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The Lense--Thirring Effect and Mach's Principle
Authors
Hermann Bondi,Joseph Samuel
Antonella Trotta,Fiona Gaughran,Yan Ding,Osuolale Tiamiyu,Thomas Palmer,Gerald Lawson,Stanley Halvorsen,Jose Alfonso Abecia,David Sherwood,Ambuj Sagar,elio desimoni,Roger Marchant,Edward Gregorich,Yasu-Taka Azuma,Sohair Abou-Elela,Sylvain Fichet,Konstantinos Karafasoulis,Manuel García León,Ivano Lippi,John Salamone,Armin Saalmüller,Galina Tereshchenko,Volker Strass,Denis Kachanov,Chimit Ochirov,Irina Davydova,Mariya Zhestkova,Natalia Migacheva,Svetlana Salugina,Jorge Casaus,Osvaldo León Córdoba,Ilonka Guenther,Alexander Sokolov,Ekaterina Nikolaeva,Catherine Klein,Замира Раджабова,Tamari Maniya,Rodion Rasulov,Javier Berdugo,Римма Терлецкая,Augustin Kasser,Jochen Uebe,D.Yu. Ovsyannikov,Patricia Mokhtarian,Alain VAUCHEZ,Peter K.N. Yu,Wayne Hodgson,Jose Hinojosa,Michael Foley,Robert Shepherd,Pak-Hing Leung,Constantino Ledesma-Montes,SIMON KAYEMBA-KAY'S,Nicholas Cook,Joshua Ko,Valery Ditlov,Elizabeth Boulding,Abi (Abebech) Belai,José Antonio Veira,Виктория Емельянова,stefan mehedinteanu,Charles Le Grand,Olga Senkevich,Etsuji Okada,Mikhail Akselrov,Claudio Pascali,Manuel Delfino,Mattanja Triemstra,Ramon Miquel,Brigitte Bloch-Devaux,Alexander Finch,Pier Simone Marrocchesi,Isidoro Ferrante,David Peterzell,Henriette van Praag,Giuseppe Vilardo,Georg Matuschek,Terrance Quinn II,George Leontaris,Mokhtar Chmeissani,Manuel Ramallo,Fomina Boriskina,Eric Chamberod,Dmitry Lebedev,Наиль Акрамов,Marina Shumikhina,Vyacheslav Lebedev,Natalia Zaikova,Darya Lakomova,Klaus Mehnert,Sanan Hamidov,Ludmila Ivashkevich,Anastasia Ponasenko,Светлана Денисова,Светлана Шугаева,Reinaldo Ruggiero,Jaume Masoliver,Poul Damgaard,Brenda Howard,Xuefeng Qian,Javier Dominguez,Christian France-Lanord,Michael Hennecke,Frans Klinkhamer,William Lee,Frederick Fuller,Anatoliy Godovalov,Vladimir Dlin,Tatyana Samsonova,Darina Gobadze,Vsevolod Misyurin,Sergey Gnusaev,Anatoly Khavkin,Brian Cummings,Catherine McCrohan,Bryan Finn,toshiyuki hamada,Bernard DUBRAY,Arpad Szucs,Harry Suehrcke,Chen-Bo Zhu,Hideto Kaba,Ali Zirh,Simon Taylor,Georgina Espinosa-Pérez,Gabor Nagy,Chih-Tien Wang,Juan F. 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arXiv:gr-qc/9607009v1 4 Jul 1996
The Lense–Thirring Effect and Mach’s Principle
Hermann Bondi† and Joseph Samuel
Raman Research Institute, Bangalore 560 080, India
Abstract
We respond to a recent paper by Rindler on the “Anti–Machian”
nature of the Lense–Thirring effect. We remark that his conclusion
depends crucially on the particular formulation of Mach’s principle
used.
† Permanent address: Churchill College, Cambridge, CB3 0DS, England
1
The Lense–Thirring Effect and Mach’s Principle
Hermann Bondi† and Joseph Samuel
Raman Research Institute, Bangalore 560 080, India
Abstract
We respond to a recent paper by Rindler on the “Anti–Machian”
nature of the Lense–Thirring effect. We remark that his conclusion
depends crucially on the particular formulation of Mach’s principle
used.
