PHY 4460 RELATIVITY

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PHY 4460RELATIVITY

• K Young, Physics Department, CUHK
• ?The Chinese University of Hong Kong

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CHAPTER 3MOVING REFERENCE FRAME I
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Objectives

• Index notation
• Galilean transformation
• M-M experiment
• Derive L transformation from c const
• Explicit form of L transformation
• Inverse transformation

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• S' moving relative to S with velocity V along x

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Galilean Transformation
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Galilean transformation
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Galilean transformation
Velocities "add"
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Michelson-MorleyExperiment
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Michelson-Morley Experiment
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• "Train" Earth V 3 ? 10?4 ms1 V/c
  10 ?4

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• There is no way to stop this "train" and compare
  with the case V 0
• Instead, compare rays parallel and perpendicular
  to direction of motion

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A sketch of the Michelson-Morley experiment
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• No effect found
• Speed of light is same in all reference frames

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No absolute motion
NO!!
YES!!
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Derivation of Lorentz Transformation
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Derivation of Lorentz transformation

• Index notation summation convention
• Basic object is an event E
• Linear assumption x' L x (4D)
• Identify an invariant
• Condition on transformation matrix

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Index Notation
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Notation
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(No Transcript)
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Summation convention

• Repeated index, 1 up, 1 down
• Sum from 0 ? 3

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Basic object

• An event E

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Linear assumption
16 coefficients Simpler notation
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Linear assumption
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Linear assumption

• Component notation

• Matrix notation

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Identify invariant

• s 2 can be negative

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Why proportional?

• Proportional

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• Proportional

• Independent of direction

• Consider reverse transformation

• Therefore (up to a sign)

• Invariance MM

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• 3D space

4D spacetime
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Metric
We can write s 2 as
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• Lowering an index

Raising an index
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The Minus Sign
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The minus sign

• h is just a way of remembering the ? sign
• Why do we need h (4?4 matrix) just to deal with
  a sign?

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Why not ?
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1 Do not hide a genuine difference

• Euclidean

Closed Finite
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• Minkowski

Open Infinite
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2 Genuine i / Fake i
Impossible to keep track!
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Constraint Arising froms 2 s '2
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Explicitly in 2D (t, x)
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• 3 conditions

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Using index notation (in general)
For 2?2case, check that these give same 3
conditions
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Using matrix notation
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Explicit Form of Lorentz Transformation
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Explicit form of L transformation

• The Lorentz transformation
• Non-relativistic limit
• Choice of units c 1
• Difference form

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Relate to relative velocity
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How to remember?
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Difference form
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Galilean limit
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Inverse Transformation
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Inverse transformation

• Invert algebraically

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Objectives

• Index notation
• Galilean transformation
• M-M experiment
• Derive L transformation from c const
• Explicit form of L transformation
• Inverse transformation

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Acknowledgment

• This project is supported in part by the Hong
  Kong University Grants Committee (UGC) Teaching
  Development Grants (TDG) 3203005 and 3201032
• I thank Prof. S.C.Liew for software
• I thank Prof. M.C.Chu and Dr. S.S.Tong for advice
• I thank Miss H.Y.Shik for design