The Influence and Divergence of Mach’s Science of Mechanics on Einstein’s General Theory of Relativity

In Science of Mechanics, Ernst Mach attacks the idea of absolute space as described by Isaac Newton in his Scholium on Time, Space, Place, and Motion wherein Newton draws a distinction between relative space and absolute space. In Science of Mechanics, Mach not only argues that Newton was wrong in some of his conclusions, but also expresses profound ideas about the purpose and proper epistemological methods of physics. It is here where Mach and Einstein most closely agree and where we can see Mach’s ideas as important precursors to Einstein’s. However, the exact place where their ideas begin to diverge is not entirely clear. I argue that Einstein’s formulation of what Einstein calls Mach’s Principle was crucial to the development of certain ideas, such as gravitational laws having to be relational laws or distant mass impacting local inertia, and principles, such as Principle of Equivalence of Gravitational and Inertial Mass, which turn out to be fundamental to the development of Einstein’s General Theory of Relativity. However, I also argue that it is not clear if Mach himself intended to posit a new physical law as Einstein takes him to be doing or if Mach’s Principle was a conclusion Einstein came to himself after pondering the explorations of inertia and rotation in Mach’s Science of Mechanics and then misattributed to Mach. I do not intend to resolve the question of whether or not Mach intended to posit Mach’s Principle, only to point out that the vagueness of Mach’s own description of these ideas suggests the possibility that Mach and Einstein’s ideas are less congruent than Einstein took them to be. Furthermore, it is clear that even if they do agree on Mach’s Principle, Mach’s ideas do not extend to a complete agreement with Einstein’s General Theory of Relativity. Nevertheless, I argue that some of Mach’s ideas were in line with Einstein’s and that at very minimum, Mach’s exploration of relative motion was crucial to inspiring many of Einstein’s ideas which lead to General Relativity.

For Newton, relative space is defining your position or motion with respect to some object. Absolute space can be considered something like true or objective space: “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable” (Newton, 27). An example of relative space is something like a single room: if we define something’s position with respect to the room, this is only a description of the thing’s relative position or its position in relative space. As such, relative space is moving with respect to absolute space, so while we can describe the object’s motion with respect to the space defined by the room, we could also describe the room’s position with respect to absolute space. Likewise if we describe the change in that object’s position with respect to the room, we have described its relative motion; absolute motion is change in relative position. Absolute motion would be the change in position of an object with respect to absolute space. If something were always at rest in absolute space, it would constitute part of absolute space and it would not constitute relative space. 

Newton argues that absolute space must exist because there is something incomplete about relative motion. If we consider the example of an object in a classroom, telling me where it is in the room does not mean much in any absolute or objective sense unless you tell me how the room is moving with respect to the solar system. Still, you would need to tell me how the solar system is moving with respect to the galaxy, and so on and so forth. Newton’s idea is thus that as long as I am only giving my description with respect to some relative space, then my description is incomplete. Newton’s thought is that only something like absolute space could complete this description and therefore absolute space must exist. This idea is often referred to as the “bucket theory” because absolute space can be imagined as something like the static bucket in which every relative space and object resides. Newton’s thought is that without absolute space, we only have incomplete descriptions of motion and in order to complete those descriptions, we need absolute space. These understandings of space and changes in position as having both a relative nature and absolute nature also translate to Newton’s understanding of time. While we divide years into months and months into days, these are only relative times. They are relative to a larger, objective, absolute standard of time. 

While it is understandable why the regress created by the notion of relative space would lead Newton to posit the existence of absolute space, one might wonder if the idea of there being a static, objective, and immutable bucket in which all other spaces exist is nothing more than philosophical baggage. Mach’s analysis of absolute space in Science of Mechanics is meant to “​​examine the point on which Newton, apparently with sound reasons, rests his distinction of absolute and relative motion” (Mach, 233). Mach directly argues against absolute space on the basis that Newton’s conception of absolute motion “[starts] ab initio from the idea of absolute space. But if we take our stand on the basis of facts, we shall find we have knowledge only of relative spaces and motions” (Mach, 234).

In Philosophiae Naturalis Principia Mathematica, Newton attempted to show, through his bucket experiment, that there is a way to tell if something is rotating with respect to the absolute space. He argues that this is done by measuring the apparent forces that come about only with absolute rotation. Newton imagines a bucket with water in it. He wants us to notice that if we begin to rotate the bucket, the water will be still at first, but eventually the walls of the bucket will slowly begin to inform the motion of the water. Newton argues that we will observe this in the form of the water beginning to rise up the sides of the bucket and curve and this is caused by the rotation creating centrifugal forces. Newton argues that through this experiment he has shown that only if the water is rotating with respect to absolute space will there be centrifugal forces. Newton’s point is that when the bucket began to rotate with respect to the water, the water was still at first. He believes that this means that the water was still with respect to absolute space while the bucket was moving with respect to absolute space and so no centrifugal forces were produced. 

