Special Relativity 1. Quiz 1.22 (10 minutes) and a few comments on quiz So far we know that Special Relativity is valid for all speeds. But it is only needed when speeds are close to the speed of light. In our everyday life, Newtonian mechanics is still a very good theory. Newtonian mechanics is a special case of Special Relativity at low speeds. 3. Examples when Special Relativity is needed: In particle physics, we deal with particles of high energy and hence high speeds. 4. Topics in Special Relativity in this course:  Inertial frame of reference and the definition of an “event”.  The Lorentz Transformation equations of spatial coordinates or time.  The Doppler effect: transformation of spatial coordinates and time.  Velocity transformation: the derivative of coordinates with respect to time.  Momentum and Energy, a step into dynamics. Today

Inertial frame of reference and the definition of an “event” The form of each physics law is the same in all inertial frame of reference. The form of each physics law is the same in all inertial frame of reference. All inertial frames of reference are relative. There is no motion that is absolute. All inertial frames of reference are relative. There is no motion that is absolute. An “event” in physics is defined by the coordinates of this event. The coordinate system contains spatial and time information. An “event” in physics is defined by the coordinates of this event. The coordinate system contains spatial and time information.

The speed of light in vacuum is a universal constant Light (or electromagnetic wave) travels through vacuum with a constant speed c = 3×10 8 m/s. Light (or electromagnetic wave) travels through vacuum with a constant speed c = 3×10 8 m/s. The Michelson-Morley (Albert Abraham Michelson, , German-born American physicist, 1907 Nobel Prize; Edward Williams Morley, American chemist and physicist) experiment ruled out the existence of ether, a postulated medium for EM wave to move through. The Michelson-Morley (Albert Abraham Michelson, , German-born American physicist, 1907 Nobel Prize; Edward Williams Morley, American chemist and physicist) experiment ruled out the existence of ether, a postulated medium for EM wave to move through. Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect ether flow. The device he designed, later known as an interferometer, sent a single source of white light through a half-silvered mirror that was used to split it into two beams traveling at right angles to one another. After leaving the splitter, the beams traveled out to the ends of long arms where they were reflected back into the middle on small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference based on the spent time to transit the arms. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. If the aether were stationary relative to the sun, then the Earth’s motion would produce a fringe shift one twenty-fifth the size of a single fringe. The lack of existence of ether also proved that there is no absolute reference frame for light.

The Lorentz Transformation equations of spatial coordinates or time Linear (one dimensional) case. Object moves from the origin with velocity u in frame S and u’ in frame S’ reaches coordinate x in frame S and x’ in frame S’, after time t. Frame S’ overlaps with frame S at t = 0, and moves with a velocity v with respect to frame S. Linear (one dimensional) case. Object moves from the origin with velocity u in frame S and u’ in frame S’ reaches coordinate x in frame S and x’ in frame S’, after time t. Frame S’ overlaps with frame S at t = 0, and moves with a velocity v with respect to frame S. Transform of the spatial coordinates of an event at time t from frame S to frame S’: Transform of the spatial coordinates of an event at time t from frame S to frame S’: Classical Galilean transformation: Classical Galilean transformation: Special Relativity: Special Relativity: Discussion: when v is very small, the two transformation formulas agree.

The Lorentz Transformation equations of spatial coordinates or time Transform of the time stamp (coordinate) of the event from frame S to frame S’: Transform of the time stamp (coordinate) of the event from frame S to frame S’: Classical Galilean transformation: Classical Galilean transformation: Special Relativity: Special Relativity: Now can you do all the transformation from frame S’ to frame S? Discussion: when v is very small, the two transformation formulas agree.

