# The elusive quest to fulfill Mach’s Principle

### Paul Halpern

The origin of inertia is a longstanding mystery. The notion that objects at rest remain at rest, and that moving objects continue at a constant speed along a straight line, unless acted on by a net external force is not immediately obvious. Aristotle famously speculated that all motion required something like a force to keep going. Even today it is commonplace for those inexperienced with the laws of physics to make the same conceptual error, equating any kind of motion, including constant motion, with the omnipresence of a force. Those with such an erroneous belief mistakably think that when force starts, movement starts.

Any serious look at the workings of sports, however, would put such an erroneous belief to rest. Imagine someone seizing a bowling ball or a bocce ball and continuing to apply a force until it reaches its target, rolling it along the ground the entire way. Or a football star who always kicks the ball continuously along the ground until he and the ball reach the goal. Clearly those constant force scenarios would look ridiculous and likely violate the rules of those respective games. In bowling, for example, one is supposed to let the ball go, and have it roll on its own. In doing so, one trusts in inertia’s capacity to get it travelling (at least its centre of mass) at a constant velocity in a straight line – if one ignores the slowing-down effect of friction, that is.

Arguably, inertia emerged as a universal physical principle
when the brilliant 16^{th}-17^{th} century Florentine thinker
Galileo Galilei (who would later be referred to as the father
of modern science) bravely refuted the respected Aristotle
by devising a clever thought-experiment. He envisioned a
system of two inclined planes, one slanted downward and
another slanted upward, in succession, like the shape of a
“V”. If a ball is released at a certain height from the downward
plane, assuming no friction, it would descend that plane,
ascend the second plane and reach exactly the same height
as its initial value. At the same time, it would travel a certain
horizontal distance along the second plane as well. Now
imagine slowly decreasing the angle of the second plane.
The ball would still attempt to reach the same height, but,
because of the reduced angle, need to travel a greater and
greater horizontal distance to do so. In the ultimate, with the
angle of the second plane set to zero (perfectly horizontal),
the ball would travel an infinite distance at a constant
speed, seeking in vain to rise to the original height. Galileo
realised the bold implication of his thought-experiment: in
contradiction to Aristotle’s teachings, nothing needs to act
on an object to keep it moving in a straight line at a constant
speed forever.

Galileo’s independence of mind, as demonstrated by his break with the ideas of the venerated Aristotle, along his remarkable capacity to generate illustrative thought- experiments, represent two of the many reasons historians consider him the vanguard of the scientific revolution. His treatment of motion established a new standard for scientific inquiry, involving formulating and testing hypotheses, rather than taking ancient literature (and scripture) to be the principal authority unless clearly proven otherwise. In other words, Galileo pioneered the scientific method, the approach to acquiring knowledge that we continue to use today. In doing so, he rendered all of nature ripe for unprejudiced exploration, including the development of a more sophisticated understanding of the state of inertia.

## 1 Newton and his bucket experiment

Inertia is clearly not a force itself. On the contrary, it
represents the *lack* of force. In essence, it is the most natural
state of a body. Hypothetically, in deepest space, far from
the gravitational influence of any massive objects such
as stars and planets, that’s how we’d expect objects to
behave. A broken-down, unpowered spaceship, under such
circumstances, would simply keeping cruising indefinitely in a
straight line at a constant rate.

Isaac Newton (see fig. 1) recognized the importance of the state of inertia to classical physics. His principles of motion relied on a solid definition of inertia that was independent of reference frame. That led him to a dilemma: how does one distinguish in an objective way an inertial frame from an accelerated frame. Someone embedded within an accelerating frame, such as sitting on a horse galloping faster and faster, might wrongly assert that they are at rest or moving at constant speed (from their perspective), and incorrectly perceive that the outside world is accelerating relative to them. Newton’s concern about that situation was expressed in his “bucket experiment:”

If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about in the contrary way, and while the cord is untwisting itself, the vessel continues for some time this motion; the surface of the water will at first be plain, as before the vessel began to move; but the vessel by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little, and ascend to the sides of the vessel, forming itself into a concave figure.... This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative.... It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space in which these motions are performed, do by no means come under the observations of our senses.

