# Spacetime and Spin

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## The Many Faces of Spin

Many of nature's deepest mysteries come in threes. Why does space have
three spatial dimensions (ones that we can see, anyway)? Why are there three
fundamental dimensions in physics (mass **M**, length **L** and time **T**)? Why three fundamental
constants in nature (Newton's gravitational constant *G*, the speed
of light *c* and Planck's constant *h*)? Why three generations
of fundamental particles in the standard model (e.g. the up/down,
charm/strange and top/bottom quarks)? Why do black holes have only three
properties—mass, charge and spin? Nobody knows the answers to these
questions, nor how or whether they may be connected. But some have sought
for clues in the last-named of these properties: *spin*.

We are all familiar with rotation in the macroscopic world of tops,
ballet dancers, planets and galaxies. Spin in the microscopic world is
subtler, and obeys rules that are at once familiar (e.g. conservation of
angular momentum) and bizarrely counter-intuitive (e.g. quantization
and half-integer spin for fermions, which in the macroscopic world would
correspond to objects that rotate through 720 rather than 360 degrees
before returning to their original states).
More abstract still are quantities like "isospin", which is analogous to
ordinary spin in some ways but governs the behavior of the strong and weak
nuclear forces (rotation through 180 degrees of isospin, for instance,
converts a proton into a neutron), and torsion, a mathematical term related
to the intrinsic twist of spacetime (this appears in some extensions
of general relativity, but Einstein himself set it to zero in general
relativity for reasons of logical economy). Are there connections between
these manifestations of spin in the worlds of the large and small? Do they
hint at the direction in which Einstein's theory of gravity might need to be
extended in order to unify it with the other forces of nature? A generation
of physicists since Einstein have thought about these questions, and they
are part of the reason what makes Gravity Probe B so important, not just as
another test of general relativity, but as a source of new insights about
spacetime itself. Nobel laureate C.N. Yang wrote in a letter to NASA
Administrator James M. Beggs in 1983 that general relativity, "though
profoundly beautiful, is likely to be amended ... whatever [the] new
geometrical symmetry will be, it is likely to entangle with spin and
rotation, which are related to a deep geometrical concept called torsion ...
The proposed Stanford experiment [Gravity Probe B] is especially interesting
since it *focuses on the spin*. I would not be surprised at all if
it gives a result in disagreement with Einstein's theory."

## Gravito-Electromagnetism

In general situations, space and time are so inextricably bound together
in general relativity that they are hard to separate. In special cases,
however, it becomes feasible to perform a "3+1 split" and decompose the
metric of four-dimensional spacetime into a scalar time-time component,
a vector time-space component and a tensor "space-space"
component. When gravitational fields are weak and velocities are low
compared to *c*, then this decomposition takes on a particularly
compelling physical interpretation: if we call the scalar component a
"gravito-electric potential" and the vector one a "gravito-magnetic potential",
then these quantities are found to obey almost exactly the same laws as
their counterparts in ordinary electromagnetism! (Although little-known
nowadays, the idea of parallels between gravity and electromagnetism is
not a new one, and goes back to Michael Faraday's experiments with
"gravitational induction" beginning in 1849.) One can construct a
"gravito-electric field" **g** and a "gravito-magnetic field **H** from the divergence and curl of the scalar and vector
potentials, and these fields turn out to obey equations that are identical
to Maxwell's equations and the Lorentz force law of ordinary electrodynamics
(modulo a sign here and a factor of two there; these can be chalked up to
the fact that gravity is associated with a spin-2 field rather than the
spin-1 field of electromagnetism). The "field equations" of
gravito-electromagnetism turn out to be of great value in interpreting
the predictions of the full theory of general relativity for spinning test
bodies in the field of a massive spinning body such as the earth — just
as Maxwell's equations govern the behavior of electric dipoles in an
external magnetic field. From symmetry considerations we can infer that
the earth's gravito-electric field must be radial, and its
gravito-magnetic one dipolar, as shown in the diagrams below:

