Gravitomagnetism The Myth and the Legend

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GravitomagnetismThe Myth and the Legend
Eric L. Michelsen
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My One Sentence
Gravity includes a velocity-independent force
(Newton) and a velocity-dependent force
(gravitomagnetic), closely analogous to the
electric and magnetic fields in EM

• Purely relativistic effect, not in Newtonian
  gravity

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Topics
Prerequisite some familiarity with General
Relativity helps

• Einsteins Theory of Gravity
• Metric Theories of Gravity
• The Gravitomagnetic Term
• Precessing Gyroscopes
• Gravity Probe B
• Lunar Orbital Perturbations
• The Controversy

• Papers
• 1 Gravitomagnetic Effects, M. L. Ruggiero et.
  al., arXivgr-qc/0207065v2, 7/9/2004.
• 2 The Role of Gravitomagnetism on Gyroscopes
  and the Moon, Tom Murphy, UCSD, as yet
  unpublished.
• 3 Lunar Laser Ranging A Comprehensive Probe
  of Post-Newtonian Gravity, K. Nordtvedt,
  arXivgr-qc/0301024, 1/7/2003.

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Gravitomagnetism

• The orphan child of gravitational physics
• Renewed interest with launch of Gravity Probe B
• Lorentz invariance implies a gravitomagnetic
  field 1, p3
• Any theory that combines Newtonian gravity
  together with Lorentz invariance in a consistent
  way, must include a gravitomagnetic field, which
  is generated by mass current. 1
• Dr. ONeil says that a 1/r2 force law is not
  Lorentz invariant
• Since Lorentz transformation includes velocity,
  any 1/r2 force must be accompanied by a
  source-velocity-dependent field
• Given the structure of the Lorentz
  transformation, the velocity dependent field must
  be a Biot-Savart-like magnetic field (to within a
  constant factor).
• Aka frame-dragging

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GR A Metric Theory of Gravity

• A metric theory of gravity defines a metric
  tensor field throughout all space
• The metric tensor field describes the shape
  (curvature) of space
• All physics, gravitational and otherwise, occurs
  in the physical spacetime described by the metric
  tensor field.
• The only dynamic field is the metric tensor field
  (loosely analogous to the EM field in
  electromagnetics)

Field Equation
sources of field
spacetime of all physics
mass/ energy
metric tensor field
(smoothly curved manifold)

• R?? and R are nonlinear functions of g??
• Just about the simplest metric theory of gravity
  there is

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The Metric Tensor Field

• The metric tensor field quantifies intervals,
  frame independent measures of the separation
  between two events.
• In an inertial frame (flat space), the
  squared-interval is the squared-distance between
  two events, minus the squared-distance light
  travels in the time between the events

(c dt)2
dx2
-ds2

• In general, the metric tensor field defines the
  dot product of any two vectors

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Metric Theories of Gravity

• By definition Will, 1993, a metric theory of
  gravity defines a metric tensor field throughout
  all space
• But other unobservable fields may be defined
• Their only purpose is to define the metric tensor
  through field equations
• In the end, only the metric tensor field affects
  observable physics

Field Equations
sources of fields

• Fields
• metric tensor field
• scalar field
• other fields

spacetime of all physics
mass/ energy
Field equations relate all the fields, to define
the all-important metric tensor field.
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A Perturbing Thought

• Nonlinear equations are hard to solve
• Use perturbation theory

• h just makes the equations simpler

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The Gravitomagnetic Term

• Use perturbation theory to compute the
  weak-field, non-relativistic perturbation to the
  metric

• Compare to EM (tensor vs. vector)

• Can jump right to gravity waves but lets not.

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The Gravitomagnetic Field

• Use the perturbed metric to compute the equations
  of motion. (Solve the geodesic
  equation.)Gravitomagnetic term

ai
vi
rij
source of field
vj

• Left hand rule
• Compare to Biot-Savart

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Where Did the Tensor Go?

• To order, (1/c2), only the first row and column
  of h are significant

• Reduces equations to vectors (rank-1 tensors)

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Gravitational Maxwells Equations

• Valid for weak field, non-relativistic speeds
• Imply propagating waves gravity waves
• Factors of 2 are remnants of rank-2 tensor wave
  equation, and spin 2 gravitons

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Gravitomagnetically Precessing Gyroscopes

• Use the solar system barycentric frame
• Source of gravitomagnetic field is earths spin
• Precession at poles is same direction as earth
  spin
• This is not geodetic precession gravitomagnetism
  is much smaller

ai
vi
L
?
precession
BG
L
vi
ai
BG
BG
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Gravity Probe B

• Equatorial precession opposite direction of earth
  spin
• Partially cancels GPB signal total precession
  ¼ polar precession
• Dipole approximation no good altitude 640 km
  0.1 R?
• Dipole never much good if far enough for dipole,
  effect is too small to see
• Do the integral 42 mas/y is the published number

L
polar orbit
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Lunar Orbital Effects

• Source of gravitomagnetic field is earths orbit
  around sun
• Spin of the earth is negligible
• We decompose the lunar velocity into two
  components
• V Lunar motion around sun earths motion
  around sun
• u Lunar motion around earth

u
vmoon V u
Sun
V
Earth
V
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Lunar Orbit Perturbations

• Velocities both objects orbit the sun at 30 km
  /s
• Lorentz contraction should contract tangential
  size, but not elongate??

a
to sun
BG
V
D
V
BG 0
BG 0
orbital elongation cos 2D 5 meters
V
BG
a
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Lunar Orbit Perturbations Part Deux

• Velocity moon orbits earth at 1 km/s

to sun
u
BG
a
D
V
BG 0
BG 0
orbital offset cos D 5 meters
u
BG
a
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The Controversy

• Lunar Laser Ranging (LLR) confirms the
  gravitomagnetic term to 0.1
• Gravity Probe B will confirm it with a different
  method to only 1
• Word-of-mouth claims say there is more to GPB
  than just the gravitomagnetic effect
• But 2 did the math, and recovers the published
  value of 42 mas/y
• Is GPB new physics?
• Most all of the 1/c2 order, post-Newtonian terms
  in the N-body equations of motion motional,
  gravitomagnetic, non-linear, inductive, etc.
  contribute to the measured details of the lunar
  orbit, so LLR achieves near-completeness as a
  gravity experiment and probe. 3, p1
• Possible confusion due to Suns quadrupolar tidal
  field, which produces cos 2D term (but 90o out of
  phase). 3, p3