The LARES Mission: An Opportunity to Teach General Relativity
Frame Dragging and Lense-Thirring Effect
A. Paolozzi
1,2
, C. Paris
2,1
, G. Sindoni
1
and A. Tartaglia
3
1
School of Aerospace Engineering and DIAEE, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy
2
Centro Fermi, Via Panisperna 89, 00184 Rome, Italy
3
Applied Science and Technology Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy
Keywords:
LARES, General Relativity, Frame Dragging, Lense-Thirring Effect, SLR.
Abstract:
LARES is an Italian Space Agency mission devoted to test frame-dragging, a prediction of general relativity.
On February 2012 the satellite has been successfully put in orbit with the qualification flight of VEGA, the
new European Space Agency launcher. Basic concepts of general relativity are becoming more and more
familiar because of the part they play in science fiction movies. But frame-dragging (more formally known
as the Lense-Thirring effect), is so peculiar that it is a relatively unknown effect. The idea of this paper is to
start from the description of the experiment and then to push some parameters of the experiment to extreme
values in order to magnify the effects of relativity. This approach will provide not only the students and
general people but also professionals not strictly specialized in general relativity, with increased interest in
gravitational theories.
1 INTRODUCTION
Many concepts of special and general relativity,
though difficult to understand in detail, are known to
be true even by non-specialists. Deflection of light
by a mass is an example. The phenomenon of grav-
itational waves is another. It is even possible, via
analogies, to explain that space and time are indeed
one single four-dimensional entity called spacetime.
More difficult is the idea that time is not absolute but
flows in different ways depending on the state of mo-
tion of the observer and on the strength of the grav-
itational field. Time, being relative to the observer,
has been used in several science fiction movies so that
many people are aware of this effect even though it re-
mains an area for experts. Frame-dragging (Ciufolini,
2010) is another intriguing prediction of general rel-
ativity, but known only to specialists. The name
“frame-dragging” was given by Einstein in a private
communication with Ernst Mach (Einstein, 1913) be-
fore the general theory of relativity was published
(Einstein, 1915). But the first mathematical deriva-
tion of this effect was performed few years later by
Josef Lense and Hans Thirring, two Austrian physi-
cists, that derived it in (Lense and Thirring, 1918).
Later a generalization was performed by Roy Kerr
that found an exact solution to the very complex non
linear equations of general relativity in the case of a
rotating mass (Kerr, 1963). However until few years
ago the smallness of this effect made its direct mea-
surement impossible. We need to come to the year
1997 to have the first measurement with the LAGEOS
satellites (Ciufolini et al., 1997) that was later con-
firmed with an accuracy of about 10 % (Ciufolini and
Pavlis, 2004), and, using a different spacecraft (Grav-
ity Probe B), with an accuracy of 19 % (Everitt et al.,
2011). There are also other proposed experiments to
measure the frame-dragging such as GINGER (Bosi
et al., 2011)(DiVirgilio et al., 2014) planned in the
next few years. The LARES satellite, put in orbit with
the new VEGA launcher on the 13th February 2012
(Paolozzi and Ciufolini, 2013) (Paolozzi et al., 2015),
is expected to measure the Lense-Thirring effect with
an unprecedented accuracy of about 1 % (Ciufolini
et al., 2011)(Ciufolini et al., 2012a)(Ciufolini et al.,
2012b). While the equivalence principle, at the foun-
dation of general relativity, has a fundamental role
also in classical Galilei-Newton mechanics, gravita-
tional waves and frame-dragging have no counterparts
in classical theories. The use of science fiction and
relevant simulations could be seen as a way to involve
students and non specialists in science and technol-
ogy. In the following we will describe frame drag-
ging, recalling some basic principles of relativity, and
343
Paolozzi A., Paris C., Sindoni G. and Tartaglia A..
The LARES Mission: An Opportunity to Teach General Relativity - Frame Dragging and Lense-Thirring Effect.
DOI: 10.5220/0005498503430348
In Proceedings of the 7th International Conference on Computer Supported Education (CSEDU-2015), pages 343-348
ISBN: 978-989-758-108-3
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
the LARES experiment. Then we will push some pa-
rameters of the experiment to extreme values so as to
magnify the effects of relativity so well exploited in
science fiction.