Faraday | Radial field lines | Dipolar field lines |

These facts allow one to derive the main predictions of general relativity
that are of relevance to Gravity Probe B, simply by replacing the electric
and magnetic fields of ordinary electrodynamics by **g** and **H** respectively (for an illuminating discussion see
Kip Thorne's contribution to *Near Zero: New Frontiers of Physics*,
1988). Based on this analogy the term "gravito-magnetic effect" is sometimes
used interchangeably with "frame-dragging" (or with "Lense-Thirring effect";
see below). However any such identification must be treated with care because
the distinction between gravito-magnetism and gravito-electricity is
frame-dependent, just like its counterpart in Maxwell's theory.
This means that observers using different coordinate systems (as, for
example, one centered on the earth and another on the barycenter of the
solar system) may disagree on the relative size of the effects they are
discussing. Gravito-electromagnetism has already been *indirectly* observed in the solar system for some time, since general relativistic
corrections are routinely used in, for instance, updating the ephemeris of
planetary positions, and
gravito-electromagnetic fields are nothing more than a necessary limit of
Einstein's gravitational field in situations where gravity is weak and velocities
are low. This is different from measuring a gravito-electromagnetic
phenomenon like frame-dragging *directly*, which is one of the two
primary goals of the Gravity Probe B mission.

## Geodetic Effect

The geodetic effect provides us with a sixth test of general relativity (after the three classical tests plus Shapiro delay and the binary pulsar), and it is the first one to involve the spin of the test body. The effect arises in the way that angular momentum is transported through a gravitational field in Einstein's theory. Einstein's friend and colleague Willem de Sitter (1872-1934), who was instrumental in making general relativity known abroad, began to study this problem when the theory was less than a year old. He found that the earth-moon system would undergo a precession in the field of the sun, a special case now referred to as the de Sitter or "solar geodetic" effect (although "heliodetic" might be more descriptive). De Sitter's calculation was extended to rotating bodies such as the earth by two of his countrymen: in 1918 by the mathematician Jan Schouten (1883-1971) and in 1920 by the physicist and musician Adriaan Fokker (1887-1972).

De Sitter | Schouten | Fokker |

In the framework of the gravito-electromagnetic analogy, the geodetic
effect arises partly as a spin-orbit interaction between the spin of
the test body (the gyroscope in the case of GP-B) and the "mass current" of
the central body (the earth). This is the exact analog of Thomas precession
in electromagnetism, where the electron experiences an induced magnetic field
(in its rest frame) due to the apparent motion of the nucleus.
In the gravitomagnetic case, the orbiting gyroscope feels the massive earth
whizzing around it (in its rest frame) and experiences an induced *gravito*magnetic torque, causing its spin vector to precess.
This spin-orbit interaction accounts for one third of the total geodetic
precession; the other two thirds arise due to space curvature alone and
cannot be interpreted gravito-electromagnetically. They can, however, be
understood geometrically. Model flat space as a 2-dimensional sheet, as shown
in the diagram below (left).

A gyroscope's spin vector (arrow) points at right angles to the plane of its motion, and its direction remains constant as the gyroscope completes a circular orbit. If, however, we fold space into a cone to simulate the effect of the presence of the massive earth (right), then we must remove part of the area of the circle (shaded) and the gyroscope's spin vector no longer lines up with itself after making a complete circuit (green and red arrows). The difference between these two directions (per orbit) makes up the other two thirds of the geodetic effect. In the case of Gravity Probe B this is sometimes referred to as the "missing inch" argument because space curvature shortens the circumference of the spacecraft's orbital path around the earth by 1.1 inches. In polar orbit at an altitude of 642 km the total geodetic effect (comprising both the spin-orbit and space curvature effects) causes a precession in the north-south direction of 6606 milliarcsec/yr — an angle so small that it is comparable to the average angular size of the planet Mercury as seen from earth.

