time dilation experiment

Atomic Clocks Experiment Reveals Time Dilation At The Smallest Scale Ever

In his theory of general relativity , Einstein predicted something called time dilation : the notion that two clocks under two different gravitational pulls will always tick at different speeds.

The effect has been observed in many experiments since, but now scientists have recorded it at the smallest scale seen so far.

The result was achieved using ultra-precise atomic clocks just a millimeter (0.04 inches) apart – about the width of a sharp pencil tip. Collecting 90 hours of data gave the team a reading that was 50 times more precise than any previous similar measurement.

And of course the smaller and more precise the scale, the more we rely on quantum mechanics to explain what's going on. The researchers are hoping that their new readings open up a way to learning more about how the curvature of spacetime – what we experience as gravity – affects the characteristics of particles according to quantum physics.

"The most important and exciting result is that we can potentially connect quantum physics with gravity, for example, probing complex physics when particles are distributed at different locations in the curved space-time," says physicist Jun Ye from the University of Colorado Boulder.

In this experiment, the researchers used what's known as an optical lattice , a web of laser light used to trap atoms in place so they can be observed. It's a technique used for the latest generation of atomic clocks, offering more precision in timekeeping through the laser light waves, and these lattices can be used for quantum simulations too.

Here, the two atomic clock readings were taken from the same cloud of atoms, in a highly controlled energy state. In fact, the atoms ticked between two energy levels in perfect synchronization for 37 seconds, a record in terms of quantum coherence (that is, keeping quantum states stable) – and that stability is essential for these measurements.

That enabled the scientists to take their readings at two separate points, measuring the redshift across the cloud of about 100,000 ultracold strontium atoms. The redshift shows the change in the frequency of the atoms' radiation along the electromagnetic spectrum – or in other words, how quickly the atomic clock is ticking.

While the difference in redshift across this tiny distance was just 0.0000000000000000001 or so, that's in line with predictions made by general relativity. Those differences can make a difference when you get out to the scale of the entire Universe, or even when you're dealing with systems that need to be ultra-accurate, such as GPS navigation.

"This is a completely new ballgame, a new regime where quantum mechanics in curved space-time can be explored," says Ye .

"If we could measure the redshift 10 times even better than this, we will be able to see the atoms' whole matter waves across the curvature of space-time. Being able to measure the time difference on such a minute scale could enable us to discover, for example, that gravity disrupts quantum coherence, which could be at the bottom of why our macroscale world is classical."

Part of what makes this time dilation research so exciting is that it points the way towards atomic clocks that are even more precise in the future, giving scientists a blueprint that can be refined to take measurements on smaller and smaller scales.

Atomic clocks have come a long way in the last few decades, and there's plenty more to come.

The research has been published in Nature .

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time dilation

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• University of Central Florida Pressbooks - Time Dilation
• CORE - Derivation of Gravitational time dilation from principle of equivalence and special relativity
• Nature - Quantum clocks observe classical and quantum time dilation
• University of Iowa Pressbooks - Simultaneity And Time Dilation
• Khan Academy - Time dilation
• Live Science - What is time dilation?
• Physics LibreTexts - Time Dilation
• IOPscience - Time dilation in quantum systems and decoherence
• The University of Mississippi High Energy Physics Group - Time Dilation

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time dilation , in the theory of special relativity , the “slowing down” of a clock as determined by an observer who is in relative motion with respect to that clock. In special relativity, an observer in inertial (i.e., nonaccelerating) motion has a well-defined means of determining which events occur simultaneously with a given event. A second inertial observer, who is in relative motion with respect to the first, however, will disagree with the first observer regarding which events are simultaneous with that given event. (Neither observer is wrong in this determination; rather, their disagreement merely reflects the fact that simultaneity is an observer-dependent notion in special relativity.) A notion of simultaneity is required in order to make a comparison of the rates of clocks carried by the two observers. If the first observer’s notion of simultaneity is used, it is found that the second observer’s clock runs slower than the first observer’s by a factor of Square root of √ (1 − v 2 / c 2 ) , where v is the relative velocity of the observers and c equals 299,792 km (186,282 miles) per second—i.e., the speed of light . Similarly, using the second observer’s notion of simultaneity, it is found that the first observer’s clock runs slower by the same factor. Thus, each inertial observer determines that all clocks in motion relative to that observer run slower than that observer’s own clock.

A closely related phenomenon predicted by special relativity is the so-called twin paradox . Suppose one of two twins carrying a clock departs on a rocket ship from the other twin, an inertial observer, at a certain time, and they rejoin at a later time. In accordance with the time-dilation effect, the elapsed time on the clock of the twin on the rocket ship will be smaller than that of the inertial observer twin—i.e., the non-inertial twin will have aged less than the inertial observer twin when they rejoin.

The time-dilation effect predicted by special relativity has been accurately confirmed by observations of the increased lifetime of unstable elementary particles traveling at nearly the speed of light. The clock paradox effect also has been substantiated by experiments comparing the elapsed time of an atomic clock on Earth with that of an atomic clock flown in an airplane . The latter experiments, furthermore, have confirmed a gravitational contribution to time dilation, as predicted by the theory of general relativity .

The Time Dilation Experiment: How Physicists Prove Its Real

As a team of physicists, we are fascinated by the concept of time dilation. It is a fundamental aspect of Einstein's Theory of Relativity that describes how time can appear to pass differently for two observers in different frames of reference. This theory has been proven experimentally time and time again, and today we want to take you through some of the most compelling experiments that have been conducted to demonstrate this phenomenon.

The first thing we need to understand is what time dilation actually means. In simple terms, it refers to the fact that time appears to move slower for an observer who is moving relative to another observer who is stationary. This may sound counterintuitive, but it has been demonstrated repeatedly through carefully designed experiments. These experiments not only help us better understand the nature of our universe but also have practical implications in fields such as GPS technology and space travel. So let's dive into the exciting world of physics and explore some fascinating examples of how physicists prove that time dilation is real!

