The Principle of Relativity
Motion is relative; for example, when you toss a ball in the air while flying in a jet plane, it simply returns to your hand, as the jet appears to be stationary from your perspective while the landscape moves beneath you.
Teenage Einstein questioned the prevailing scientific beliefs that light waves followed different rules than material objects, particularly regarding the principle of inertia Scholars of his time posited that light waves were vibrations of a mysterious substance known as ether, suggesting that the speed of light should be viewed relative to this ether This notion contradicted the long-standing idea that motion is relative, introducing the concept of an absolute frame of reference in relation to the ether, which was deemed superior to moving frames.
Experiments, however, failed to detect this mysterious ether.
The concept of ether suggested that it permeated all physical objects, including window glass and the human eye, allowing light to travel as a wave However, understanding and defining the properties of ether proved to be quite challenging.
Light can also travel through a vacuum (as when sunlight comes to the earth through outer space), so ether, it seemed, was immune to vacuum pumps.
Einstein questioned the existence of the ether, concluding that if it couldn't be detected or manipulated, it might as well not exist This raised the question of what the consistent speed of light, measured at 3×10^8 meters per second, truly meant He pondered whether it was possible to conceptually ride alongside a beam of light, where its speed would appear zero, yet all experiments consistently indicated that light's speed remained constant at 3×10^8 m/s.
Einstein established that the speed of light is governed by the laws of physics, particularly those related to electromagnetic induction, leading to the conclusion that it remains constant across all frames of reference This groundbreaking insight positioned both light and matter as equal, as they both adhere to universal physical laws applicable in every frame of reference, embodying the principle of relativity.
Experiments don’t come out different due to the straight-line, constant-speed motion of the apparatus This includes both light and matter.
This is almost the same as Galileo’s principle of inertia, except that
Section 1.1 The Principle of Relativity 15 c / Albert Michelson, in 1887, the year of the Michelson-Morley experiment. d / George FitzGerald, 1851-
1901. we explicitly state that it applies to light as well.
It is challenging to comprehend how a dog running away at 5 m/s relative to the sidewalk appears to have a speed of 2 m/s from my perspective when I chase it at 3 m/s This situation aligns with our understanding of motion, where the dog exhibits different speeds in different frames of reference However, it raises the question of how a beam of light maintains the same speed, regardless of whether someone is pursuing it.
The Michelson-Morley experiment of 1887 revealed the puzzling constancy of the speed of light Conducted by Albert A Michelson and Edward W Morley, the experiment utilized a sophisticated apparatus to detect any differences in the speed of light beams traveling in east-west and north-south directions, considering the Earth's motion around the Sun at approximately 110,000 km/hour Operating under the ether hypothesis, they anticipated that the speed of light would vary relative to the ether as the Earth moved Specifically, they expected that a light beam directed westward would travel slower due to the Earth moving towards it However, their results showed no significant change in the speed of light, defying their expectations and challenging the prevailing theories of the time.
Although the Michelson-Morley experiment occurred nearly two decades before Einstein published his first paper on relativity in 1905, he likely learned about it only after submitting his work At that time, Einstein was employed at the Swiss patent office, which kept him distanced from the prevailing developments in physics.
Einstein proposed a groundbreaking solution to the puzzling behavior of light waves, which do not conform to traditional velocity addition and subtraction rules in relative motion He suggested that space and time are perceived differently by observers in varying frames of reference, leading to the conclusion that an appropriate distortion of time and space could account for the constant speed of light.
The controversy surrounding the Michelson-Morley experiment persisted for 40 years, with the experimenters themselves uncertain about the implications of their findings in a moving frame Earlier physicists, influenced by Newton's concepts of absolute space and time, failed to consider a radical shift in understanding until Einstein George FitzGerald proposed that the experiment's negative result could be explained by the contraction of physical objects due to Earth's motion through the ether, while Hendrik Lorentz developed the necessary mathematics However, neither recognized that the true distortion involved space and time, not merely physical objects.