2 FRAME-DRAGGING AND
LENSE-THIRRING EFFECT
We are so accustomed to very common physical phe-
nomena that we do not realize the deep concepts they
involve. The centrifugal force, or more generally any
inertial force, is a good example. Everyone has felt
those forces in a non-uniform motion. A car mov-
ing along a curve or accelerating induces an “appar-
ent force”: the inertial force or, in the first case, more
specifically the centrifugal force. Those forces ex-
ist in the body fixed reference frame, which is not
an inertial frame. But the next question is: what are
the inertial reference frames? According to the stan-
dard definition they correspond to observers not sub-
ject to any force. General relativity generalizes that
definition stating that all “freely falling” observers
are equivalent to inertial observers, but the free fall
depends on the mass-energy distribution in the uni-
verse. In fact masses distort spacetime, a fact mathe-
matically embodied in the elements of the metric ten-
sor, i.e. the mathematical tool expressing distances
and, more generally, the geometry of spacetime. The
warps in spacetime in turn determine the paths of
freely falling objects in the gravitational field. These
paths are called geodesics in space-time. A person
(“an observer”) falling along a geodesic path is lo-
cally indistinguishable from a person in an inertial
reference frame. A massive body distorts spacetime,
furthermore, a rotating massive body will produce an
additional spacetime deformation. Since the Earth ro-
tates, it will produce such an additional spacetime dis-
tortion, though very weakly because it rotates very
slowly and is not very massive. In Newtonian me-
chanics a perfectly spherical and homogeneous body
will generate a gravitational field in the same way as
if the entire mass was concentrated in the centre of
the sphere. The orbital plane of a satellite will re-
main fixed, in the simplified hypothesis of absence of
non conservative forces, with respect to distant stars,
which in classical mechanics form an inertial refer-
ence frame. But if the body rotates, general relativity
predicts that also the orbital plane rotates very slowly.
The classical angular momentum
J of a body is a
vector defined as I
ω , where I represents the mass
property of the body (more rigorously called moment
of inertia) and
ω the angular velocity vector which is
oriented as shown in Figure 1.
Figure 1: Representation of angular velocity (left) and of
angular momentum (right).
2.1 Lense-Thirring Effect
A graphical representation of what happens to the or-
bital plane of a satellite in classical and in general rel-
ativity theory is represented in Figure 2.
Figure 2: Orbital planes in a reference frame fixed with re-
spect to distant stars. (Top) classical mechanics; (bottom)
general relativity. The red arrow shows the Lense-Thirring
nodal shift.
Mathematically, by representing the unit vector of
J with
ˆ
J, the angular velocity of the precession
of the orbital plane of a satellite is (in the approxima-
tion of slow motion and weak field) (Chandrasekhar,
1983):
LT
=
2GJ
c
2
a
3
(1 e
2
)
3/2
ˆ
J (1)
where
LT
is the Lense-Thirring precession of an
orbital plane, a and e the semimajor axis and the ec-
centricity of the orbit, c the speed of light and G the
gravitational constant.
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2.2 Frame-dragging
More generally frame-dragging can be evaluated, for
a generic point of coordinates (a,θ), where θ is the co-
latitude. The longitude is not present because frame-
dragging is identical for points of any longitude.
The precession is given by (Tartaglia, 2000):
=
R
S
AC
a
3
+ aA
2
+ R
S
A
2
sin
2
θ + aA
2
cos
2
θ(1 +
A
2
a
2
)
ˆ
J
(2)
where
A =
J
Mc
(3)
J = Iω (4)
and
R
S
=
2GM
c
2
(5)
is the Schwarzschild radius and M the mass of the
central body. The Schwarzschild radius can be con-
sidered a radius inside which a mass is so concen-
trated that nothing can escape from it, not even light.
3 LARES SPACE EXPERIMENT
LARES is a passive satellite of the Italian Space
Agency (ASI) put in orbit on the 13 February 2012.
The position of the satellite is obtained by means of
about 50 laser stations belonging to the International
Laser Ranging Service (Pearlman et al., 2002). The
satellite was designed to minimize the effects of non
gravitational perturbations (Paolozzi et al., 2011) and
in particular thermal thrust (Ciufolini et al., 2014).
Alternative designs were studied earlier in (Ciufolini
et al., 2003) (Bosco et al., 2007). The main objec-
tive of the LARES mission is to measure the Lense-
Thirring effect, and to improve its previous measure-
ment (Ciufolini, 2010) (Ciufolini et al., 2012c) by one
order of magnitude (Ciufolini et al., 2010). The fea-
sibility of this goal has been demonstrated in (Ciu-
folini et al., 2013a) and by the recent data analysis
performed in (Ciufolini et al., 2012b) (Ciufolini et al.,
2013b).
The fact that the Earth is neither spherical nor ho-
mogeneous causes the orbital plane to rotate as shown
in Figure 3 where the actual orbits of LARES, deter-
mined by laser ranging, are drawn. The experimental
value obtained for LARES orbit precession is of 1.706
degrees per day, as can be approximately verified in
Figure 3 using a protractor. An entire rotation of 360
degrees occurs in 211 days. The Lense-Thirring effect
is instead much smaller: only about 0.118/year. That
is, about 19 million times smaller than the classical
effect.
