# Exploring Einstein's Thought Experiments: A Comprehensive Overview

Written on

Einstein, the renowned German physicist, utilized visualized thought experiments, referred to as *Gedankenexperiments*, to deepen his comprehension and convey complex physical theories. His career was marked by numerous thought experiments. Notably, as a young man, he envisioned pursuing light beams, which, alongside his imaginative scenarios involving trains and lightning, led to significant insights into special relativity. While developing general relativity, he contemplated various scenarios, including a person in free fall from a rooftop, accelerating elevators, and blind beetles navigating curved surfaces. His discussions with Niels Bohr even incorporated fictional devices meant to challenge Heisenberg's uncertainty principle. Additionally, Einstein anticipated the idea of quantum entanglement while investigating how two particles could briefly correlate their states before diverging.

## Introduction

An experiment typically entails logical reasoning and hypothetical situations to investigate a theory or concept. Scientists often employ imaginary or idealized settings to understand the implications of various laws or theories. Unlike physical experiments, thought experiments do not yield new empirical data but rely instead on deductive or inductive reasoning. Extraneous details can mislead readers into perceiving these scenarios as actual experiments; however, these can often be stripped away, revealing a simplified argument. Philosopher John D. Norton posits that a successful thought experiment equates to a strong argument, while a flawed one does not.

Sometimes, superfluous details serve to engage a reader's intuition, transforming a straightforward argument into a hypothetical scenario. Galileo's renowned assertion that all objects fall at the same rate, regardless of their mass, is hailed as one of the pivotal thought experiments in modern science. Contrary to some beliefs, Galileo did not physically demonstrate this concept but articulated it logically in his work *Discorsi e dimostrazioni matematiche* (1638). His ability to visualize the practical implications of theoretical concepts was evident during his time at the patent office, resulting in his writings being rich in vivid and practical details, distinguishing them from those of contemporaries like Lorentz and Maxwell.

## Special Relativity

In his later years, Einstein recalled a paradox he faced as a sixteen-year-old. To resolve this paradox, he understood that if he pursued a beam of light in a vacuum, he would perceive it as an electromagnetic field at rest, despite its spatial oscillations. His equations, along with personal experiences, did not support this observation as Maxwell had indicated. Einstein grasped that an observer at rest would witness events unfolding according to consistent laws, raising questions about how one could identify their uniform and rapid motion without external cues. This paradox planted the seeds for the special theory of relativity.

Einstein's youthful reflections are frequently referenced, although Norton suggests they may have been colored by hindsight after decades. Historical accuracy poses challenges for Einstein's recollections. Numerous sources indicate that he did not encounter Maxwell’s theory until 1898 during his university years, while he claimed to have conceived his thought experiment in late 1895 or early 1896 as a student at the Aarau Gymnasium. Aether theory adherents could have easily taken part in the thought experiment if they adhered to its principles. Einstein's declaration that "there appears to be no such thing…that has ever been experienced" would not have posed a challenge but rather been accepted as a fact. Aether theorists might interpret Einstein’s understanding of “nor” as erroneous, as they might view light's speed as not a universal limit.

Contrary to popular belief, aether theories align with thought experiments. Nonetheless, Einstein seemed to intuitively sense an inconsistency in the scenario. In terms of optics, he believed that the laws of relativity should hold true. His early thought experiment gained significance with his maturation. According to Einstein, Maxwell's equations apply to all observers in inertial motion, indicating a unique speed of light that remains constant, regardless of the observer's velocity. This constant light speed contradicts Newtonian mechanics and Einstein's interpretation of light based on Maxwell's equations. Earlier in this article, we noted that Einstein employed early thought experiments to test the validity of physical theories, providing a strong counterargument to his emission theories developed prior to 1905.

## Magnet and Conductor

Einstein's 1905 paper on special relativity emphasized that Maxwell's electrodynamics could not be applied to moving objects as conventionally understood. For example, the electromagnetic interaction between a magnet and a conductor moving relative to one another would not produce any effect. Magnets and conductors in independent motion are generally treated as distinct scenarios. When a magnet generates an electric field while stationary, the conductor, in turn, induces a current. Conversely, no electric field arises when the magnet is stationary while the conductor moves. However, if the relative motion is identical in both scenarios, the electromotive force generated within the conductor will produce electrical currents with the same magnitude and direction as in the previous case.