Experimental detection (or non-detection) of the geodetic effect will
place new and independent limits on alternative theories of gravity known as
"metric theories" (loosely speaking, theories that respect Einstein's
equivalence principle). These theories are characterized by the Eddington
or Parametrized Post-Newtonian (PPN) parameters β and γ, which are
both equal to one in general relativity. The geodetic effect is proportional
to (1+2γ)/3, so a confirmation of the Einstein prediction at the level
of 0.01% would translate into comparable constraint on γ — more
stringent than all but the most recent Shapiro time-delay test based on data
from Cassini. Gravity Probe B observations of geodetic precession could also impose new constraints on other "generalizations of general relativity" such as the scalar-tensor theories pioneered by Carl Brans and Robert Dicke in 1961 (see Kamal Nandi *et al*, 2001). Another such class of theories incorporates torsion into Einstein's theory; examples have been proposed by Kenji Hayashi and Takeshi Shirafuji (1979),
Leopold Halpern (1984) and Yi Mao *et al.* (2006). Another is based
on extending the theory to higher dimensions; constraints on such theories arising from the geodetic effect have been discussed by Dimitri Kalligas *et al.* (1995) and Hongya Liu and James Overduin (2000). The most recent kind of generalization
involves violations of Lorentz invariance, the conceptual foundation of
special relativity; implications of such theories for Gravity Probe B
have been worked out by Quentin Bailey and Alan Kostelecky (2006).

## Frame-Dragging Effect

Lense

Thirring

The frame-dragging effect, the seventh test of general relativity and the
second one to involve the spin of the test body, reveals most clearly the
Machian aspect of Einstein's theory. In fact, it is curious that Einstein
did not work out this effect himself, given that he had obtained explicit
frame-dragging effects in all his previous attempts at gravitational field
theories, and that he still regarded Mach's principle as the philosophical
pillar of general relativity in 1918. Whatever the reason, it was not until
that year that the general-relativistic frame-dragging formula was derived
by Hans Thirring (1888-1976) and Josef Lense (1890-1985), after whom the
effect is now usually named. In an ironic twist, Thirring had not intended
to do calculations at all; he had wanted to build a
frame-dragging *experiment* (a cylindrical version of Föppl's
flywheel experiment) and only settled for theoretical work after he was unable
to arrange the necessary financing (see Herbert Pfister's contribution to *Mach's Principle: From Newton's Bucket to Quantum Gravity*, 1995).
Thirring's initial result described the gravitational field inside a rotating
cylinder; his second calculation (with Lense) involved the field outside a
slowly rotating solid sphere and forms the basis for experimental tests
such as Gravity Probe B. Both results are "Machian" in the sense that the
inertial reference frame of a test particle is strongly influenced by the
properties of the larger mass (the cylinder or sphere). This is completely
unlike Newtonian dynamics, where a test particle's inertia is defined only
by its motion with respect to "absolute space" and is unaffected by the
distribution of matter. In fact, with the right parameters it is possible
for a large mass in general relativity to completely "screen" the background
geometry, so that a test particle feels only the reference frame defined
by that mass. This phenomenon is known as "total" or "perfect dragging" of
inertial frames (more on this below).

Frame-dragging in realistic experimental situations is not nearly that
strong and the utmost ingenuity has to be exercised to detect it at all.
Analyzed in terms of the gravito-electromagnetic analogy, the effect arises
due to the spin-spin interaction between the gyroscope and rotating central
mass, and is perfectly analogous to the interaction of a magnetic dipole **μ** with a magnetic field **B** (the basis of nuclear Magnetic Resonance Imaging or MRI). Just as a torque **μ×B** acts in the magnetic case, so a gyroscope
with spin **s** experiences a torque proportional to **s×H** in the gravitational case. For Gravity Probe B,
in polar orbit 642 km above the earth, this torque causes the gyroscope
spin axes to precess in the east-west direction by a mere 39 milliarcsec/yr
— an angle so tiny that it is equivalent to the average angular width of the
dwarf planet *Pluto* as seen from earth.

## Genesis of GP-B

As the calculations of de Sitter, Schouten and Fokker became more widely
known, particularly through Arthur Eddington's influential textbook *The Mathematical Theory of Relativity* (1923), experimentalists
began to take interest. P.M.S. Blackett (1897-1974) considered looking for
the de Sitter effect with a laboratory gyroscope in the 1930s, but concluded
(rightly) that the task was hopeless with existing technology. To see what
makes the problem so challenging, consider the gyroscope rotor shown below.
The de Sitter effect and frame-dragging around the earth are both of order
~10 milliarcsec/yr, so to measure either of them with 1% accuracy requires
that all unmodeled precessions on this rotor (known technically as the "drift
rate") add up to *less than 0.1 milliarcsec/yr*, or 10^{-18} rad/s. (See video clip "How smalll is 1/10th of a milliarcsecond?" at right.)

