Michael Soffel

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The Earth rotates, but about what?

• Michael Soffel

Dresden Technical
University
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The two classical answers
the stellar compass
the inertial compass
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Newtons absolute space
In Newtons theory these are simply two ways to
recover absolute space
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The stellar compass

• stellar astrometry -gt
• (quasi-inertial) celestial reference
  system
• Necessary corrections
• proper motions
• parallax
• aberration

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Astrometry accuracies
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VLBI and the ICRS
21-m VLBI antenna Wettzell, Germany
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ICRF Source distribution
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Quasars as reference objects
Problems - Frequency dependent radio core
position (e.g., 1823568) - structure and
variability
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Uncertainties in position
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Structure and variablities cannot be modelled
Blandford-Znajek model of a quasar engine
electric currents
interstellar medium
Spinning black hole
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Astrometry relativity becomes crucial
Relativity aberration cannot be described
correctly without it!
Annalen der Physik und Chemie Jg. 17, 1905 1905
birth of SRT
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Speed of light is independent upon the velocity
of the light-source
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Otherwise the appearance of binary systems
would be very different Multiple images of one
and the same star could be seen
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Nach der Äthertheorie sollte Raumschiff A
eine größere Lichtgeschwindigkeit messen als B
By means of an optical interferometer Michelson
and Morley proved that the speed of light is also
independent upon the velocity of the observer!
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Principle of a Michelson Interferometer
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The famous experiment of Michelson and Morley,
1887 Cleveland, Ohio
Erwartete Verschiebung bei 360 Grad Drehung 0,04
Streifen
Das Interferenzmuster hängt nicht von
der Ausrichtung der Plattform im Raum ab
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One consequence is clock at rest shows ?t , a
moving observer would measure
0
a moving clock appears to be slowed down
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A. Einstein also the appearance of space
depends upon the velocity of the observer
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SRT aberration
Lorentz transformation gives correct Special
Relativistic aberration formula

• For an observer on the Earth or on a typical
  satellite
• Newtonian aberration, (v/c) ?20?
• relativistic aberration, (v/c) ? 4 mas
• second-order relativistic aberration ? 1 ?as

2
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Gravitational light-deflection as consequence of
Einsteins GRT
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Einsteins theory of gravity extends the
Newtonian one to make it compatibel with
SRT and the equivalence principle
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Newtons law of gravity
F - G M m r/r
3
3

• G G (r) ?
• G G (t) ?
• experiments provide stringent limits
• for the 5th force and G-dot

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Weak equivalence principle apart from tidal
forces all uncharged test bodies fall at the
same rate
inertial mass gravitational mass
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Earth and Moon in free-fall towards Sun (LLR)
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A. Einstein Gravity can be understood as
effect of space-time curvature
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Gravitation als Phänomen der Krümmung von
Raum und Zeit
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Gravitational light deflection

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Gravitational light deflection

• The principal effects due to the major bodies of
  the solar system in ?as
• The maximal angular distance to the bodies where
  the effect is still gt1 ?as

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Gravitational light deflection moons, minor
planets

• A body of mean density ? produces a light
  deflection not less than ?
• if its radius

Pluto 7 Charon 4 Titania 3 Oberon
3 Iapetus 2 Rea 2 Dione 1 Ariel
1 Umbriel 1 Ceres 1
Ganymede 35 Titan 32 Io 30 Callisto
28 Triton 20 Europe 19
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Newtonian astrometry
physically preferred global inertial coordinates
observables are directly related to the inertial
coordinates
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Newtonian astrometry details

• Scheme
• aberration
• parallax
• proper motion
• All parameters of the model are defined
• in the preferred global coordinates

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Astrometry in GRT
no physically preferred coordinates
observables have to be computed as coordinate
independent quantities
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Astrometry and reference frames in GRT
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GRT metric as fundamental object

• Pythagorean theorem in 2-dimensional Euclidean
  space

• length of a curve

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Metric tensor special relativity

• special relativity, inertial coordinates

• The constancy of the velocity of light in
  inertial coordinates

can be expressed as where
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Metric tensor and reference systems

• In relativistic astrometry the
• BCRS (Barycentric Celestial Reference System)
• GCRS (Geocentric Celestial Reference System)
• Local reference system of an observer
• play an important role.
• All these reference systems are defined by
• the form of the corresponding metric tensor.

