The Equivalence Principle
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Though it rarely inspires science to illumine the mind,
soul-warming Art offered me this treasure to find.
In Nineteen-Eighty's vernal season
down a black hole to a wonderland of Reason
my White Rabbit led me to truth sublime
and showed me how gravity deforms space and time.
Younger sister of the classical Greek Muses,
her fantasy in logic's service she uses.
In blue minidress, red cape and boots,
she took me to examine Reality's roots.
Falling through warped space and cosmic fire,
the spirit of beauty entered the stellar pyre.
From within the black star came inspiration's glint
when the girl of steel struck Reality's flint,
showered sparks into my imagination,
and lit off a creative conflagration.
Ideas, like flames, illuminated Plato's cave
revealing in full glory the knowledge that she gave.
So to that Eternal Feminine in comic-strip guise
who revealed gravity's secret to my mind's eyes.
to that angelic dream who breaks not my solitude,
to that fairychild from the stars I owe my gratitude,
This is where the Map of Physics began for me. This is where I began to see the possibility of deducing all of the laws of physics from a simple set of axioms. While I developed my own version of General Relativity from my insight I also remembered a minor topic that often shows up in discussions of the theory -- Mach's principle. Physicists usually present it by describing how it accounts for bodies possessing inertial mass; to wit, bodies possess inertial mass because other bodies exist. Thus, I possess mass because Earth exists: Earth possesses mass because the surrounding stars exist: the surrounding stars possess mass because the galaxies exist: the galaxies possess mass because.... There the process must stop because nothing exists outside the Universe to give the galaxies their mass. I saw then that the conventional argument had it wrong: bodies possess mass because nothing exists outside the Universe: mass must exist as a property of bodies so that the conservation laws pertaining to linear and angular momenta can properly describe a necessarily immobile Universe.
But now I want to show you the version of General Relativity that I devised from my inspiration. It led me directly to what I call the Schwarzschild Transformation, but before I get into that I want to discuss the foundation of General Relativity, Einstein's equivalence principle.
Since November of 1915, when Albert Einstein published his theory of General Relativity, we have had, in effect, two almost completely separate theories of Relativity. Special Relativity, the simpler of the two, requires little of us mathematically. When he presented it to the world in the 1905 Sep 26 issue of Annalen der Physik, Einstein employed fairly elementary algebra and Euclidean geometry in its mathematical formulation; thus, it can be taught in a high-school level course in physics. General Relativity, the other Relativity, the vastly more difficult one, requires of its students a firm mastery of Riemannian geometry and tensor calculus; consequently, it is taught only in the universities and larger colleges and only at the graduate level.
That fact has long struck me as rather strange. After all, General Relativity is purported to be a modified version of Special Relativity, one that accommodates the space-warping effects of gravity. We should expect it to be more difficult, of course, but not that much more difficult. As it turns out, General Relativity is actually not that much more difficult than is Special Relativity. Through an odd stroke of luck I have discovered a version of General Relativity that's only slightly more complicated than is Special Relativity; indeed, I derive its mathematical formulation directly from the Lorentz Transformation.
I predicated the devising of this simple version of General Relativity on a slight conceptual shift. Physicists usually call General Relativity Einstein's theory of gravity, as though it were a replacement for Newton's theory of gravity. If, instead, we regard General Relativity as a theory of relativistic motion in a Newtonian gravitational field, we can readily modify the equations of the Lorentz Transformation into transformation equations that correspond to the metric equations devised by Karl Schwarzschild and Roy P. Kerr. Instead of Einstein's geometric approach, which has gravity warping the structure of space and time, I use a kinematic approach, which has gravity deforming inertial reference frames.
According to his own recollection, Albert Einstein was inspired to conceive the basis for General Relativity one morning in 1907 while he was sitting at his desk in the Swiss Patent Office reading the newspaper. Einstein told Abraham Pais, one of his graduate students, "Then there occurred to me the happiest thought of my life in the following form. The gravitational field has only a relative existence in a way similar to the electric field generated by magnetoelectric induction. Because for an observer falling freely from the roof of a house there exists -- at least in his immediate vicinity -- no gravitational field. Indeed, if the observer drops some bodies then these remain relative to him in a state of rest or of uniform motion, independent of their particular chemical or physical nature (in this consideration the air resistance is, of course, ignored). The observer therefore has the right to interpret his state as 'at rest'."
Apparently a report of a carpenter who had fallen off a roof and, fortunately, landed in a pile of sand had caught Einstein's attention and engaged his imagination. The carpenter had told the reporter something incredible, something not only astonishing to the average person but all the more so to the trained physicist: he claimed that in the brief time of his fall he saw his tools floating before him as if they were weightless. How can that be? Einstein wondered. Weightlessness is a property of matter occupying inertial frames of reference, which are unaccelerated, and yet the carpenter and his tools were clearly accelerating, under Earth's gravity, toward Earth's surface at 9.81 meters (32.2 feet) per second per second. Einstein resolved the apparent contradiction in the carpenter's claim by conceiving the Equivalence Principle, which states that
a system falling freely in a gravitational field is equivalent to an identical system occupying an inertial frame of reference and a system constrained to move at any unaccelerated speed in a gravitational field is equivalent to an identical system being accelerated through a series of inertial frames in empty space.