This introductory paragraph outlines Michael Faraday's findings from 1831. According to the Lorentz force, a wire moving through a magnetic field generates a motional electromagnetic force (EMF), while the Maxwell-Faraday equation elucidates how a transformer produces an electromagnetic field that changes in response to a varying magnetic field. James Clerk Maxwell recognized this distinction in his paper titled “On Physical Lines of Force.” Part II of this work elaborates on each phenomenon separately. Despite Einstein's assertion that this distinction was well-known, there is no evidence that his contemporaries viewed the difference between motional EMF and transformer EMF as peculiar or indicative of misunderstandings. Maxwell concluded that Faraday's laws of induction are contingent on the relative motion between the magnet and the conductor, based on the magnitude and direction of the current. In theoretical treatment, no difference exists between the motion of the conductor and that of the magnet.

However, Einstein's contemplation of this experiment marked a significant moment in the evolution of special relativity. While the equations governing the two scenarios differ fundamentally, discerning whether the magnet or conductor is in motion proves impossible. In a review of the unpublished *Fundamental Ideas and Methods of the Theory of Relativity* (1920), Einstein expressed discomfort with this lack of symmetry, stating he could not accept that these scenarios should be fundamentally different. He surmised that the disparity arose from differing viewpoints rather than an inherent distinction in nature.

This thought experiment provided the foundation for a comprehensive theory of magnetism based on the relationship between magnets and conductors. Years passed before Einstein identified how to effectively address this challenge. Though his precise methodology remains unclear, he spent considerable time attempting to formulate an emission theory of light, ultimately encountering significant obstacles that led to his abandonment of the endeavor. In time, he lost hope that constructive efforts based on existing knowledge could yield true laws, becoming increasingly convinced that discovering a universal formal principle was the sole path to reliable results.

Following this realization, he developed special relativity, founded on two confident postulates: 1. The laws of physics maintain the same form in every inertial frame. 2. Light always travels at the same velocity, regardless of whether it originates from a stationary or uniformly moving body. While the second postulate resonated widely with theorists of the time, its formulation was more intuitive than the stronger version often presented in popular literature.

## Trains, Embankments, and Lightning Flashes

Einstein's journey toward special relativity is fascinating. In the early 1900s, he was a young patent officer, self-taught in physics, and somewhat disconnected from mainstream scientific research. His *Annus Mirabilis* papers, released in 1905, represent his four most significant publications. Of these, only the one on Brownian motion appears related to his earlier works. Notably, his paper titled “On the Electrodynamics of Moving Bodies” was polished and showed little indication of its developmental process. The details of its creation are primarily known through a few sentences in early correspondence and later statements by Einstein, some of which are secondhand and occasionally contradictory.

In his 1905 paper, Einstein delves deeply into the relativity of simultaneity. He elucidates how time is communicated between clocks through signal exchanges. To clarify his paper's formal presentation, he employs a thought experiment featuring a train, an embankment, and lightning. Here’s a brief overview of the thought experiment:

On the left, spectator M stands still on the embankment, while on the right, spectator M’ rides a rapidly moving train. Lightning strikes points A and B, equidistant from both M and M’, at the moment they align. M observes the lightning strikes as simultaneous, despite the train’s acceleration. According to Einstein's assumptions, M’ perceives the same speed of light as M, even with the train's acceleration. When the lightning strikes, A and B were approximately midway between M’ and A. M’ concludes that the strikes were not simultaneous, as light from B reaches him before light from A, indicating that the discharge at B occurred first.

A common belief among historians of science is that Einstein discovered the relativity of simultaneity while contemplating how to synchronize clocks using light signals, as stated in his 1905 paper and later writings. The conventions for synchronizing clocks were originally developed by telegraphers in the mid-19th century due to the increasing need for accurate timekeeping in various fields, including astronomy and surveying. During his tenure as a patent examiner, where he assessed electromagnetic and electromechanical patents, Einstein became acquainted with advancements in timekeeping. This exposure might have influenced his understanding of simultaneity as a relative concept, although this remains speculative. Later accounts reveal that Einstein referred to his experiences with light beams and experiments involving magnets and conductors as inspirations for his development of special relativity. He also highlighted the significance of the Fizeau experiment and the observation of stellar aberrations, which contributed to his thoughts on clocks, though he did not cite specific thought experiments.

When light is analyzed through Newtonian corpuscles, relativity is unnecessary for examining the Fizeau experiment and stellar aberration. Viewing light as waves traversing through aether poses challenges that could be resolved using the relativity of simultaneity. The analysis of these phenomena suggests that Einstein might have arrived at special relativity by an alternative route. The extent to which his development of simultaneity was influenced by clock synchronization or the train and embankment thought experiment remains uncertain, although he preferred to illustrate this concept using the latter.