5 See discussion question F on page 26, and homework problem 12
Section 1.1 The Principle of Relativity 17
Distortion of Time and Space
In a rocket ship equipped with a tube and mirrors, a flash of light emitted from the bottom reflects back and forth, functioning as a clock by counting its trips to measure time While this concept may seem impractical, it mirrors the principles of an atomic clock For an observer inside the rocket, the clock operates normally, akin to a passenger on a jet tossing a ball without noticing any differences However, to an observer on Earth, the light's path appears zigzagged, resulting in an increased travel distance due to the rocket's high speed relative to Earth.
According to the principle of relativity, the speed of light remains constant across different frames of reference, which means that while an earthbound observer may perceive light to travel faster, this perception changes at speeds approaching that of light Consequently, time appears to be distorted, causing a light-clock to run more slowly from the perspective of an observer on Earth.
A clock runs fastest for observers in the same motion as the clock, while it appears to run slower for those moving relative to it For example, one observer measures the light traveling a distance of cT, while another observer measures it as traveling ct In a scenario involving a spaceship, an observer on Earth perceives the time taken for light to travel from the bottom to the top of the ship as T, while the spaceship observer measures it as t The spaceship's velocity relative to Earth is denoted as v, and the light beam's path forms the hypotenuse of a right triangle, where the base is vT Both observers agree on the vertical distance traveled by the light beam, represented by height = ct, where c is the speed of light The hypotenuse, which represents the distance the light travels in the spaceship's frame, is cT By applying the Pythagorean theorem, we can relate these three quantities.
(cT) 2 = (vT) 2 + (ct) 2 , and solving forT, we find
The amount of distortion is given by the factor 1/ q
1−(v/c) 2 , and this quantity appears so often that we give it a special name,γ
What is γ when v = 0? What does this mean? Answer, p 132
Time is often perceived as absolute and universal, but it can flow differently for observers in varying frames of reference The γ factor illustrates this concept, remaining nearly constant at low speeds, and even at 20% of the speed of light, changes in γ are minimal In our daily experiences, we encounter speeds that are only a fraction of the speed of light, making the relativistic effects on time negligible This aligns with the correspondence principle, which states that new theories, like relativity, must be consistent with established laws, such as Newton's, within their applicable domains.
Section 1.2 Distortion of Time and Space 19 h / The behavior of the γ factor.
The speed of light remains constant across all frames of reference, defined as distance divided by time Altering time inevitably affects distance, ensuring the speed remains unchanged If time experiences distortion by a factor of γ, lengths must also adjust by the same ratio Consequently, an object in motion appears longest to an observer at rest relative to it, while it appears shortened in the direction of motion to other observers.
The concept of absolute time allowed for statements like, “I wonder what my uncle in Beijing is doing right now,” under the assumption that clocks in different locations, such as Los Angeles and Beijing, could be synchronized This perspective enabled a clear understanding of events occurring simultaneously, as illustrated by a flash of light hitting the back wall of a moving ship first, due to the wall approaching the light, while the front wall receives the light later as it moves away However, this leads to the realization that different observers may not agree on the simultaneity of the light flashes hitting the front and back of the ship.
We conclude that simultaneity is not a well-defined concept.
This idea may be easier to accept if we compare time with space.
In Galilean relativity, spatial points lack inherent identity, as the perception of events can vary based on the observer's frame of reference For example, if you tap your knuckles on your desk and then wait five seconds before tapping again, you may believe both taps occurred at the same location However, an observer on Mars would argue that, due to the high-speed motion of Earth, the second tap actually took place hundreds of kilometers away from the first.
According to the theory of relativity, concepts like simultaneity and "simulplaceity" lose their meaning unless the relative velocity between two reference frames is significantly lower than the speed of light For instance, in the garage's frame of reference, the bus appears to be in motion and fits inside the garage, while from the bus's perspective, the garage is the one moving and cannot accommodate the bus.