Understanding Time Dilation Theory

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Understanding the mind-bending theory of time dilation is essential for grasping the intricacies of Einstein's theory of relativity. In simple terms, time dilation can be defined as the difference in elapsed time between two events that occur at different distances from a gravitational mass or relative to each other's motion. This means that time passes slower for an object in motion or near a massive object than it does for an observer who is stationary and far away.

To understand this concept better, let's take an example. Imagine two synchronized clocks placed at different altitudes - one on top of Mount Everest and another at sea level. According to the theory of general relativity, because gravity is weaker at higher altitudes, the clock on Mount Everest would tick faster than the one at sea level. This phenomenon can be explained mathematically using equations such as Lorentz transformations and special relativity formulas.

With this understanding of time dilation, we can now delve into the concept of time dilation experiment without missing any crucial details.

](/blog/time-travel-theories/time-dilation/time-dilation-experiment-physicists-prove-real)As we delve deeper into the concept of measuring time in different ways, a mind-bending realization starts to take shape. The theory of relativity suggests that time is not constant and can be influenced by various factors, such as gravity and motion. To prove this theory, physicists have conducted numerous experiments over the years using advanced measurement techniques and observational evidence.

To further illustrate the concept of time dilation, here are some key points to consider:

• According to the theory of relativity, time passes more slowly in strong gravitational fields or at high velocities.
• This means that if two individuals were traveling at different speeds or in different gravitational fields, they would experience time differently.
• The first experimental evidence for time dilation came from the famous Hafele-Keating experiment in 1971, which involved atomic clocks being flown around the world on commercial airliners.

With these ideas in mind, let us explore how physicists were able to conduct their first time dilation experiment.

You will delve into the first demonstration of time's non-constant nature through an experiment using advanced measurement techniques and observational evidence. The first time dilation experiment was conducted by two physicists, Joseph Hafele and Richard Keating, in 1971. They flew atomic clocks on separate commercial airplanes that traveled around the world in opposite directions. This experimental setup allowed them to compare the elapsed time of one clock with respect to another.

The data collection process involved comparing the readings of the clocks after they returned from their journeys. The results showed that the clock traveling westward experienced a slower passage of time than the stationary clock on Earth, whereas the clock flying eastward experienced a faster passage of time than its counterpart on Earth. This finding provided strong evidence for Einstein's theory of relativity and proved that time dilation is not just a theoretical concept but a real phenomenon that occurs in our universe.

This groundbreaking experiment paved the way for further research into understanding how gravity affects space-time and led to more recent time dilation experiments exploring new frontiers such as black holes and neutron stars.

In recent years, there have been several groundbreaking experiments that further prove the existence of time dilation. One such experiment involved atomic clocks, which are incredibly precise timekeeping devices. By measuring the differences in time between two identical atomic clocks (one stationary and one in motion), scientists were able to observe time dilation effects predicted by Einstein's theory of relativity.

Another experiment involved observing gravitational time dilation, which occurs when an object is located near a massive body causing it to experience slower time than an observer farther away from the massive body. Scientists observed this effect by using extremely sensitive atomic clocks placed at different heights above sea level.

The results and analysis of these experiments provide even more evidence for the reality of time dilation and its importance in our understanding of physics.

You'll feel the ticking of an atomic clock in your bones as you imagine the precision and accuracy required for this experiment. Atomic clocks are the standard for measuring time with extreme accuracy, relying on the natural vibrations of cesium atoms to keep incredibly precise time. The recent atomic clock experiment conducted by physicists tested whether or not time dilation occurs at different altitudes above Earth's surface.

The test involved comparing two identical atomic clocks: one kept on the ground and another taken up to a high altitude via airplane. The results confirmed that time dilation does indeed occur, with the higher altitude clock running slightly faster than its grounded counterpart due to gravitational differences. This level of atomic clock accuracy is essential for measuring even the smallest differences in time dilation, providing crucial data for theories like Einstein's theory of relativity.

Now, let's move on to the next step where we explore how physicists conduct experiments that prove gravitational time dilation is real.

Get ready to feel the thrill of discovery as we delve into the fascinating world of gravitational time differences and how they can be measured with incredible precision. The gravitational time dilation experiment involves measuring the difference in time between two clocks placed at different altitudes in a gravitational field. As Einstein's theory of general relativity predicted, time moves slower closer to a massive object due to the curvature of space-time caused by gravity.

Experimental evidence for this effect was first observed in 1962 when atomic clocks on board airplanes flew around the Earth and were found to be out of sync with identical clocks on the ground. More recent experiments have used highly precise atomic clocks flown on airplanes or launched into space satellites to measure these effects even more accurately. These experiments have also been able to detect other factors that can affect time dilation, such as changes in velocity and gravitational waves. With this technology, physicists are able to confirm that general relativity is indeed an accurate description of our universe.

As we move onto discussing results and analysis, it's important to note that these experiments have not only provided evidence for Einstein's theory but also opened up new avenues for research into fundamental physics, including investigations into dark matter and quantum gravity.

Now we can finally see the fascinating and groundbreaking results that confirm Einstein's theory of general relativity. The gravitational time dilation experiment has provided evidence that time slows down in stronger gravitational fields, which is consistent with the predictions made by the theory. By using precision measurement techniques to compare atomic clocks at different altitudes, scientists have demonstrated that time passes more slowly closer to massive objects.

The results obtained from this experiment are statistically significant and provide strong support for Einstein's theory. They indicate that gravity affects not only space but also time, which is a fundamental concept in physics. These findings have important implications for our understanding of the universe and its behavior. As we move on to discussing the implications of time dilation, we must keep in mind how crucial these experimental results are for advancing our knowledge of physics.

So now that we understand the basics of time dilation and how it has been experimentally proven, let's look at some of its implications. First, there are practical applications for space travel: as objects near the speed of light experience less time than those at rest, astronauts on long space missions could age slower than their counterparts on Earth. Secondly, time dilation has theoretical implications for our understanding of the nature of time itself and its relationship to space. Finally, continued research and development in this area could lead to new technologies and a deeper understanding of fundamental physics.

