Newton's Bucket

and

Einstein's Ellipsoid

In the scholium at the end of Definitions, the section with which he opens his treatise Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy, 1687), Isaac Newton described an experiment that he had performed to illustrate what he meant by absolute space. To perform the experiment Newton suspended a bucket from a long cord and turned the bucket until he had twisted the cord tight. Then he put water into the bucket, waited for all of its motion to cease, and then gave the bucket a twist to set it spinning.

At first the water's surface lay flat. But as viscosity forced the water to share ever more of the bucket's rotary motion, the center of the water's surface descended and the outer part of the water's surface climbed up the bucket's side. As the water spun with the bucket its surface took on the shape of a parabola - the faster the water spun, the deeper the parabola grew.

In Newton's conception the turning of the water involved an acceleration relative to absolute space. That centripetal acceleration evoked in the water a force of inertia that acted to keep each small parcel of water moving in a straight line. Responding to that force, the water shifted in the bucket until its weight, pushing down the slope of the parabola, balanced the centrifugal force of inertia.

That experiment gave Newton evidence to support his definition of absolute space as an unchanging plenum in which bodies act in accordance with his three laws of motion. In that definition Newton validated Galileo's break with Aristotle in postulating the principle of Relativity. But it was Relativity with respect only to uniform motion in straight lines (the subject of Newton's first law of motion): with respect to accelerated motion of any kind Newtonian observers would agree as to which frame accelerated and which didn't.

In "The Foundation of the General Theory of
Relativity", which he published in Annalen der Physik (Annals of Physics) in
1916, Albert Einstein described, by way of an imaginary experiment, the need he
saw to modify the principle of Relativity. He imagined two fluid bodies floating
in deep space far from any other bodies and from each other. An observer
floating motionless next to each body sees the other body rotating at a uniform
angular velocity about the straight line that passes through the centers of both
bodies. After surveying their respective bodies the observers discover that body
S_{1} has the shape of a sphere and body S_{2} has the shape of
an oblate ellipsoid. What reason can we give, Einstein asked, for that
difference between the two bodies?

A Newtonian would say that S_{1} was not
rotating in the absolute frame and that S_{2} was displaying the effect
of rotating in absolute space. But, with reference to the Positivist doctrine of
Ernst Mach, Einstein said, "No answer can be admitted as epistemologically
satisfactory, unless the reason given is an *observable fact of experience*.
The law of causality has not the significance of a statement as to the world of
experience, except when *observable facts* ultimately appear as causes
and effects." In that light Einstein declared that Newtonian mechanics did not
give a satisfactory reason for the difference between the two bodies.

We know that the word "causality" does not denote a real thing, but, rather, denotes a relationship between pairs of events and among the bodies that participate in those events. We cannot, therefore, impute some property called "cause" to entities in order to explain our observations of Reality. Thus Einstein dismissed Newton's concept of absolute space as an improper contrivance: although Einstein's imaginary experiment shows us observable facts that signify an effect (one body spherical and the other ellipsoidal), it gives us no observable fact to signify a cause of that effect. All possible reference frames mark subsets of space that all have the same property - extent - and no other properties that would mark one as different from all the others. For Einstein that fact meant that space in itself cannot signify a cause of the difference between the bodies in his imaginary experiment, because it does not offer an observable fact that would designate an absolute frame. (That rather obsessive Copernicanism may seem a little excessive, but it does, in fact, lead to a correct description of Reality).

Instead Einstein posited the distant bodies of
the Universe as comprising the observable fact that signifies the cause of the
described effect. It was not that non-acceleration necessarily inheres in the
space R_{1}, in which the body S_{1} is at rest, and not in the
space R_{2}, in which S_{2} is at rest, but rather the purely
contingent fact that by the fundamental accident of the creation of matter the
faraway galaxies do not accelerate in R_{1} and do accelerate
centripetally in R_{2} that signifies the cause of S_{1}'s
sphericity and S_{2}'s ellipticity. Otherwise, the two spaces have no
difference between them except their relative rotation.

On that basis Einstein declared, "The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." Expanding on that statement, he then set forth his postulate of General Relativity: "The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant)." That is, the laws of physics must have the same mathematical form for all observers, regardless of any uniform motions or accelerated motions that may exist among those observers.

So who got it right? Newton or Einstein?

If I want to assert the Newtonian hypothesis as true to Reality, then I fall into a trap of my own devising. By that hypothesis I must infer that when the Universe sprang into existence and filled itself with space, that space must have included an absolute frame defining non-acceleration. But then we must ask "non-acceleration relative to what?" and the Newtonian hypothesis goes away.

Recall that from our basic axiom, that the Universe exists in a context of Absolute Nothingness, we deduced the fact, among others, that space comprises a set of inertial frames, of which none can possibly define a state of absolute rest. An absolute rest frame could only exist on the existence of some thing external to the Universe to give it reference (the difference that makes the difference, the observable fact that signifies the cause), but no such thing exists, so the absolute rest frame also does not exist.

Now we must apply that reasoning to non-inertial frames. Nothing can exist outside the Universe to set any non-inertial frame apart from the others. Further, our deduction that acceleration does not affect the speed of light gives an accelerating frame the character of an inertial frame in that an observer occupying that frame would measure the boundary of space moving away from them in all directions at the same speed and would, by measurement, appear to occupy the center of the Universe. The fact that inertial and non-inertial frames share that most fundamental fact of spatio-temporal geometry implies that Einstein did, indeed, get it right: in their fundamental natures we cannot properly distinguish inertial frames from non-inertial frames.