One of the most famous of all the so-called relativity paradoxes has to do with our incorrect feeling that simultaneity is well defined.
The concept involves driving a school bus at relativistic speeds into a standard-sized garage, a scenario where the bus would typically be too large to enter Due to the phenomenon of length contraction, as described in the theory of relativity, the bus would theoretically fit into the garage despite its normal dimensions.
In the context of the garage paradox, the distortion of time and space becomes evident when the bus is perceived differently by observers in distinct frames of reference An observer inside the garage may conclude that the bus fits due to its contracted length, while the driver views the garage as contracted, making it seem inadequate to contain the bus This paradox highlights the flawed assumption that the front and back of the bus can be considered simultaneously within the garage Observers moving at high relative speeds often disagree on the simultaneity of events; for instance, the garage observer might believe they shut the door as the bus's front bumper reaches the back wall, while the driver perceives the door closing only after passing through.
Nothing can go faster than the speed of light.
What happens if we want to send a rocket ship off at, say, twice the speed of light, v = 2c? Then γ will be 1/√
Your math teacher warned you about the serious consequences of taking the square root of a negative number, as it yields a physically meaningless result This leads to the conclusion that no object can exceed the speed of light Furthermore, traveling precisely at the speed of light is also deemed impossible for material objects, as it would result in an infinite gamma factor (γ).
Einstein resolved his initial paradox regarding riding a motorcycle alongside a beam of light by concluding that it is impossible for the motorcycle to reach the speed of light.
Many individuals, upon learning that nothing can exceed the speed of light, often contemplate ways to break this fundamental rule One common thought is that by continuously applying a force to an object over time, it could achieve constant acceleration and potentially surpass the speed of light These intriguing concepts will be explored in section 1.3.
A classic experiment illustrating time distortion involves observing the decay of muons at rest compared to those moving at a speed of 0.995c relative to the observer The results highlight the significant differences in decay rates, emphasizing the effects of relativistic speeds on time perception.
Muons are not commonly found in our environment due to their radioactive nature, as they exist for only 2.2 microseconds before transforming into an electron and two neutrinos Despite their short lifespan, muons can function as a unique type of clock, albeit one that is unpredictable Experimental data illustrates the decay rates of muons, comparing those created at rest with high-velocity muons generated in cosmic-ray showers Notably, the decay rate of high-velocity muons is observed to be stretched by a factor of approximately ten, aligning with the predictions of relativity theory, expressed as γ = 1/p.
Since a muon takes many microseconds to pass through the atmo- sphere, the result is a marked increase in the number of muons that reach the surface.
Time dilation for objects larger than the atomic scale
Dynamics
So far we have said nothing about how to predict motion in relativ- ity Do Newton’s laws still work? Do conservation laws still apply?
Yes, many definitions require modification to encompass new phenomena, such as the transformation of mass into energy and vice versa, as illustrated by the renowned equation E=mc².
The concept of motion exceeding the speed of light highlights a fundamental distinction between relativistic and nonrelativistic physics For instance, if Janet travels in a spaceship and accelerates to 0.8c (80% of the speed of light) relative to Earth, and then launches a space probe at 0.4c relative to her ship, one might mistakenly conclude that the probe is moving at 1.2c relative to Earth However, this scenario illustrates the flaws in assumptions about combining velocities in the context of relativity, emphasizing that speeds cannot simply be added together when approaching the speed of light.
The issue with Janet's reasoning is that her perception of the probe moving at 0.4c differs from that of earthbound observers, highlighting the discrepancies in time and space perception Unlike Galilean relativity, velocities do not simply add together in the same manner To illustrate this, we can express all velocities as fractions of the speed of light, which emphasizes the limitations of Galilean addition of velocities.
For a comprehensive understanding of the correct relativistic results, detailed algebraic derivations are available in my book, "Simple Nature." Here, I will present the numerical outcomes directly.