You'll be fascinated to learn that space travel could become more efficient and faster with the use of time dilation, as demonstrated by the fictional spacecraft in the movie Interstellar. The concept behind this is simple: if astronauts travel at a speed close to the speed of light, their time will slow down relative to those on Earth. This means that they can effectively age slower than their counterparts back home, allowing them to spend more time exploring and less time aging.

This has huge implications for astronaut travel, as it means that we can potentially send humans on long-duration missions without worrying about the effects of prolonged exposure to zero gravity. Furthermore, it also opens up possibilities for interstellar travel and even time travel (in theory). Of course, there are still many technical challenges that need to be overcome before we can realize these dreams, but it's an exciting prospect nonetheless. With all of this in mind, let's delve deeper into the theoretical implications of time dilation.

We can hardly contain our excitement as we explore the mind-boggling theoretical implications of time slowing down at high speeds. Philosophical considerations arise when we ponder how this phenomenon challenges our understanding of the nature of time itself. Our traditional view of time as an absolute and constant entity is shattered by the reality that it can warp and distort depending on relative motion.

The practical implications are equally fascinating. Time dilation has been observed in experiments involving atomic clocks, which have shown that even fractions of a second can make a significant difference over long distances or high velocities. This has important implications for GPS systems, where precise timing is critical for accurate location tracking. As we continue to unravel the mysteries of time dilation, future applications in fields such as space travel and telecommunications may become possible. But first, more research and development is needed to fully harness this incredible phenomenon.

You're about to discover the exciting possibilities that lie ahead in the field of researching and developing new technologies that can harness the incredible effects of time distortion at high speeds. With the confirmation of time dilation through experiments, scientists are now exploring ways to apply this phenomenon in innovative timekeeping devices and space travel. One potential application is using atomic clocks on spacecraft to accurately measure time in space, where the effects of gravity and velocity can distort time.

Technological advancements in quantum mechanics and nanotechnology are also paving the way for more precise measurements of time dilation. Researchers are experimenting with using quantum entanglement to create ultra-precise clocks that could be used for navigation or even detecting gravitational waves. As we continue to uncover more about this fascinating aspect of physics, it's clear that there are countless possibilities for future research and development in this field.

When discussing time dilation theory, it's impossible not to mention Einstein's contributions to the field of physics. His theory of relativity revolutionized our understanding of space and time, showing that they are intertwined and not absolute. Time perception is a crucial aspect of this theory, as it suggests that time can appear differently depending on one's frame of reference. This idea has been tested and proven in various experiments, including the famous Hafele-Keating experiment where atomic clocks were flown around the world to measure differences in elapsed time due to changes in velocity and gravity. Overall, Einstein's work on relativity paved the way for further exploration into the nature of time and how it relates to our physical universe.

When it comes to performing time dilation experiments, there are certainly limitations and potential drawbacks to consider. One major limitation is the accuracy of the experiment itself. In order to measure time dilation accurately, physicists must use incredibly precise instruments and methods. Even small errors in measurement could lead to inaccurate results, which could have serious implications for our understanding of the universe. Another potential drawback is that time dilation experiments can be incredibly complex and difficult to carry out. They require a great deal of planning, resources, and expertise, which may not always be available. Despite these challenges, however, time dilation experiments remain an important tool for physicists seeking to better understand the nature of time and space.

When it comes to practical implications of time dilation, physicists have developed experimental methods that help account for its effects. For instance, GPS systems rely on precise timing to determine a user's location. However, the satellites that send signals to GPS devices are in motion relative to the Earth and therefore experience time dilation. To ensure accurate timing, scientists must adjust the clocks on the satellites based on calculations of their velocity and altitude. By doing so, they can correct for the effects of time dilation and provide users with reliable location data. Overall, while time dilation can pose challenges in certain applications, physicists have found ways to mitigate its impact through careful experimentation and analysis.

Everyday examples of time dilation can be observed in our daily lives. One example is the aging process, where time appears to pass more quickly for those who are moving at higher speeds relative to a stationary observer. Experimental methods have also been used to prove the existence of time dilation, such as high-speed particle accelerators and spacecraft traveling at high velocities. These experiments have shown that time dilation is not just a theoretical concept, but a real phenomenon that occurs in extreme conditions as well as everyday situations.

Future implications of time dilation theory are vast and exciting. Technological advancements in the field will allow for more precise measurements, leading to a deeper understanding of the universe's fundamental workings. To put this into perspective, consider that the world's most accurate atomic clock loses only one second every 15 billion years due to time dilation effects. This level of precision is necessary for research in areas such as space exploration, satellite communication, and GPS technology. As we continue to push the limits of our understanding of time and space, time dilation theory will undoubtedly play a crucial role in shaping our future discoveries and innovations.

So, there you have it – time dilation is not just a theory, but a proven fact. Through various experiments conducted over the years, physicists have demonstrated that time really does slow down when an object moves at high speeds or experiences intense gravitational forces.

But what does this mean for us? Well, it has implications for everything from our GPS systems (which rely on precise timing) to our understanding of the universe itself. It's mind-boggling to think about how much we've learned through these experiments and how much more we still have yet to discover. The possibilities are endless and truly exciting.

In conclusion, time dilation is one of those concepts that can seem too abstract and outlandish to be believed at first glance. But thanks to the hard work and ingenuity of countless scientists over the years, we now know that it's real – a verified phenomenon that shapes our world in ways we're only beginning to understand. It's proof that sometimes even the wildest theories can turn out to be true – a testament to human curiosity and perseverance if ever there was one.

