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[escepticos] Inercia


A continuación mando el capítulo 1 del libro que mencionó Carlos Ungil hace
pocos días (Ciufolini & Wheeler, 'Gravitation and Inertia', Princeton
University Press, 1995). Creo que es de interés para el tema de la 'inercia
extrínseca' que venimos debatiendo. He eliminado las abundantes remisiones
bibliográficas, y también parte de las remisiones a otros capítulos de la
obra. Un comentario: lo de los 15 segundos de arco de desviación observados
por Gauss me suenan raros. estaba -Gauss- midiendo un triángulo plano, entre
tres cumbres. La masa de la Tierra es, creo, insuficiente como para generar
esa curvatura. Lo curioso es que el texto dice que eso era prueba de la
curvatura de la Tierra, pero Gauss no estaba -sus visuales eran rectas-
midiendo un triángulo esférico... Creo que a los autores se les han 'cruzado
los cables' en este comentario.



A First Tour

In this chapter we introduce some basic concepts and ideas of Einstein's
Theory of Relativity which are developed in the book.


"Gravity is not a foreign and physical force transmitted through space and
time. It is a manifestation of the curvature of spacetime." That, in a
nutshell, is
Einstein's theory.
What this theory is and what it means, we grasp more fully by looking at its
intellectual antecedents. First, there was the idea of Riemann that space,
mass how to move, must itself-by the principle of action and reaction-be af-
fected by mass. It cannot be an ideal Euclidean perfection, standing in high
mightiness above the battles of matter and energy. Space geometry must be a
participant in the world of physics. Second, there was the contention of
Mach that the "acceleration relative to absolute space" of Newton is only
erly understood when it is viewed as acceleration relative to the sole
mass there really is, the distant stars. According to this "Mach principle,"
ertia here arises from mass there. Third was that great insight of Einstein
we summarize in the phrase "free fall is free float": the equivalence
ple, one of the best-tested principles in physics, from the inclined tables
Galilei and the pendulum experiments of Galilei, Huygens, and Newton to the
highly accurate torsion balance measurements of the twentieth century, and
Lunar Laser Ranging experiment. With those three clues
vibrating in his head, the magic of the mind opened to Einstein what remains
one of mankind's most precious insights: gravity is manifestation of
Euclid's (active around 300 B.C.) fifth postulate states that, given any
line and any point not on it, we can draw through that point one and only
straight line parallel to the given line, that is, a line that will never
meet the
given one (this altemative formulation of the fifth postulate is essentially
to Proclos). This is the parallel postulate. In the early 1800s the
grew lively about whether the properties of parallel lines as presupposed in
Euclidean geometry could be derived from the other postulates and axioms,
or whether the parallel postulate had to be assumed independently. More than
two thousand years after Euclid, Karl Friedrich Gauss, János Bolyai, and
lai Ivanovich Lobachevskiy discovered pencil-and-paper geometric systems
satisfy all the axioms and postulates of Euclidean geometry except the
postulate. These geometries showed not only that the parallel postulate must
be assumed in order to obtain Euclidean geometry but, more important, that
non-Euclidean geometries as mathematical abstractions can and do exist.
Consider the two-dimensional surface of a sphere, itself embedded in the
three-dimensional space geometry of everyday existence. Euclid's system ac-
curately describes the geometry of ordinary three-dimensional space, but not
the geometry on the surface of a sphere. Let us consider two lines locally
on the surface of a sphere. They propagate on the surface as straight
as any lines could possibly be, they bend in their courses one whit neither
left or fight. Yet they meet and cross. Clearly, geodesic lines (on a
surface, a
geodesic is the shortest line between two nearby points) on the curved
of a sphere do not obey Euclid's parallel postulate.
The thoughts of the great mathematician Karl Friedrich Gauss about curva-
ture stemmed not from theoretical spheres drawn on paper but from concrete,
down-to-Earth measurements. Commissioned by the govemment in 1827 to
make a survey map of the region for miles around G6ttingen, he found that
sum of the angles in his largest survey triangle was different from 180°.
deviation from 180° observed by Gauss-almost 15 seconds of arc -was both
inescapable evidence for and a measure of the curvature of the surface of
To recognize that straight and initially parallel lines on the surface of a
can meet was the first step in exploring the idea of a curved space. Second
the discovery of Gauss that we do not need to consider a sphere or other
dimensional surface to be embedded in a three-dimensional space to define
geometry. It is enough to consider measurements made entirely within that
dimensional geometry, such as, would be made by an ant forever restricted to
live on that surface. The ant would know that the surface is curved by
that the sum of the internal angles of a large dangle differs from 180°, or
measuring that the ratio between a large circumference and its radius R
from 2pi.
Gauss did not limit himself to thinking of a curved two-dimensional surface