Janet's probe travels at 0.91c, a significant deviation from the expected speed of 1.2c This discrepancy is particularly noticeable at the edges of the data tables, where all results align with the speed of light, adhering to the principle of relativity For instance, if Janet emits a beam of light, it demonstrates this fundamental concept.
In the study of relativistic dynamics, the concept of velocity addition is crucial For velocities that are relatively small compared to the speed of light, the outcomes align closely with Galilean principles, as illustrated by the green oval in the center of the table Conversely, the blue edges of the table emphasize that all observers agree on the speed of light, asserting that a probe moving at 1.00 times the speed of light is universally recognized, rather than being calculated as 1.8 by adding 0.8 Furthermore, the correspondence principle ensures that relativistic results align with classical addition at sufficiently low velocities, which is reflected in the similarity of the tables at the center.
A flawed concept for surpassing the speed of light involves an experiment with a ping-pong ball and a baseball stacked like a snowman When dropped, the two balls separate mid-air, allowing the baseball to hit the ground and rebound before colliding with the descending ping-pong ball This unexpected interaction causes the ping-pong ball to shoot upwards at high speed, illustrating a surprising principle also observed in pedestrian accident investigations involving moving vehicles.
When a car traveling at 90 kilometers per hour collides with a pedestrian, the impact can propel the pedestrian at nearly double that speed, reaching approximately 180 kilometers per hour However, if the car were to move at 90 percent of the speed of light, the dynamics of the collision would change dramatically, raising the question of whether the pedestrian would be propelled at 180 percent of the speed of light.
To understand the speed-doubling phenomenon in collisions, we must consider the center-of-mass frame, where colliding objects approach each other, collide, and rebound with reversed velocities In this frame, the total momentum remains zero before and after the collision Observing an unequal collision from both the center-of-mass perspective and the frame where a smaller ball is initially at rest reveals the dynamics of the event, akin to the sequential frames of an old-fashioned movie camera.
The experiment utilized two cameras: one tracked the center of mass, while the other followed a small ball and maintained its speed post-collision In accordance with the principle that everything not forbidden is mandatory, each experiment yielded a single outcome that adhered to all conservation laws.
How do we know that momentum and kinetic energy are conserved in figure q/1? Answer, p 132
In this example, consider a small ball with a mass of 1 kg and a larger ball with a mass of 8 kg In the first frame of reference, the velocities are set at 0.8 m/s for the small ball and -0.1 m/s for the big ball.
In the new frame of reference where the small ball is initially at rest, all velocities from the previous table can be adjusted by adding 0.8 The updated velocities for the small and big balls are now 1.6, 0.9, and 0.7, respectively.
In this frame, as expected, the small ball flies off with a velocity,
1.6, that is almost twice the initial velocity of the big ball, 0.9.
When considering velocities measured in meters per second, the calculations align perfectly However, if these velocities are expressed in terms of the speed of light, the previous approximations become inadequate.
In the realm of relativistic dynamics, velocities must be combined using specific rules to yield accurate results For example, when two velocities of 0.8 times the speed of light are combined, the resulting speed is 0.98 times the speed of light, rather than the expected 1.6 This principle is crucial when analyzing collisions, such as an 8-kg ball traveling at 83% of the speed of light colliding with a 1-kg ball.
The balls appear foreshortened due to the relativistic distortion of space.
In Figure q/1, the large ball moves slowly, resembling how an ant on the ball would perceive the scene However, from the perspective of an observer in frame r, both balls appear to be moving at nearly the speed of light after their collision This results in a foreshortening effect on both balls and the distance between them, making it seem as though the small ball is not distancing itself from the large ball quickly.
Randomness Isn’t Random
Einstein's aversion to randomness stems from his belief in determinism, which he associated with a divine order, reflecting the Enlightenment view of the universe as a vast clockwork mechanism set in motion by a Creator This perspective influenced many pioneers of quantum mechanics, who explored potential connections between physics and philosophical concepts.
