chapter 1physics.uwyo.edu/~teyu/class/chapter1.pdfe&m wave (light) and maxwell eqs. light is...
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Chapter 1Relativity 1
Classical Relativity โinertial vs noninertial reference frames
Inertial Reference Frames
๐ฅโฒ = ๐ฅ โ ๐ฃ๐ก; ๐ฆโฒ = ๐ฆ; ๐งโฒ = ๐ง; ๐กโฒ = ๐ก
๐ข๐ฅโฒ = ๐ข๐ฅ โ ๐ฃ; ๐ข๐ฆ
โฒ = ๐ข๐ฆ; ๐ข๐งโฒ = ๐ข๐ง
Any reference frame that moves at constant velocity with respect to an inertial frame is also an inertial frame. Newtonโs laws of mechanics are invariant in all reference systems connected by Galilean transformation.
Galilean transformation:
Classical Relativity โinertial vs noninertial reference frames
๐นโฒ โ ๐น ๐นโฒ โ ๐น
Maxwell Eqs.
๐น = ๐๐ธ =๐2๐๐
๐ฆ1๐นโฒ = ๐๐ธ + ๐ ๐ฃ ร ๐ต =
๐2๐๐
๐ฆ1โ
๐0๐๐ฃ2๐
2๐๐ฆ1
๐นโฒ โ ๐นInertial Reference Frames, butโฆ
Maxwell Eqs. Do not follow Galilean transformation
E&M wave (light) and Maxwell Eqs.
Light is E&M wave, well understood by Maxwell Eqs., has a velocity of light: ๐ =1
๐0๐0= 3 ร 108๐/๐
In 19th century, all waves are understood as needing media to propagate: ether (media of E&M wave)
Michelson-Morley Experiment
How to understand the experiment
Michelson-Morley Experiment - Meaning
โข If ether exist, the relative velocity of the earth and the ether has a upper limit of 5 km/s (Michelson-Morley in 1887); or 1.5 km/s (Georg in 1930); and 15 m/s (recently)
โข The speed of light (E&M wave) is the same in all inertial reference system.
โข This implies, there must be some relativity principle that apply to E&M as well as to mechanics. This principle should be converged back to Galilean transformation in certain conditions.
Einsteinโs Postulates
โข Postulate 1: The laws of physics are the same in all inertial reference frames.
โข Postulate 2: The speed of light in a vacuum is equal to the value c, independent of the motion of the source.
โข Fact: Speed of light (no media) is independent of the inertial reference frames.
Events and Observers
โข Events: Physical Event is something that happens.
โข Observers: Someone or something that see/detect the events in a certain inertial reference frame.
โข Information needs time to propagate. What is โsimultaneityโ?โข The spatially separated events simultaneous in one reference frame are not,
in general, simultaneous in another inertial frame moving relative to the first.
โข Clocks synchronized in one reference frame are not, in general, synchronized in another inertial frame moving relative to the first.
Thought Experiments
Lorentz transformation
๐ฅโฒ = ๐ฅ โ ๐ฃ๐ก; ๐ฆโฒ = ๐ฆ; ๐งโฒ = ๐ง; ๐กโฒ = ๐ก
๐ฅ = ๐ฅโฒ + ๐ฃ๐ก; ๐ฆ = ๐ฆโฒ; ๐ง = ๐งโฒ; ๐ก = ๐กโฒ
Galilean transformation:
Lorentz transformation: should satisfy Einsteinโs postulates and converge back to Galilean when v is small.
๐ฅโฒ = ๐พ ๐ฅ โ ๐ฃ๐ก
๐ฅ = ๐พ ๐ฅโฒ + ๐ฃ๐ก๐พ โ 1 ๐คโ๐๐ ๐ฃ/๐ โ 0
๐กโฒ = ๐พ ๐ก +1 โ ๐พ2
๐พ2
๐ฅ
๐ฃ
Obtaining g
๐ฅ2 + ๐ฆ2 + ๐ง2 = ๐2๐ก2
A flash of light in S at t = 0 at origin while defining the same point in Sโ as the origin in Sโ at tโ = 0.