floating in a flat three-dimensional universe. In an 1824 letter to
Ferdinand Karl
Schweikart, he dared to conceive that space itself is curved: "Indeed I have
therefore from time to time in jest expressed the desire that Euclidean
would not be correct." He also wrote: "Although geometers have given much
attention to general investigations of curved surfaces and their results
cover a
significant portion of the domain of higher geometry, this subject is still
so far
from being exhausted, that it can well be said that, up to this time, but a
portion of an exceedingly fruitful field has been cultivated" (Royal Society
G6ttingen, 8 October 1827). The inspiration of these thoughts, dreams, and
hopes passed from Gauss to his student, Bernhard Riemann.
Bemhard Riemann went on to generalize the ideas of Gauss so that they
could be used to describe curved spaces of three or more dimensions. Gauss
had found that the curvature in the neighborhood of a given point of a
two-dimensional space geometry is given by a single number: the Gaussian
curvature. Riemann found that six numbers are needed to describe the
of a three-dimensional space at a given point, and that 20 numbers at each
are required for a four-dimensional geometry: the 20 independent components
of the so-called Riemann curvature tensor.
In a famous lecture he gave 10 June 1854, entitled *On the Hypotheses
That Lie at the Foundations of Geometry*, Riemann emphasized that the truth
about space is to be discovered not from perusal of the 2000-year-old books
Euclid but from physical experience. He pointed out that space could be
irregular at very small distances and yet appear smooth at everyday
At very great di stances, he also noted, large-scale curvature of space
might show
up, perhaps even bending the universe into a closed system like a gigantic
He wrote: "Space [in the large] if one ascribes to it a constant curvature,
is nec-
essarily finite, provided only that this curvature has a positive value,
small. . .. It is quite conceivable that the geometry of space in the very
does not satisfy the axioms of [Euclidean] geometry. . . . The curvature in
three directions can have arbitrary values if only the entire curvature for
sizeable region of space does not differ greatly from zero. . . . The
which distinguish space from other conceivable triply-extended magnitudes
only to be deduced from experience."
But as Einstein was later to remark, "Physicists were still far removed from
such a way of thinking: space was still, for them, a rigid, homogeneous
thing, susceptible of no change or conditions. Only the genius of Riemann,
solitary and uncomprehended, had already won its way by the middle of the
last century to a new conception of space, in which space was deprived of
rigidity, and in which its power to take part in physical events was
as possible."
Even as the 39-year-old Riemann lay dying of tuberculosis at Selasca on Lake
Maggiore in the summer of 1866, having already achieved his great mathemat-
ical description of space curvature, he was working on a unified description
electromagnetism and gravitation. Why then did he not, half a century before
Einstein, arrive at a geometric account of gravity? No obstacle in his way
greater than this: he thought only of space and the curvature of space,
Einstein discovered that he had to deal with spacetime and the spacetime
vature. Einstein could not thank Riemann, who ought to have been still
A letter of warm thanks he did, however, write to Mach. In it he explained
how mass there does indeed influence inertia here, through its influence on
the enveloping spacetime geometry. Einstein's geometrodynamics had trans-
muted Mach's bit of philosophy into a bit of physics, susceptible to
prediction, and test.
Let us bring out the main idea in what we may call poor man's language.
Inertia here, in the sense of local inertial frames, that is the grip of
here on mass here, is fully defined by the geometry, the curvature, the
structure of
spacetime here. The geometry here, however, has to fit smoothly to the
of the immediate surroundings; those domains, onto their surroundings; and
on, all the way around the great curve of space. Moreover, the geometry in
each local region responds in its curvature to the mass in that region.
every bit of momentum-energy, wherever located, makes its influence felt on
the geometry of space throughout the whole universe-and felt, thus, on
right here.
The bumpy surface of a potato is easy to picture. It is the two-dimensional
analogue of a bumpy three-sphere, the space geometry of a universe loaded
irregularly here and there with concentrations and distributions of
energy. If the spacetime has a Cauchy surface, that three-geometry once
known-mathematical solution as it is of the so-called initial-value problem
geometrodynamics -the future evolution of that geometry follows
straightforwardly and deterministically.
In other words, inertia (local inertial frames) everywhere and at all times
totally fixed, specified, determined, by the initial distribution of
energy, of mass and mass-in-motion. The mathematics cries out with all the
force at its command that mass there does determine inertia here.
We will enter into the mathematics of this initial-value problem-so thor-
oughly investigated in our day- in chapter 5. Of all the contributions made
to inertia here by all the mass and mass-in-motion in the whole universe,
fractional contribution made by a particular mass at a particular distance
is of
the order of magnitude

