Gravitiomagnetism

I wrote a paper for astrophysics class. Just copy-pasted from the pdf, so some formatting issues that i'm too lazy to fix right now. Here it is:



image: NCSA



Gravitomagnetism

Eric R. Andersen, Astrophysics, Spring 2007, Knox College


The technology required to measure the dragging of inertial
frames around massive spinning objects as predicted by general
relativity is now available. The theoretical basis of
gravitomagnetism in the context of general relativity is
summarized, focusing on the behavior of gyroscopes in the weakfield
and low-velocity limits. Selected recent observations of the
geodetic and Lense-Thirring effects in distant x-ray binaries, the
solar system, earth-orbiting satellites, and terrestrial
environments are reviewed.

I. INTRODUCTION
The theory of general relativity (GR), predicts that the
spacetime near a massive spinning body will be dragged
around the object, displaying a gravitational field with
magnetic-like components. This post-Newtonian effect of
gravitomagnetism, a.k.a. “the dragging of inertial frames”, can
be approximated by the solution of Lense and Thirring [1].
Accurate measurements of the Lense-Thirring (LT) effect are
currently at the frontier of experimental tests of GR. Direct
measurement of the effect is difficult, owing to its extreme
smallness in ordinary terrestrial and astronomical systems.
Recently, attempts to measure the effect have been made by
employing a variety of approaches. Systems exhibiting
measurable gravitomagnetic effects have been observed in
astronomical objects, earth-orbiting satellites, and even in the
laboratory.


II. THE DRAGGING OF INERTIAL FRAMES
A. The Kerr Metric
An exact solution to Einstein’s field equations which
describes the spacetime geometry due to a spinning mass was
discovered by Roy Kerr in 1963 [2]. The metric can be
written in Boyer-Lindquist (BL) coordinates (t,r,θ,φ) as

ds2 = -(1-2Mr/Σ) dt2 - 2(2Mr/Σ) a sin2θ dt dφ + (Σ/) dr2
+ Σ dθ2 + (A/Σ) sin2θ dφ2, (1)

where

Σ ≡ r2 +a2 cos2θ ,
 ≡ r2 +a2 – 2Mr,
A ≡ (r2 +a2)2 –  a2 sin2θ ,

and M, a are real parameters. In these coordinates, the axial
symmetry and time independence of the metric is made
explicit [3]. In the case M2>a2, the Kerr metric (1) describes a
rotating black hole with mass M and angular momentum
J=Ma. This spacetime has a number of notable features; a
ring-singularity shrouded by a pair of nested spherical event
horizons, and an ellipsoidal “ergoregion” wherein even light
must rotate along with the spinning mass [1]. One notes that
the metric (1) in BL coordinates contains coordinate
singularities on its axis of symmetry and at the event horizons.

Figure 1 The features of a Kerr black hole (Adapted from nrumiano [11] ).

The general covariance of the field equations allows one to
use a different coordinate system in order to avoid the
coordinate singularities. Kerr coordinates remove the
singularity at the horizon via an infinite compression of
coordinate time and an infinite untwisting of null geodesics in
the neighborhood of the outermost horizon [1]. Kerr-Schild
coordinates remove the axial singularity, and are used in [4] to
calculate the effects of the Kerr geometry on a gyroscope in
polar orbit around a spinning mass. In the context of
experimental measurements of frame-dragging that can be
made using terrestrial satellites, weak-field and low-velocity
assumptions are valid, and a first-order approximation in (a/r)
suffices for making predictions. Still, it should be noted that
more exact treatments of the situation are possible, and for
modeling extreme cases like black holes, such approaches may
be preferable [8] [9].