BCRS
GCRS
Local RS of an observer
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Barycentric Celestial Reference System
The BCRS is a particular reference system in the
curved space-time of the Solar system

• One can
• use any
• but one
• should
• fix one

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Barycentric Celestial Reference System

• The BCRS
• adopted by the International Astronomical Union
  (2000)
• suitable to model high-accuracy astronomical
  observations

relativistic gravitational potentials t TCB
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Barycentric orientation of spatial axes
IAU-GA 2006, Prag orientation of spatial BCRS
axes given by the ICRF
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equations of light propagation

• The equations of light propagation
• in the BCRS

• Relativistic corrections to
• the Newtonian straight line

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Geocentric Celestial Reference System
The GCRS is adopted by the International
Astronomical Union (2000) to model physical
processes in the vicinity of the Earth A The
gravitational field of external bodies is
represented only in the form of a
relativistic tidal potential. B The internal
gravitational field of the Earth coincides with
the gravitational field of a corresponding
isolated Earth.
internal inertial tidal external potentials
T TCG
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observables proper direction

• To describe observed directions (angles) one
  should introduce spatial
• reference vectors moving with the observer
  explicitly into the formalism

• Observed angles between incident light rays and
  a spatial reference vector
• can be computed with the metric of the local
  reference system of the observer

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The standard astrometric model
observed (topocentric) BCRS related to light
ray BCRS auxiliary quantities
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Sequences of transformations
(1) aberration (2) gravitational deflection (3)
coupling to finite distance (4) parallax (5)
proper motion
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Orientation of GCRS-axes
BCRS and GCRS are connected by a
4-dimensional space-time transformation
(generalized Lorentz-transformation) GCRS
kinematically non-rotating w.r.t. the
BCRS i.e., no rotation between x(BCRS) and
X(GCRS)
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Earths rotation and the stellar compass
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• The DSX-papers provide
• a new relativistic definition of the rotation
  vector L of the Earth
• as member of a gravitational N-body problem in
  the GCRS
• corresponding rotational equations of motion
• d/dT L D (post-Newtonian
  torque)
• The Earth rotates with respect to the GCRS if
  stellar
• compass provides the basic reference
• The connection to the quasars is only indirectly
  via the
• BCRS

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Some final remarks concerning the stellar compass
and the ICRS
Presently matter outside the solar system is not
taken into account
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• One might continue with a hierarchy of systems
• GCRS (geocentric celestial reference system)
• BCRS (barycentric)
• GaCRS (galactic)
• LoGrCRS (local group) etc.
• each systems contains tidal forces due to
• system below
• BUT
• expansion of the universe has to be taken into
  account
• if cosmic distances (redshifts etc.) are
  considered

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BCRS for a non-isolated system
Tidal forces from the next 100 stars their
quadrupole moment can be represented by two
fictitious bodies
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The universe appears to be homogeneous and
isotropic at large scales
-4
??/? lt 10 for R gt 1000 (Mpc/h)
(O.Lahav, 2000)
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a homogeneous and isotropic universe is
completely described by a cosmic scale factor a(t)
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BCRS and the expansion of the universe
work has been done (S.Klioner M.S.) to
include the cosmic expansion in the BCRS
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Gravitational lensing

• Gravitational light deflection caused by the
  gravitational fields
• generated outside the solar system
• microlensing by stars of our Galaxy,
• gravitational waves from compact sources,
• primordial (cosmological) gravitational waves,
• binary companions,
• Microlensing noise could be
• a crucial problem
• for going well below 1 microarcsecond