To put this in simple terms, imagine an intelligent insect sitting just inside of the rim of a bucket filled nearly to the top with water. Suppose that pail is attached to a strong rope, tied to the limb of the tree, and then spun. As the bucket rotates, its water molecules wish to travel in a straight line, due to inertia. Therefore, they start to build up along the side of the pail. As a result, the surface of the water no longer looks flat, but rather has a meniscus: a curved upper surface.

The smart insect notices the meniscus, however, and reaches an odd conclusion. Surmising that the bucket is perfectly still, she thinks that a force has struck it from above and lifted up some of its water. Meanwhile, the rest of the world is spinning around the bucket, for some curious reason. In short, she thinks that the bucket on which she resides is in a state of inertia, and the outside world is accelerating!

From the bucket argument, Newton realised that inertial frames could not be defined arbitrarily and subjectivity. In that case, anyone and anything could claim to be in an inertial state. Therefore, he concluded that inertia must be compared to an objective framework in the universe – which he called “absolute space” and “absolute time.” Independent of the stars and the planets, that fixed framework, Newton believed, allowed for a rigorous definition of uniform motion (versus acceleration), and therefore formed the bedrock of the concept of inertia.

## Mach's Principle emerges

Yet, given the vagueness of absolute space and time, many thinkers found
Newton’s approach rather unsatisfactory. In particular, in his 1883 treatise
*The Science of Mechanics*, the Austrian philosopher and physicist Ernst Mach
(see fig. 2) called for a markedly different approach to the concept of inertia.
Eschewing the idea of fixed, abstract frameworks, he posited that inertia
stemmed from a physical principle – based on the mutual interactions of
distant bodies. As he wrote:

Instead, now, of referring a moving body... to a system of coordinates, let us view directly its relation to the bodies of the universe, by which alone such a system of coordinates can be determined. Bodies very remote from each other, moving with constant direction and velocity with respect to other distant fixed bodies, change their mutual distances proportionately to the time. We may also say, all very remote bodies – all mutual or other forces neglected – alter their mutual distances proportionately to those distances.... It is manifestly much simpler and clearer to regard the two bodies as independent of each other and to consider the constancy of their direction and velocity with respect to other bodies.

Mach never perfected his scheme to explain inertia by means of mutual interactions of distant bodies. Yet its tangibility, in contrast to the abstractness of Newton’s scheme, had a certain appeal. With his mechanics treatise a popular textbook, Mach influenced an entire generation of thinkers.

Amongst those strongly influenced by Mach was the young Albert Einstein.
From about 1901 to 1904, when he lived in Bern, Switzerland and worked
as a patent officer, Einstein participated in a reading and discussion group,
*“Akademie Olympia”*, along with his friends Conrad Habicht and Maurice
Solovine. The works of Mach formed one of the foci of their discussions.

A key problem Einstein was wrestling with at the time was the discrepancy between the Newtonian framework for motion, and the prediction of Maxwell’s equations of light’s invariant speed in a vacuum. That is, while for material objects moving at constant velocities relative to a fixed framework, their relative velocity would be the vector difference between those velocities (for example, two bikers moving in the same direction at exactly the same speed would perceive each other’s motorbike as being at rest), light does not work in the same way. No matter how fast one travels in trying to race a beam of light (as Einstein imagined) it would still seem to be moving at the vacuum speed of light, not at a smaller relative velocity.

Bravely, in trying to resolve the problem, Einstein set aside a popular idea of his day, that light requires a medium, called the “luminiferous aether”, through which it travels. If such a substance existed, one could calculate relative speeds of light with regard to the “aether wind”: the motion of Earth through the aether. However, by the time Einstein set his mind to the task there was no direct physical evidence for the aether, and, in fact, key experiments, such as the Michelson-Morley interferometry experiment, conducted by Albert Michelson and Edward Morley in 1887, that seemed to negate its existence. In essence, Einstein chose to be neutral on whether or not aether exists, and instead sought a completely different way of resolving the question of how the constancy of the speed of light might be maintained for relative frameworks.