† Permanent address: Churchill College, Cambridge, CB3 0DS, England
1
1 Introduction
In a recent paper, Rindler [1] has analysed the Lense–Thirring effect [2, 3, 4]
and concluded that the result is anti-Machian. Rindler uses a particular
interpretation of Mach’s principle. We wish to stress here that Rindler’s
interpretation is only one amongst many. Indeed, the literature on this topic
is so diffuse that we think it desirable to set out a list of interpretations that
come to mind. Our list is far from exhaustive, but it is long enough to make
numbering different versions necessary.
We begin with Mach0, which is the basis of the whole idea: The universe,
as represented by the average motion of distant galaxies [5] does not appear
to rotate relative to local inertial frames.
We illustrate this point by a modern version of Newton’s famous bucket
experiment: the Sagnac effect. This effect provides an operational method
for an observer to decide, by local measurements, if she is rotating. Consider
an astronaut in an enclosed spaceship with angular velocity ω. The astronaut
takes a closed circular fibre optic tube at rest with respect to the spaceship
and sends two rays of monochromatic laser light in opposite directions around
the tube. These rays are made to interfere [6] after each ray has gone round
once. If the spaceship is rotating, the corotating ray will take longer to come
around than the counter–rotating one, leading to an arrival time difference,
which can be observed as a fringe shift. The time difference is given by:
∆t = −4Aω/c2, where ω is the angular velocity of the spaceship and A is the
area enclosed by the tube. Using the Sagnac effect, one can by experiments
internal to the spaceship, so arrange the angular velocity of the spaceship that
the Sagnac shift (defined as ∆t/2) vanishes. A frame at rest with respect to
such a spaceship is called a locally non rotating frame. Sit in this frame, look
2
In a recent paper, Rindler [1] has analysed the Lense–Thirring effect [2, 3, 4]
and concluded that the result is anti-Machian. Rindler uses a particular
interpretation of Mach’s principle. We wish to stress here that Rindler’s
interpretation is only one amongst many. Indeed, the literature on this topic
is so diffuse that we think it desirable to set out a list of interpretations that
come to mind. Our list is far from exhaustive, but it is long enough to make
numbering different versions necessary.
We begin with Mach0, which is the basis of the whole idea: The universe,
as represented by the average motion of distant galaxies [5] does not appear
to rotate relative to local inertial frames.
We illustrate this point by a modern version of Newton’s famous bucket
experiment: the Sagnac effect. This effect provides an operational method
for an observer to decide, by local measurements, if she is rotating. Consider
an astronaut in an enclosed spaceship with angular velocity ω. The astronaut
takes a closed circular fibre optic tube at rest with respect to the spaceship
and sends two rays of monochromatic laser light in opposite directions around
the tube. These rays are made to interfere [6] after each ray has gone round
once. If the spaceship is rotating, the corotating ray will take longer to come
around than the counter–rotating one, leading to an arrival time difference,
which can be observed as a fringe shift. The time difference is given by:
∆t = −4Aω/c2, where ω is the angular velocity of the spaceship and A is the
area enclosed by the tube. Using the Sagnac effect, one can by experiments
internal to the spaceship, so arrange the angular velocity of the spaceship that
the Sagnac shift (defined as ∆t/2) vanishes. A frame at rest with respect to
such a spaceship is called a locally non rotating frame. Sit in this frame, look
2
up at the sky and note that the distant galaxies are still. Mach’s principle
(Mach0) is the experimental observation that the inertial frame defined by
local physics (zero Sagnac shift) coincides with the frame in which the distant
objects are at rest.
Mach0 is an experimental observation and not a principle. One could
interpret Mach’s writings as a suggestion to construct a theory in which
Mach0 appears as a natural consequence. But Mach’s writings have been
variously interpreted. Our purpose here is to list a number of interpretations
of Mach’s principle and view them in the light of currently accepted theories
in an effort to refine and clarify the idea.
