The Bulging Earth

According to Newtonian mechanics the spinning of the Earth on its
axis should cause it to bulge at the equator.  This is the content
of Proposition 18 of Book III of the Principia, and in Proposition
19 Newton estimates that the equitorial diameter differs from the
polar diameter by about 1 part in 230.  (Modern estimates are about
1 part in 297.4.)  One of the confirmations of Newton's theory was
provided by the astronomer Giovanni Domenico Cassini, director of 
the Paris observatory, who found that the planet Jupiter is wider
at its equator than from pole to pole.  

Beginning with Giovanni Domenico Cassini (1625-1712), the position 
of Director of the Paris observatory was held by the Cassini family
for four generations.  Giovanni was succeeded by his son Jacques 
(1677-1756), who was succeeded by HIS son Cesar Francois (1714-1784),
who was succeeded by HIS son Jacques Dominique (1748-1845).  Each
of these descendents of Giovanni were born at the Paris observatory,
and their dates indicate heriditary longevity, living to the ages of
87, 79, 70, and 98 respectively.

Ironically, the Cassinis Giovanni and Jacques became involved in a 
controversy over Newton's prediction of the Earth's equatorial bulge.
Their geodesy measurements, extending from Dunkerque in the North 
to Spain in the South, indicated that one degree of latitude was 
shorter near the poles than near the equator, whereas if the Earth 
bulges at the equator it should be the other way around (because the 
radius of curvature of the surface in the latitudinal direction 
should be be smaller at the equator than at the poles).  From these 
measurements the Cassinis and their supporters concluded that the 
Earth must be elongated along the polar axis, and narrower at the 
equator - just the opposite of Newton's prediction.  

The controversy was "settled" by the expedition led by P.L.M de
Maupertius in 1736, who went to Lapland to make measurements near
the arctic circle.  These measurements brilliantly confirmed Newton's
theory, very much as Eddington's expedition to Africa to view the 
Solar eclipse in 1919 confirmed Einstein's theory... although it's 
interesting to note that in both cases it was later determined that 
the errors in measurements were so great that neither of these 
confirmations was really very solid.  In the late 1920's Leinberg 
found that Maupertius' has accumulated errors - all in the same 
direction - so that his results strongly confirmed the equatorial 
bulge, but if he had accumulated these same errors in the opposite 
direction his results would have been totally inconclusive.  

One wonders how often this has happenned, i.e., a scientific issue 
has been "settled" by an experiment that is later shown to have been
inconclusive - even if the conclusion was fortuitiously correct.
It's hard to avoid the impression that experimentalists are often
influenced by their preconceived beliefs about what answer MUST be
correct.  After all, how could the spinning Earth possibly NOT bulge 
at the equator?  And Eddington was certainly convinced of GR prior
to his expedition.  In any case, Voltaire (a avid supporter of Newton)
was inspried to composed a couplet making light of Maupertius and his 
far-flung measurements that roughly translates as

   He confirmed with great effort at distant places 
   What Newton knew without ever leaving home.

Of course, it's one thing to predict that a spinning planet will 
bulge at the equator, and quite another to provide a quantitative 
estimate of the magnitude of this bulge.  This is actually a somewhat
tricky problem, because the Earth's actual shape is quite complicated
(when considered in close detail), it's density is not uniform, and 
its gravitational field is not simple.  To approach this difficult
problem, Newton's basic idea was that the Earth must rise toward the
equator by an amount that compensates very nearly for the increasing
centripetal tendancies because, as Newton put it

    ...if our earth were not a little higher around the equator 
    than at the poles, the seas would subside at the poles and,
    by ascending in the region of the equator, would flood
    everything there.

The implication is that the Earth's surface must be (very nearly) 
an equi-potential surface, because otherwise the waters would slide
sideways in the direction of the lower potential.  Of course, this
compensation is not merely coincidental, because the entire Earth
was once spinning in a molten state, so we would expect it to
have solidified with an equatorial bulge.  Admittedly the rotation
rate of the Earth has been reduced since then, but the Earth is
still maleable enough to response (over millions of years) to the
changing forces and maintain a roughly equi-potential surface.

From a relativistic standpoint, the natural paths of objects tend
to veer in the direction of slower proper time, which is why particles
fall in the direction of massive objects where the rate of proper time
is lower.  In effect, the total "potential" of an object in any state
of motion and in any gravitational field can be defined as the
derivative dtau/dt, where tau is the object's proper time and t is
the time coordinate far from any gravitating bodies.  Hence, the
traditional cute answer to the question of whether ideal clocks run
faster at the poles or at the equator is that they run at the same
rate, almost by definition, because the surface of the Earth is an 
equi-potential (i.e., equi-time-rate) surface.  