In Science of Mechanics, Mach argues that Newton has failed to show what he thinks he has. Mach argues that that bucket argument only shows that no centrifugal forces are created when the water rotates with respect to the bucket. Mach points out that Newton has no idea how the water would act if we increased the width and depth of the bucket until they are extremely large. Instead of absolute motion, Mach proposes a complete kind of relativism where all motions only have meaning with reference to other bodies. I can say the bucket is moving but only with respect to the bucket or the earth or some larger structure but not in any absolute sense. Any idea of absolute motion we might have is simply a result of the asymmetrical reference systems between small bodies that we take to be in motion and much larger and much heavier bodies that we take to be still.

Mach’s analysis begins with the idea that physiological space is different from geometrical space. While Mach did not invent the idea of physiological space and is not the first person to discuss it, this concept does serve as an important starting point for the general views articulated in his Science of Mechanics. In Mach’s view, our cognitive structure determines physiological space which is entirely psychological as well as non-uniform, highly bounded, and finite. We do not intentionally construct physiological space in our minds; rather, physiological space comes from unconscious adaptation. Geometrical space, however, is homogeneous, unbounded, and infinite and is constructed intellectually as opposed to unconsciously. Mach not only thinks these two kinds of space are different things, but he also thinks geometrical space is an abstraction on physiological space and that neither of these is absolute space. As physiological space originates from unconscious biological need, it has no inherent metric. Geometrical space, however, originates from physiological space and intellect.

Mach’s problem with geometrical space as a candidate for absolute space is that it comes from physiological space which is itself rooted in psychology, so geometrical space cannot be objective or absolute space. Mach argues that while we might believe we are measuring absolute space or time when we measure some interval in miles or seconds, Mach points out that measurement rests on comparison. Mach argues that when we take ourselves to be making measurements, we are not making any kind of objective measurement of absolute geometrical space, instead we are making a comparison of our spatial sensations. Mach tells us we do not ever make any absolute measurements because we always need to choose a standard on which to base these measurements and the standard we choose will always be based on physiological comparison. Mach argues that physics is deeply beholden to biological and psychological origins and as such, relative conceptions of space and time cannot be the basis of some absolute version: 

But we do not measure mere space; we require a material standard of measurement, and with this the whole system of manifold sensations is brought back again. It is only intuitional sense-presentations that can lead to the formulation of the equations of physics, and it is precisely in such presentations that the interpretation of these equations consists. (Mach, 23)

This idea, that Newtonian physics was weighed down and inhibited by faulty concepts like absolute space or time which it has been unreflectively assuming to exist, is the first window into the influence Mach’s ideas had on Einstein’s. Einstein is deeply affected by Mach’s epistemological reflection and approach. Einstein agrees with Mach that physics was trapped by the presupposition of concepts that reflection might destroy and Mach’s philosophical ideas about how physics should develop and proceed:

Physical science does not pretend to be a complete view of the world; it simply claims that it is working toward such a complete view in the future. The highest philosophy of the scientific investigator is precisely this toleration of an incomplete conception of the world and the preference for it rather than an apparently perfect, but inadequate conception. (Mach, 235)

Mach’s epistemological methods and arguments against Newton explicitly disrupted long-standing prevailing paradigms, particularly with regard to debunking absolute space, which laid the groundwork for Einstein’s theories of relativity. Einstein’s theories align with Mach’s ideas in that they reflect a flat out rejection of absolute space (a concept which had for a long time been taken as a given) and an epistemological approach to physics which rigorously challenged even those concepts thought to be objective and fundamental. 

Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as “necessities of thought,” “a priori givens,” etc. The path of scientific advance is often made impassable for a long time through such errors. For that reason, it is by no means an idle game if we become practiced in analyzing the long commonplace concepts and exhibiting those circumstances upon which their justification and usefulness depend, how they have grown up, individually, out of the givens of experience. By this means, their all-too-great authority will be broken. They will be removed if they cannot be properly legitimated, corrected if their correlation with given things be far too superfluous, replaced by others if a new system can be established that we prefer for whatever reason. (Einstein, 72)

Both Mach and Einstein reflect a methodology which prefers an incomplete theory which nevertheless rests on facts than a complete theory which is only complete by necessitating non-existent but convenient concepts. 