B
Figure 3 The predicted geodetic and frame-dragging of the gyroscopes in
Gravity Probe B [10].
B. The Lense-Thirring Approximation
In the case that the parameter a is small, (0<(a/r)<<1), the
Kerr metric assumes the form,
ds2 = – (1 – 2M/r) dt2 + (1 – 2M/r)-1 dr2
+ r2 (dθ2 + r2 sin2θ dφ2 ) – 4M (a/r) sin2θ dt dφ (2)
which is the metric of Lense and Thirring [5]. Analysis of the
behavior of a gyroscope placed in a circular polar orbit in this
spacetime gives, in the weak-field and slow-motion limit, two
distinct precessional terms. The rate of the gyroscope’s spin
precession relative to the distant stars, averaged over one
period, is calculated to be,
π a (M/r3)1/2 (3)
normal to the orbital plane and,
3π (M/r) (4)
in the plane of the orbit [4]. These two precession periods are
due to the LT effect and the geodetic effect (also known as de
Sitter – Fokker precession), respectively. It should be noted
that, in addition to these, GR predicts other precession
phenomena associated with the intrinsic angular momentum of
an orbiting particle, namely, Thomas precession and Schiff
precession [6]. Substituting values for a typical earth-satellite
experiment, one obtains ~8 arcsec/yr for the geodetic effect,
and ~0.05 arcsec/yr for the LT effect [4]. By an appropriate
change of coordinates, the geodetic effect can be recast as
frame-dragging; in this reference frame, both effects can be
seen as aspects of a common phenomenon--gravtiomagnetism
[7].
III. EXPERIMENTAL EVIDENCE FOR FRAME-DRAGGING
Frame-dragging can help to explain quasi-periodic
oscillations (QPOs) seen in the light-curves of certain X-ray
binary systems [8]. Low-mass X-ray binaries that contain a
neutron star (LMBXNs) and black hole binaries (BHBs) can
have an accretion disk which is tilted with respect to the orbital
plane. High luminosity X-ray emission comes primarily from
the innermost edge of the disk [8]. The accretion disk
undergoes precession due to frame-dragging, thereby
modulating the light curve. For LMBXNs and BHBs such as
Cyg X-1, the Lense-Thirring approximation is sufficient to
explain the observed low-frequency QPOs therein observed,
while for rapidly-spinning systems such as GRS 1915+105, a
more exact description of the frame-dragging effect such as
that found by Wilkins [9] is required to account for the
observed QPO’s centroid frequencies [8]. The agreement of
the calculated disk precession rates with observed x-ray
oscillations in BHBs is compelling evidence of relativistic
frame dragging in the strong-field and rapid-rotation limits.
Furthermore, the observed QPO frequency can help to
determine the physical parameters (e.g. spin) of BHBs [8].
The Lense Thirring effect has been measured by observing
the orbit of geodetic satellites. The LAGEOS and LAGEOS 2
satellites are small geodetic satellites which are covered by
retroreflective mirrors to allow precise laser-ranging. These
satellites describe nearly circular, complementary nonequatorial
earth orbits at an altitude of about 12,200 km [14].
By combining the range data from the two satellites, the nodal
variations due to the earth’s quadrupole moment can be
eliminated, allowing the geodetic and frame-dragging effects
due to the earth’s rotation to be measured [14].
Figure 2 The orbits of LAGEOS and LAGEOS 2. (Reproduced from
Cuifolini and Pavlis [14] ).
To accurately measure the frame-dragging effect, Ciufolini
and Pavlis combined the orbital data from LAGEOS and
LAGEOS 2 with a high-resolution gravity model of the earth,
EIGEN-GRACE02S [14]. The frame-dragging thus measured
agreed with the Lense-Thirring effect with a reported
uncertainty of +/- 10% [14].
Amongst the various tests of GR which have been attempted
in recent times, none has received as much attention in the
popular press as Gravity Probe B. A way to measure framedragging
with earth-orbiting gyroscopes was proposed in 1959
by Leonard Schiff and Robert Cannon of Stanford University
[10]. The experiment is conceptually simple, but technically
quite challenging. The basic idea was to put a satellite into a
polar orbit around the earth, and measure the deviations of the
spin-axes of gyroscopes relative to the distant stars. Their goal
was to measure the effect with an uncertainty of less than 1%.
In order to achieve the desired resolution, incredible
engineering hurdles needed to be overcome. The telescope
used to keep the satellite pointed at a fixed point in the sky
needed unprecedented resolution. The gyroscopes had to be