Eastern and Western religious and philosophical thought, but every educated person has a different concept of religion and philosophy.
Bertrand Russell remarked, “Sir Arthur Eddington deduces religion from the fact that atoms do not obey the laws of mathematics Sir
James Jeans deduces it from the fact that they do.”
Russell's humorous remark suggests a misconception about mathematics and randomness, highlighting the need for a nuanced understanding of this complex topic It is essential to avoid the oversimplification that "everything is random" because certain events are fundamentally impossible, as dictated by the laws of classical and quantum physics Conservation laws—such as those governing mass, energy, momentum, and angular momentum—remain valid, making the creation of energy from nothing not just improbable, but outright impossible.
An effective analogy for understanding the role of randomness in evolution highlights two key ideas from Darwin: first, that species evolve through random genetic variations, and second, that natural selection preserves beneficial changes while eliminating maladaptive ones Critics often focus solely on the randomness of these variations, likening the emergence of complex organisms like Homo sapiens to a whirlwind assembling a jumbo jet from junk However, this reasoning overlooks the deterministic constraints that govern random processes, akin to how it is just as improbable for an atom to violate energy conservation as it is for chimpanzees to conquer the world.
A Economists often behave like wannabe physicists, probably because it seems prestigious to make numerical calculations instead of talking about human relationships and organizations like other social scientists.
Economists often attempt to model economic behavior using mechanical metaphors, likening market supply and demand to a self-adjusting machine and viewing individuals as economic automatons focused solely on wealth maximization However, this raises the question: what evidence exists to support the idea of randomness in economics, challenging the notion of mechanical determinism?
Section 2.1 Randomness Isn’t Random 45 a / The probability that one wheel will give a cherry is 1/10.
The probability that all three wheels will give cherries is
Calculating Randomness
You should also realize that even if something is random, we can still understand it, and we can still calculate probabilities numerically.
Physicists are akin to skilled bookmakers, as they can calculate probabilities with greater accuracy than novices However, despite their expertise, they cannot foresee the outcome of any specific event, much like a bookmaker unable to predict the result of an individual horse race.
When playing a 25-cent slot machine with three wheels, each having a 10% chance of landing on a cherry, the odds of winning $100 by getting all three wheels to show cherries can be calculated Despite the randomness of each spin, the probability of winning on any single attempt is determined to be 1 in 1,000, or 0.001, reflecting the mathematical likelihood of this outcome.
A probability of 0 indicates an impossible event, while a probability of 1 signifies a certain event Expressing odds as a ratio of 0 to 1 simplifies calculations and enhances clarity compared to traditional statements like “The odds are 999 to 1.”
Each trial in games of chance, such as slot machines and craps, is statistically independent, meaning past outcomes do not influence future results Many gamblers mistakenly believe that previous losses increase their chances of winning in subsequent plays, leading them to think that a slot machine is "due" for a payout or that they should follow a particular shooter in craps This misconception of correlation in games of chance can affect betting strategies and overall gambling behavior.
Craps players often believe in a positive correlation between winning rolls, suggesting that a successful roll increases the likelihood of future wins This phenomenon, referred to as being "hot," implies that players may feel more confident and optimistic about their chances as they continue to roll favorable numbers.
My method of calculating the probability of winning on the slot machine was an example of the following important rule for calcu- lations based on independent probabilities:
Independent probabilities mean that the outcome of one event does not affect the outcome of another For example, when drawing a coin from your pocket containing a nickel and a dime, the chance of selecting the nickel is 0.5 If you replace the coin and draw again, the probability remains 0.5 for the nickel Therefore, the likelihood of drawing the nickel twice in a row is 0.25.
When drawing two coins without replacing the first coin, the outcome of the first draw directly influences the second draw In this scenario, it is impossible to pull out the same coin twice, resulting in a probability of zero for drawing a nickel on both attempts Thus, the trials are not independent, and the law of independent probabilities does not apply, contrasting with the 0.25 probability seen in independent trials.