[fractional contribution
by a given mass, *there*
to the determination
of the direction of
axes of the local gyroscopes,
the compass of inertia, *here*]

is of order of

[mass, *there*]/[distance, *there* to *here*]

In this rough measure of the voting power, the "inertia-contributing power"
any object or any concentration of energy, its mass is understood to be
in the same geometric units as the distance.
Does this whole idea of voting fights and inertia-contributing power make
sense? It surely does so if the total voting power of all the mass there is
the whole universe adds up to 100%. But does it? Let us run a check on
the closed Friedmann model universe (chap. 4)? There the total amount of
mass is of the order of ~ 6 x 10^56 g (see § 4.2). This amount, translated
into geometric units by way of the conversion factor 0,742 x 10^-28 cm/g, is
~ 4,5 x 10^28 cm of mass. It is much harder to assign an effective distance
which that mass lies from us, and for two reasons. First, distances are
ing with time. So at what time is it that we think of the distance as being

If we got into all the subtleties of that question we would be fight back
at the full mathematics of the so-called initial-value problem (see chap.
mathematics sufficiently complicated to have persuaded us to short circuit
it in
this introduction by our quick and rough poor man's account of inertia. This
problem of "when" to measure distances let us therefore resolve by taking
at the instant of maximum expansion, when they are not changing.
What is the other problem about defining an effective distance to all the
mass in the universe? Simple! Some lies at one distance, some at another.
What separation does it make sense to adopt as a suitable average measure
of the distance between inertia here and mass there? It is surely too much
accept as an average the mileage all the way around to the opposite side of
three-sphere, the most remote bit of mass in the whole universe ! As a start
let us
nevertheless look at that maximum figure a minute. It is half of the
of the three-sphere. In other words, it is pi times the radius of the
at maximum expansion, ~ 10 to 20 * 10^9 yr * 3 x 10^7 s/yr * 3 x 10^10 cm/s
2 x 10^28 cm, or ~ 6 x 10^28 cm. It is too big, we know, but surely not
too big, not fivefold too big, conceivably not even twice as great as a
figure for an effective average distance from here to mass there. Then, we
why not adopt for that fuzzy average distance figure the very amount of mass
figure that we have for the three-sphere model universe, ~ 4,5 x 10^28 cm?
figure is surely of the right order of magnitude, and in our rough and ready
of figuring inertia, gives the satisfying 100% signal of "all votes in" when
the mass is counted. This is the poor man's version of the origin of
Now for inertia-determination in action! Mount a gyroscope on frictionless
gimbals or, better, float it weightless in space to eliminate the gravity
force that
here on Earth grinds surface against surface. Picture our ideal gyro as
sitting on
a platform at the North Pole with the weather so cloudy that it has not one
at the distant stars. Pointing initially to the flag and flagpole at a comer
of the
support platform, will the gyro continue to point that way as a 24 hour
gradually bums away? No. The clouds do not deceive it. It does not see the
to which its spin axis points, but to that star it nevertheless continues to
point as
the day wears on. Earth turns beneath the heedless gyro, one rotation and
another as the days go by. That is the inertia-determining power of the mass
spread throughout space, as that voting power is seen in its action on the
Wait! As conscientious workers against election fraud, let us inspect the
that each mass casts. Where is the one signed "Earth"? Surely it was
entitled to
vote on what would be the free-float frame at the location of the gyro. The
tabulator sheepishly confesses. "I know that Earth voted for having the
frwne of
reference turn with it. However, its vote had so little inertia-determining
compared to all the other masses in the universe that I left it out." We
reply, "you did wrong. Let us see how much damage you have donel' Yes, the
voting power of Earth at the location of the gyro is small, but it is not
zero. It
is of the order of magnitude of