Figure 3: Real orbits of LARES six days apart one from
the other as seen from the north pole in an inertial reference
frame. Units in the axis are meters.
4 LIMIT CASES
In this section we will examine limit cases in which
the relativity effects are magnified. We will suppose
first the Earth rotating with a surface peripheral ve-
locity not far from the speed of light. We will then
consider LARES orbiting around a black hole with
the same “size” of the Earth. We shall discover that
LARES is too close to the event horizon of the black
hole and it would spiral down. However the orbit of
LARES will be considered stable also in this case.
4.1 Earth
Frame-dragging would be more pronounced if the
Earth angular momentum would increase. Consider-
ing the Earth undeformable and unbreakable, let us
imagine a limit case in which the Earth would rotate,
so that the surface almost reaches the speed of light.
In Figure 4 equations 1 and 2 are compared. In
particular the effect of the colatitude θ on frame drag-
ging is small even at the speed of light. Values of θ
are reported in the top left corner of the table.
So, without making significant errors, we can, for
the sake of simplicity, assume θ = 90 degrees (i.e. a
point of the equator). In this case equation 2 yields
19.4 degrees/year. So summarizing we see that frame
dragging is very small around the Earth, no matter
how fast it rotates: for an Earth equator rotating at
TheLARESMission:AnOpportunitytoTeachGeneralRelativity-FrameDraggingandLense-ThirringEffect
345
Figure 4: Frame-dragging for different values of θ and Lense-Thirring effect as a function of Earth peripheral speed (i.e. the
speed of points located in the Earth equator).
almost the speed of light, the effect on the LARES or-
bit would be only 19.4 degrees/year i.e., smaller than
the classical effect shown in Figure 3 due to the Earth
oblateness, which amounts to 623 degrees/year.
4.2 Rotating Black Hole
Suppose now to have a rotating black hole (techni-
cally it is called a Kerr black hole) whose inner event
horizon has the size of the Earth. This means that
the mass of the black hole, M
bh
, would be (using the
Schwarzschild radius formula):
M
bh
=
R
s
c
2
2G
=
Rc
2
2G
= 4.310
33
kg (6)
i.e. about 2160 solar masses. If the black hole
would not rotate, a small object (like LARES) could
not revolve in a stable orbit at the same distance at
which it rotates now. In fact the lowest possible cir-
cular orbit should have a radius of not less than 3 R
S
(i.e. a semimajor axis of about 20000 km), whereas
now LARES orbits at 7820 km from the center of the
Earth. In practice LARES would fall into the black
hole in a certain amount of time which is not the case
to calculate here. Just to pursue our example this as-
pect will be neglected and LARES will be considered
in a stable orbit around the black hole (Figure 5).
Let us first recall that for a black hole it is not pos-
sible to separate the angular velocity from the moment
of inertia. In fact a black hole destroys all the infor-
mation falling inside it; all that can be “felt”, in the
Figure 5: A LARES satellite orbiting a black hole of the
“size” of the Earth.
surroundings outside the event horizon, is the mass,
the angular momentum and the charge. There is an
upper limit for the total angular momentum i.e. the
ratio A/R
S
can be at most 1/2, which means that the
highest possible angular momentum would be:
J
max
= G
M
2
c
(7)
The application of equation 2 in the case of the
black hole under concern with a value of J ranging
from zero to J
max
provides the curves reported in Fig-
ure 6 parameterized with the value of the colatitudes
θ listed in the box in the top left corner. The values
of frame-dragging in this example are extraordinarily
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346
Figure 6: Frame-dragging for different values of θ (in the box on the upper left corner) as a function of the black hole angular
momentum.
high. At almost the maximum value of J, frame drag-
ging would be 16x10
9
deg/year i.e. the nodes of the
orbits would revolve at a rate of 1.4 rev/s.
5 CONCLUSIONS
The phenomenon of frame dragging has been de-
scribed using the LARES mission. The parameters of
the experiment have been pushed to unrealistic val-
ues, with the purpose of magnifying the effects of
relativity. An hypothetical LARES orbiting a rotat-
ing black hole, with event horizon of the same size as
the Earth has also been considered. The aim of the
study is mainly to attract the attention and interest of
non specialists to a tiny effect of general relativity, as
the frame dragging is. It is shown that the dragging
(or Lense-Thirring) effect becomes really important
only in the extreme case of a central rotating black
hole. Despite this fact the LARES mission will be
able to measure it around the Earth. The reader can
thus understand how delicate the experiment is and
what level of accuracy is needed to bring it to success.
ACKNOWLEDGEMENTS
The authors greatly acknowledge the Italian Space
Agency for supporting the LARES mission un-
der contracts I/043/08/0, I/043/08/1, I/034/12/0 and
I/034/12/1, and the International Laser Ranging Ser-
vice for tracking the satellite and providing the laser
ranging data.
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