption 2: The speed of light in vacuum is a constant.

5.1.1.1.2.3. Assumption 3: The relationship between x & t in two frames is linear for the 4 dimensions

5.1.1.1.3.     Einstein showed that space (length) and time are not each invariant but transform as a 4 dim. vector

5.1.1.1.3.1. This 4-vector of space-time described an event for one observer & related it to another observer

5.1.1.2. Mathematical

5.1.1.2.1.     Lorentz Contraction: One can then show that length is contracted by  L = L₀ (1-v²/c²)^1/2  

5.1.1.2.1.1. where L is the observed length and  L₀ is the length in its own rest frame

5.1.1.2.2.     Time Dilation: One can also show that time is expanded by  t = t₀ /(1-v²/c²)^1/2 

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

5.1.1.2.2.2. These effects are only about 1% when one gets to a tenth of the speed of light: v/c =1/10

5.1.1.2.2.3. Below that relativity is essentially negligible. Yet effects explode near v=c.

5.1.1.2.3.     The old formula for KE = p²/(2m) is now replaced by: (E/c)² - P_x,² - P_y² - P_z² = m²c² = E²/c² - P²

5.1.1.2.3.1. Now using E²/c² - P² = m²c² to solve for E we get

5.1.1.2.3.2. which is the famous Einstein equation

5.1.1.2.4.     In relativity neither mass nor energy is separately conserved but only their combination via E=mc²

5.1.1.2.5.     The negative sign was ignored for 20 years until it was shown to correspond to ‘antimatter’

5.1.1.2.5.1. Antimatter is identical to matter except of opposite charge and it annihilates corresponding matter   

5.1.1.2.6.     Next we solve E²/c² - P² = m²c² for m (choose units with c=1): giving 3 cases:

5.1.1.2.6.1. E>p giving m >0 and v<c   This is ordinary matter and must move slower than c

5.1.1.2.6.2. E=p giving m = 0 and v=c  These massless particles, such as photons, always have v=c

5.1.1.2.6.3. E<p giving m imaginary and thus v>c are called tachyons and must move faster than light

5.1.1.2.6.4.  

5.1.1.3. Advanced

5.1.1.3.1.     The Lorentz transformation derived: x’ = L x where x = (ct, x, y, z) = (x⁰, x¹, x², x³) = x^m  

5.1.1.3.1.1. This set of four ‘coordinates’ of an event, is a 4 dimensional vector under L 

5.1.1.3.1.2. A sphere of light, ct=r must be seen the same by all observers thus c²t²-r² = invariant

5.1.1.3.1.3. Compute this in two dimensions to get (x’⁰, x’¹) = (L⁰₀, L⁰₁,/ L¹₀, L¹₁) (x⁰, x¹) then

5.1.1.3.1.4. One obtains (L⁰₀, L⁰₁,/ L¹₀, L¹₁) = (chj, shj / shj , chj  ) where th j = v/c

5.1.1.3.1.4.1.     because of ch²j - sh²j = 1   (compare to cos²q + sin²q = 1)

5.1.1.3.1.4.2.         

5.1.1.3.2.     The scalar product, defining the metric properties of the space is A B = g_mn A^mBⁿ where 

5.1.1.3.3.     The metric for this invariant is g_mn  is defined by g_mm = (+1, -1,-1,-1) and g_mn  =0 off diagonal

5.1.1.3.4.     Thus dt² = g_mn  dx^m dxⁿ is invariant and is called the proper time: dt² = c² dt² - dr²  

5.1.1.3.4.1.1.     because it gives the invariant time interval on a clock on the particle that is moving

5.1.1.3.4.2. As time is part of a 4 vector, we cannot effectively use it to take derivatives and must use

5.1.1.3.4.2.1.     dt  thus giving  a 4-vector velocity of v^m = c dx^m /dt (note that ‘c’ give it dimensions of vel)

5.1.1.3.4.2.2.     and one can verify that the invariant length of this vector is always c : g_mn v^mvⁿ = c² 

5.1.1.3.5.     The 4-vector momentum is thus defined as mass times velocity:  p^m  = m v^m then  g_mn p^mpⁿ = m²c² 

5.1.1.3.5.1. Thus energy & momentum form a 4 vector: (E/c, P_x, P_y, P_z) =P^m and transform like dx^m

5.1.1.3.6.     When g_mn p^mpⁿ = m²c² is written out it becomes: (E/c)² - P_x,² - P_y² - P_z² = m²c² = E²/c² - P²

5.1.1.3.6.1. This is the relativistic equation relating energy, momentum and mass that replaces E= p²/(2m)

5.1.1.3.6.2. Now using E²/c² - P² = m²c² to solve for E we get

5.1.1.3.6.2.1.       which is the famous Einstein equation

5.1.1.3.6.3. Next we solve E²/c² - P² = m²c² for m (choose units with c=1): giving 3 cases:

5.1.1.3.6.3.1.     E>p giving m >0 and v<c   This is ordinary matter and must move slower than c

5.1.1.3.6.3.2.     E=p giving m = 0 and v=c  These massless particles, such as photons, always have v=c

5.1.1.3.6.3.3.     E<p giving m imaginary and thus v>c are called tachyons and must move faster than light

5.1.1.3.6.4. Although m<0 (negative mass) is shown here as possible, it has never been observed

5.1.1.3.6.4.1.     It would result in gravitational repulsion rather than attraction with regular m>0 matter.