Precession of a gyroscope due to a nearby celestial body's rotation affecting spacetime
In
general relativity
,
Lense?Thirring precession
or the
Lense?Thirring effect
(
Austrian German:
[?l?ns?
?t?r?ŋ]
; named after
Josef Lense
and
Hans Thirring
) is a
relativistic
correction to the
precession
of a
gyroscope
near a large rotating mass such as the Earth. It is a
gravitomagnetic
frame-dragging
effect. It is a prediction of general relativity consisting of
secular
precessions of the longitude of the
ascending node
and the
argument of pericenter
of a test particle freely orbiting a central spinning mass endowed with
angular momentum
.
The difference between
de Sitter precession
and the Lense?Thirring effect is that the de Sitter effect is due simply to the presence of a central mass, whereas the Lense?Thirring effect is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense?Thirring precession.
According to a 2007 historical analysis by Herbert Pfister,
[1]
the effect should be renamed the
Einstein
?Thirring?Lense effect.
The Lense?Thirring metric
[
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]
The gravitational field of a spinning spherical body of constant density was studied by Lense and Thirring in 1918, in the
weak-field approximation
. They obtained the metric
[2]
[3]
where the symbols represent:
- the
metric
,
- the flat-space
line element
in three dimensions,
- the "radial" position of the observer,
- the
speed of light
,
- the
gravitational constant
,
- the completely antisymmetric
Levi-Civita symbol
,
- the mass of the rotating body,
- the
angular momentum
of the rotating body,
- the
energy?momentum tensor
.
The above is the weak-field approximation of the full solution of the
Einstein equations
for a rotating body, known as the
Kerr metric
, which, due to the difficulty of its solution, was not obtained until 1965.
The Coriolis term
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The frame-dragging effect can be demonstrated in several ways. One way is to solve for
geodesics
; these will then exhibit a
Coriolis force
-like term, except that, in this case (unlike the standard Coriolis force), the force is not fictional, but is due to frame dragging induced by the rotating body. So, for example, an (instantaneously) radially infalling geodesic at the equator will satisfy the equation
[2]
where
- is the time,
- is the
azimuthal angle
(longitudinal angle),
- is the magnitude of the angular momentum of the spinning massive body.
The above can be compared to the standard equation for motion subject to the
Coriolis force
:
where
is the
angular velocity
of the rotating coordinate system. Note that, in either case, if the observer is not in radial motion, i.e. if
, there is no effect on the observer.
Precession
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The frame-dragging effect will cause a
gyroscope
to
precess
. The rate of precession is given by
[3]
where:
- is the
angular velocity
of the precession, a vector, and
one of its components,
- the angular momentum of the spinning body, as before,
- the ordinary flat-metric
inner product
of the position and the angular momentum.
That is, if the gyroscope's angular momentum relative to the fixed stars is
, then it precesses as
The rate of precession is given by
where
is the
Christoffel symbol
for the above metric.
Gravitation
by Misner, Thorne, and Wheeler
[3]
provides hints on how to most easily calculate this.
Gravitomagnetic analysis
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]
It is popular in some circles to use the
gravitomagnetic
approach to the
linearized field equations
. The reason for this popularity should be immediately evident below, by contrasting it to the difficulties of working with the equations above. The linearized metric
can be read off from the Lense?Thirring metric given above, where
, and
. In this approach, one writes the linearized metric, given in terms of the gravitomagnetic potentials
and
is
and
where
is the gravito-electric potential, and
is the gravitomagnetic potential. Here
is the 3D spatial coordinate of the observer, and
is the angular momentum of the rotating body, exactly as defined above. The corresponding fields are
for the gravito-electric field, and
is the gravitomagnetic field. It is then a matter of substitution and rearranging to obtain
as the gravitomagnetic field. Note that it is half the Lense?Thirring precession frequency. In this context, Lense?Thirring precession can essentially be viewed as a form of
Larmor precession
. The factor of 1/2 suggests that the correct gravitomagnetic analog of the
gyromagnetic ratio
is (curiously!) two. This factor of two can be explained completely analogous to the electron's g-factor by taking into account relativistic calculations.