September 23, 2010

How Time Flies: Ultraprecise Clock Rates Vary with Tiny Differences in Speed and Elevation

Newly developed optical clocks are so precise that they register the passage of time differently at elevations of just a few dozen centimeters or velocities of a few meters per second

By John Matson

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If you have ever found yourself cursing a noisy upstairs neighbor, take solace in the fact that he or she is aging faster than you are. Albert Einstein's theory of general relativity predicts that clocks at different gravitational potentials will tick at different rates—a clock at higher elevation will tick faster than will a clock closer to Earth's center. In other words, time passes more quickly in your neighbor's upstairs apartment than it does in your apartment. To complicate matters, the theory of special relativity, which preceded general relativity by a decade, predicts a similar effect for clocks in motion— a stationary clock will tick faster than a moving clock . This is the source of the famous twin paradox: Following a round-trip journey on a spaceship traveling at some exceptionally high velocity, a traveler would return to Earth to find that her twin sibling is now older than she is, because time has passed more slowly on the moving ship than on Earth. Both of these so-called time dilation effects have been verified in a number of experiments throughout the decades, which have traditionally depended on large scales of distance or velocity. In one landmark 1971 test Joseph Hafele of Washington University in Saint Louis and Richard Keating of the U.S. Naval Observatory flew cesium atomic clocks around the world on commercial jet flights, then compared the clocks with reference clocks on the ground to find that they had diverged, as predicted by relativity . But even at the speed and altitude of jet aircraft, the effects of relativistic time dilation are tiny—in the Hafele–Keating experiment the atomic clocks differed after their journeys by just tens to hundreds of nanoseconds. Thanks to improved timekeeping, similar demonstrations can now take place at more mundane scales in the laboratory. In a series of experiments described in the September 24 issue of Science , researchers at the National Institute of Standards and Technology (NIST) in Boulder, Colo., registered differences in the passage of time between two high-precision optical atomic clocks when one was elevated by just a third of a meter or when one was set in motion at speeds of less than 10 meters per second. Again, the effects are minuscule: It would take the elevated clock hundreds of millions of years to log one more second than its counterpart, and a clock moving a few meters per second would need to run about as long to lag one second behind its stationary counterpart. But the development of optical clocks based on aluminum ions, which can keep time to within one second in roughly 3.7 billion years, allows researchers to expose those diminutive relativistic effects. "People usually think of it as negligible, but for us it is not," says lead study author James Chin-wen Chou , a postdoctoral research associate at NIST. "We can definitely see it." The NIST group's optical clocks use lasers to probe the quantum state of aluminum ions held in radio-frequency traps. When the laser's frequency is just right, it resonates with a transition between quantum states in the aluminum ion whose frequency is constant in time. By constantly tuning the laser to drive that aluminum transition, an interaction that only occurs in a tiny window near 1.121 petahertz (1.121 quadrillion cycles per second), the laser's frequency can be stabilized to an exquisitely sensitive degree, allowing it to act as the clock's pendulum. "If we anchor the frequency of the oscillator—in our case, laser light—to the unchanging, stable optical transition in aluminum, the laser oscillation can serve as the tick of the clock," Chou explains. To put the sensitivity of the optical clocks in perspective, Chou notes that the two timekeepers in the study differed after a height change of a mere step on a staircase—never mind the entire floor separating you from your noisy neighbor—or with just a few meters per second of motion. "If you push your daughter on a swing, it's about that speed," he says. In the past, such relativistic experiments have involved either massive scales of distance or velocity, or else oscillations so fast that their ticks cannot be reliably counted for timing purposes, says Holger Müller , an atomic physicist at the University of California, Berkeley. "It's an enormous achievement that you can build optical clocks so good that you can now see relativity in the lab," he says. Müller has used atom interferometry to make precision measurements of relativistic effects, measurements that rely not on counting individual oscillations but on tracking the interference between two waves. (The frequencies of such waves, which oscillate tens of billions of times faster than the petahertz laser in an aluminum clock, are simply too high to monitor and count.) It is a process akin to striking two tuning forks to listen to the pulsations of their interference, without actually measuring how many times each fork vibrates. In that sense atom interferometers are pendulums without clockwork, so although they can make physical measurements with great precision, they cannot be used to keep time. "The new work operates on familiar scales of distance and velocity, with clocks that can be used for universal timing applications," Müller says. "They see the effects of general and special relativity, and that makes relativity something you can kind of see and touch."

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https://www.nist.gov/news-events/news/2010/09/nist-pair-aluminum-atomic-clocks-reveal-einsteins-relativity-personal-scale

NIST Pair of Aluminum Atomic Clocks Reveal Einstein's Relativity at a Personal Scale

Clocks tick a little faster if you're a foot higher, NIST experiments have shown, confirming predictions from Einstein's general theory of relativity.

BOULDER, Colo. – Scientists have known for decades that time passes faster at higher elevations—a curious aspect of Einstein's theories of relativity that previously has been measured by comparing clocks on the Earth's surface and a high-flying rocket.

Now, physicists at the National Institute of Standards and Technology (NIST) have measured this effect at a more down-to-earth scale of 33 centimeters, or about 1 foot, demonstrating, for instance, that you age faster when you stand a couple of steps higher on a staircase.

Described in the Sept. 24 issue of Science , the difference is much too small for humans to perceive directly—adding up to approximately 90 billionths of a second over a 79-year lifetime—but may provide practical applications in geophysics and other fields.

Similarly, the NIST researchers observed another aspect of relativity—that time passes more slowly when you move faster—at speeds comparable to a car travelling about 20 miles per hour, a more comprehensible scale than previous measurements made using jet aircraft.

NIST scientists performed the new "time dilation" experiments by comparing operations of a pair of the world's best experimental atomic clocks. The nearly identical clocks are each based on the "ticking" of a single aluminum ion (electrically charged atom) as it vibrates between two energy levels over a million billion times per second. One clock keeps time to within 1 second in about 3.7 billion years (see NIST's Second 'Quantum Logic Clock' Based on Aluminum Ion is Now World's Most Precise Clock ) and the other is close behind in performance. The two clocks are located in different laboratories at NIST and connected by a 75-meter-long optical fiber.

NIST's aluminum clocks—also called "quantum logic clocks" because they borrow logical decision-making techniques from experimental quantum computing—are precise and stable enough to reveal slight differences that could not be seen until now. The clocks operate by shining laser light on the ions at optical frequencies, which are higher than the microwave frequencies used in today's standard atomic clocks based on the cesium atom. Optical clocks could someday lead to time standards 100 times more accurate than today's standard clocks.