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The rotation of the universe
There are theoretical models where the
universe rotates intrinsically (non-zero
vorticity as in Gödels universe) about every
dynamically non-rotating observer Use the
quadrupole-anisotropy of the CMBR to get upper
limits!
-12
lt 10 rev. since Big Bang (Collins
Hawking)
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GRT and the inertial compass
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GRT and the inertial compass
Leon Foucault, 1851 Pantheon, Paris
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A modern version of the Foucault pendulum
a laser gyro
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The laser-gyro in Wettzell, Germany
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Cerodur groundplate 16 m
beam recombiner
laser excitation
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Status Ring Lasers very sensitive to
rotations Example G Regional deformations ?
strain effects ? transient signals in laser
measurements Reasons for deformations
atmospheric pressure gradients, wind loading,
mountain torques
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• Example
• Earth Rotation Rate measured in Wettzell (red
  line)
• Difference in atmospheric pressure between
  Wettzell
• and Medicina (black line)

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The problem of inertia in GRT
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Inertial frames
Das Problem in Newtons theory inertial frames
are determined by absolute space
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Dragging of inertial frames
In GRT locally inertial systems rotate with
respect to the fixed stars
A torque free gyro is dragged by the rotating
Earth (Lense-Thirring effect)
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Lense-Thirring effect in the motion of
satellites precession of orbit in space
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Frame dragging Experimentally detected in the
motion of satellites by I.Ciufolini
Lageos I (II) nodal drift 20 ?as/rev.
Ignazio Ciufolini
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The geodetic precession
A torque-free gyro, moving with the Earth
precesses w.r.t. the quasar-sky because of its
motion about the Sun. This geodetic precession
amounts to
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? (3/2c ) v x ?U ? 1.98 /cen.
GP
E
ext
If the Earth is considered in rotation w.r.t.
GCRS (stellar compass) the geodetic
precession/nutation will be in the PN-matrix
(even for zero ellipticity!)
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Frame dragging and the geodetic
precession Gravity-Probe B (GP-B) the longest
lasting experiment In history (start 1959
end 2006 ??)
(Francis Everitt)
Francis Everitt
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Gravity-Probe B
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R 2 cm 2000 Hz
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Successful launch April 20, 2004 from
Vandenberg Air Force Base
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• The fundamental problem with GP-B
• the Lense-Thirring effect has been seen already
  in
• satellite dynamics
• also the geodetic precession has been observed
  in the lunar
• orbit (LLR) possibly also in Earths rotation
  (Krasinsky)

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Geodetic precession in the lunar orbit

• the lunar orbit presents a gyroscope
• the geodetic precession leads to the de-Sitter
  Fokker
• precession
• ? 1.98/cen. (1 h)
• Shapiro et al. (1988)
• LLR data (1970 1986)
• simult. Determination of 335 parameters
• 250 EOP
• mass of Jupiter (10)
• h 0.019 ? 0.010

.
M
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Further LLR-measurements

• Müller et al. 91 analysis of 6300 LLR data (1969
  - 1990)
• Among the estimated parameters are
• geocentric station coordinates
• coordinates of 4 retroreflectors
• lower lunar potential-coefficients
• n-dot moon
• libration angles at initial epoch
• EOPs
• parameters of gravitational theories G-dot
• h 0.002 ? 0.01 0.002

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Earths rotation w.r.t. an inertial compass
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Earths rotation w.r.t. an inertial compass
A dynamically non-rotating geocentric system,
i.e., a local geocentric inertial-system (GIRS),
precesses w.r.t. the GCRS because of the
geodetic precession/nutation. If the Earth is
considered to be in rotation w.r.t. GIRS, the
geodetic precession/nutation will not appear in
the P-N matrix!
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Conclusions
Answers to the question about what the Earth
rotates are complicated and subtle. Both obvious
answers, about a kinematically non-rotating
(stellar compass) or a dynamically non-rotating
(inertial compass) geocentric system, involve
detailed knowledge about our cosmic
neighbourhood and the large scale structure of
the universe.
The End