## Relativistic Center-of-Mass Theorem

A core belief of Einstein's is the equivalence of mass and energy. This understanding evolved further in subsequent decades, aided by Einstein and other eminent physicists such as Max Planck, Gilbert N. Lewis, Richard C. Tolman, Max von Laue (who provided a comprehensive proof of M0 = E0/c based on the stress-energy tensor in 1911), and Paul Dirac (who explored negative solutions in his 1928 formulation of the energy-momentum relation, eventually predicting the existence of antimatter in 1930).

Einstein's relativistic center-of-mass theorem from 1906 clarifies this concept. Henri Poincaré, in the early days of modern physics, noted a contradiction. By applying Maxwell’s equations, one could theoretically create a reactionless drive by adhering to the principle of action and reaction. This device could move without expelling any propellant, seemingly violating momentum conservation. Poincaré proposed that electromagnetic energy behaves like a fluid with specific density, capable of generating and destroying momentum by absorbing and releasing energy. These movements, he argued, would displace the center of mass and disrupt momentum conservation. Nevertheless, Einstein demonstrated that Poincaré's explanation was unavoidable. His alternative reasoning was that the equivalence of mass and energy resolves the paradox both automatically and satisfactorily. Einstein presented a distinct derivation of mass-energy equivalence, illustrated through a thought experiment that mirrors Poincaré’s abstract mathematical argument.

In his lifetime, Einstein envisioned a stationary, sealed tube suspended in space, with mass M and length L. On one end, a device emits light energy (photons) in a vertical direction. The momentum of the radiation can be calculated by dividing by c, the speed of light. Since the system exhibits no momentum, it moves in the opposite direction at a velocity v calculated by dividing negative energy E by mass M and the speed of light c. The length of the tube divided by c determines the time interval (delta t) within which the radiation reaches the other end. Delta x is defined as negative energy divided by length L divided by mass M multiplied by the square of the speed of light, leading to a stop.

A weightless shuttle mechanism k transfers energy from the right side of the tube to the left by returning to its original position, resulting in only leftward displacement. Repetition of this process is feasible. According to mechanics principles, an object cannot move without external force if the center of mass remains stationary. Einstein's relativity asserts that if energy is transferred from left to right, shuttle mechanism k cannot be massless. This inconsistency can be resolved by dividing the speed of light by its inertia m.

Neither Einstein's 1905 mass-energy equivalence formula nor his 1906 center of mass theorem can be deemed definitive. An illustration of this is the center-of-mass thought experiment. Since step (b) does not surpass the speed of light, the force exerted on the tube cannot exceed it. Photons are yet to initiate movement when they strike the right wall in step (c). According to Ohanian, von Laue’s (1911) derivation of M0 = E0/c stands as the first truly definitive proof.

## Impossibility of Faster-Than-Light Signaling

The composition law for velocities, coupled with Einstein’s observations, suggests that signaling at speeds exceeding that of light is unattainable. In his material band concept, signals could theoretically be transmitted faster than light, symbolized by W. For A to send a signal to B, A is positioned adjacent to the material band, which moves at speed v in the negative x-direction, separated from B by a distance L. The speed of the signal from A to B is calculated as (W-v) divided by (1 minus (Wv divided by the speed of light squared)). The signal travels from point A to B within a specified time frame, represented by T. Using L, we can divide (1 minus Wv divided by the square of light’s speed) by (W — v).

## General Relativity

### Falling Painters and Accelerating Elevators

In a 1920 unpublished review, Einstein elucidated the origin of the equivalence principle. In 1907, he summarized his work on special relativity and began attempting to reconcile Newton's gravitational theory with his own. The methods he devised did not satisfy him as they were based on unfounded physical assumptions. At this juncture, he made a groundbreaking realization: gravitational fields and magneto-electric induction produce analogous electric fields. An observer in free fall from a rooftop perceives their surroundings as devoid of a gravitational field. As the observer gazes around, objects remain stationary or move uniformly in relation to them, irrespective of their chemical or physical properties, justifying their interpretation of being "at rest."

This startling realization propelled Einstein into an eight-year endeavor that culminated in his most celebrated achievement. Over time, many authors have embellished the narrative of the falling man. Various retellings depict him as a painter, with the tale suggesting that Einstein's inspiration arose when he witnessed a painter tumble from a nearby building while working at the patent office. However, Einstein did not clarify why witnessing such a tragic incident would be deemed one of the happiest moments of his life in this version.