But we do distinguish inertial frames from non-inertial ones, as Newton's bucket experiment and Einstein's imaginary observation of two fluid bodies demonstrate. However, we now know that the truth of that statement comes about not because any necessary property is inherent in the nature of space, as Newton would have it, but because of a contingent property that came out of the accidents of the origin of the Universe. Instead of coming into existence in an already defined absolute frame, as Newton would have it, matter came into existence in a fully relativistic space, as Einstein had it, and then that part of space, those frames, in which the matter's average acceleration equaled zero became the de facto frame of non-acceleration. Thus, in accordance with Mach's principle, the distant galaxies, which evolved out of that matter, do indeed mark the frame of non-acceleration.

Now we must ask how the distant galaxies enforce that fact, how they made the water rise up the side of Newton's bucket and how they would transform one but not the other of Einstein's fluid bodies into an ellipsoid. Like Einstein, I dislike spooky actions at a distance, so I want an answer to that question in terms of mechanism by way of some kind of forcefield. In the 1950's Dennis Sciama provided just such an answer.

Sciama's hypothesis has matter filling space with the gravitational analogue of the magnetic vector potential, the gravitonic field. Immersed in that field, a body of mass M forced to accelerate would encounter an opposing force

(Eq'n 1)

which corresponds to what Newton called the force of inertia, the force with which the body resists an externally imposed acceleration. If existence makes Sciama's hypothesis true to Reality, then we expect to find that

(Eq'n 2)

for any given body moving at the velocity
**v** and forced to accelerate at the rate d**v**/dt.

In deducing the existence and description of the
magnetic field, we saw the magnetic vector potential emerge naturally from the
moving electrostatic field. We can use the same pattern of emergence to devise a
description of the gravitational vector potential, the mathematical expression
of the gravitonic field, from the moving gravitostatic field.
At a distance r from a minuscule element of mass dm moving with velocity **
v** we have

(Eq'n 3)

in which G represents the Newtonian
gravitational constant (G=6.672x10^{-11} newton-meter^{2}/kilogram^{2}).

To calculate the gravitational vector potential of the
whole cosmos we begin by subdividing all of space into a set of coaxial
cylindrical shells. Each shell has radius R and thickness dR. For convenience we
assert that the axis of each cylindrical shell coincides with our x-axis. Thus
we have cylinders whose lengths vary from
Δx=2R_{0} to
Δx=0 extending
away from the x-axis from R=0 to R=R_{0}=13.7 billion lightyears. The
length of each shell extends from x=-
to x=+, the
formula reflecting our assumption into our premises of the premise that space
has the shape of a sphere.

Next we divide each cylindrical shell into a set
of rings, each of thickness dx. Each ring thus has volume 2πRdRdx. We fill each
ring with matter at the average density of the Universe,
ρ=9.7x10^{-27}
kilogram per cubic meter, which figure includes the density due to baryonic
(ordinary) matter, dark matter, and dark energy. Each ring thus represents an
element of mass,

(Eq'n 4)

We want to calculate to gravitational vector potential at the point R=0, x=0 (the point at the center of our spherical space. A ring of radius R at ordinate x will generate a gravitational potential at that field point in the amount

(Eq'n 5)

in which

(Eq'n 6)

the distance from every part of the ring to the field point. To convert that number into the increment that the ring's mass adds to the cosmic gravitational vector potential, we multiply it by some undetermined velocity v and divide it by the square of lightspeed, so we have

(Eq'n 7)

Finally we calculate the overall cosmic
gravitational
vector potential by integrating the contributions from the rings comprising a
disc of radius ,
integrating the contributions from the discs that lie between x=0 and x=R_{0}
(comprising one hemisphere of space), and then doubling the result to include
the contributions from the other hemisphere. We thus get

(Eq'n 8)

If the cosmic gravitational vector potential accounts
fully for inertia, then that result should equal precisely **A**_{g}=1.000**v**
(in accordance with Equation 2). But if we contemplate the relative magnitudes
of the numbers that we put into the calculation at the end of Equation 8, we
must feel astonishment that the result came as close as it did to the ideal
amount. We have a temptation to believe that such a close result validates
Sciama's hypothesis (after all, James Clerk Maxwell justified his theory that
light comprises an electromagnetic wave on a calculation of the speed of
electromagnetic propagation that matched Armand Fizeau's measurement of the
speed of light about as poorly as our result matches the requirement of the
Sciama hypothesis) and I can offer a reason to yield to that temptation.

That reason boils down to a statement that the above calculation does not represent the gravitational vector potential in the real Universe, but rather represents that potential in a crude approximation to the real Universe. I did not, for example, shape the calculation to acknowledge the fact that the mass density of the Universe must increase as we carry our calculation closer to the boundary of space. And I did not incorporate an accurate description of space and time, which you can see reflected in my tacit representation of the boundary of space as a spherical surface rather than as a point.

Thus I know that I will have to revisit this calculation in another essay, after we have worked out the proper description of space and time across the full width of the Universe and of the distribution of mass within it. For now, though, I simply restate the postulates of General Relativity (and hope to upgrade their status to that of theorems later):

1. The laws of physics must be of such a nature that we must express them in equations that apply to all systems of spatio-temporal coordinates, regardless of any motion, uniform or accelerated, among those systems.

2. A ray of light moves at the same speed for all observers, regardless of any motions, uniform or accelerated, between any two of those observers.

And, based on the implication that inertia originates in the gravitational analogue of the magnetic vector potential, we have the equivalence principle;

3. A system under constant acceleration is indistinguishable from a system held stationary in a uniform gravitational field.

Now we need only work out the consequences of
those postulates.

habg