In the special theory of relativity published in 1905, Einstein abolished the concepts of absolute space and absolute time. Rather, he gave an important role to the “co-moving framework” in any physical observation. That framework is defined as the coordinate system assigned to the object under measurement, that travels along with that object, as opposed to coordinate systems assigned to outside observers who might be moving at different relative velocities. For example, if an observer on Earth is taking measurements of lengths and times pertaining to a high-speed spaceship, the co-moving framework is that of the spaceship. In Einstein’s original construct, none of the frameworks are accelerating. (Later, Hermann Minkowski expressed the theory in a four-dimensional vector formalism, making it possible to define acceleration in special relativity.) Therefore, by making assumptions about inertial frameworks without explaining their origin, Einstein did not address Mach’s Principle at that point.

Equipped with the four-dimensional innovation of Hermann Minkowski, and the differential geometry of mathematicians such as Bernhard Riemann (fig. 3) and Gregorio Ricci-Curbastro, Einstein then set out to extend relativity to explain the force of gravitation in a local manner (rather than Newtonian action at a distance) and to include the ability to select coordinate systems and frameworks in a general fashion. Superseding the Newtonian theory of gravitation was necessary because special relativity sets the speed of light as the maximum rate of interactions, and Newtonian gravitation transpires instantly – a stark contradiction. Therefore, Einstein sought a local field theory. He came to realize that such a theory also needed to be coordinate system independent; that is, the physics cannot depend upon the arbitrary choice of coordinate systems. Those requirements set limits on viable theories.

## 3 On the equivalence of mass

An important step forward for Einstein was his development of the “equivalence principle” in 1907: namely the proposition that gravitating and accelerating frameworks locally behave identically. For example, the physical situation inside a rocket accelerating upward at 9.8 meters per second is indistinguishable from the circumstance in which the rocket is subject to Earth’s gravitational force. That equivalence stems from the important natural identity that an object’s gravitational mass (the parameter that governs its gravitational attraction to other bodies) is precisely equal to its inertial mass (the parameter that governs how much it accelerates under the influence of any net force). Because gravitational force is proportional to gravitational mass, and Newton’s second law of motion states that force is equal to acceleration times inertia mass, the equality of gravitational mass and inertial mass implies that gravitational acceleration is independent of mass. In other words, all objects accelerate in a gravitational field at the same rate, regardless of their masses, as long as there are no other forces (such as air resistance) affecting them.

Einstein was certainly not the first scientist to note the constancy of gravitational acceleration. In fact, centuries earlier, Galileo pointed out that light objects, such as feathers, and heavy objects, such as stones, released from a window near the top of a tower, would accelerate at the same rate, and reach the ground at the same time. If Einstein simply stated that massive falling objects speed up at the same pace, in the absence of air resistance, he would have been reiterating Galileo’s observation. However, he went well beyond that. Einstein made use of the equality of gravitational and inertial mass to formulate that local geometry – the fabric of space-time itself – governs local dynamics, by offering a local definition of inertial frameworks.

In a thought-experiment Einstein imagined a man falling off a roof. If he let go an object from his hand – a coin, for example – it would fall alongside him at the same rate. If he looked only at that coin, therefore, he would perceive it to be at rest. That is, it would seem to be in an inertial state. If, instead of falling off a roof, he was trapped with the released object in a freely falling elevator, and could not see outside, he could well conclude that its interior was in an inertial state. Moreover, any physics experiment he performed locally within the confines of that elevator would yield the same outcome as it would in any inertial framework. Therefore, Einstein concluded, one might define a local inertial framework anywhere in space by imagining it riding in a free-falling elevator.