The aluminum clocks can detect small relativity-based effects because of their extreme precision and high "Q factor"—a quantity that reflects how reliably the ion absorbs and retains optical energy in changing from one energy level to another—says NIST postdoctoral researcher James Chin-Wen Chou, first author of the paper.

"We have observed the highest Q factor in atomic physics," Chou says. "You can think about it as how long a tuning fork would vibrate before it loses the energy stored in the resonating structure. We have the ion oscillating in sync with the laser frequency for about 400 thousand billion cycles."

The NIST experiments focused on two scenarios predicted by Einstein's theories of relativity. First, when two clocks are subjected to unequal gravitational forces due to their different elevations above the surface of the Earth, the higher clock—experiencing a smaller gravitational force—runs faster. Second, when an observer is moving, a stationary clock's tick appears to last longer, so the clock appears to run slow. Scientists refer to this as the "twin paradox," in which a twin sibling who travels on a fast-moving rocket ship would return home younger than the other twin. The crucial factor is the acceleration (speeding up and slowing down) of the travelling twin in making the round-trip journey.

NIST scientists observed these effects by making specific changes in one of the two aluminum clocks and measuring the resulting differences in the two ions' relative ticking rates, or frequencies.

In one set of experiments, scientists raised one of the clocks by jacking up the laser table to a height one-third of a meter (about a foot) above the second clock. Sure enough, the higher clock ran at a slightly faster rate than the lower clock, exactly as predicted.

The second set of experiments examined the effects of altering the physical motion of the ion in one clock. (The ions are almost completely motionless during normal clock operations.) NIST scientists tweaked the one ion so that it gyrated back and forth at speeds equivalent to several meters per second. That clock ticked at a slightly slower rate than the second clock, as predicted by relativity. The moving ion acts like the traveling twin in the twin paradox.

Such comparisons of super-precise clocks eventually may be useful in geodesy, the science of measuring the Earth and its gravitational field, with applications in geophysics and hydrology, and possibly in space-based tests of fundamental physics theories, suggests physicist Till Rosenband, leader of NIST's aluminum ion clock team.

NIST scientists hope to improve the precision of the aluminum clocks even further, as much as 10-fold, through changes in ion trap geometry and better control of ion motion and environmental interference. The aim is to measure differences in timekeeping well enough to measure heights to an accuracy of 1 centimeter, a performance level suitable for making geodetic measurements. The paper suggests that optical clocks could be linked to form a network of "inland tidal gauges" to measure the distance from the earth's surface to the geoid (the surface of the earth's gravity field that matches the global mean sea level). Such a network could be updated far more frequently than current techniques.

The research was supported in part by the Office of Naval Research.

C.W. Chou, D.B. Hume, T. Rosenband and D.J. Wineland. Optical Clocks and Relativity. Science . Sept. 24, 2010.

5.3 Time Dilation

Learning objectives.

By the end of this section, you will be able to:

• Explain how time intervals can be measured differently in different reference frames.
• Describe how to distinguish a proper time interval from a dilated time interval.
• Describe the significance of the muon experiment.
• Explain why the twin paradox is not a contradiction.
• Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return ( Figure 5.4 ). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in Figure 5.4 . Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

Time Dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in Figure 5.4 to each other, then express each distance in terms of the time of travel (respectively either Δ t Δ t or Δ τ Δ τ ) of the pulse in the corresponding reference frame. The resulting equation can then be solved for Δ t Δ t in terms of Δ τ . Δ τ .

The lengths D and L in Figure 5.4 are the sides of a right triangle with hypotenuse s . From the Pythagorean theorem,

The lengths 2 s and 2 L are, respectively, the distances that the pulse of light and the spacecraft travel in time Δ t Δ t in the earthbound observer’s frame. The length D is the distance that the light pulse travels in time Δ τ Δ τ in the astronaut’s frame. This gives us three equations:

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

Then we rearrange to obtain

Finally, solving for Δ t Δ t in terms of Δ τ Δ τ gives us

This is equivalent to

where γ γ is the relativistic factor (often called the Lorentz factor ) given by

and v and c are the speeds of the moving observer and light, respectively.

Note the asymmetry between the two measurements. Only one of them is a measurement of the time interval between two events—the emission and arrival of the light pulse—at the same position. It is a measurement of the time interval in the rest frame of a single clock. The measurement in the earthbound frame involves comparing the time interval between two events that occur at different locations. The time interval between events that occur at a single location has a separate name to distinguish it from the time measured by the earthbound observer, and we use the separate symbol Δ τ Δ τ to refer to it throughout this chapter.

Proper Time

The proper time interval Δ τ Δ τ between two events is the time interval measured by an observer for whom both events occur at the same location.

The equation relating Δ t Δ t and Δ τ Δ τ is truly remarkable. First, as stated earlier, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. A proper time interval Δ τ Δ τ for an observer who, like the astronaut, is moving with the apparatus, is smaller than the time interval for other observers. It is the smallest possible measured time between two events. The earthbound observer sees time intervals within the moving system as dilated (i.e., lengthened) relative to how the observer moving relative to Earth sees them within the moving system. Alternatively, according to the earthbound observer, less time passes between events within the moving frame. Note that the shortest elapsed time between events is in the inertial frame in which the observer sees the events (e.g., the emission and arrival of the light signal) occur at the same point.

This time effect is real and is not caused by inaccurate clocks or improper measurements. Time-interval measurements of the same event differ for observers in relative motion. The dilation of time is an intrinsic property of time itself. All clocks moving relative to an observer, including biological clocks, such as a person’s heartbeat, or aging, are observed to run more slowly compared with a clock that is stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light ( v < < c ) , ( v < < c ) , then v 2 / c 2 v 2 / c 2 is extremely small, and the elapsed times Δ t Δ t and Δ τ Δ τ are nearly equal. At low velocities, physics based on modern relativity approaches classical physics—everyday experiences involve very small relativistic effects. However, for speeds near the speed of light, v 2 / c 2 v 2 / c 2 is close to one, so 1 − v 2 / c 2 1 − v 2 / c 2 is very small and Δ t Δ t becomes significantly larger than Δ τ . Δ τ .