The gravitomagnetic analog of the
Lorentz force
in the non-relativistic limit is given by
where
is the mass of a test particle moving with velocity
. This can be used in a straightforward way to compute the classical motion of bodies in the gravitomagnetic field. For example, a radially infalling body will have a velocity
; direct substitution yields the Coriolis term given in a previous section.
Example: Foucault's pendulum
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]
To get a sense of the magnitude of the effect, the above can be used to compute the rate of precession of
Foucault's pendulum
, located at the surface of the Earth.
For a solid ball of uniform density, such as the Earth, of radius
, the
moment of inertia
is given by
so that the absolute value of the angular momentum
is
with
the angular speed of the spinning ball.
The direction of the spin of the Earth may be taken as the
z
axis, whereas the axis of the pendulum is perpendicular to the Earth's surface, in the radial direction. Thus, we may take
, where
is the
latitude
. Similarly, the location of the observer
is at the Earth's surface
. This leaves rate of precession is as
As an example the latitude of the city of
Nijmegen
in the Netherlands is used for reference. This latitude gives a value for the Lense?Thirring precession
At this rate a
Foucault pendulum
would have to oscillate for more than 16000 years to precess 1 degree. Despite being quite small, it is still two orders of magnitude larger than
Thomas precession
for such a pendulum.
The above does not include the
de Sitter precession
; it would need to be added to get the total relativistic precessions on Earth.
Experimental verification
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]
The Lense?Thirring effect, and the effect of frame dragging in general, continues to be studied experimentally. There are two basic settings for experimental tests: direct observation via satellites and spacecraft orbiting Earth, Mars or Jupiter, and indirect observation by measuring astrophysical phenomena, such as
accretion disks
surrounding
black holes
and
neutron stars
, or
astrophysical jets
from the same.
The
Juno
spacecraft's suite of science instruments will primarily characterize and explore the three-dimensional structure of Jupiter's polar
magnetosphere
,
auroras
and mass composition.
[4]
As Juno is a polar-orbit mission, it will be possible to measure the orbital
frame-dragging
, known also as Lense?Thirring precession, caused by the
angular momentum
of Jupiter.
[5]
Results from astrophysical settings are presented after the following section.
Astrophysical setting
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]
A star orbiting a spinning
supermassive black hole
experiences Lense?Thirring precession, causing its orbital
line of nodes
to precess at a rate
[6]
where
- a
and
e
are the
semimajor axis
and
eccentricity
of the orbit,
- M
is the mass of the black hole,
- χ
is the dimensionless spin parameter (0 <
χ
< 1).
The precessing stars also exert a
torque
back on the black hole, causing its spin axis to precess, at a rate
[7]
where
- L
j
is the
angular momentum
of the
j
-th star,
- a
j
and
e
j
are its semimajor axis and eccentricity.
A gaseous
accretion disk
that is tilted with respect to a spinning black hole will experience Lense?Thirring precession, at a rate given by the above equation, after setting
e
= 0 and identifying
a
with the disk radius. Because the precession rate varies with distance from the black hole, the disk will "wrap up", until
viscosity
forces the gas into a new plane, aligned with the black hole's spin axis (the "
Bardeen?Petterson effect
").
[8]
Astrophysical tests
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]
The orientation of an
astrophysical jet
can be used as evidence to deduce the orientation of an
accretion disk
; a rapidly changing jet orientation suggests a reorientation of the accretion disk, as described above. Exactly such a change was observed in 2019 with the black hole X-ray binary in
V404 Cygni
.
[9]
Pulsars
emit rapidly repeating radio pulses with extremely high regularity, which can be measured with microsecond precision over time spans of years and even decades. A 2020 study reports the observation of a pulsar in a tight orbit with a
white dwarf
, to sub-millisecond precision over two decades. The precise determination allows the change of orbital parameters to be studied; these confirm the operation of the Lense?Thirring effect in this astrophysical setting.
[10]
It may be possible to detect the Lense?Thirring effect by long-term measurement of the orbit of the
S2 star
around the supermassive black hole in the center of the
Milky Way
, using the GRAVITY instrument of the
Very Large Telescope
.
