Physics 2D Lecture Slides Jan 14 Vivek Sharma UCSD Physics
“Fixing” (upgrading) Newtonian mechanics • To confirm with fast velocities • Re-examine – Spacetime : X,Y,Z,t – Velocity : V x , V y , V z – Momentum : P x , P y , P z – (Proper) Rest Mass – Acceleration: a x , a y , a z – Force – Work Done & Energy Change – Kinetic Energy – Mass IS energy • Nuclear Fission • Nuclear Fusion • Making baby universes Learning about the first three minutes since the beginning of the universe
Lorentz Transformation Between Ref Frames Inverse Lorentz Transformation Lorentz Transformation = γ − = γ + x ' ( x v t ) x ( x ' v t ) = = y ' y y y ' = = z ' z z z ' v x v x ' = γ − = γ + t ' t t t ' 2 2 c c As v → 0 , Galilean Transformation is recovered, as per requirement Notice : SPACE and TIME Coordinates mixed up !!!
Lorentz Transform for Pair of Events S S’ ruler x X ’ x 2 x 1 Can understand Simultaneity, Length contraction & Time dilation formulae from this Time dilation: Bulb in S frame turned on at t 1 & off at t 2 : What ∆ t’ did S’ measure ? two events occur at same place in S frame => ∆ x = 0 ∆ t’ = γ ∆ t ( ∆ t = proper time) Length Contraction: Ruler measured in S between x 1 & x 2 : What ∆ x’ did S’ measure ? two ends measured at same time in S’ frame => ∆ t’ = 0 ∆ x = γ ( ∆ x’ + 0 ) => ∆ x’ = ∆ x / γ ( ∆ x = proper length)
Lorentz Velocity Transformation Rule − ' ' ' x x dx = = In S' frame, u 2 1 S and S’ are measuring x' − ' ' ' t t dt ant’s speed u along x, y, z 2 1 axes v = γ − = γ − ' dx ( dx v d t ) , dt ' ( dt dx ) 2 c − dx vdt S’ = S u , divide by dt' v x' v − dt dx u 2 c − u v = u x x' v u − 1 x 2 c = − For v << c, u u v x' x (Gali lean Trans. Restor ed)
Does Lorentz Transform “work” ? Two rockets travel in opposite directions y’ S’ An observer on earth (S) S y -0.85c measures speeds = 0.75c B 0.7c And 0.85c for A & B A respectively x’ What does A measure as O’ x B’s speed? O Place an imaginary S’ frame on Rocket A ⇒ v = 0.75c relative to Earth Observer S Consistent with Special Theory of Relativity
Velocity Transformation Perpendicular to S-S’ motion v = = γ − dy ' dy , dt ' ( dt dx ) Similarly 2 c dy ' dy Z component of = = ' u y v dy ' γ − Ant' s velocity ( dt dx ) 2 c transforms as divide by dt on R H S u u = ' u z y = ' u z v y v γ − γ − (1 c u ) (1 u ) x x 2 2 c There is a change in velocity in the ⊥ direction to S-S' motion !
Inverse Lorentz Velocity Transformation Inverse Velocity Transform: + u v = u x ' x vu + 1 x ' 2 c As usual, ' u = y u replace y v v ⇒ - v γ + ' (1 u ) x 2 c ' u = u z z v γ + ' ( 1 u ) x 2 c
Example of Inverse velocity Transform Biker moves with speed = 0.8c past stationary observer Throws a ball forward with speed = 0.7c What does stationary observer see as velocity of ball ? Speed of ball relative to Place S’ frame on biker stationary observer Biker sees ball speed u X ? u X’ =0.7c
Can you be seen to be born before your mother? A frame of Ref where sequence of events is REVERSED ?!! u D S S S’ m I arrive in SF o r f f f o e k a t I ( x t , ) ( , ) x t 2 2 1 1 ' ' ( x t , ) ' ' ( , ) x t 2 2 1 1 γ ∆ v x ∆ = − = ∆ − ' ' t ' t t t 2 1 2 c ∆ < For what value of v can t ' 0
I Cant ‘be seen to arrive in SF before I take off from SD u S S’ ( x , t ) ( x , t ) 2 2 1 1 ' ' ( x , t ) ' ' ( x , t ) 2 2 1 1 γ ∆ v x ∆ = − = ∆ − ' ' t ' t t t 2 1 2 c ∆ < For what value of v can t ' 0 ∆ ∆ = v x v x v u ∆ < ⇒ ∆ < ⇒ t ' 0 t 1 < ∆ 2 2 2 c c t c v c ⇒ > ⇒ > v c : Not al lowe d c u
Relativistic Momentum and Revised Newton’s Laws � � Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform = and the Special theory of relativity: Example : p mu Watching an Inelastic Collision between two putty balls S P = mv –mv = 0 P = 0 Before V=0 1 2 v v 1 2 After − − − − v v v v 2 v V v = = = = = = − ' ' v 1 0, v 2 , V ' v 1 v v 2 v v V v 2 v − − − 1 1 1 1 + 1 1 2 2 2 c c c 2 c − 2 mv ' = ' + ' = ' = = − p mv m v , p 2 mV ' 2 mv before 1 2 after 2 v + 1 S’ 2 c ' ' p p ≠ before after v 1 ’=0 1 2 V’ 1 2 v 2 ’ Before After
Definition (without proof) of Relativistic Momentum � � With the new definition relativistic � mu = = γ p mu momentum is conserved in all frames − 2 1 ( / ) u c of references : Do the exercise New Concepts Rest mass = mass of object measured In a frame of ref. where object is at rest 1 γ = − 2 1 ( / ) u c u is velocity of the object NOT of a referen ce frame !
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