Half-Life of a Muon

There is considerable experimental evidence that the equation Δ t = γ Δ τ Δ t = γ Δ τ is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons . The half-life (amount of time for half of a material to decay) of a muon is 1.52 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval Δ τ . Δ τ . This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an earthbound observer ( Δ t Δ t ) varies with velocity exactly as predicted by the equation Δ t = γ Δ τ . Δ t = γ Δ τ . The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.

Before we present the first example of solving a problem in relativity, we state a strategy you can use as a guideline for these calculations.

Problem-Solving Strategy

• Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity v .
• Identify exactly what needs to be determined in the problem (identify the unknowns).
• Make certain you understand the conceptual aspects of the problem before making any calculations (express the answer as an equation). Decide, for example, which observer sees time dilated or length contracted before working with the equations or using them to carry out the calculation. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
• Determine the primary type of calculation to be done to find the unknowns identified above (do the calculation). You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.

Note that you should not round off during the calculation . As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem solution, but do not use a rounded number in a subsequent calculation. Also, check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, which has few everyday examples to provide experience with what is reasonable. But you can look for velocities greater than c or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).

Example 5.1

Time dilation in a high-speed vehicle.

• Identify the knowns: Δ τ = 1 s; v = 5830 m/s. Δ τ = 1 s; v = 5830 m/s.
• Identify the unknown: Δ t . Δ t .
• Express the answer as an equation: Δ t = γ Δ τ = Δ τ 1 − v 2 c 2 . Δ t = γ Δ τ = Δ τ 1 − v 2 c 2 .
• Do the calculation. Use the expression for γ γ to determine Δ t Δ t from Δ τ Δ τ : Δ t = 1 s 1 − ( 5830 m/s 3.00 × 10 8 m/s ) 2 = 1.000000000189 s = 1 s + 1.89 × 10 −10 s. Δ t = 1 s 1 − ( 5830 m/s 3.00 × 10 8 m/s ) 2 = 1.000000000189 s = 1 s + 1.89 × 10 −10 s.

Significance

Example 5.2, what speeds are relativistic.

• Identify the known: Δ τ Δ t = 1 1.01 . Δ τ Δ t = 1 1.01 .
• Identify the unknown: v/c .
• Express the answer as an equation: Δ t = γ Δ τ = 1 1 − v 2 / c 2 Δ τ Δ τ Δ t = 1 − v 2 / c 2 ( Δ τ Δ t ) 2 = 1 − v 2 c 2 v c = 1 − ( Δ τ / Δ t ) 2 . Δ t = γ Δ τ = 1 1 − v 2 / c 2 Δ τ Δ τ Δ t = 1 − v 2 / c 2 ( Δ τ Δ t ) 2 = 1 − v 2 c 2 v c = 1 − ( Δ τ / Δ t ) 2 .
• Do the calculation: v c = 1 − ( 1 / 1.01 ) 2 = 0.14 . v c = 1 − ( 1 / 1.01 ) 2 = 0.14 .

Example 5.3

Calculating δ t δ t for a relativistic event.

As we will discuss later, in the muon’s reference frame, it travels a shorter distance than measured in Earth’s reference frame.

• Identify the knowns: v = 0.950 c , Δ τ = 2.20 μ s. v = 0.950 c , Δ τ = 2.20 μ s.
• Express the answer as an equation. Use: Δ t = γ Δ τ Δ t = γ Δ τ with γ = 1 1 − v 2 c 2 . γ = 1 1 − v 2 c 2 .
• Do the calculation. Use the expression for γ γ to determine Δ t Δ t from Δ τ Δ τ Δ t = γ Δ τ = 1 1 − v 2 c 2 Δ τ = 2.20 μ s 1 − ( 0.950 ) 2 = 7.05 μ s. Δ t = γ Δ τ = 1 1 − v 2 c 2 Δ τ = 2.20 μ s 1 − ( 0.950 ) 2 = 7.05 μ s. Remember to keep extra significant figures until the final answer.

Example 5.4

Relativistic television, strategy for (a).

• Identify the knowns: v = 6.00 × 10 7 m/s; d = 0.200 m . v = 6.00 × 10 7 m/s; d = 0.200 m .
• Identify the unknown: the time of travel Δ t . Δ t .
• Express the answer as an equation: Δ t = d v . Δ t = d v .
• Do the calculation: t = 0.200 m 6.00 × 10 7 m/s = 3.33 × 10 −9 s. t = 0.200 m 6.00 × 10 7 m/s = 3.33 × 10 −9 s.

Strategy for (b)

• Identify the knowns (from part a): Δ t = 3.33 × 10 −9 s; v = 6.00 × 10 7 m/s; d = 0.200 m. Δ t = 3.33 × 10 −9 s; v = 6.00 × 10 7 m/s; d = 0.200 m.
• Identify the unknown: τ . τ .
• Express the answer as an equation: Δ t = γ Δ τ = Δ τ 1 − v 2 / c 2 Δ τ = Δ t 1 − v 2 / c 2 . Δ t = γ Δ τ = Δ τ 1 − v 2 / c 2 Δ τ = Δ t 1 − v 2 / c 2 .
• Do the calculation: Δ τ = ( 3.33 × 10 −9 s ) 1 − ( 6.00 × 10 7 m/s 3.00 × 10 8 m/s ) 2 = 3.26 × 10 −9 s. Δ τ = ( 3.33 × 10 −9 s ) 1 − ( 6.00 × 10 7 m/s 3.00 × 10 8 m/s ) 2 = 3.26 × 10 −9 s.

Check Your Understanding 5.2

What is γ γ if v = 0.650 c ? v = 0.650 c ?

