Tests of the Equivalence Principle and the Classic Weak Field Tests

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Tests of the Equivalence Principle and the
Classic Weak Field Tests

• Compartmentalizing General Relativity into
  testable components
• Tests of the Equivalence Principle
• Post-Newtonian Parameterization
• The classic solar system tests
• Deflection of light
• Precession of perihelion of Mercury
• Shapiro delay

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The Metric Tensor

• Definition of (3d) metric tensor gij the
  distance between points (x1, x2, x3) and
  (x1?x1,x2?x2,x3?x3)
• Can simply generalize this to 4-d spacetime
• Geodesics are those paths which minimize the
  distance between two points A and B
• For a given matter distribution, metric is
  determined by Einsteins equation (ten coupled
  partial differential equations)

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Example metrics

• Normal (flat) 3-d space in cartesian coordinates
• Flat spacetime of Special Relativity in cartesian
  coordinates
• Non-spinning, uncharged black hole (Schwarzschild
  metric) in Schwarzschild coordinates

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• Einsteins Equivalence Principle
• Weak EP (grav. acc. same for all bodies)
• Local Lorentz Invariance (constancy of speed of
  light)
• Local Positional Invariance (universality of
  grav. redshift)

Spacetime Metric
Mass/Energy distribution

• Einsteins field equations
• Assume matter couples to metric in simplest
  possible way (with no other fields involved).

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III.5 Tests of the Einstein Equivalence
Principle (EEP)

• Weak Equivalence Principle (WEP)
• Dates back to Galileo (or earlier!)
• Statement the acceleration experienced by an
  object in a given gravitational field is
  independent of the objects mass or composition.
• Inertial and gravitational mass are equivalent
• To test this, we need a formulism for describing
  deviations from the WEP.

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• Weak Equivalence Principle formulism
• Consider an object as a collection of masses
  bound together by various interactions (weak,
  strong, electromagnetic)
• Suppose that some part of the binding energy
  associated with these interactions violates the
  WEP
• If we measure the acceleration of two bodies with
  different composition, we can define

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• Classic Experiment of Eotvos
• Used torsional balance to determine that ?lt10-8
• Roll, Krotkov, Dicke (Princeton)
• Improved Eotvos method and obtained tighter limit
  ?lt10-11

The confrontation between GR and Experiment
C.M.Will
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• Local Lorentz Invariance
• Are laws of physics really the same in any
  inertial reference frame?
• Hard to test in general there might always be
  some other law of physics that violates LLI that
  we hadnt though of.
• But lets think about violations of LLI for
  electromagnetic phenomena
• e.g., suppose that speed of light c is
  different to limiting velocity of particles c?1
  (in appropriate units)
• Can define parameter that measures this deviation

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• Classic experiment of Michelson Morley using
  interferometry
• Hughes-Drever experiments probe c through
  effect on energy levels in atomic nucleus
• Can now directly time one-way trips of light
  signal

The confrontation between GR and Experiment
C.M.Will
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• Local Positional Invariance
• Is physics independent of location?
• Test through gravitational redshift
• Suppose two identical clocks are placed at
  different locations in gravitational field weve
  seen that there is a fractional shift of
  frequency
• But if the physics governing the clocks depends
  on position, then we can write

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• Pound-Rebka-Snider tower experiment (Harvard)
• frequency shift of gamma-ray emission line from
  Fe-57
• Null redshift experiment
• Synchronization of three H-maser clocks (used
  changes in solar grav. potential)

The confrontation between GR and Experiment
C.M.Will
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III.6 Post-Newtonian Theory

• Lets move on to testing the other side of GR
  Theory the effect that matter and energy have on
  the spacetime metric
• Very useful approach has been the so-called
  post-Newtonian parameterization
• Essentially amounts to doing a Taylor expansion
  of the full equations which determine the metric
  and (carefully) keeping the leading order terms
• GR predicts definite coefficients for the
  expansion
• Alternative theories of gravity (which connect
  the spacetime metric to the matter/energy in more
  complex ways) predict deviations from these
  coefficents

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III.7 Light deflection by the Sun

• GR predicts that a light-ray grazing the Sun is
  deflected by 1.75 arcsec
• In terms of post-Newtonian parameter
• Write the deflection as
• In GR, ?1
• Measuring ?? gives us constraint on ?

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• Classic expedition by Eddington to observe solar
  eclipse
• First confirmation of GR
• Made Einstein instantly famous!
• Modern measurements of light-bending possible
  from VLBI observations of quasars get GR result
  to 0.2.

The confrontation between GR and Experiment
C.M.Will
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III.8 Shapiro delay

• Spacetime curvature increase time taken for a
  photon to propagate from point A to B
• Within our solar system, spacetime curvature from
  Sun dominates so events that occur on far side
  of Sun are late

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