However, the explicit influence of Mach’s ideas on Einstein’s General Theory of Relativity is much deeper than simply a general agreement about proper epistemological methods and certain concepts being outdated. In Mach’s Science of Mechanics, Mach develops a hypothesis which Einstein calls “Mach’s Principle” which describes how celestial bodies maintain a frame of reference. Mach’s principle posits that the large-scale distribution of matter determines absolute rotation, which distinguishes local inertial frames from rotating reference frames. Mach imagines something like standing in a field and looking up at stars on a clear night. He asks us to notice how when we are still, the stars are static and our arms are hanging still at our sides. If we begin to spin, however, we’ll notice that the stars will appear to blur and spin above us and our arms will be pulled upwards. The point Mach is trying to make is that there is a reason our arms are pushed up when the stars appear to spin and hang freely when the stars appear still. Through this example, Mach concludes that there must be a physical law which describes the relation between the motion of the stars and local inertial frames such that when the stars appear to me to spin, I feel a centrifugal force acting on me (this is what pushes my arms up). Put simply, very distant mass has an effect on inertia very closeby. More broadly, Mach’s Principle suggests that the larger structure of the universe determines local physical laws. 

Mach’s Principle is important to the development of Einstein’s General Theory of Relativity because a theory of relative motion which eliminates the existence of absolute motion leaves a question of how to measure the inertia of a body. This is a problem that Einstein needed to find a way to resolve. If the relativist argues that the inertia must be measured with respect to something else, how would one measure the motion of a particle which existed entirely alone in the universe? While one might feel that there ought to still be some kind of way to understand the motion state of the imagined particle, Mach’s Principle appears to suggest that without reference to any other object, a particle’s state of motion does not mean anything: “the investigator must feel the need of... knowledge of the immediate connections, say, of the masses of the universe. They will hover before him as an ideal insight into the principles of the whole matter, from which accelerated and inertial motions will result in the same way” (Mach, 233). From Mach’s Principle, Einstein gets the idea that inertia comes from interactions between bodies. The idea that gravitational theories must be relational theories is crucial to Einstein’s General Theory of Relativity and it is Einstein’s theory which brought Mach’s Principle into mainstream physics. 

However, while it is clear that Mach’s and Einstein’s ideas about relativity align in several ways, there are important differences which distinguish one from the other. Firstly, it is unclear if Mach necessarily meant what Einstein took him to mean. While Einstein explicitly interprets Mach as attempting to formulate a new physical law, it is not clear that Mach had this thought or was intending to do this. While Mach himself only claims to be proposing a "redescription of motion in space as experiences that do not invoke the term space," Einstein attributes the entire invention of a physical mechanism which determines physical behavior between heavy bodies and the inertia of bodies in such a way that heavy and distant bodies are responsible for the majority of inertial forces (Mach, 233). Whether Mach intended to propose a whole physical law is unclear; what is clear, however, is that Einstein’s interpretation of Mach lead him to an understanding of total relativity such that he came to understand that there exists a profound connection between gravitation and relational theories or large scale far-away structures and local physical laws. Importantly, however, Einstein’s interpretation of Mach’s Principle, that inertia comes from the interactions of bodies, is not a fundamental assumption of Einstein’s General Theory of Relativity. Mach never clearly formulated a quantitative physical theory which explained a mechanism by which large celestial objects have such an effect on local physical laws. To this end, it is clear that any such physical theory developed by Einstein is not explicitly identical to what Mach describes in Science of Mechanics. However, while Einstein’s General Theory of Relativity does not rest on Mach’s Principle, Einstein did deduce from Mach’s Principle, a deep understanding of the Principle of Equivalence of Gravitational and Inertial Mass and this principle is fundamental to Einstein’s General Theory of Relativity. 

Mach himself claimed to disagree with Einstein’s General Theory of Relativity which further reveals how while the two thinkers had profoundly related ideas when it came to certain aspects of how relative motion should work in a universe with no absolute space, their ideas still diverge significantly from one another when we move past Mach’s Principle; and, even in the case of Mach’s Principle, it is still unclear if Einstein’s formulation of it is actually what Mach intended to say. I argue that while Mach’s ideas were influential to Einstein and did contribute to the development of Einstein’s General Theory of Relativity, the vagueness of Mach’s formulation of what Einstein calls Mach’s Principle in Science of Mechanics deeply complicates the question of how closely Mach’s Principle, described by Einstein as the positing of a physical law, truly aligns with Mach’s ideas or if Mach was simply exploring relative motion without himself reaching the conclusions Einstein reached from reading Mach’s exploration. I would also argue, however, that Mach’s arguments against Newton and absolute space are a place where Mach and Einstein agree and Mach’s work in arguing against Newton was an important precursor to relativistic theories. Furthermore, it is clear that the two physicists shared a profound belief that epistemology was crucial to physics and that holding on to outdated concepts in an effort to complete a theory is detrimental to physics. Both Mach and Einstein saw physics as an ever-growing and changing field which needs to constantly review and challenge even its most fundamental assumptions and find comfort in being yet incomplete. This is exactly what they both did.

Works Cited

Mach, Ernst. The Science of Mechanics: A Critical and Historical Account of Its Development. Merchant Books, 2007.

Einstein, Albert. Relativity : the Special and General Theory: Original. 2017.

Newton, Isaac. Scholium on Time, Space, Place, and Motion.