With the help of Swiss mathematician Marcel Grossmann and others, Einstein spent several years combining the equivalence principle with the power of differential geometry to craft a revolutionary approach to the understanding of gravitation: the general theory of relativity, released in 1915. In it, gravitation is expressed solely as a feature of the geometry of the space-time manifold: the four-dimensional fabric that represents the dynamic arena of all of physical reality. Mass, energy (other than gravitational), and momentum pertaining to all substances in the universe manifest themselves as an entity called the stress-energy tensor. (A tensor is a mathematical entity that transforms in a predictable way under coordinate transformations and rotations.) Through Einstein’s field equations, that tensor is equated with another mathematical entity that represents geometry. In particular, it encapsulates the curvature of space-time at each point. Finally, the factors representing curvature dictate the components of the metric tensor – a mathematical shorthand for how distances and durations are measured – in any given region. In essence, a metric tensor is a generalization of the Pythagorean theorem to non-Euclidean geometry, that shows how the shortest distance between two points might be a curve, rather than a straight line. In short, the matter and energy in a region cause it to bend, which in turn affects how objects move most efficiently. For example, the Sun’s mass warps the region of the Solar System, forcing the planets to travel along elliptical paths. Contrast this explanation with Newtonian gravity, which, rather unrealistically, imagines gravitational forces as invisible strings pulling planets along their paths. The general theory of relativity has the distinct advantages of being local, rather than action at a distance. Moreover, by identifying gravitation as a fundamental property of the fabric of space-time itself, rather than something added as a separate feature of nature, general relativity seems more natural. Most importantly, it passed several key experimental tests, including matching the known precession of the orbit of Mercury, and successfully predicting the angle by which the light of distant stars bends due to the massive influence of the Sun – tested during several expeditions during total solar eclipses.

General relativity has the remarkable feature that at any point in space-time, a coordinate transformation is possible to a locally inertial framework – represented by what is called Minkowski space-time – the flat manifold associated with special relativity. In other words, at any point, those laws of physics that require the state of inertia are well-defined. The theory thereby incorporates the equivalence principle, allowing for free- falling frameworks at any point that essentially serve as inertial reference frames. Newtonian absolute space and time, therefore, are not just abolished; they are superseded by a more useful, local definition of inertia.

It is interesting to consider Newton’s “bucket
experiment” within the context of general
relativity. Every point in the bucket – including
an intelligent insect resting on it, as stated in
our imaginative example – could well have
claimed to be *locally* situated in an inertial
framework. Globally, however, because of
spatial rotation, one would have to connect
those local frames by means of differential
geometry, leading to a distorted metric with
non-zero curvature. That’s the outside picture
one would see by observing the bucket from
afar. Thus, the physics is defined through
internal connections, not through reference to
any absolutes in space, or even to the distant
stars (as in Einstein’s interpretation of Mach).

n 1918, Austrian physicists Joseph Lense and Hans Thirring calculated the subtle distortion of the space-time metric in any rotating object’s vicinity – a phenomenon present in general relativity, but not in Newtonian physics, called the Lense-Thirring Effect. Remarkably that effect, also known as “frame-dragging”, is so minute, even for bodies in the Solar System such as Earth, it would only be detected in 2004 and 2005 by the extraordinarily precise gyroscopes of the Gravity Probe B satellite mission, headed by C. W. Francis Everitt.

## 4 Einstein’s cosmic quest

Despite no clear connection between Einstein’s local definition of inertia, and his interpretation of Mach’s Principle that inertia is linked to the influence of distant astral bodies (or to the distribution of remote massive objects in general), Einstein persisted in trying to prove that Mach’s Principle was a natural consequence of the cosmological solutions found in general relativity. As he wrote in 1920, in response to an article by Ernst Reichbächer:

I myself am of Mach’s opinion, which can be formulated in the language of the theory of relativity thus: all the masses in the universe determine the [gravitational] field.... In my opinion, inertia is in the same sense a communicated mutual action between the masses of the universe, just like the actions that the Newtonian theory considers as gravitational actions.