[11]
The star orbits with a period of 16 years, and it should be possible to constrain the angular momentum of the black hole by observing the star over two to three periods (32 to 48 years).
[12]
See also
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]
References
[
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]
- ^
Pfister, Herbert (November 2007). "On the history of the so-called Lense?Thirring effect".
General Relativity and Gravitation
.
39
(11): 1735?1748.
Bibcode
:
2007GReGr..39.1735P
.
CiteSeerX
10.1.1.693.4061
.
doi
:
10.1007/s10714-007-0521-4
.
S2CID
22593373
.
- ^
a
b
Ronald Adler; Maurice Bazin; Menahem Schiffer (1965). "Section 7.7".
Introduction to General Relativity
. McGraw-Hill Book Company.
ISBN
0-07-000423-4
.
- ^
a
b
c
Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1973). "Chapter 19".
Gravitation
. W. H. Freeman.
ISBN
0-7167-0334-3
.
- ^
"Juno Science Objectives"
.
University of Wisconsin-Madison
. Archived from
the original
on October 16, 2008
. Retrieved
October 13,
2008
.
- ^
Iorio, L. (August 2010). "Juno, the angular momentum of Jupiter and the Lense?Thirring effect".
New Astronomy
.
15
(6): 554?560.
arXiv
:
0812.1485
.
Bibcode
:
2010NewA...15..554I
.
doi
:
10.1016/j.newast.2010.01.004
.
- ^
Merritt, David
(2013).
Dynamics and Evolution of Galactic Nuclei
. Princeton, NJ:
Princeton University Press
. p. 169.
ISBN
978-1-4008-4612-2
.
- ^
Merritt, David
; Vasiliev, Eugene (November 2012). "Spin evolution of supermassive black holes and galactic nuclei".
Physical Review D
.
86
(10): 102002.
arXiv
:
1205.2739
.
Bibcode
:
2012PhRvD..86j2002M
.
doi
:
10.1103/PhysRevD.86.022002
.
S2CID
118452256
.
- ^
Bardeen, James M.; Petterson, Jacobus A. (January 1975).
"The Lense?Thirring Effect and Accretion Disks around Kerr Black Holes"
.
The Astrophysical Journal Letters
.
195
: L65.
Bibcode
:
1975ApJ...195L..65B
.
doi
:
10.1086/181711
.
- ^
James C. A. Miller-Jones; Alexandra J. Tetarenko; Gregory R. Sivakoff; Matthew J. Middleton; Diego Altamirano; Gemma E. Anderson; Tomaso M. Belloni; Rob P. Fender; Peter G. Jonker; Elmar G. Kording; Hans A. Krimm; Dipankar Maitra; Sera Markoff; Simone Migliari; Kunal P. Mooley; Michael P. Rupen; David M. Russell; Thomas D. Russell; Craig L. Sarazin; Roberto Soria; Valeriu Tudose (29 April 2019).
"A rapidly changing jet orientation in the stellar-mass black-hole system V404 Cygni"
(PDF)
.
Nature
.
569
(7756): 374?377.
arXiv
:
1906.05400
.
Bibcode
:
2019Natur.569..374M
.
doi
:
10.1038/s41586-019-1152-0
.
PMID
31036949
.
S2CID
139106116
.
- ^
"Space-time is swirling around a dead star, proving Einstein right again"
.
Space.com
. 2020-01-30.
- ^
Eisenhauer, Frank; et al. (March 2011). "GRAVITY: Observing the Universe in Motion".
The Messenger
.
143
: 16?24.
Bibcode
:
2011Msngr.143...16E
.
- ^
Grould, Marion; Vincent, Frederic H.; Paumard, Thibaut; Perrin, Guy (2016).
"Detection of relativistic effects on the S2 orbit with GRAVITY"
.
Proceedings of the International Astronomical Union
.
11
(S322). Cambridge University Press (CUP): 25?30.
doi
:
10.1017/s174392131601245x
.
ISSN
1743-9213
.
External links
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