We do have at our disposal two well established theories of space, time,
gravity, matter and motion –Newton’s and Einstein’s– both experimentally
succesfull in their respective domains of validity. Newton’s holds that space
and time are absolute. Einstein’s holds that space time geometry is affected
by matter. There is no question (as Rindler observes) that these experimen-
tally successful theories are here to stay regardless of whether they satisfy
any of the rather philosophical criteria embodied in Mach’s principle.
2 Versions of Mach’s Principle
Recent discussions of Mach’s Principle, including this one, have greatly ben-
efitted from the 1993 Conference organised by J. Barbour and H. Pfister and
the excellent book [7] resulting from it. A glance at the book (note especially
J. Barbour’s list on page 530) will show that there have been numerous in-
terpretations of Mach’s writings. For an authoritative account of the history
of Machian ideas, the reader is referred to [7].
We now list a few versions of Mach’s principle which appear in the lit-
3
(Mach0) is the experimental observation that the inertial frame defined by
local physics (zero Sagnac shift) coincides with the frame in which the distant
objects are at rest.
Mach0 is an experimental observation and not a principle. One could
interpret Mach’s writings as a suggestion to construct a theory in which
Mach0 appears as a natural consequence. But Mach’s writings have been
variously interpreted. Our purpose here is to list a number of interpretations
of Mach’s principle and view them in the light of currently accepted theories
in an effort to refine and clarify the idea.
We do have at our disposal two well established theories of space, time,
gravity, matter and motion –Newton’s and Einstein’s– both experimentally
succesfull in their respective domains of validity. Newton’s holds that space
and time are absolute. Einstein’s holds that space time geometry is affected
by matter. There is no question (as Rindler observes) that these experimen-
tally successful theories are here to stay regardless of whether they satisfy
any of the rather philosophical criteria embodied in Mach’s principle.
2 Versions of Mach’s Principle
Recent discussions of Mach’s Principle, including this one, have greatly ben-
efitted from the 1993 Conference organised by J. Barbour and H. Pfister and
the excellent book [7] resulting from it. A glance at the book (note especially
J. Barbour’s list on page 530) will show that there have been numerous in-
terpretations of Mach’s writings. For an authoritative account of the history
of Machian ideas, the reader is referred to [7].
We now list a few versions of Mach’s principle which appear in the lit-
3
erature. Each statement of Mach’s principle, will be accompanied by a dec-
laration of the theoretical framework in which it is intended to apply. Two
levels of compatibility will be considered: Does the particular statement of
Mach’s Principle make sense in the theory, and secondly, is it satisfied by
it? We use the letters N and E to refer to Newtonian and Einsteinian space
time. Even within Einstein’s theory there is a further dichotomy– is one dis-
cussing cosmology (the whole universe) or an isolated system embedded in
an asymptotically flat space time? This distinction is made by the notation
EA for asyptotically flat spacetimes and EC for relativistic Cosmologies. Our
purpose in compiling this list is to draw attention to the diversity of ideas
that pass under the guise of “Mach’s principle”. (Page numbers refer to [7]
unless otherwise indicated.)
• Mach1: Newton’s gravitational constant G is a dynamical field. (Makes
sense in N, EA, EC.) Mach1 is not true in N or E. This version applied
to Einstein’s theory has led to Brans–Dicke Theory[8, 9].
• Mach2: An isolated body in otherwise empty space has no inertia (pp
11,39,181, 185). (Makes sense in N, EA, EC.) Neither Newtonian nor
Einsteinian gravity satisfy this version. In both theories the motion of
an isolated body is determined and not arbitrary.
• Mach3: local inertial frames are affected by the cosmic motion and
distribution of matter (p92). (Makes sense in N, EA, EC [10] .) This
version is closest to the bucket experiment. In this form, Newton’s
theory is in clear conflict with Mach3. Einstein’s theory is not (see
section 4 below).