After Einstein completed his masterful general theory of relativity, he soon attempted to develop a finite, stable model of the universe as a whole that would demonstrate how Mach’s Principle sets inertia locally. The reason he chose a finite model was to prove that the boundary conditions for the cosmos – namely its mass distribution on its periphery – connected with the metric tensor components at every point in space-time. Such components, when transformed into local coordinates, would lead to a “flat” Minkowski space- time at every point, corresponding to the local experience of inertia. It is in some sense like weighing down a carpet at its edges, noting that it becomes fixed in its overall shape, and placing a flat coin at each point to show that even bumpiness looks flat when viewed from a certain angle on a small scale. Given that, in general relativity, such a local definition of inertia is always possible, it was unclear how invoking Mach’s Principle would make a physical difference. Nonetheless, he persisted.

Einstein’s desire for a stable model stemmed from his belief, that while Newton’s notions of absolute space and time were incorrect, his general picture of the dynamics of the universe as a realm of stars that move amidst an overall fixed background was correct. Therefore, he expected that a cosmological solution should be unchanging over time.

Einstein also insisted that a cosmological solution must be homogeneous (the same from point to point) and isotropic (the same, looking out from Earth or any vantage point, in all directions). That meant that it would be uniform spatially, as well as temporally. There are three types of homogeneous, anisotropic geometries associated with non-Euclidean manifolds in three spatial dimensions: a hypersphere, with positive curvature like a ball, a hyperboloid, with negative curvature like a saddle, and a hyperplane, with zero curvature like an endless, flat table. Einstein chose positive curvature, with the expectation that it would give him a finite solution, like the surface of a globe, for which anything travelling in one direction for a long enough time would eventually circle back and reach its starting point. Attempting to fulfill Mach’s principle – in which inertia represents the combined influence of remote massive objects – required a finite universe because an endless array of distant bodies would wrongly produce an infinite effect. As a hypersphere, space would be finite, but lacking borders – ideal for that purpose.

When Einstein applied a uniform distribution of mass, and
a hyperspherical geometry to his field equations, however,
he found a major dilemma: none of the solutions were
stable. Rather, even a slight perturbation would cause them
either to collapse or to expand. Later, the results of work by
Vesto Slipher, Georges Lemaître, Edwin Hubble, and others
would indicate that cosmic growth was a reasonable model
– supported by the recession of galaxies. But at that time,
Einstein though that a static cosmos was the only viable case.
To rectify the situation and assure stability, he decided to
take a bold and unusual step: adding an extra factor to the
geometric side of his field equations, called the “cosmological
constant”, that would serve simply to balance any instabilities
caused by the material-induced actions of his original theory.
That *ad hoc* change fortuitously produced static cosmological
solutions. Clearly, it was an inelegant fix, like adding
scaffolding to support a beautiful architecture structure that
threatens to topple. Nevertheless, Einstein was horrified that,
without the cosmological constant, his theory would have no
natural stable solutions. Moreover, without such stability, his
goal of incorporating Mach’s Principle seemed unobtainable.
After all, for Machian inertia to be steady and predictable, the
distribution of matter and energy in the universe similarly
needed to be uniform.

Soon thereafter, Dutch physicist Willem de Sitter (see fig. 4) called Einstein’s assumptions into question. He discovered a second, independent solution of Einstein’s equations of general relativity, as modified by the inclusion of a cosmological constant. Rather than being filled with material of constant density, as Einstein had done, de Sitter started with vacuum conditions. His goal, in leaving out the matter and energy, was to see if inertia was possible in the absence of any massive bodies, save a single test object. Remarkable, de Sitter found that a vacuum universe with a cosmological constant, though lacking mass, has its own dynamics. In plotting the behaviour of space over time, one discovers exponential growth – an astonishing result. Moreover, in stark contradiction to Mach’s Principle, even in an empty universe, inertia exists.

As de Sitter writes in his paper: “To the question: If all
matter is supposed not to exist, with the exception of one
material point which is to be used as a test-body, has then
this test-body inertia or not? The school of Mach requires the
answer *No*. Our experience however very decidedly gives the
answer *Yes*, if by ‘all matter’ is meant all ordinary physical
matter: stars, nebulae, clusters”.