What does
this requirement mean for our gyroscope? Precession Ω is related to
torque τ by Ω=τ/(*I*ω) where *I* =
(2/5)*mr ^{2}* is the moment of inertia and ω=v/

*r*is the angular velocity. Inhomogeneities of size δ

*r*produce torques of order τ=

*maδr*where

*a*is the tangential acceleration. Combining these expressions gives a drift rate of Ω = (5/2)(

*a*/v)(δ

*r/r*). Assuming a spin speed of v~1000 cm/s and accelerations comparable to those on the surface of the earth (

*a~g*), the rotor must be homogeneous to within (

*δr/r*) < 10

^{-17}to attain a drift rate less than 10

^{-18}rad/s! Such homogeneities are utterly unattainable on earth. In space, however, it is possible —

*just*possible, with a great deal of work — to suppress unwanted accelerations on a test body by as much as eleven orders of magnitude, to

*a*~10

^{-11}

*g*. If this can be done, then the gyro rotor need only be homogeneous to one part in 10

^{6}, rather than 10

^{17}— a level that can be achieved, with great effort, using the best materials on earth.

Considerations of this kind led two people to take a new look at gyroscopic
tests of general relativity shortly after the dawn of the space age.
George E. Pugh and Leonard I. Schiff (1915-1971) hit independently
on the key ideas within months of each other. Pugh was stimulated by a talk
given by Huseyin Yilmaz proposing a satellite test to distinguish his
alternative theory of gravity from Einstein's, while Schiff was likely inspired
at least in part by an advertisement for a new "Cryogenic Gyro ... with the
possibility of exceptionally low drift rates" in *Physics Today* magazine (see Francis Everitt's contribution to *Near Zero: New Frontiers of Physics*, 1988). Pugh's paper,
published in a Pentagon memorandum in November 1959, is now
recognized as the birth of the concept of **drag-free motion**.
This is a critical element of the Gravity Probe B mission, whereby any one
of the gyroscopes can be isolated from the rest of the experiment and
protected from all non-inertial forces; the rest of the spacecraft is then
made to "chase after" the reference gyro by means of helium boiloff vented
through a revolutionary porous plug and specially designed thrusters.
In this way unmodeled accelerations on all the gyros, such as those resulting
from the effects of solar radiation pressure and atmospheric friction on the
spacecraft, can be reduced from *a*~10^{-8}*g* to below
10^{-11}*g* as required. (See animation clip "Drag-Free Motion" below.)

Blackett | Pugh in 2007 | Schiff ~1970 | Drag-free motion |

The drag-free control system is only one of the innovations that made Gravity Probe B possible. The experiment depends on monitoring the precession of near-perfect gyroscopes relative to a fixed reference direction such as the line of sight to a distant guide star. But how is one to find the spin axis of a perfectly spherical, perfectly homogeneous gyroscope suspended in vacuum? This is the "readout problem"; another, closely related problem is how to spin up such a gyroscope in the first place. Various possibilities were considered in the early days, until 1962 when Francis Everitt and William Fairbank hit on the idea of exploiting what had until then been a small but annoying source of unwanted torque in magnetically levitated gyroscopes. Spinning superconductors develop a magnetic moment, known as the **London moment**, which is proportional to spin speed and always aligned with the spin axis. If the rotors were levitated electrically instead of magnetically, this tiny effect could be used to tell where their spin axes were pointed. (Measuring it would of course require magnetic shielding orders of magnitude beyond anything available in 1962, another story in itself.) Thus was born the London moment readout, which in its modern incarnation uses SQUIDs (Superconducting QUantum Interference Devices) as magnetometers. So sensitive are these devices that they register a change in spin-axis direction of 1 milliarcsec in five hours of integration time. (See animation clip "London Moment Readout" below.)