Time dilation and length contraction Proper time and proper length: Proper time and proper length: Proper time: time interval (coordinate difference) of two events measured in the frame when objects are at rest. Proper time: time interval (coordinate difference) of two events measured in the frame when objects are at rest. Proper length: spatial coordinate difference of two events (length between two events) measured in the frame when objects are at rest. Proper length: spatial coordinate difference of two events (length between two events) measured in the frame when objects are at rest. Still assume frame S and S’. We assume u’ =0. So time and length measured in frame S’ are the proper time and length. Still assume frame S and S’. We assume u’ =0. So time and length measured in frame S’ are the proper time and length. Time dilation: Time dilation: Length contraction: Length contraction:

Examples Work on the blackboard on examples 2.1 to 2.5. Work on the blackboard on examples 2.1 to : transfer from S’ to S. Use (2-13a and b) to get delta_t. delta_x’=-2m, delta_t’=0, then delta_t can be calculated. 2.1: transfer from S’ to S. Use (2-13a and b) to get delta_t. delta_x’=-2m, delta_t’=0, then delta_t can be calculated. 2.2: (a) in S frame, t=d/v gives 10.2 us. (b) choose S’ to be fixed on muon, then v = 0.98c, that gives. life time is delta_t_0, this gives the life time of muons in frame S, to be 11.1 us. (c) The 3 km is measured in S, and that is the proper length. When muon measures this length, use formula : (a) in S frame, t=d/v gives 10.2 us. (b) choose S’ to be fixed on muon, then v = 0.98c, that gives. life time is delta_t_0, this gives the life time of muons in frame S, to be 11.1 us. (c) The 3 km is measured in S, and that is the proper length. When muon measures this length, use formula : S is on the ground, S’ is on the plane. The length of the plane measured when parked is the proper length. The time measured inside the plane is the proper time, while the time measured on the ground of d/v is not. 2.3: S is on the ground, S’ is on the plane. The length of the plane measured when parked is the proper length. The time measured inside the plane is the proper time, while the time measured on the ground of d/v is not. 2.4: From one finds and v. Use the coordinates transformation to answer the rest questions. 2.4: From one finds and v. Use the coordinates transformation to answer the rest questions. 2.5: S is on the ground, S’ is on Anna’s spaceship. The distance 40 ly is the proper length, not the one Anna sees. The time 30 yr is what Anna counts in her spaceship and is the proper time. So v = proper length/proper time will lead to an equation with v in. Solve it to get the required speed. 2.5: S is on the ground, S’ is on Anna’s spaceship. The distance 40 ly is the proper length, not the one Anna sees. The time 30 yr is what Anna counts in her spaceship and is the proper time. So v = proper length/proper time will lead to an equation with v in. Solve it to get the required speed.

Review questions In a particle physics experiment a particle called pion is used to hit a target. The particle pion has a lifetime of 2.6E-8 second and is accelerated to a speed 0.99 c with respect to the linear accelerator. A straight beam pipe is used to transport the pions to the target. What would be the maximum length of the beam pipe? What is the length of the beam pipe the pions see? In a particle physics experiment a particle called pion is used to hit a target. The particle pion has a lifetime of 2.6E-8 second and is accelerated to a speed 0.99 c with respect to the linear accelerator. A straight beam pipe is used to transport the pions to the target. What would be the maximum length of the beam pipe? What is the length of the beam pipe the pions see?

Preview for the next class Text to be read: Text to be read: In chapter 2: In chapter 2: Sections 2.5 to 2.7. Sections 2.5 to 2.7. Section 2.8 if the material interests you. Section 2.8 if the material interests you. Questions: Questions: How do people measure speed of stars when they are moving towards/away from us? How do people measure speed of stars when they are moving towards/away from us? Two particles on a track for head-on collision move with a speed of c with respect to lab. What is the approaching speed one particle “sees” the other? Two particles on a track for head-on collision move with a speed of c with respect to lab. What is the approaching speed one particle “sees” the other? Photon has no rest mass, but in particle physics, we often quote a moment and an energy of a photon. Please explain. Photon has no rest mass, but in particle physics, we often quote a moment and an energy of a photon. Please explain.

Homework 2 1. A very concise summary on what we discussed in the class of 8.24? 2. Problem 20 on page Problem 21 on page Problem 33 on page 64.