Clearly, de Sitter’s result was a blow to Einstein’s hopes of incorporating Mach’s Principle into general relativity. While at first Einstein tried to cast doubt on the physical validity of de Sitter’s solution, he soon came to realize that Mach’s Principle was not a natural consequence of his masterful theory of gravitation.

## 5 Beyond Einstein: attempts to modify his masterpiece

In the final decades of his life, Einstein focused
more on attempted unified field theories, trying to
unite gravitation with electromagnetism, as well as
humanitarian projects, than on general relativity itself.
He said little in those later years about Mach’s Principle,
save admitting that he could never get it to work.
Nevertheless, the idea was not forgotten. In particular,
Cambridge physicist Dennis Sciama took up the banner,
and decided to attempt his own way of incorporating
Mach’s Principle into gravitational theory. The result
was his 1952 paper, “On the Origin of Inertia”, published
in the prestigious journal *Monthly Notices of the Royal
Astronomical Society*.

Sciama begins his paper with a bold claim: “As Einstein has pointed out, general relativity does not account satisfactorily for the inertial properties of matter, so that an adequate theory of inertia is still lacking. This paper describes a theory of gravitation which ascribes inertia to an inductive effect of distant matter”.

Surprisingly, Sciama’s paper goes on to try and construct a theory of gravitation and inertia virtually from scratch, rather than starting with the general theory of relativity. He proposes a novel kind of gravitational field, based on the overall material distribution of the universe, that, for objects in their rest frame, precisely balances out the standard gravitational field, based on the local density of matter and energy, lending that object inertial behaviour without needing to invoke the equivalence principle. In other words, Sciama aspired to explain the equivalence principle via a Machian explanation, and then build a new gravitational theory, superseding general relativity from there. Sciama admitted in his article that his model was rudimentary. Also, he noted that it required a much higher density of matter in the universe than was observed at that time based on visible material (in that era the idea of dark matter – invisible, but gravitating – was a little-known theory advanced by Swiss astronomer Fritz Zwicky, so Sciama did not take that unseen material into account).

Around that time, American theoretical physicist John Wheeler began to promote the study of general relativity at Princeton University. He taught some of the first classes on the subject (for which in one case, he brought his students to Einstein’s house for a visit), and began to do some of the first fundamental research in years. Soon thereafter, his Princeton colleague, experimental physicist Robert H. Dicke (see fig. 5), whose original specialty was studying radiation, went on sabbatical at Harvard University during the academic year 1954-1955, and learned there about Mach’s Principle. That led him to think about a modification of Einsteinian gravitational theory in which the gravitational constant is replaced with a varying scalar field.

As Dicke later recalled in an oral history interview with Martin Harwitt:

I got interested in astrophysics and geophysics because I thought these
subjects provided a tool for getting at some of the questions which my
interest in general relativity were bringing up.... [T]he conclusion that I
reached that Mach’s principle probably was not properly incorporated in
general relativity, which led naturally to adding a scalar field to the tensor
field, so the scalar-tensor field approach to gravitation came out of this
problem that I solved *vis-a-vis* Mach’s principle. In that framework you have
the requirement that the gravitational constant is not a real constant but it’s
a function of coordinates. The cosmological solution varies with time, which
carries with it obvious geophysical and astrophysical implications, if true. This
looked like a means of providing the tools for getting at this question.

Returning to Princeton, he and Wheeler began to engage in fruitful discussions about the nature of general relativity. Each prominently attended the 1957 Chapel Hill Conference on the Role of Gravitation in Physics, a meeting seen as helping inaugurate a renaissance in the study of gravitational theory.