London Moment readout | Dan Debra, Bill Fairbank, Francis Everitt and Bob Cannon with a model of Gravity Probe B, 1980 |

These are only two pieces of an experiment so beautifully intricate that
it is as much a work of art as it is science and technology. Many of its
key features reflect a guiding principle of physics experimenters through
the ages, namely to turn *obstacles into opportunities*. How, for
instance, can one meaningfully compare the gyroscope spin-axis direction
(which is read out in volts) with the position of the guide star
(which comes from an onboard telescope in radians)? The answer
is to exploit nature's own calibration in the form of **stellar
aberration**. This phenomenon, an apparent back-and-forth motion of
the guide star position due to the orbit of the earth around the sun, is
entirely Newtonian and inserts "wiggles" into the data whose period and
amplitude are exquisitely well known (to give a sense of the precision of
the experiment, the calibration requires terms of *second*, as well
as first order in the earth's speed v/c). What about the fact that the guide
star has an unknown "proper motion" large enough to obscure the predicted
relativity signal? This allows the experiment to be designed in a classic
"double-blind" fashion; a separate team of astronomers uses VLBI (Very
Long-Baseline radio Interferometry) to monitor the movements of the guide
star *itself*, relative to even more distant quasars. Only at the conclusion
of the experiment are the two sets of data to be compared; this helps to
prevent the physicists from "finding what they want to see."

For many more such examples, see the Unique Technology Challenges & Solutions page in the Technology tab. Gravity Probe B (or the Stanford Relativity Gyroscope Experiment, as it was known until 1971) received its first NASA funding in March 1964. The photograph above shows several of the early project leaders with a model of the spacecraft circa 1980: Dan Debra (a propulsion expert), Fairbank (the experimental low-temperature physicist par excellence), Everitt and Bob Cannon (a gyroscope specialist). See the History & Management section of the Mission page for more details.

## Astrophysical Significance

When Gravity Probe B was originally conceived, frame-dragging was seen
as being of more theoretical than practical interest. To be sure, experimental
confirmation of the Einstein (i.e. Lense-Thirring) prediction would place
another independent constraint on alternative metric theories of gravity.
Frame-dragging precession is proportional to the combination of PPN parameters
(γ+1+α_{1}/4)/2 where γ describes the warping of
space and α_{1} is known as a "preferred-frame" parameter
that allows for a possible dependence on motion relative to the rest frame
of the universe (it takes the value zero in general relativity). However,
frame-dragging is such a small effect in the solar system that experimental
bounds it places on these parameters are not likely to be competitive with
those from other tests. Confirmation of the Einstein (i.e. Lense-Thirring)
prediction at the 1% level, for example, would translate into a comparable
bound on γ and would not significantly constrain α_{1}.

This situation has changed dramatically since the 1980s. Physicists now
see frame-dragging as the gravitational analog of magnetism, and
astrophysicists invoke this gravitomagnetic field as the engine and alignment
mechanism for the vast and otherwise incomprehensible jets of gas and magnetic
field ejected from quasars and galactic nuclei like the radio source NGC 6251,
as shown above left. We know that these jets act as power sources for quasars and
other strong extragalactic radio sources and that they are generated by compact,
supermassive objects (almost certainly black holes) inside galactic nuclei,
as illustrated above right. The megaparsec distance scale in the radio image above left implies that
these compact objects are capable of holding the jet direction constant over
timescale as long as ten million years. Black holes can only do this by means
of their gyroscopic spin, and they can only communicate the direction of that
spin to the jet via their gravitomagnetic field **H**.
Such a field will cause an accretion disk to precess around the black hole,
and that precession combined with the disk's viscosity should drive the
inner region of the disk into the hole's equatorial plane, resulting in only
two preferred directions for the jets: the north and south poles of the black
hole. This phenomenon, known as the Bardeen-Petterson effect (diagram at left), is widely believed to be the physical mechanism responsible for jet
alignment.

Gravitomagnetism is also thought to explain the generation of the
astounding energy contained in these jets in the first place. The event
horizon of the black hole acts like a "gravitomagnetic battery", driving
currents around closed loops like that shown in the diagram at left: up the
magnetic field lines from the horizon to a region where the magnetic field is
weak, across the field lines there, and then back down the field lines to the
horizon and through the "battery" where the gravitomagnetic potential of the
black hole interacts with the tangential component of the ordinary magnetic
field **B** to produce a drop in electric potential (see Kip
Thorne's contribution to *Near Zero: New Frontiers of Physics*, 1988).
This phenomenon, known as the Blandford-Znajek mechanism, effectively draws
on the immense gravitomagnetic, rotational energy of the supermassive black
hole and converts it into an outgoing stream of ultra-relativistic charged
particles. Gravity Probe B has thus become a crucial test of the mechanism
that powers the most violent explosions in the universe.