The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than the astronaut’s earthbound twin. This is often known as the twin paradox . Imagine the astronaut moving at such a velocity that γ = 30.0 , γ = 30.0 , as in Figure 5.7 . A trip that takes 2.00 years in her frame would take 60.0 years in the earthbound twin’s frame. Suppose the astronaut travels 1.00 year to another star system, briefly explores the area, and then travels 1.00 year back. An astronaut who was 40 years old at the start of the trip would be would be 42 when the spaceship returns. Everything on Earth, however, would have aged 60.0 years. The earthbound twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut in Figure 5.7 . Because motion is relative, the spaceship would seem to be stationary and Earth would appear to move. (This is the sensation you have when flying in a jet.) Looking out the window of the spaceship, the astronaut would see time slow down on Earth by a factor of γ = 30.0 . γ = 30.0 . Seen from the spaceship, the earthbound sibling will have aged only 2/30, or 0.07, of a year, whereas the astronaut would have aged 2.00 years.

The paradox here is that the two twins cannot both be correct. As with all paradoxes, conflicting conclusions come from a false premise. In fact, the astronaut’s motion is significantly different from that of the earthbound twin. The astronaut accelerates to a high velocity and then accelerates opposite to the motion to view the star system. To return to Earth, she again accelerates and decelerates. The spacecraft is not in a single inertial frame to which the time dilation formula can be directly applied. That is, the astronaut twin changes inertial references. The earthbound twin does not experience these accelerations and remains in the same inertial frame. Thus, the situation is not symmetric, and it is incorrect to claim that the astronaut observes the same effects as her twin. The lack of symmetry between the twins will be still more evident when we analyze the journey later in this chapter in terms of the path the astronaut follows through four-dimensional space-time.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, because gravity and accelerations were involved as well as relative motion.

Check Your Understanding 5.3

a. A particle travels at 1.90 × 10 8 m/s 1.90 × 10 8 m/s and lives 2.10 × 10 −8 s 2.10 × 10 −8 s when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

b. Spacecraft A and B pass in opposite directions at a relative speed of 4.00 × 10 7 m/s . 4.00 × 10 7 m/s . An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?

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Time Dilation and Length Contraction

Hsc physics syllabus.

investigate the evidence, from Einstein’s thought experiments and subsequent experimental validation, for time dilation `t=t_o/sqrt((1-v^2/c^2))` and length contraction `l=l_osqrt((1-v^2/c^2))`, and analyse quantitatively situations in which these are observed, for example:

• observations of cosmic-origin muons at the Earth’s surface
• atomic clocks (Hafele–Keating experiment)
• evidence from particle accelerators
• evidence from cosmological studies

Special Relativity – Time Dilation & Length Contraction

Experimental evidence for time dilation & length contraction.

Time dilation and length contraction are not postulates but the implication of the constant nature of light's speed, as proposed by Einstein. 

Simultaneity

• The concept of simultaneity – occurrence of simultaneous events is not absolute. This means two simultaneous events may be so for one observer, they may not occur at the same time for another.
• A simple thought experiment can be used to clarify this understanding

Figure 1: Thought experiment demonstrating the effect of special relativity on simultaneity. Credit to Lumen Learning .

Observer A is sitting on a train carriage moving at a substantially fast velocity v . Two sources of light are located at the front and back end of the carriage respectively, equidistant from observer A. When they are turned on simultaneously, observer A receives both stimuli at the same time as the distance between the source and observer remains the same.

Observer B, however, does not observe the sources of light simultaneously. This is because during the time light travels from their sources to observer B, the carriage would have displaced by a certain distance (to the right). This displacement causes the distance between sources of light to observer B to become different. 

By Einstein’s second postulate of special relativity (speed of light is constant), the light emitted from the rear end of the carriage will take a shorter time to reach observer B. Light emitted from the front end of the carriage will take a longer time to reach observer B. 

What is Time Dilation?

• The non-absolute nature of simultaneity gives rise to time dilation (one of the three relativistic consequences of the constancy of light)
• The understanding of time dilation can be consolidated and quantitatively analysed using another thought experiment.

Figure 2: Thought experiment demonstrating the effect of special relativity on time. Time becomes dilated (longer) when an object is travelling at relativistic speeds as measured by a stationary observer. Credit to   Lumen Learning .

Suppose an astronaut wants to measure the time taken for a beam of light to travel back and forth the width ( D ) of the spaceship, by reflecting off a mirror. The time taken ( t 0 ) would be the total distance divided by the velocity of light:

`t_o=(2D)/c` 

However, to an observer outside the spacecraft, the distance travelled by light is longer. To calculate this distance, we need to use Pythagoras’ theorem to obtain:

`s=sqrt(D^2+((vt)/2)^2)` 

Since the distance travelled observed by the person outside the spacecraft is 2s , the time taken for light to travel to and from the mirror in the cabin is: 

By substituting  s  with the expression we derived previously:

Since the time ( t 0 ) taken by light observed by the astronaut inside the spacecraft is:

`t_0=(2D)/c`

Therefore by substituting:

`t^2=t_o^2+(v^2*t^2)/c^2`

`t^2-(v^2*t^2)/c^2=t_o^2`

`t^2(1-v^2/c^2)=t_o^2`

Make t 2 the subject of equation:

`t^2=t_o^2(1-v^2/c^2)`

Finally, square root both sides of equation:

`t=t_o/sqrt(1-v^2/c^2)` 

• where t is the time observed by an observer with relative motion to the event being observed. E.g. person outside the spacecraft.
• t 0 is the proper time observed by an observer at rest relative to the event being observed. E.g. astronaut moving at the same velocity as the spacecraft.

Length Contraction

Distance depends on the observer’s relative motion. Since distance is the product of time and speed, shorter time entails a shorter distance covered. 

The velocity of a particular object relative to an observer at rest is the proper length divided by the dilated time. Proper length l 0 is the distance between two points measured by an observer who is at rest relative to both of the points.

As such, for an observer who is at rest relative to the moving object (moving at the same speed), velocity is defined as:

`v=l/t_o`  

In this case, the relative velocity in both reference frames are equal: 

`l_o/t=l/t_o`

Make  l the subject of equation:

 `t_o/t=sqrt((1-v^2/c^2))`

By substitution:

`l=l_osqrt((1-v^2/c^2))`

• where l  is the length measured by an observer moving at the same speed as the object travelling the observed length/distance.
• l o is the length measured by an observed at rest relative to the start and end points of the observed length/distance

An object travelling a relativistic velocity experiences length contraction in the dimension of its movement. Its length does not contract in all dimensions.