Experiments on radioactive decay reveal that the likelihood of a nucleus decaying over a specific time period is independent of the behavior of other nuclei and unaffected by the duration it has remained stable This independence is logical, as nuclei are isolated within their atoms, preventing any physical influence among them Additionally, the uniformity of atoms suggests that even those that have not decayed for an extended period do not possess unique characteristics; no detectable differences have been found among atoms Consequently, an atom's experiences throughout its lifetime do not alter its fundamental properties.
The law of independent probabilities states that to find the likelihood of two independent events A and B occurring together, we multiply their probabilities In contrast, when considering the probability of either event A or event B happening, particularly if they are mutually exclusive, we simply add their probabilities together: P(A) + P(B) For example, in bowling, if a player has distinct outcomes, the chances of achieving one or the other can be calculated using this method.
30% chance of getting a strike (knocking down all ten pins) and a
20% chance of knocking down nine of them The bowler’s chance of knocking down either nine pins or ten pins is therefore 50%.
It does not make sense to add probabilities of things that are not mutually exclusive, i.e., that could both happen Say I have a
90% chance of eating lunch on any given day, and a 90% chance of eating dinner The probability that I will eat either lunch or dinner is not 180%.
Section 2.2 Calculating Randomness 47 b / Normalization: the proba- bility of picking land plus the probability of picking water adds up to 1.
When randomly selecting a point on a globe, there is a 70% likelihood of choosing a location in an ocean and a 30% chance of selecting a land area Since water and land are mutually exclusive outcomes, their probabilities sum to 100% This principle applies universally; when all possible outcomes are categorized into mutually exclusive events, their probabilities must total 1, or 100% This fundamental characteristic of probabilities is referred to as normalization.
One effective way to manage randomness is through averaging outcomes Casinos understand that over time, your wins will closely align with the total number of games played multiplied by the winning probability For instance, if you play a game with a winning probability of 0.001 and invest $2500 for 10,000 plays in a week, you can expect to win approximately 10 times, resulting in earnings of about $1000 Consequently, the casino stands to profit around $1500 from your gameplay This illustrates the fundamental principle of calculating averages in gambling scenarios.
When performing N identical and statistically independent trials with a success probability of P for each trial, the expected number of successful outcomes is N multiplied by P As the number of trials increases, the relative error in this estimation decreases, leading to a more accurate prediction of success.
The statement that the rule for calculating averages gets more and more accurate for larger and largerN (known popularly as the
The law of averages serves as a principle that links classical and quantum physics, illustrating that the power output of a nuclear power plant is predictable due to the vast number of atoms involved This means that while atomic behavior may appear random, such randomness averages out over large quantities of atoms, leading to a deterministic view of physics prior to the development of methods for individual atomic study.
We can achieve great precision with averages in quantum physics c / Why are dice random? self-check A
Which of the following things must be independent, which could be in- dependent, and which definitely are not independent?
(1) the probability of successfully making two free-throws in a row in basketball
(2) the probability that it will rain in London tomorrow and the probability that it will rain on the same day in a certain city in a distant galaxy
(3) your probability of dying today and of dying tomorrow
Newtonian physics serves as an accurate framework for understanding the motion of dice, as it is fundamentally deterministic However, despite this predictability, the outcome of rolling dice is perceived as random due to the complex interactions and variables involved in the process, which make it impossible to predict the exact result in practice.
B Why isn’t it valid to define randomness by saying that randomness is when all the outcomes are equally likely?
The sequence of digits 121212121212121212 appears nonrandom, while 41592653589793 seems random; however, the latter is actually the decimal representation of pi, starting from the third digit A notable story involves the Indian mathematician Ramanujan, who was a self-taught prodigy When a friend visited him and commented that the cab number 1729 seemed dull, Ramanujan countered that it was, in fact, quite interesting as it is the smallest number expressible as the sum of two cubes in two distinct ways.
The Argentine author Jorge Luis Borges wrote a short story called “The