## Cosmological Significance

Star trails at night

Here is a simple experiment that almost anyone can perform on a clear
night: pirouette freely around while looking up at the stars. You will
notice two things: one, that the stars seem to spin around in the sky, and
two, that your arms are pulled upwards by centrifugal force. Are these
phenomena connected in some physical way? Not according to Newton. For him,
centrifugal force is a consequence of accelerating (i.e. rotating) with
respect to absolute space; it has no *physical* origin (and is
therefore often called a "fictitious force" in elementary physics classes).
It is, furthermore, a coincidence that the stars above us are at rest with
respect to this same absolute space. We look upward, in effect, from two
fundamentally different reference frames: one defined by our local sense of
inertia, and the other defined by the global rest frame of the universe at
large. Why should these two reference frames happen to coincide? Newton
did not try to answer this question.

We know that the concept of absolute space(time) is retained in general
relativity, so we might have expected that the same coincidental alignment
of our local inertial frame with that of the global matter distribution
would carry over to Einstein's theory as well. Astonishingly, however,
it does not. If general relativity is correct, then there are strong
indications that our local "compass of inertia" *has no choice* but
to be aligned with the rest of the universe — the two are linked by
the frame-dragging effect. These indications do not come from experiment,
but from theoretical calculations similar to that performed by Lense and
Thirring. The calculations show that general-relativistic frame-dragging
goes over to "perfect dragging" *when the dimensions of the large mass
(its size and density) become cosmological*. In this limit, the
distribution of matter in the universe appears sufficient to define the
inertial reference frame of observers within it. For a particularly clear and simple explanation of how and why this happens, see *The Unity of the Universe* (1959) by Dennis Sciama. Had Mach lived 10 years
longer, he could have predicted the existence of the extragalactic universe
based on observations that the stars in the Milky Way rotate around a common
center!

To put the cosmological significance of frame-dragging in concrete
terms, imagine that the earth were standing still and that the rest of the
universe were rotating around *it*: would its equator still bulge?
Newton would have said "No". According to standard textbook physics the
equatorial bulge is due to the rotation of the earth with respect to
absolute space. On the basis of Lense and Thirring's results, however,
Einstein would have had to answer "Yes"! In this respect general relativity
is indeed more relativistic than its predecessors: it does not matter whether
we choose to regard the earth as rotating and the heavens fixed, or the other
way around: the two situations are now dynamically, as well as kinematically
equivalent. The early calculations were flawed in many ways, but the
phenomenon of perfect dragging has persisted in most subsequent, more
sophisticated treatments, notably those of Helmut Hönl and Heinz
Dehnen (1962, 1964), Dieter Brill and Jeffrey Cohen (1966) and Herbert
Pfister and Karlheinz Braun (1986). Pfister sums up current thinking
this way in *Mach's Principle: From Newton's Bucket to Quantum Gravity* (1995): "Although Einstein's theory of gravity does not, despite its name
'general relativity,' yet fulfil Mach's postulate of a description of nature
with only relative concepts, it is quite successful in providing an intimate
connection between inertial properties and matter, at least in a class of not
too unrealistic models for our universe. Perhaps against majority expectation,
this connection is instantaneous in nature. Furthermore, general relativity
has brought us nearer to an understanding of the observational fact that the
local inertial compass is fixed relative to the most distant cosmic objects,
but there is surely desire for still deeper understanding." Thus does
direct detection of frame-dragging by Gravity Probe B gain new importance:
it will shine experimental light on what has heretofore been a theoretical
mystery, namely the **origin of inertia**. For some, this is
perhaps the most beautiful and profound manifestation of spin in Einstein's
spacetime: it binds us here to the universe out there, in such a way that
you, standing at night under the stars on a planet known as earth, cannot
turn so much as around without feeling a tug from the rest of the universe.

*James Overduin, January 2008*

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