Unfortunately the cute answer is not entirely correct, for several 
reasons.  First, the Earth's surface is not perfectly equi-potential, 
because it isn't perfectly maleable.  Second, actual clocks respond 
differently to variations in gravitational vs centripetal acceleration.
Even assuming an equi-potential surface, the effective weight of a 
given mass is less at the equator than at the poles (inversely 
proportional to the radii).  Indeed one of the first confirmations 
of the equatorial bulge was the observation of the astronomer Jean
Richer (1630-1696) that pendulum clocks run about 2.5 minutes slower
per day at Cayenne in French Guiana (at the Northern end of South 
America) about 5 degrees North of the equator than at Paris, which 
is nearly 49 degrees North of the equator.  Nevertheless, an IDEAL
clock (one that does not rely on the local vertical weight of an
object), such as a cesium clock, would run at the same rate at all
points on an equi-potential surface.

On the other hand, we should be careful to specify just what we mean
by "potential" in a relativistic context.  In arbitrary spacetimes 
it can be tricky to define a meaningful potential, but of course when 
dealing with problems involving terrestial phenomena we can model the
Earth as an isolated body in an asymptotically flat spacetime, 
possessing a nice time global coordinate "t" (as in the Schwarzschild
coordinates).  In such a spacetime it is reasonable to define the 
potential at any given point of any given worldline is simply dtau/dt,
which makes the relation between potential and "time dilation" 
tautological.

Having made this definition, we can show that a non-viscous rotating
blob of matter (possessing some natural self-repulsion) in equilibrium
will have a shape whose surface has constant dtau/dt.  An arm-waving 
"proof" is just to point out that each part on the surface of the 
blob would slide tangentially if dtau/dt at any neighboring location
was greater than at its current location, because it wants to maximize
its dtau/dt (to get as close to a geodesic as it can, under the 
constraint of its repulsion from the surrounding material).  This 
is essentially just a re-statement of Newton's argument about the
distribution of the seas.  [I'd be interested in seeing a rigorous 
proof of the exact equipotentialness of the surface, particularly at
intermediate latitudes of an oblate spheroid, where the centripetal 
acceleration is not parallel to the gravitational "acceleration".  
Since we don't have an exact solution of the field equations for an
oblate spheroid, a rigorous proof would presumably have to be based 
on more fundamental considerations.]

Assuming the equi-potential surface has constant proper time, we can
begin by noting that a disproportionate fraction of the Earth's mass 
is located in its core, so we will not be too far wrong if we assume
(at first) a spherically symmetrical gravitational field.  In other 
words, assume that the shape (and rotation) of the Earth's outer 
mantle and surface doesn't have much effect on the field.  For a 
particle with constant radial coordinate r and constant latitude 
q, the proper time tau is given by the line element

    (dtau)^2  =  [1 - 2m/r](dt)^2  -  [r^2 sin(q)^2](dp)^2

where p is the longitude and t is the Schwarzschild time coordinate
(which corresponds to the proper time of a sufficiently distant 
static observer).  Dividing through by (dt)^2, setting w = dp/dt, 
and using the expansion (1-x)^(1/2) ~ 1 - x/2  gives

      dtau/dt  =   1  -  m/r  -  (1/2) r^2 sin(q)^2 w^2

At the North pole we have q=0 so this is just 1 - m/r where r is 
the distance from the North pole to the center of the Earth.  At 
the equator we have q = pi/2, and distance to the center of the 
Earth is R = r+h, where h is the height above the Earth's nominally
spherical surface representing our first approximation of the 
equatorial bulge.  This gives dtau/dt = 1 - m/R - (1/2) R^2 w^2.  
Subtracting the proper time rate at the pole from the proper time 
rate at the equator gives

 (dtau/dt)_e - (dtau/dt)_p  =  mh/[r(r+h)] - (1/2)(r+h)^2 w^2

The Earth's bulge is relatively slight, so the first-order effect
dominates, and the break-even height h is approximately r^4 w^2/(2m).
This is the height above the surface of a perfectly spherical Earth
at the equator such that the rate of an ideal clock equals the rate
of ideal clocks at the poles on the Earth's surface.  The height for
other latitudes q can be computed from the previous expression.

