Special Relativity II-Classical Physics-Handouts, Lecture notes of Classical Physics

Download Special Relativity II-Classical Physics-Handouts and more Lecture notes Classical Physics in PDF only on Docsity! PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 135 Summary of Lecture 40 – SPECIAL RELATIVITY II ( ) 2 1 2 1 2 v1. Recall the Lorentz Transformation: v and . Suppose we take the space interval between two events , and the time interval . Then, these intervals will b x x t t t x c x x x t t t γ γ ⎛ ⎞′ ′= − = −⎜ ⎟ ⎝ ⎠ Δ = − Δ = − ( )2 ve seen in S as and v . Now consider two particular cases: a) Suppose the two events occur at the same place (so 0) but at different times (so 0). Not t t x x x t c x t γ γ⎛ ⎞′ ′ ′Δ = Δ − Δ Δ = Δ − Δ⎜ ⎟ ⎝ ⎠ Δ = Δ ≠ ( )e that in S they do not occur at the same point: 0 v ! b) Suppose the two events occur at the same time (so 0) but at different places (so 0). Note that in S they are not simul x t t x γ′ ′Δ = − Δ Δ = ′Δ ≠ 2 vtaneous: 0 . 2. As seen in the frame S, suppose a particle moves a distance in time . Its velocity is then (in S-frame). As seen in the S -frame, meanwhile, it has move t x c dx dt u dxu dt γ ⎛ ⎞′Δ = − Δ⎜ ⎟ ⎝ ⎠ ′= ( ) ( ) 2 2 22 d a distance v where v and the time that has elapsed is . The velocity v / v v in S -frame is therefore . Thiv vv 1 / 1 dx dx dx dt dt dt dx c dx dtdx dx dt uu udt dx dtdt dx c cc γ γ γ γ ′ ⎛ ⎞′ ′= − = −⎜ ⎟ ⎝ ⎠ −′ − −′ ′ = = = = ′ ⎛ ⎞ − −−⎜ ⎟ ⎝ ⎠ 2 s is the Einstein velocity addition rule. It is an easy exercise to solve this for u in terms of , v . v1 3. Note one very interestin u uu u c ′ ′ + = ′ + g result of the above: suppose that a car is moving at speed v and it turns on its headlight. What will the speed of the light be according to the observer on the ground? If we use the Galilean 2 transformation result, the answer is v+c (wrong!). But v v v using the relativistic result we have and . In other v v v1 1 words, the speed of the source makes no differenc c c cu c u c cc c c c + + +′ = = = = = ++ + 2 e to the speed of light in your frame. Note that if either or v is much less than , then reduces to the familiar result: v v, which is the Galilean velocity addition rule. v1 u c u uu uu c ′ −′ = → − − docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 136 x ct rocket photon body at rest u c< u c> u c= ct xpresent 0t =present 0t = your future your past your world line 4. The Lorentz transformations have an interesting property that we shall now explore. Take the time and space intervals between two events as observed in frame S, and the corresponding quantit ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 22 2 2 ies as observed in S . We will now prove that the quantities defined respectively as and are equal. Let's start with : v / 2v I c t x I c t x I I c t x c t x c t xγ ′ ′ ′ ′ ′= Δ − Δ = Δ − Δ ′ ′ ′= Δ − Δ = Δ + Δ − Δ Δ( ( ) ( )2 2 v 2vx t t x− Δ − Δ + Δ Δ ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) 2 22 2 2 2 2 2 2 2 1 v 1 v / 1 v / This guarantees that all inertial observers measure the same speed of light !! 5. If t c t x c c c t x I = − Δ − Δ − − = Δ − Δ = he time separation is large, then 0 and we call the interval If the space separation is large, then 0 and we call the interval If 0 and we call the interval I timelike. I spacelike. I lightl > < = : a) If an interval is timelike in one frame, it is timelike in all other frames as well. b) If interval between two events is timelike, their time ordering is absolute. ike. Note c) If the interval is spacelike the ordering of two events depends on the frame from which they are observed. 6. It is sometimes nice to look at things graphically. Here is a graph of position versus time for objects that move with different speeds along a fixed direction. First look at an object at rest. Its position is fixed even though time (plotted on the vertical axis) keeps increasing. Then look at the rocket moving at constant speed (which has to be less than c), and finally a photon (which can only move at c). 7. The trajectory of a body as it moves through space- time is called its world-line. Let's take a rocket that is at 0 at 0. It moves with non-constant speed, and that is why its world-li x t= = ne is wavy. A photon that moves to the right will have a world line with slope equal to +1, and that to the right with slope -1. The upper triangle (with positive) is called the future t light cone (don't forget we also have and ). The lower light cone consists of past events. y x docsity.com