To frame his hypothetical scenario, Einstein contemplated a situation of free fall in space within a large enclosed chest or elevator. Free fall gives an individual the sensation of weightlessness, causing objects removed from their pockets to float alongside them. In this scenario, an "Einstein rope" is affixed to the ceiling of the chamber. Subsequently, a strong force continuously pulls the rope, accelerating the chamber in an "upward" direction. The observations made by the man align with those in a uniform gravitational field within the chamber. The question Einstein posed was whether the conclusions reached by the man should be disregarded as incorrect. Rather than responding, he asserted that this thought experiment supports the addition of accelerating frames of reference to the general theory of relativity, thus motivating a broader relativity postulate.

It is widely recognized that gravitational mass and inertial mass are interconnected, a relationship Einstein addressed in his thought experiment. Gravitational mass determines an object's attraction to others, while inertial mass dictates its acceleration. Historically, these two types of mass appeared equal, yet the underlying reason for this equality remained elusive until Einstein's thought experiment. He proposed that it is impossible to experimentally discern whether a specific coordinate system is accelerated or if the observed effects arise from gravity. This realization led to the derivation of the equivalence principle, which posits that gravitational mass equates to inertial mass. Consequently, light rays follow curved paths due to the gravitational field, as demonstrated by Einstein's thought experiment involving an accelerating observer.

### Early Applications of the Equivalence Principle

Through his studies of kinematics—examining the motion of objects devoid of force considerations—Einstein laid the groundwork for Special Relativity. As a geometric interpretation of special relativity, Einstein's former mathematics instructor, Hermann Minkowski, introduced the concept of spacetime to the Göttingen Mathematical Society in late 1907. Initially, Einstein dismissed Minkowski's geometric interpretation as unnecessary academic theorizing. The kinematic analysis provided a clearer understanding of general relativity than geometric analysis for Einstein. In his 1907 publication *Jahrbuch*, he noted that gravity influences light propagation and impacts clocks. After revising his theory in 1911, he recognized that his predictions could be experimentally validated. Einstein's 1911 paper, alongside other scientists, proposed that as an object's energy content increases, its inertial mass correspondingly rises. Calculating the inertial mass involves dividing an increase in energy by the square of the speed of light.

As gravitational mass escalates, so too does inertial mass. Einstein sought to establish whether a correspondence existed between the two. Furthermore, he aimed to determine the proportional increase of gravitational mass relative to the rise in inertial mass. He argued that the existence of this correspondence is indisputable due to the equivalence principle.

Einstein's scenario demonstrated the equivalence of energy and gravity. An energy source, designated S2, is positioned at a height h above a receiver, denoted S1, within a uniform gravitational field with a force per unit mass of g. As energy departs from S2, it results in energy entering S1. Assuming this system is devoid of gravity, it can also be described as experiencing constant acceleration g along a positive z-axis, employing the concept of equivalence. In these conditions, S2 maintains a fixed distance h from S1. S1 receives light from S2, which takes approximately h/c to traverse (where c signifies the speed of light). The velocity of S1 increases during transmission by g*h/c relative to its speed when the light was emitted. Consequently, S1 receives more energy, E1, than E. The equation for E1 can be approximated as:

E1 = E * (1 + velocity/c)

For a system in a gravitational field that is not accelerating, gh is replaced by phi, representing the difference in gravitational potential between S1 and S2. Thus, E1 equals E plus E divided by c squared divided by gravitational potential difference. The energy E1 received by S1 exceeds E2 emitted by S2 due to gravitational potential energy divided by c squared in the gravitational field for mass E. A defined amount of energy possesses both gravitational and inertial mass when divided by c squared.

The following cyclic process outlines Einstein's proposition regarding gravitational and inertial mass energies: (a) Position the light source h above the receiver S1 within a uniform gravitational field. There are two points, S2 and S1, where a movable mass can be located. S2 transmits electromagnetic energy E to S1. A mass M descends to S1, releasing Mgh or work in the process. This results in the generation of M’. The mass is then lifted back to S2, adding work M’gh to the equation. Consequently, S2 encompasses the energy carried by the mass. Therefore, the difference between raising and lowering the mass, M’gh — Mgh, must equal Egh/c². If this equation is not satisfied, perpetual motion machines become feasible. Consequently, M’ — M = E/c². This process predicts a similar increase in inertial mass as dictated by special relativity. Einstein contemplated a scenario where a continuous electromagnetic beam was transmitted from S2 to S1 at frequency v2 (measured at S2). According to S1, frequency v1 = v2(1 + ?/c²).