[mass of Earth/radius of Earth]~= 0,44 cm/6,4*10^8 cm~= 0,69*10^-9

roughly only one-billionth as much influence as all the rest of the universe
together. "It was not only that its voice was so weak that I threw out its
the vote tabulator interrupts to say. "The free-float frame of reference
that Earth
wanted the gyro axis to adhere to was so little different from the frame
for the gyro by the faraway stars that I could not believe that Earth really
its own wishes seriously. It wants the gyroscope axis to creep slowly around
in a twenty-four hour day rather than keep pointing at one star? Who cares
about that peanut difference?" "Again you are wrong," we tell him. "You do
know our friends the astronomers. For a long time there was so little that
could measure about a faraway star except its direction that they developed
art of measuring angles to a fantastic discrimination. With Very Long
Interferometry one can measure the angular separation of
the distant quasars down to less than a tenth of millisecond of arc and the
of turning of Earth down to less than I milliarcsec per year. Content to
wait a
day to see the effect of Earth on the direction of the spin axis of the
gyro? No.
Figure our friends as watching it for a whole 365-day year. Do you know how
many milliseconds of arc the axis of the gyro would tum through in the
of a whole year, relative to the distant stars, if it followed totally and
the urging of Earth? Four-hundred and seventy-three billion milliarcseconds!
And that is the tuming effect which you considered so miniscule that you
away Earth's ballot! We are here to see justice done."
The corrected turning rate of the gyro, relative to the distant stars, is of
order of magnitude of

[voting power of Earth]*[rate of turn desired by Earth]=[0,698 billionth of
total voting power]*[473 billion milliarcsec per year]=[330 milliarcsec per

Rough and ready though the reasoning that leads to this estimate, it gives
right order of magnitude. However, nobody has figured out how to operate on
Earth's surface a gyroscope sufficiently close to friction-free that it can
the predicted effect.
What to call this still undetected phenomenon? "Dragging," Einstein called
it in his 1913 letter of thanks to Mach, already two years before he arrived
the final formulation of his geometric theory of gravity; and
*frame-dragging* the
effect is often still called today following the lead of Jeffrey Cohen (see
ref. 24). Others often call it the *Lense-Thirring effect*, after the 1918
paper of
Hans Thirring and J. Lense, that gave a quantitative analysis of it (see
W. de Sitter 1916 and §§ 3.4.3 and 6.10 below). Today still other words are
used for that force at work on the gyro which causes the slow tuming of its
*gyrogravitation* or gravitomagnetism (see fig. 1.2). Such a new name is
justified for a new force. This force differs as much from everyday gravity
a magnetic force differs from an electric force. Magnetism known since Greek
times was analyzed (William Gilbert, *De Magnete*, 1600) long after
It took even longer to recognize that an electric charge going round and
round in a circuit produces magnetism (H.C. Oersted, 1820). Gravitomagnetism
or gyrogravitation, predicted in 1896-1916 but still not brought to the
light of the day, is produced by the motion of mass around and around in a
circle. How amazing that a new force of nature should be enveloping us all
and yet sitll stand there undetected!
 Active work is in progress to detect gravitomagnetism. The detecting
gyroscopes will be otrbiting Earth. The one gyroscope is a quartz spehre
smaller than a hand, carried along with an electrostatic suspension and a
sensitive readout devices, the whole under development by the Stanford group
of C. W. F.
Everitt and his associates for more than 20 years.
The other gyroscope is huge, roughly four Earth radii in diameter. It
of a sphere of 30 cm of radius covered with reflecting mirrors, going round
and round in Earth orbit every 3,758 hours. An axis perpendicular to that
bit we identify with the gyroscope axis. Such a satellite,
LAGEOS, already exists. It was launched in 1976. Its position is regularly
read with a precision better than that of any other object in the sky, or an
certainty of about a centimeter. For it the predicted gravitomagnetic
is 31 milliarcsec per year,  quite enough to be detected if there were not
mously larger precessions at work that drown it out, about 126° per year.
devastating competition arises from the nonsphericity of the Earth.
the possibility exists to launch another LAGEOS satellite going around at
other inclination, so chosen that the anomalies in Earth's shape give a
orbit tuming but the Earth's angular momentum gives the same Lense-Thirring
frame-dragging (Ciufolini 1984). In other words the two LAGEOS satel-
lites, with their supplementary inclinations, will define an enormous
unaffected by the multipole mass moments of the Earth but only a%ected by
Lense-Thirring drag.  If this is done the small "difference" between
the "large" tuming rates will yield a direct measure of a force of nature
new to
man. Still more important, we will have direct evidence that mass there
inertia here!