nearly perfect spheres. The effects of the earth’s magnetic
3
field and electrostatic effects had to be virtually eliminated,
and the satellite’s orbit needed to be tracked to within a
centimeter. With funding from NASA and the Air Force
commencing in 1965, the engineering obstacles were
ponderously and systematically tackled [10]. Finally in April
of 2004, Gravity Probe B was launched.
Figure 3 The predicted geodetic and frame-dragging of the gyroscopes in
Gravity Probe B [10].
The spacecraft took data for about one year before its
supply of liquid helium ran out; the data are still being
analyzed. Preliminary data confirm the geodetic effect, but the
magnitude of frame-dragging has yet to be confirmed within
the desired uncertainty [10]. Unexpected dampening of the
polhode motion of the gyroscopes was observed during the
mission. The anomaly has been ascribed to unexpected
appearances of electrostatic patches on the gyroscopes and
rotors. Models have been constructed to account for such
effects; improved results incorporating these are expected in
late 2007 [10].
An anomalous gravitomagnetic moment has been observed
in the vicinity of rapidly-spinning low-Tc superconductors by
scientists at the European Space Agency [12] [13]. The
laboratory work was performed at the ARC Seibersdorf
Reseach organization in Seibersdorf Austria. With a Niobium
superconductor, the application of a tangential acceleration to
the disk of 1500 rad/s2 resulted in an acceleration field outside
of the disk of 100μg, 30 orders of magnitude larger than
predicted by classical general relativity. Signal to noise ratios
of the accelerometers were better than 3:1, and differential
signals (derived with reference accelerometers) were used for
bias removal [13]. The effect was only observed when the
superconductor was below its critical temperature. High-Tc
superconductors did not display the effect. A possible
theoretical explanation of these observations is offered in [12],
attributing the effect to a nonzero photon mass within the bulk
superconductor. If confirmed, this would be the first
measurement of a gravitational field produced in the
laboratory, outside of conventional Cavendish-style
experiments.
IV. CONCLUSION
The dragging of inertial frames, once a purely theoretical
prediction, is now firmly established as empirical fact. Framedragging
accounts for the rates of QPOs in LMBXNs and
BHBs, has been observed by orbiting satellites, and may be
present (in highly magnified form) for spinning
superconductors. The extreme smallness of the effect as
predicted by general relativity continues to complicate
attempts at direct measurement in classical systems. However,
the anticipated success of Gravity Probe B in reducing the
uncertainty in the measured effect may help to narrow the field
of possible metric theories of gravity within the parameterized
post-Newtonian framework [1]. The tantalizing first reports of
a gravitomagnetic London moment in rotating
superconductors, if confirmed, may provide for terrestrial
confirmation of general relativistic effects.
ACKNOWLEDGMENT
The author would like to thank Knox College and its
physics department for their continued grace and hospitality.
REFERENCES
[1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, (Freeman,
San Francisco, 1973), pp.1117-1120, 1119f, 879ff, 893-896, 1120
[2] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963).
[3] R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967).
[4] D. Tsoubelis, A. Economou, and E. Stoghianidis, Phys. Rev. D 36, 1045
(1987).
[5] J. Lense and H. Thirring, Phys. Z. 19, 156 (1918).
[6] I. Cuifolini, Phys. Rev. Lett. 56, 278 (1986).
[7] N. Ashby and B. Shahid-Saless, Phys. Rev. D 42, 1118 (1990).
[8] W. Cui, S. N. Zhang, and W. Chen, ApJ 492, L53 (1998).
[9] Wilkins, D. C., Phys. Rev. D 5, 814 (1972).
[10] The Gravity Probe B Website, http://einstein.stanford.edu/
[11] http://nrumiano.free.fr/Estars/int_bh2.html
[12] C. J. de Matos and M. Tajmar,
http://arxiv.org/ftp/cond-mat/papers/0602/0602591.pdf
[13] M. Tajmar1, F. Plesescu1, K. Marhold1, and C. J. de Matos,
http://esamultimedia.esa.int/docs/gsp/Experimental_Detection.pdf
[14] L. Ciufolini and E. C. Pavlis, Nature 431, 958 (2004)

Eric R. Andersen (b. 1972), grew up in the city of Galesburg, IL. He
received a bachelor of arts degree in physics and art from Knox College in
1997. His research interests include theoretical physics, philosophy, and farout
conspiracy theories.