• Mach4: The universe is spatially Closed (p 79). (Makes sense only in
4
laration of the theoretical framework in which it is intended to apply. Two
levels of compatibility will be considered: Does the particular statement of
Mach’s Principle make sense in the theory, and secondly, is it satisfied by
it? We use the letters N and E to refer to Newtonian and Einsteinian space
time. Even within Einstein’s theory there is a further dichotomy– is one dis-
cussing cosmology (the whole universe) or an isolated system embedded in
an asymptotically flat space time? This distinction is made by the notation
EA for asyptotically flat spacetimes and EC for relativistic Cosmologies. Our
purpose in compiling this list is to draw attention to the diversity of ideas
that pass under the guise of “Mach’s principle”. (Page numbers refer to [7]
unless otherwise indicated.)
• Mach1: Newton’s gravitational constant G is a dynamical field. (Makes
sense in N, EA, EC.) Mach1 is not true in N or E. This version applied
to Einstein’s theory has led to Brans–Dicke Theory[8, 9].
• Mach2: An isolated body in otherwise empty space has no inertia (pp
11,39,181, 185). (Makes sense in N, EA, EC.) Neither Newtonian nor
Einsteinian gravity satisfy this version. In both theories the motion of
an isolated body is determined and not arbitrary.
• Mach3: local inertial frames are affected by the cosmic motion and
distribution of matter (p92). (Makes sense in N, EA, EC [10] .) This
version is closest to the bucket experiment. In this form, Newton’s
theory is in clear conflict with Mach3. Einstein’s theory is not (see
section 4 below).
• Mach4: The universe is spatially Closed (p 79). (Makes sense only in
4
EC.) We do not know if Mach4 is true.
• Mach5: the total energy, angular and linear momentum of the universe
are zero (p237).(Makes sense in N, EA, EC.) It is not true in N and
EA. In EC it is claimed [11] that the total angular momentum of a
closed universe must vanish.
• Mach6: Inertial mass is affected by the global distribution of matter (pp
91,249). Makes sense in (N, EA, EC). Is not true in any of them. Hoyle
and Narlikar [12] proposed a theory in which implements Mach6.
• Mach7: If you take away all matter, there is no more space [13]. Makes
sense in (N,EA,EC). Not true in any of them.
• Mach8: Ω = 4πρGT 2 is a definite number of order unity (p475). (Here,
ρ is the mean density matter in the universe and T is the Hubble time.
Makes sense in EC only.) Ω does seem to be of order unity in our present
universe, but note that of all EC models, only the Einstein–DeSitter
makes this number a constant, if Ω is not exactly one. Making a theory
in which this approximate equality appears natural is a worthwhile and
ongoing effort (eg inflationary cosmologies).
• Mach9: The theory contains no absolute elements ([14]. (Makes sense
in N, EA and EC) This version is clearly explained by J¨urgen Ehlers
in [7] p 458. The elements (fields, for example) appearing in the theory
can be divided into dynamical (those that are varied in an Action prin-
ciple) and absolute (those that are not). The Action principle leads to
equations for the dynamical fields to satisfy. The absolute elements are
predetermined and unaffected by the dynamics.
5
• Mach5: the total energy, angular and linear momentum of the universe
are zero (p237).(Makes sense in N, EA, EC.) It is not true in N and
EA. In EC it is claimed [11] that the total angular momentum of a
closed universe must vanish.
• Mach6: Inertial mass is affected by the global distribution of matter (pp
91,249). Makes sense in (N, EA, EC). Is not true in any of them. Hoyle
and Narlikar [12] proposed a theory in which implements Mach6.
• Mach7: If you take away all matter, there is no more space [13]. Makes
sense in (N,EA,EC). Not true in any of them.
• Mach8: Ω = 4πρGT 2 is a definite number of order unity (p475). (Here,
ρ is the mean density matter in the universe and T is the Hubble time.
Makes sense in EC only.) Ω does seem to be of order unity in our present
universe, but note that of all EC models, only the Einstein–DeSitter
makes this number a constant, if Ω is not exactly one. Making a theory
in which this approximate equality appears natural is a worthwhile and
ongoing effort (eg inflationary cosmologies).