๐ฅโฒ2 + ๐ฆโฒ2 + ๐งโฒ2 = ๐2๐กโฒ2
๐พ =1
1 โ๐ฃ2
๐2
=1
1 โ ๐ฝ2
Lorentz transformation
๐ฅโฒ = ๐พ ๐ฅ โ ๐ฃ๐ก
๐กโฒ = ๐พ ๐ก โ๐ฃ
๐2 ๐ฅ
๐พ =1
1 โ๐ฃ2
๐2
=1
1 โ ๐ฝ2
๐ฆโฒ = ๐ฆ
๐งโฒ = ๐ง
๐ฅ = ๐พ ๐ฅโฒ + ๐ฃ๐กโฒ
๐ก = ๐พ ๐กโฒ +๐ฃ
๐2 ๐ฅโฒ
๐ฆ = ๐ฆโฒ
๐ง = ๐งโฒ
Example
โข The arrivals of two cosmic ray m mesons (muons) are recorded by detectors in the laboratory, one at time ๐ก๐ at location ๐ฅ๐ and the second at time ๐ก๐ at location ๐ฅ๐ in the laboratory reference frame, S in Fig. 1-17. What is the time interval between those two events in system Sโ, which moves relative to S at speed v?
Fig. 1-17
Relativistic velocity transformations
๐ขโฒ๐ฅ =๐ข๐ฅ โ ๐ฃ
1 โ๐ฃ๐ข๐ฅ
๐2
๐ข๐ฅ =๐ขโฒ๐ฅ + ๐ฃ
1 +๐ฃ๐ขโฒ๐ฅ๐2
๐ขโฒ๐ฆ =๐ข๐ฆ
๐พ 1 โ๐ฃ๐ข๐ฅ
๐2
๐ขโฒ๐ง =๐ข๐ง
๐พ 1 โ๐ฃ๐ข๐ฅ
๐2
๐ข๐ฆ =๐ขโฒ๐ฆ
๐พ 1 +๐ฃ๐ขโฒ๐ฅ๐2
๐ข๐ง =๐ขโฒ๐ง
๐พ 1 +๐ฃ๐ขโฒ๐ฅ๐2
Example
โข Suppose that two cosmic-ray protons approach Earth from opposite directions as shown in Fig. 1-18a. The speeds relative to Earth are measured to be ๐ฃ1 = 0.6๐ and ๐ฃ2 = โ0.8๐. What is Earthโs velocity relative to each proton, and what is the velocity of each proton relative to the other?
Fig. 1-18
Spacetime Diagrams
1D in space3D in space
Worldlines in Spacetime Diagrams
โข Find the speed u of particle 3 in the figure to the right.
โข The speed of light is the limit of the moving speed of particle. In spacetime diagram, the dashed line (speed of light) limit the particlesโ trajectory in spacetime diagram at any point.
Two inertial reference systems in spacetime diagram
๐ฅโฒ = ๐พ ๐ฅ โ ๐ฃ๐ก
๐กโฒ = ๐พ ๐ก โ๐ฃ
๐2 ๐ฅ
xโ axis: tโ = 0
๐กโฒ = ๐พ ๐ก โ๐ฃ
๐2 ๐ฅ = 0
๐๐ก = ๐ฝ๐ฅ
tโ axis: xโ = 0
๐ฅโฒ = ๐พ ๐ฅ โ ๐ฃ๐ก = 0
๐๐ก =1
๐ฝ๐ฅ
Spacetime Diagram for Train
If v = 0.5 c, the two flashes of light are simultaneous in S. What is the time interval in Sโ?
Time Dilation
โ๐กโฒ =2๐ท
๐
โ๐ก =
2 ๐ท2 + ๐ฃโ๐ก2
2
๐
โ๐ก =2๐ท
๐
1
1 โ๐ฃ2
๐2
โ๐ก = ๐พโ๐กโฒ = ๐พ๐
๐ is โproper timeโ, shortest measurable time in all frames, which could be done when the measurements of the two events are at the same location in this frame.
