Special RelativitySpecial Relativity1.1. Quiz 1.22 (10 minutes) and a few comments on quiz Quiz 1.22 (10 minutes) and a few comments on quiz

0. 0. 2.2. So far we know that Special Relativity is valid for all So far we know that Special Relativity is valid for all

speeds. But it is only needed when speeds are close speeds. But it is only needed when speeds are close to the speed of light. In our everyday life, Newtonian to the speed of light. In our everyday life, Newtonian mechanics is still a very good theory. Newtonian mechanics is still a very good theory. Newtonian mechanics is a special case of Special Relativity at mechanics is a special case of Special Relativity at low speeds.low speeds.

3.3. Examples when Special Relativity is needed:Examples when Special Relativity is needed:In particle physics, we deal with particles of high In particle physics, we deal with particles of high energy and hence high speeds.energy and hence high speeds.

4.4. Topics in Special Relativity in this course:Topics in Special Relativity in this course: Inertial frame of reference and the definition of an Inertial frame of reference and the definition of an

“event”.“event”. The Lorentz Transformation equations of spatial The Lorentz Transformation equations of spatial

coordinates or time.coordinates or time. The Doppler effect: transformation of spatial The Doppler effect: transformation of spatial

coordinates and time.coordinates and time. Velocity transformation: the derivative of coordinates Velocity transformation: the derivative of coordinates

with respect to time.with respect to time. Momentum and Energy, a step into dynamics. Momentum and Energy, a step into dynamics.

Today

Inertial frame of reference Inertial frame of reference and the definition of an and the definition of an

“event”“event” The form of each physics law is the The form of each physics law is the

same in all inertial frame of reference.same in all inertial frame of reference. All inertial frames of reference are All inertial frames of reference are

relative. There is no motion that is relative. There is no motion that is absolute.absolute.

An “event” in physics is defined by the An “event” in physics is defined by the coordinates of this event. The coordinates of this event. The coordinate system contains spatial coordinate system contains spatial and time information. and time information.

The speed of light in The speed of light in vacuum is a universal vacuum is a universal

constantconstant Light (or electromagnetic wave) travels through vacuum with a constant Light (or electromagnetic wave) travels through vacuum with a constant speed speed cc = 3×10 = 3×1088 m/s. m/s.

The Michelson-Morley (Albert Abraham Michelson, 1852-1931, German-The Michelson-Morley (Albert Abraham Michelson, 1852-1931, German-born American physicist, 1907 Nobel Prize; Edward Williams Morley, born American physicist, 1907 Nobel Prize; Edward Williams Morley, American chemist and physicist) experiment ruled out the existence of American chemist and physicist) experiment ruled out the existence of ether, a postulated medium for EM wave to move through. 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 The Lorentz Transformation equations of spatial equations of spatial coordinates or timecoordinates or time Linear (one dimensional) case. Object moves Linear (one dimensional) case. Object moves

from the origin with velocity from the origin with velocity uu in frame S and in frame S and u’u’ in frame S’ reaches coordinate in frame S’ reaches coordinate xx in frame S in frame S and and x’x’ in frame S’, after time in frame S’, after time tt. Frame S’ . Frame S’ overlaps with frame S at overlaps with frame S at tt = 0, and moves = 0, and moves with a velocity with a velocity vv with respect to frame S. with respect to frame S.

Transform of the spatial coordinates of an Transform of the spatial coordinates of an event at time event at time tt from frame S to frame S’: from frame S to frame S’: Classical Galilean transformation:Classical Galilean transformation:

Special Relativity: Special Relativity:

x' x vt

vx' x vt

2

1

1v

v,

c

Discussion: when v is very small, the two transformation formulas agree.

The Lorentz Transformation The Lorentz Transformation equations of spatial equations of spatial coordinates or timecoordinates or time

Transform of the time stamp Transform of the time stamp (coordinate) of the event from frame (coordinate) of the event from frame S to frame S’:S to frame S’: Classical Galilean transformation:Classical Galilean transformation:

Special Relativity: Special Relativity:

t ' t

2v

vt' x t

c

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 Time dilation and length contractioncontraction

Proper time and proper length:Proper time and proper length: Proper time: time interval (coordinate difference) Proper time: time interval (coordinate difference)

of two events measured in the frame when objects of two events measured in the frame when objects are at rest.are at rest.

Proper length: spatial coordinate difference of two Proper length: spatial coordinate difference of two events (length between two events) measured in events (length between two events) measured in the frame when objects are at rest. the frame when objects are at rest.

Still assume frame S and S’. We assume Still assume frame S and S’. We assume u’u’ =0. So =0. So time and length measured in frame S’ are the time and length measured in frame S’ are the proper time and length.proper time and length.

Time dilation:Time dilation:

Length contraction:Length contraction:

proper 0v v vt t ' t t

proper 0

v v v

L LL'L

ExamplesExamples Work on the blackboard on examples 2.1 to 2.5.Work on the blackboard on examples 2.1 to 2.5.

2.1: transfer from S’ to S. Use (2-13a and b) to get delta_t. 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.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 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 fixed on muon, then v = 0.98c, that gives . life time is . life time is delta_t_0, this gives the life time of muons in frame S, to 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 be 11.1 us. (c) The 3 km is measured in S, and that is the proper length. When muon measures this length, use proper length. When muon measures this length, use formula 2-16. formula 2-16.

2.3: S is on the ground, S’ is on the plane. The length of 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 plane measured when parked is the proper length. The time measured inside the plane is the proper time, The time measured inside the plane is the proper time, while the time measured on the ground of d/v is not. while the time measured on the ground of d/v is not.

2.4: From one finds and v. Use the coordinates 2.4: From one finds and v. Use the coordinates transformation to answer the rest questions.transformation to answer the rest questions.

2.5: S is on the ground, S’ is on Anna’s spaceship. The 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. distance 40 ly is the proper length, not the one Anna sees. The time 30 yr is what Anna counts in her spaceship and The time 30 yr is what Anna counts in her spaceship and is the proper time. So v = proper length/proper time will is the proper time. So v = proper length/proper time will lead to an equation with v in. Solve it to get the required lead to an equation with v in. Solve it to get the required speed. speed.

2

1

1v

v,

c

0

v

LL

v

Review questionsReview questions

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

Preview for the next Preview for the next classclass

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 How do people measure speed of stars when

they are moving towards/away from us?they are moving towards/away from us? Two particles on a track for head-on collision Two particles on a track for head-on collision

move with a speed of 0.999move with a speed of 0.999cc with respect to with respect to lab. What is the approaching speed one lab. What is the approaching speed one particle “sees” the other? particle “sees” the other?

Photon has no rest mass, but in particle Photon has no rest mass, but in particle physics, we often quote a moment and an physics, we often quote a moment and an energy of a photon. Please explain. energy of a photon. Please explain.

Homework 2Homework 2

1.1. A very concise summary on what A very concise summary on what we discussed in the class of 8.24?we discussed in the class of 8.24?

2.2. Problem 20 on page 62.Problem 20 on page 62.

3.3. Problem 21 on page 62.Problem 21 on page 62.

4.4. Problem 33 on page 64. Problem 33 on page 64.