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MP2013_04-SR_Spacetime&LorentzTransformations

ES302: Modern Physics 近代物理(电子类)
何谷峰 gufenghe@https://www.doczj.com/doc/ac14899163.html, Tel: 34207045
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09.22 Day 4: Spacetime Addition of Velocities Lorentz Transformations
“The only reason for time is so that everything doesn’t happen at once.” - Albert Einstein Tuesday: Relativistic Momentum & Energy
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Last time: ? Time dilation and length contraction ? Lorentz transformations
Today: ? Spacetime ? Addition of velocities ? Relativistic momentum
Next time:
Relativistic momentum and energy
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Time dilation in moving frames
h
v · Δt’/2
Ethel Ricky Ethel and Ricky measure the time interval:
2h Dt ' = g, c
g =
1 v2 1- 2 c
But Lucy measured Δt = 2h/c !! 4

Time dilation in moving frames
Lucy measures: Δt Ethel and Ricky: Δt’ = γΔt, with 1 g = 2 v 1- 2 c Δt’ = γΔt ≥ Δt For Lucy time seems to run slower!
(Lucy is moving relative to Ethel and Ricky)
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Length contraction in moving frames
Speeds are the same (both refer to the relative speed). And so
L¢ L L | v |= = = Dt ¢ Dt gDt '
L L¢ = g
Length in moving frame Length in stick’s rest frame (proper length)
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Length contraction is a consequence of time dilation (and vice-versa).

Lorentz Transformations
If S’ is moving with speed v in the positive x direction relative to S, then the coordinates of the same event in the two frames are related by:
Galilean transformation (classical) Lorentz transformation (relativistic)
Dx ¢ = Dx - vDt Dy¢ = Dy Dz¢ = Dz Dt ¢ = Dt
Dx ¢ = g (Dx - vDt) Dy¢ = Dy Dz¢ = Dz v Dt ¢ = g (Dt - 2 Dx) c
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Spacetime Diagrams (1D in space)
In Classical Physics: v x x
Δx Δt
v = Δx/Δt t
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Spacetime Diagrams (1D in space)
c ·t In Modern Physics : x x
t
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Spacetime Diagrams (1D in space)
c· t In Modern Physics :
object moving with 0x
object moving with 0>v>-c
c·t
object at rest at x=1 -2 -1 0 1 2
c·t
object moving with v = -c. x=0 at time t=0 -2 -1 0 1 2
x
x
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Recall:
Lucy plays with a fire cracker in the train. Ricky watches the scene from the track.
L
R v Lucy
Ricky
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Example: Lucy in the train
ct
Light reaches both walls at the same time. Light travels to both walls
L
x R Lucy concludes: Light reaches both sides at the same time In Lucy’s frame: Walls are at rest
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Example: Ricky on the tracks ct
Ricky concludes: Light reaches left side first. L x R
In Ricky’s frame: Walls are in motion
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S
... -3 -2 -1 0 1 2 3 ...
S’
... -3 -2 -1 0 1 2 3 ...
v=0.5c
Frame S’ is moving to the right at v = 0.5c. The origins of S and S’ coincide at t=t’=0. Which shows the world line of the origin of S’ as viewed in S? A ct B ct C ct D ct
x
x
x
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x

Frame S’ as viewed from S
These angles are equal ct ct’ x’ x This is the space axis of the frame S’ This is the time axis of the frame S’
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Frame S’ as viewed from S
Euclidean space through circular angle φ
Minkowski spacetime through hyperbolic angle φ
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Spacetime Interval
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Distance in Galilean Relativity
The distance between the blue and the red ball is:
2 2 (3m) + (4m) = 25m = 5m
If the two balls are not moving relative to each other, we find that the distance between them is “invariant” under Galileo transformations.
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Remember Lucy?
h Lucy
Event 1 – firecracker explodes Event 2 – light reaches detector Distance between events is h
h = cDt
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Remember Ricky?
cΔt’ h
Δx’ Event 1 – firecracker explodes Event 2 – light reaches detector Distance between events is cΔt’ But distance between x-coordinates is Δx’ and: (cΔt’)2 = (Δx’)2 + h2 We can write h 2 = (cDt ¢)2 - (Dx¢)2 And Lucy got h 2 = (cDt )2 - (Dx )2 since
Dx = 0
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