In 1961, working with Carl Brans, Dicke decided to draw upon a modified version of Sciama’s idea to find a way of combining local and remote gravitational influences, with the goal of generating inertia by means of a field theory. Instead of fine-tuning the mass distribution of the universe to produce the wanted effect, they decided to make the gravitational constant variable over space and time. Replacing the constant in Einstein’s field equations with a scalar field, they developed what is generally called “Brans-Dicke theory” or “Jordan-Brans-Dicke” theory (German physicist Pascual Jordan had earlier proposed a similar idea), and sometimes also “scalar-tensor theory”.

One benefit of making such a bold change to general relativity was the introduction of testable predictions. If the strength of gravitation varies over time, for example, Brans and Dicke pointed out, certain astrophysical and geophysical properties might vary over the eons. Also, the equivalence principle itself might not be as robust as Einstein proposed; there might be circumstances in which inertial mass and gravitational mass differ. Interestingly, however, standard general relativity – rather than the Brans-Dicke alternative – has passed all known experimental tests so far.

Independent of Brans and Dicke, two other researchers began to explore a
Machian generalization of general relativity: Cambridge physicist Fred Hoyle
(see fig. 6) and his student Jayant Narlikar. The Hoyle-Narlikar model attempts
to establish the long-distance gravitational influences that are supposed to
ensure stable inertia by turning to a model of electromagnetic interactions
from the 1940s developed by Wheeler and Richard Feynman. The Wheeler-Feynman absorber theory of electromagnetism removes the intermediary
exchange field between charged particles, and replaces it with direct action at
a distance. For the purposes of matching their theory with known properties
of charged particles, the two researchers calculated that the action must be
an equal combination of advanced (backward-in-time) and retarded (forward-
in-time) signals. Hoyle and Narlikar attempted to apply the same idea to
gravitation, rendering Mach’s Principle as a kind of action at a distance that
generates inertial mass for each particle. That is, without such signals, particles
wouldn’t have mass. Hoyle’s presentation of their theory
attracted press interest, including the following report in
*Time* magazine:

When he presented his new gravitation theory to a packed meeting of Britain’s venerable Royal Society, he modestly described his work, done in collaboration with Indian Mathematician Jayant V. Narlikar, as a slight extension of Einstein’s theory of general relativity. “We are clearly aware”, he explained, “that in putting forward still another idea we may be like small boys trying to steal apples”.

Far from a slight extension of Einstein’s work, Hoyle’s apple stealing is more ambitious larceny. His new theory stems from the Mach Principle, that the mass of every object in the universe is affected by its interaction with every other object. Einstein tried to incorporate the Mach principle in his own scheme of the universe and admittedly failed. Hoyle claims to have succeeded.

Ironically, it was a fellow student from Cambridge, young
Stephen Hawking, a *protégé* of Sciama, who found flaws with
the Hoyle-Narlikar model. For expanding universes, Hawking
noted, there would be an imbalance between the advanced
and the retarded signals. Because unlike electric charges,
all masses have the same sign, Hawking calculated that the
advanced gravitational fields would generally be infinite, and
the retarded signals, finite, resulting in a discrepancy.
He concluded that, under ordinary physical circumstances,
Mach’s Principle would therefore not be achievable by means
of that theory.

In conclusion, we see how a highly speculative notion by Ernst Mach became encapsulated by Albert Einstein as a proposed principle of nature, intended to explain why objects maintain their velocities in the absence of a net force by means of a deep connection with the mass distribution of the universe as a whole. While Einstein never explicitedly achieved his goal, it helped motivate him to find novel ways of understanding gravitation, inertia, and acceleration. His quest inspired Willem de Sitter to construct a novel and important cosmological solution in general relativity. It also motivated several physicists, including Dennis Sciama, Robert Dicke, Carl Brans, Fred Hoyle, and Jayant Narlikar, to pursue alternatives to standard general relativity, with the aim of making Mach’s Principle the established explanation for inertia. These theories motivated numerous experiments to check their predictions. Ironically the result was that standard general relativity was bolstered more than ever. It has passed every known test so far. Einstein’s masterful theory stands strong, with the motivated factor of trying to establish Mach’s Principle now mainly a historical curiosity.