• Mach9: The theory contains no absolute elements ([14]. (Makes sense
in N, EA and EC) This version is clearly explained by J¨urgen Ehlers
in [7] p 458. The elements (fields, for example) appearing in the theory
can be divided into dynamical (those that are varied in an Action prin-
ciple) and absolute (those that are not). The Action principle leads to
equations for the dynamical fields to satisfy. The absolute elements are
predetermined and unaffected by the dynamics.
5
Newton’s theory does not satisfy Mach9 (space and time are abso-
lute) and neither does EA (asymptotic flatness introduces an absolute
element–the flat metric at infinity). EC does satisfy Mach9 [15]. From
the point of view of invariance groups (J.L Anderson, A. Trautmann,
quoted on p 468 [7]) Mach9 is the requirement that the invariance group
of the theory is the entire diffeomorphism group of spacetime. Viewed
in this light Mach9 is just the principle of general covariance.
• Mach10: Overall rigid rotations and translations of a system are unob-
servable. (This version makes sense only in N; In Einsteinian spacetime
one has no idea what a rigid rotation is anymore than one knows what
a rigid body is.) This is not satisfied in Newtonian theory. If one insists
on the principle and constructs a theory which satisfies it, one is led [16]
to a class of models (called “relational” by Barbour and Bertotti [16]).
There is considerable literature on these models [7, 17]. We spend a
few words on these models and their connection with Newonian theory.
Relational Models: Let xi
a , i = 1, 2, 3, a = 1...N be the positions of
N particles in Newtonian spacetime and pia their conjugate momenta.
The Hamiltonian H(x, p) determines the time evolution of (xi
a, pia) via
Hamilton’s equations. The transformation
xi
a(t) → Ri
j (t)xj
a(t)
pia(t) → Rj
i (t)pja(t), (1)
where Ri
j (t) is an arbitrary time dependent rotation matrix maintains
the distance relations between the N particles. If a model is relational
[16], such a transformation is unobservable, like a “gauge transforma-
tion” in electrodynamics. From Dirac’s theory of constrained systems
6
lute) and neither does EA (asymptotic flatness introduces an absolute
element–the flat metric at infinity). EC does satisfy Mach9 [15]. From
the point of view of invariance groups (J.L Anderson, A. Trautmann,
quoted on p 468 [7]) Mach9 is the requirement that the invariance group
of the theory is the entire diffeomorphism group of spacetime. Viewed
in this light Mach9 is just the principle of general covariance.
• Mach10: Overall rigid rotations and translations of a system are unob-
servable. (This version makes sense only in N; In Einsteinian spacetime
one has no idea what a rigid rotation is anymore than one knows what
a rigid body is.) This is not satisfied in Newtonian theory. If one insists
on the principle and constructs a theory which satisfies it, one is led [16]
to a class of models (called “relational” by Barbour and Bertotti [16]).
There is considerable literature on these models [7, 17]. We spend a
few words on these models and their connection with Newonian theory.
Relational Models: Let xi
a , i = 1, 2, 3, a = 1...N be the positions of
N particles in Newtonian spacetime and pia their conjugate momenta.
The Hamiltonian H(x, p) determines the time evolution of (xi
a, pia) via
Hamilton’s equations. The transformation
xi
a(t) → Ri
j (t)xj
a(t)
pia(t) → Rj
i (t)pja(t), (1)
where Ri
j (t) is an arbitrary time dependent rotation matrix maintains
the distance relations between the N particles. If a model is relational
[16], such a transformation is unobservable, like a “gauge transforma-
tion” in electrodynamics. From Dirac’s theory of constrained systems
6
[18, 19], it follows that the transformations (1) must be generated by
first class constraints. The generator of overall rotations of the system
is the total angular momentum:
Ji = Σaǫijkxaj pak.
Thus the system is subject to the constraints
φi(x, p) := Ji − Ci ≈ 0,
where Ci are constants. The requirement that the constraints be first
class in the sense of Dirac [18] forces the constants Ci to vanish.