In this book we will trace out the meaning and consequences of Einstein's
1915 battle-tested and still standard picture of gravity as manifestation of
curvature of four-dimensional spacetime.

I weigh all that is.
Nowhere in the universe
Is there anything over which
I do not have dominion.
As spacetime,
As curved all-pervading spacetime,
I reach everywhere.
My name is gravity.

What of the other forces of nature?
Every other force-the electric force that rules the motion of the atomic
tron, the weak nuclear force that govems the emission of electrons and
from radioactive nuclei, and the strong nuclear force that holds together
the con-
stituents of particles heavier than the electron-Remands for its
today's researches argue, a geometry of more than four dimensions, perhaps
as many as ten. The extra six dimensions are envisaged as curled up into an
ultrasmall cavity, with one such cavity located at each point in spacetime.
tiny world admits of many a different organ-pipe resonance, many a different
vibration frequency, each with its own characteristic quantum energy. Each
these modes of vibration of the cavity shows itself to us of the larger
world as
a particle of a particular mass, with its own special properties.
Such ideas are pursued by dozens of able investigators in many leading cen-
ters today, under the name of "grand unified field theories (GUTS)" or
theories." They open up exciting prospects for deeper understanding. How-
ever, this whole domain of research is in such a turmoil that no one can
with any confidence any overarching final view, least of all any final
as to how gravity fits into the grand pattem.
The theories of the unification of forces with greatest promise today all
this striking feature that they, like the battle-tested, but simpler and
older Ein-
stein gravitation theory, build themselves on the boundary of a boundary
principle, though in a higher dimensional version; for example, "the eight-
dimensional boundary of the nine-dimensional boundary of a ten-dimensional
region is automatically zero." Hidden to the uninstructed student of modern
field theory is this vital organizing power of the boundary principle (see §
Hidden to the casual observer of a beautiful modern building is the steel
work that alone supports it; and hidden in Einstein's great account of
gravity, hidden until Élie Cartan's penetrating insight brought it to light,
the unfolding of it all, from the grip of spacetime on mass to the grip of
on spacetime, and from the automatic conservation of momentum-energy-
without benefit of gears and pinions or any sophisticated manual to instruct
nature what to dw-to all the rich garden of manifestations of gravity in na-
ture, the unfolding of all this from "the one-dimensional boundary of the
two-dimensional boundary of a three-dimensional region is zero" and "the
dimensional boundary of the three-dimensional boundary of a four-dimensional
region is zero."
Nowhere more decisively than in the fantastic austerity of this organizing
idea-the boundary principle-does nature instruct man that it is at heart
simple-and that some day we will see it so.