In this equation, Einstein made an intriguing observation: it seems perplexing that the number of light waves received at S1 could differ from those emitted at S2. The upward journey from S2 to S1 must generate wave crests. However, this question presupposes absolute time, while clocks do not operate at the same speed across different gravitational potentials. According to the equivalence principle, gravity slows time. Any gravitational theory adhering to Einstein’s principle of equivalence, including Newtonian gravity, can utilize the argument he presented concerning gravitational time dilation. Experiments such as those conducted by Pound-Rebka demonstrate gravitational time dilation, illustrating that general relativity and Newtonian gravitation cannot be differentiated. Einstein's 1911 paper highlighted the potential for light rays to curve under the influence of a gravitational field. At that time, his theory was incomplete, leading to predictions about the bending of light that later evolved into the comprehensive theory of general relativity.

## Non-Euclidean Geometry and the Rotating Disk

In 1912, while advancing the kinematics of general relativity, Einstein encountered a challenge. Recognizing the need to surpass his existing mathematical tools, he began exploring new avenues. His examination of the relativistic rotating disk led to a breakthrough, as emphasized by Stachel. The special relativity analysis of rigid bodies by Max Born and Paul Ehrenfest in 1909 outlined the concept of a rigid rotating disk. Observers positioned at the edge of rotating disks experience a perceived force known as "centrifugal force," often regarded as "fictitious" or "pseudo." By 1912, Einstein began to believe that centrifugal force and gravitation were interconnected. The equivalence principle suggests that system K exhibits the same gravitational field as a stationary system with a static gravitational field.

In this illustration, two circular disks, labeled A and B, are presented. From an inertial reference frame, disk A has a diameter of 10 units. Its circumference, equivalent to its diameter, is depicted by 311.4 rulers. Disk B, on the other hand, shares the same diameter but rotates rapidly. Consequently, an observer not spinning with disk B would observe the rulers contracting along the circumference, necessitating a greater number of rulers. It is crucial to note that the scenario does not explicitly mention moving disk A to create disk B. According to Born's definition, a "rigid" disk cannot rotate because its circumference, not its radial surfaces, experiences pressures induced by material contractions. Einstein's demonstration of how a gravitational field influences measuring rods in a non-Euclidean manner is illustrated by the rapidly rotating disk. While articulating the non-Euclidean conception of space and time he envisioned, Einstein acknowledged his lack of mathematical proficiency. Seeking assistance, he approached Marcel Grossmann, who possessed a strong mathematical background. In Grossmann’s library, Einstein discovered a review article by Ricci and Levi-Civita concerning absolute differential calculus (tensor calculus), published in 1914 and 1913 when they developed their first generalized theory of gravitation. By integrating curved spacetime into Minkowski’s geometry of relativity, Einstein expanded upon Minkowski’s geometric approach.

Explore Einstein’s thoughts in “Exploring Einstein’s Thought Experiments.” He discusses a series of thought experiments that significantly advanced our understanding of the universe. This exploration traces Einstein's quest through the examination of a beam of light, revealing a thought experiment that played a pivotal role in the development of special relativity. This revolutionary theory reshaped our perceptions of space, time, and reality. The article also delves into the intriguing relationship between conductors and magnets, highlighting the evolution of electromagnetic induction since Einstein's thought experiments. The unique scenario involving trains, embankments, and lightning flashes illustrates the relativity of simultaneity and its implications for event synchronization. Furthermore, the article presents a thought experiment demonstrating momentum and energy conservation in relativistic systems, extensively exploring a substantial fraction of light’s speed.

Additionally, the article tackles the concept of faster-than-light communication, which Einstein’s causality experiment ultimately dismissed. Cosmological speed limits provide a foundation for understanding the fundamental laws of nature. The theories surrounding falling painters and accelerating elevators are also discussed, showcasing experiments that reveal the indistinguishability of gravitational forces and accelerations. Moreover, the article emphasizes the early applications of the equivalence principle while exploring how the theory of general relativity developed. Through this groundbreaking theory, we gain a deeper understanding of how spacetime curves, revolutionizing gravitational research. Finally, the article uncovers a profound connection between Einstein’s rotating disk thought experiment and non-Euclidean geometry, illustrating how traditional notions of space and geometry were challenged. This realization led to the understanding that spacetime is dynamic and influenced by mass and energy, showcasing Einstein’s exceptional intellect and profound insights. This foundational experiment has significantly shaped and guided modern physics.