In that principle Einstein found the means to extend his theory of Special Relativity, which applies only to unaccelerated motions, and develop a more general theory of Relativity, one that would apply to all possible motions and would thus comprise a description of the relationship between time and space applicable everywhere in the Universe.
But an astute observer might think that Einstein is making a shady move on us here. This Equivalence Principle reflects a tacit assumption that gravitational mass and inertial mass are identical to each other. Further Einstein did not try to deduce that identity from any of his previously established theorems. He simply asserted it as a postulate. In all fairness, though, I would think that he would have to know what gravity is before he could obtain the identity between gravitational mass and inertial mass by deduction.
It seems a minor point, but it's actually important to this theory. If that postulate were not true to Reality, then bodies with different compositions would accelerate at different rates in the same gravitational field because the masses that determine how the bodies respond to a force would differ from the masses that determine the force exerted upon them by the gravitating body. However, experiments and observations made beginning in the Twentieth Century have, in fact, upheld the Equivalence Principle. For a time, until we can devise a true theory of gravity (one that tells us what gravity is and not merely what it does), we must accept the fact that in this matter we must let the axiomatic-deductive method yield way to the empirical-inductive method.
As for the experiments themselves, they were, in principle, very much like the experiment that is attributed to Galileo, the one in which he is alleged to have dropped balls of different compositions from the Leaning Tower of Pisa. Of course that tower experiment was not nearly sensitive enough to put the proposition to a proper proof. The first properly sensitive experiment was conducted in 1922 under the leadership of Baron Roland Eötvös and its result upheld the identity between inertial mass and gravitational mass. So we shall accept the Equivalence Principle as true to Reality and hope to deduce it from axioms later.
One of Einstein's math teachers, Hermann Minkowski, provided Einstein's next major inspiration toward General Relativity in 1908. Minkowski had shown that Special Relativity can be regarded as describing a Euclidean geometry in four dimensions and had gone so far as to suggest that the concepts of space and time as separate entities would fade away and be replaced by a kind of fusion of the two (more properly, a con-fusion, with all that that word connotes), which four-dimensional continuum he called "spacetime" (actually "raumzeit": he was speaking German at the time). (I have not used the word spacetime in this monograph largely because it is generally used more to cultivate mystical mindfuzz and less to promote scientific clarity.) Einstein took Minkowski's description as a cue to reconceive the Equivalence Principle, to envision a reference frame marked by a gravitationally accelerated object as actually being an inertial frame, but one that was warped out of true by the presence of the gravitating body. And then Einstein knew that he would have to describe that deformed Euclidean entity with a non-Euclidean geometry.
Euclid's geometry, the geometry created by the classical Greeks, is the science of the description and deduction of relationships among figures drawn upon a perfectly flat plane (whence the name plane geometry). That's the geometry upon which Rene Descartes drew his coordinate grid in the early part of the Seventeenth Century and, through it, worked out the means to translate algebraic equations into geometric figures and vice versa. Mathematicians call the resulting fusion of basic algebra and plane geometry analytic geometry and its extension into three dimensions is what physicists have used to work out the laws of classical Newtonian physics and what Einstein used to work out the laws of Special Relativity. In the Nineteenth Century mathematicians began exploring non-Euclidean geometries; that is, geometries whose figures are drawn upon curved surfaces, such as the surface of a sphere (with positive curvature) or a saddle (with negative curvature). Of the results of those explorations, Einstein chose that of Georg Friedrich Bernhard Riemann (1826-1866) as the analogue of analytic geometry in which he would work out General Relativity. Because Riemannian geometry can describe space of any curvature (including zero, which is the realm of Special Relativity), it seemed ideally suited to describing a truly general theory of Relativity.
Unfortunately, the number-manipulating math that Riemann was obliged to fuse with non-Euclidean geometry to create his analogue of analytic geometry makes General Relativity the most difficult theory in physics to learn and to apply to an understanding of natural phenomena. It's called tensor calculus and where the fundamental objects of basic algebra are formulae made of letters representing numbers the fundamental objects of tensor calculus are multi-dimensional arrays of algebraic formulae and operators (entities that transform algebraic formulae into other algebraic formulae, much as algebraic formulae transform numbers into other numbers). But with that math, however difficult it is to use, Einstein was able to complete his theory of General Relativity in November of 1915. The centerpiece of that theory is the equation that relates the tensor describing the four-dimensional curvature of space and time to the tensor describing the distribution of matter and energy in the region of space and time under study.
In his version of General Relativity Einstein represented gravity as a warping of space and time in the vicinity of a massive body. That especially intimate relationship between gravity and the structure of space and time carries a weird implication concerning the nature of gravity. In accordance with that implication, in an essay following this presentation, I offer a bizarre, but workable hypothesis of how gravity might work.
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