Example
โข Elephants have a gestation peiod of 21 months. Suppose that a freshly impregnated elephant is placed on a spaceship and sent toward a distant space jungle at v = 0.75 c. If we monitor radio transmissions from the spaceship, how long after launch might we expect to hear the first squealing trumpet from the newborn calf?
Example
โข Elephants have a gestation peiod of 21 months. Suppose that a freshly impregnated elephant is placed on a spaceship and sent toward a distant space jungle at v = 0.75 c. If we monitor radio transmissions from the spaceship, how long after launch might we expect to hear the first squealing trumpet from the newborn calf?
Length Contraction
โ๐ฅโฒ = ๐ฟ๐ = ๐ฅโฒ2 โ ๐ฅโฒ1 measured when โ๐กโฒ = 0
โ๐ฅ = ๐ฟ = ๐ฅ2 โ ๐ฅ1 measured when โ๐ก = 0
๐ฅโฒ2 = ๐พ ๐ฅ2 โ ๐ฃ๐ก2
๐ฅโฒ1 = ๐พ ๐ฅ1 โ ๐ฃ๐ก1
โ๐ฅ =1
๐พโ๐ฅโฒ ๐ฟ =
1
๐พ๐ฟ๐
๐ฟ๐ is โproper lengthโ, longest measurable length in all frames, which could be done when the measurements of the two ends are at rest in this frame.
Example
โข A stick that has a proper length of 1 m moves in a direction parallel to its length with speed v relative to you. The length of the stick as measured by you is 0.914 m. What is the speed v?
Muon decay
๐ ๐ก = ๐0๐โ๐ก/๐
๐ = 2๐๐
๐ฃ = 0.998๐
Proper time:
๐ป = 9000๐Proper length (height):
โ = 600๐
๐โฒ = 30๐๐
Non-relativistic Relativistic
N=30.6 N= 3.68ร 107
๐0 = 108
Spacetime Interval
โ๐ 2 = ๐โ๐ก 2 โ โ๐ฅ2 + โ๐ฆ2 + โ๐ง2
Timelike interval
Spacelike interval
Lightlike interval
๐โ๐ก 2 > โ๐ฅ2 + โ๐ฆ2 + โ๐ง2
๐โ๐ก 2 < โ๐ฅ2 + โ๐ฆ2 + โ๐ง2
๐โ๐ก 2 = โ๐ฅ2 + โ๐ฆ2 + โ๐ง2
Doppler Effect
๐ =1 + ๐ฝ
1 โ ๐ฝ๐0
๐ =1 โ ๐ฝ
1 + ๐ฝ๐0
approaching
receding
Useful approximations
๐ โ 1 + ๐ฝ ๐0
๐ โ 1 โ ๐ฝ ๐0
approaching
receding
โ๐ = ๐0 โ ๐ โ โ๐ฝ๐0
โ๐ = ๐0 โ ๐ โ ๐ฝ๐0
Example
โข The longest wavelength of light emitted by hydrogen in the Balmerseries (see Chapter 4) has a wavelength of ๐0 = 656๐๐. In light from a distant galaxy, this wavelength is measured as๐ = 1458๐๐. Find the speed at which the galaxy is receding from Earth, assuming the shift to be due to Doppler effect.
Twin Paradox
โข Twin: Homer and Ulysses. Ulysses traveled from Earth in a ship with 0.8 c away from Earth and then come back to Earth with -0.8 c. Homer see Ulysses 10 years after the launch.
Pole and Barn Paradox
โข Runner with 10 m pole in his hand, moving toward a 5 m barn with front door opened and rear door closed. Farmer (same frame with barn) sees the pole is entirely enclosed in the barn at certain moment.
Headlight Effect
๐๐๐ ๐ =๐๐๐ ๐โฒ + ๐ฝ
1 + ๐ฝ๐๐๐ ๐โฒ
๐๐๐ ๐ =โ๐ฅ
โ ๐๐ก
๐๐๐ ๐โฒ =โ๐ฅโฒ
โ ๐๐กโฒ