The extended Hamiltonian in the sense of Dirac is
HE (x, p) = H(x, p) + ωiJi,
where ωi are arbitrary functions. While we have only dealt with overall
rotations in (1), one can similarly deal with arbitrary translations and
arbitrary time reparametrizations. Relational models can be thus de-
rived from Newtonian Hamiltonian mechanics by imposing constraints
on the phase space so that the total angular momentum, momentum
and Energy vanish.
These relational models are clearly distinct from Newtonian theory.
For instance, Newtonian theory admits solutions with nonzero angular
momentum (like the solar system in an otherwise empty universe) while
relational models do not permit such solutions.
3 Rindler’s Criticism
We now briefly summarise Rindler’s argument. Consider the earth in an
otherwise empty universe. Let O be a reference frame rigidly attached
7
first class constraints. The generator of overall rotations of the system
is the total angular momentum:
Ji = Σaǫijkxaj pak.
Thus the system is subject to the constraints
φi(x, p) := Ji − Ci ≈ 0,
where Ci are constants. The requirement that the constraints be first
class in the sense of Dirac [18] forces the constants Ci to vanish.
The extended Hamiltonian in the sense of Dirac is
HE (x, p) = H(x, p) + ωiJi,
where ωi are arbitrary functions. While we have only dealt with overall
rotations in (1), one can similarly deal with arbitrary translations and
arbitrary time reparametrizations. Relational models can be thus de-
rived from Newtonian Hamiltonian mechanics by imposing constraints
on the phase space so that the total angular momentum, momentum
and Energy vanish.
These relational models are clearly distinct from Newtonian theory.
For instance, Newtonian theory admits solutions with nonzero angular
momentum (like the solar system in an otherwise empty universe) while
relational models do not permit such solutions.
3 Rindler’s Criticism
We now briefly summarise Rindler’s argument. Consider the earth in an
otherwise empty universe. Let O be a reference frame rigidly attached
7
to the earth. Suppose that a gyroscope G is taken around the earth in
the equatorial plane along a circle of radius r with a constant clockwise
angular velocity Ω. To keep track of orientations, we suppose the earth
and the gyroscope marked with cross hairs (as in Fig.1 of Rindler).
We arrange that the orientation of G relative to the earth’s is constant
during the motion. (Rindler uses the Schwarzschild metric outside the
earth to compute α the precession rate of the gyroscope. We choose
the radius r to set α to zero. It simplifies the argument.)
Now view the situation from the point of view of an observer O′, who
rotates rigidly relative to O with constant clockwise angular velocity
Ω. O′ sees the earth rotating anticlockwise with angular velocity Ω,
the centre of the gyroscope at rest. Notice however, that the gyroscope
(which was not rotating with respect to O) now rotates anticlockwise
with angular velocity Ω relative to O′. Thus the gyroscope rotates in
the same sense as the earth.
It follows from Mach10 that a rotating body in otherwise empty space
makes the local compass of inertia take up all of the body’s angular
velocity. Applied to the earth, which is not in empty space but in the
universe, one would expect that the effect of the earth on the gyroscope
should be considerably diluted by the effect of the rest of the universe.
Thus one would expect that the local compass of inertia would take up
a small positive fraction of the earth’s angular velocity. The sign of this
effect is everywhere positive unlike the sign of the Lense–Thirring effect.
This is the basis for Rindler’s conclusion that the Lense–Thirring effect
is Anti–Machian.
8
the equatorial plane along a circle of radius r with a constant clockwise
angular velocity Ω. To keep track of orientations, we suppose the earth
and the gyroscope marked with cross hairs (as in Fig.1 of Rindler).
We arrange that the orientation of G relative to the earth’s is constant
during the motion. (Rindler uses the Schwarzschild metric outside the
earth to compute α the precession rate of the gyroscope. We choose
the radius r to set α to zero. It simplifies the argument.)
Now view the situation from the point of view of an observer O′, who
rotates rigidly relative to O with constant clockwise angular velocity
Ω. O′ sees the earth rotating anticlockwise with angular velocity Ω,
the centre of the gyroscope at rest. Notice however, that the gyroscope
(which was not rotating with respect to O) now rotates anticlockwise
with angular velocity Ω relative to O′. Thus the gyroscope rotates in
the same sense as the earth.