However, if we fill up the the envelope of equal-time-rate height h(q)
with matter having the same density as the Earth, the equatorial time
rate will be slowed back down somewhat.  This means we need to raise
the elevation of the equatorial clock by an additional amount to make
its rate equal the polar rate.  Since the ring of equatorial matter 
produces, at a nearby point, only about half the gravitation of a 
complete spherical shell, and since the sensitivity of clock rate to
height is roughly unchanged, we may assume that the additional increase
in elevation is only about half the original elevation h.  If we then
fill up the region to h + h/2 we will again reduce the equatorial
clock rate, so we will need to further increase the elevation by
half as much again, and so on.  This sequence of approximations leads
to the convergent series for the total equatorial bulge

          b  =  h + h/2 + h/4 + h/8 + ...

                           r^4 w^2
                =  2h  =   -------
                              m

The mass of the Earth (in geometrical units) is m = 0.00443 meters,
the angular speed of the Earth is w = (2.424)10^-13 rad/meter, and 
the nominal radius of the Earth is about r = (6.38)10^6 meters, so 
we estimate that the Earth's equatorial radius exceeds its polar
radius by approximately 22,000 meters.  The compares remarkably well
with the best modern measurements of the Earth's actual equatorial 
bulge, which give the currently accepted value of about 21,476 meters
(which is 1 part in 297.0).

The formula should also be roughly applicable to the other planets.
For example, the planet Jupiter has a radius 10.79 times the Earth's
radius, it's mass is 318.1 times the Earth's mass, and it's rotational
speed is 2.421 times the Earth's rotational speed.  Therefore, using
the expression  b = r^4 w^2 / m  we would estimate that the difference
between the polar and equatorial radii of Jupiter is about 250 times
the difference for the Earth, which gives about 5.2 million meters.
Current measurements of Jupiter indicate that the actual difference 
is 4.3 million meters, so our rough formula is about 20% high.  This
is probably attributable to the greater variation in density of 
Jupiter, whose outer layers are gaseous.  As a result, the sequence
of corrections may give a geometric factor of something like
(1 + 1/3 + 1/9 + ...) = 3/2 instead of 2 as in the case of the 
Earth with its rocky mantle.

Denoting the constant "geometric factor" by k = 1/(1-f), we can 
express our basic equation for the bulge of a spinning planet in 
the form
                                      
           b         r^3 w^2         3 w^2
          ---  =   k -------  =  -------------
           r           2m        8pi (1-f) rho


where rho = m/(4/3 pi r^3) is the density.  The factor f accounts for
variations in density, and is about 1/2 for the Earth and slightly 
greater than 1/3 for Jupiter.  This equation is consistent with
Newton's statement that

   If a planet is larger or smaller than the earth, while its
   density and periodic time of daily rotation remain the same,
   the ratio of centrifugal force to gravity will remain the
   same, and therefore the ratio of the diameter between the
   poles to the diameter at the equator will also remain the
   same.  But if the daily motion is accelerated or retarded
   in any ratio, the centrifugal force will be increased or
   decreased in that same ratio squared, and therefore the
   [ratio of] the diameters will be increased or decreased 
   nearly in the same squared ratio.  And if the density of
   a planet is increased or decreased in any ratio, the gravity
   toward the planet will also be increased or decreased in
   the same ratio, and the difference between the diameters
   in turn will be decreased in the ratio of the increase in
   the gravity or will be increased in the ratio of the 
   decrease in gravity.

By the way, to quantify the magnitude of the differences in time
rates that result in these bulges, notice that if the Earth was 
perfectly spherical with radius r, then a clock at the equator 
located at an elevation of 21 kilometers would have a rate exceeding
the rate at the pole by (1.074)10^-12 sec/sec, which gives about 
92.8 extra nanoseconds per day.  This is the net effect of 
+(2.278)10^-12 sec/sec due to high elevation and -(1.204)10^-12 
sec/sec due to rotation.

It's also interesting to note that on the equator we have the general
equation for any radius r

           dtau/dt  =  1 - m/r - (1/2) r^2 w^2

Differentiating this with respect to r gives

                d^2 tau        m
                -------   =   ---  -  w^2 r
                 dt dr        r^2

The first term on the right corresponds to the classical "acceleration
of gravity" and the second term is the centripetal acceleration v^2/r,
so we see that the mixed second derivative d^2tau/(dt dr)  can be
identified with the net "acceleration".