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 9GRAVITY AS SPACETIME CURVATURE
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Objectives

• New theory of gravitation
• Principle of equivalence
• Gravitational redshift
• Tidal gravitational force
• Spacetime curvature

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Objectives

• Math of curve space
• differential geometry
• Theory of gravity
• motion of particles
• generation of curvature

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New Theoryof Gravitation
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New theory of gravitation

• Why we need new theory
• What are its main features
• Compare EM

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Why New Theoryof Gravitation
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Newtonian theory is action-at-a-distance
Signals propagate instantaneously Actually
delayed by t r/c
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From Coulomb to Maxwell
Q
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Concept of field
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Field introduces delay!
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Mass-energy equivalence

• energy should also gravitate
• momentum should also gravitate

Source mass
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New Theory
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New theory

• Should be a field theory
• Whose source is energy and momentum
• count components
• There are additional feature
• principle of equivalence
• non-linearity

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Count components (EM)

• Charge 1

• Source Jm (r, J)
• Field Am (f, A)

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Count components (gravity)

• Energy 1

• Momentum 3

• Total 16

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• Energy density T00 Energy flux T0j

• Momentum density Ti0 Momentum flux Tij

• Source Tmn
• Field gmn

?

• But nonlinear!

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Look for theory of gravity

• Tensor theory gmn

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Principle of Equivalence
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Principle of equivalence

• All objects fall at same acceleration mg mi
• Uniform g equivalent to accelerating observer
• Experimental basis

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Principle of equivalence

• In Millikan oil drop experiment, oil drops fall
  at different rates
• a (q/m)E (Ignoring gravity)
• q measure of coupling to force
• m measure of inertia

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• In free fall under gravity, situation is
  different
• Experimental fact
• acceleration is the same for all particles
• high precision, 1 part in 1012 or better

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• Newton's second law F mi a
• Law of gravity F mg g

Compare F qE

• Hence a (mg/mi)g
• Thus mg mi for all objects
• up to a constant multiple
• which can be set to unity

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Different ways of stating principle of equivalence

• All objects accelerate at exactly the same rate a
  g
• The ratio mg/mi is exactly 1 for all objects
• Uniform gravity is equivalent to an accelerating
  observer

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Uniform gravity is equivalent to an accelerating
observer

• Get rid of g by accelerating in same direction
• "Create" effective g by accelerating in opposite
  direction
• Concept of pseudoforce

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Get rid of g by acceleratingin same direction
S
S'
Same acceleration (free fall)
No acceleration "No gravity"
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"Create" effective g by accelerating in opposite
direction
S
S'
No gravity Accelerating observer
Seems to be gravity (pseudoforce) in opposite
direction
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Gravitational redshift

• Experimental fact photons climbing "upwards" are
  redshifted

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Gravitational redshift

• First explanation energy also interacts with
  gravity (E mc2)
• Second explanation
• regard the experiment as being done in a lift
  that is accelerating upwards
• by the time the photon is detected, the observer
  has a relative velocity
• Apply SR

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First ExplanationE mc2
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NOT a strictly valid argument

• Photon at z 0

• Effective "mass"

• Work done

• Energy at z h

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• Related to w'

Correct to 1st order in F
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Typical numbers

• Climb out of gravitational field of earth

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Relies onPrinciple of Equivalence

• All forms of matter/energy interacts with
  gravitational field in same way

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Second Explanation g ? a
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Ignore 2nd order
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Can We Get Rid of All g?
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Tidal gravitational force

• Inhomogeneous g is "real" effect
• The two main properties of tides

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A
B
C
Co-move with B
Cannot eliminate!
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A
B
C
Co-move with B
Cannot eliminate!
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Inhomogeneous partTidal gravitational force
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Why this name?

• Tides are caused by the inhomogenous part

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Tides
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Naive picture

• Once a day

• Ms/Rs2 gtgt Mm/Rm2 Solar gtgt lunar

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Correct picture
In co-moving frame see only difference
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Tides

• Depends on

• Hence lunar gtgt solar
• Twice a day

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Spacetime Curvature
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Spacetime curvature

• Explain differential motion by saying that
  spacetime is curved

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No g
Stay parallel
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Uniform g
Stay parallel
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g decreases with height

• In an inhomogeneous
• grav field, paths that are
• nearby
• originally parallel (in spacetime)
• will eventually deviate

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Analogy in curved space

• Two lines that are originally parallel deviate
  from each other
• There is a force pushing them apart?

• Space is curved?

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Light rays

• No g

Uniform g
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Gravitational redshift ? spacetime is curved
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Situation in spacetime

• Two paths that are originally parallel deviate
  from each other
• There is a force pushing them apart
• Spacetime is curved?

Principle of Equivalence! Compare electric force!
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Look for theory of gravity

• Tensor theory gmn

• Base on spacetime curvature

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Nonlinear
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Compare with EM

• Q ? E
• E ? more Q?
• No, E is NOT changed

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Heuristic
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Look for theory of gravity

• Tensor theory gmn
• Base on spacetime curvature

• Nonlinear (automatic)

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Objectives

• New theory of gravitation
• Principle of equivalence
• Gravitational redshift
• Tidal gravitational force
• Spacetime curvature

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Objectives

• Math of curve space
• differential geometry
• Theory of gravity
• motion of particles
• generation of curvature

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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