It follows from Mach10 that a rotating body in otherwise empty space
makes the local compass of inertia take up all of the body’s angular
velocity. Applied to the earth, which is not in empty space but in the
universe, one would expect that the effect of the earth on the gyroscope
should be considerably diluted by the effect of the rest of the universe.
Thus one would expect that the local compass of inertia would take up
a small positive fraction of the earth’s angular velocity. The sign of this
effect is everywhere positive unlike the sign of the Lense–Thirring effect.
This is the basis for Rindler’s conclusion that the Lense–Thirring effect
is Anti–Machian.
8
4 The Lense-Thirring effect as Machian
We now show that one can arrive at the opposite conclusion from Rindler’s
by using a different version of Mach’s Principle. We use the often employed
exact analogy between rotation in General Relativity and magnetic fields [20]
to deduce that the slight influence of a spinning body on the rotation of the
near-by compass of inertia goes with that of the body near the poles and in
the opposite sense in the equatorial plane.
The Lense–Thirring effect: Consider a stationary spacetime i.e one with a
timelike Killing vector ξ: ∇aξb +∇bξa = 0. One can adapt the time coordinate
to ξ so that ξ = ∂/∂t and the metric assumes the form:
ds2 = g00(dt + Aidxi)2 − γij dxidxj ,
where Ai = g0i/g00. The coordinate transformations that preserve this form
include t → t + α(xi), which physically represents the resetting of clocks.
Under such transformations Ai transforms as Ai → Ai + ∇iα like the vector
potential in electrodynamics. Consequently its curl Fij := ∂iAj − ∂j Ai is
invariant and represents rotation of the spacetime (more geometrically, the
failure of ξ to be hypersurface orthogonal). It is easily seen that a stationary
Sagnac tube will measure a Sagnac shift of ∮ Aidxi. A locally nonrotating
Sagnac tube (one that measures zero Sagnac shift) will appear to rotate
as viewed from infinity. The angular velocity of rotation has the spatial
distribution of a dipole magnetic field and reverses sign between the equator
and the poles. As we show below this is exactly what one expects from Mach’s
principle (Mach3).
If one applies Mach’s Principle in the form Mach3 to understanding ro-
tation in General Relativity, one sees that the prediction of Mach3 agrees
9
We now show that one can arrive at the opposite conclusion from Rindler’s
by using a different version of Mach’s Principle. We use the often employed
exact analogy between rotation in General Relativity and magnetic fields [20]
to deduce that the slight influence of a spinning body on the rotation of the
near-by compass of inertia goes with that of the body near the poles and in
the opposite sense in the equatorial plane.
The Lense–Thirring effect: Consider a stationary spacetime i.e one with a
timelike Killing vector ξ: ∇aξb +∇bξa = 0. One can adapt the time coordinate
to ξ so that ξ = ∂/∂t and the metric assumes the form:
ds2 = g00(dt + Aidxi)2 − γij dxidxj ,
where Ai = g0i/g00. The coordinate transformations that preserve this form
include t → t + α(xi), which physically represents the resetting of clocks.
Under such transformations Ai transforms as Ai → Ai + ∇iα like the vector
potential in electrodynamics. Consequently its curl Fij := ∂iAj − ∂j Ai is
invariant and represents rotation of the spacetime (more geometrically, the
failure of ξ to be hypersurface orthogonal). It is easily seen that a stationary
Sagnac tube will measure a Sagnac shift of ∮ Aidxi. A locally nonrotating
Sagnac tube (one that measures zero Sagnac shift) will appear to rotate
as viewed from infinity. The angular velocity of rotation has the spatial
distribution of a dipole magnetic field and reverses sign between the equator
and the poles. As we show below this is exactly what one expects from Mach’s
principle (Mach3).
If one applies Mach’s Principle in the form Mach3 to understanding ro-
tation in General Relativity, one sees that the prediction of Mach3 agrees
9
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