Limitation of Special Relativity

Relativistic effects due to special relativity have two limitations:

• These effects are negligible when a frame of reference is not moving at a relativistic speed.
• Relativistic effects can occur in non-inertial frames of reference but in these scenarios, they are not only attributed to special relativity. Effects due to general relativity must be considered in non-inertial frames of reference. As a result, only relativistic effects in inertial frames of reference are entirely due to special relativity.

Evidence for Time Dilation & Length Contraction

Experiments involving muons.

Muons are cosmic ray particles formed in Earth’s atmosphere. When formed, they travel at velocities near the speed of light towards Earth's surface. However, due to their extremely short half life, majority of muons would decay before reaching the surface. 

In an experiment conducted at Mount Washington, quantities of muons were measured at the summit and bottom of the mountain. These numbers were then compared.

A greater proportion of muons was detected at sea level compared to what was originally predicted by considering the muons' average velocity and half-life at rest (1.5  µ s) 

Evidence for Time Dilation

This observation can be accounted for using time dilation. When muons are moving near the speed of light (0.98 c ), their half-life is increased when measured by a stationary observer on Earth. This dilated half-life as observed by an Earth-bound observer allows muons to reach the surface of Earth before decaying.

Evidence for Length Contraction

This observation can also be accounted for using length contraction. In the  muons' frame of reference, the distance between the top of Mount Washington and sea level becomes shorter due to length contraction.

Therefore, in the same amount of lifetime before decaying, more muons can reach Earth's surface.

Hafele-Keating Experiment

In 1971, Joseph Hafele and Richard Keating deonstrated time dilation using  caesium-beam atomic clocks.

Twelve clocks were used in total. Four clocks were flown on a plane in an eastward direction, four were flown in a westward direction, and the last four remained on Earth. After being flown twice around Earth, the times on the three groups of clocks were compared.

The atomic clocks, flown eastward, moved slower (as observed from an Earth-bound observer) and consequently 'lost time'. In other words, a shorter time elapsed on these clocks compared to those on Earth.

In contrast, the atomic clocks flown westward, moved faster and gained time. In other words, a longer time elapsed on these clocks compared to those on Earth.

This experiment is not the best to discuss as evidence for special relativity as the effects of general relativity also affects the experimental data. However, after the effect of general relativity are accounted for, the observed differences in time were consistent with predictions made using time dilation equations. 

Evidence of Special Relativity from Particle Accelerators  

In 2014, an experiment was conducted to demonstrate time dilation using lithium ions travelling at 0.338 c in a particle accelerator. 

The time interval between excitation of electrons in lithium ions and their return to ground state was measured when lithium ions are travelling at 0.338 c and at rest. 

Physicists found that the interval was longer for moving lithium ions compared to those at rest, as measured by a stationary observer in the laboratory. This difference in time was consistent with time dilation. 

Previous section:  Evidence for Einstein's Postulates

Next section:  Relativistic Momentum and Energy-Mass Equivalence

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Light Clock - Time Dilation

Understanding why moving clocks run slow....

Einstein's Special Relativity states that the laws of physics are the same in all inertial frames, specifically the same on a steadily moving train as they are on the ground. But the laws predict that the speed of light, called c , has a precise value, close to three hundred thousand kilometers per second. This means two different observers, looking at the same moving blob of light, will say that's how fast it's moving relative to them , even though they are moving relative to each other!

A major consequence of this apparent paradox is that time flows at different rates in the two frames : a moving clock is observed to run slow. Here's a primitive clock: a blob of light bouncing back and forth between two fixed mirrors. If the clock is moving relative to us, the light must go further on a zigzag path—but its speed must be the same, so it takes longer, and the clock runs slow.

It's all fully explained in this lecture !

The reality of this effect is well illustrated by the GPS, which depends on very accurate clocks in moving satellites. The system yields seriously incorrect results if the relativistic clock slowing is ignored. There is in addition a general relativistic effect of comparable magnitude, discussed here .

A thought experiment on time dilation

Imagine a person in a train with a torch (Figure 1). They shine the beam of the torch across the carriage and time how long it takes to return to them. Very simply it is just the distance the light travels (twice the width of the carriage (d)) divided by the speed of light (c). Someone on the embankment by the train will also agree with the measurement of the time that the light beam takes to get back to the person with the torch after reflecting from the mirror. They will both say that the time (t) is 2d/c. Now consider what happens as the train moves at a constant speed along the track. The person in the train still considers that the light has gone from the torch, straight across the carriage and returned to them. It has still travelled a distance of 2d and if the speed of light is c the time (t) it has taken is 2d/c.

However to the person on the embankment this is not the case. They will observe the light beam moving a distance given by the equation: Distance travelled by light according to an observer on the bank = 2[d 2 +s 2 ] 1/2 because the train has moved along a distance s while the light beam crosses the train and returns to the observer. Now in classical physics, pre relativity, we would now say that since the light beam has moved further in the same time it must be moving faster, in other words we have to "add" the speed of the train to the speed of the light. But the theory of relativity does not allow us to do this. It says that the speed of light is constant. So we must alter something else. The "something else" is the time, we have to assume that the light has had longer to travel the greater distance � in other words more time has passed for the observer on the bank than for the observer in the carriage. This is called time dilation . We will call the time for the "stationary" observer on the embankment to. We can prove just how much longer by the following piece of algebra. If the train moves at a speed v we have: Time taken (t o ) = 2s/v = 2 [d 2 +s 2 ] 1/2 /c but t = 2d/c Therefore: 4[d 2 + s 2 ]/c 2 = t o 2 and so: 4[c 2 t 2 /4 + v 2 t o 2 /4] = c 2 t o 2 Giving:   c 2 t 2 + v 2 t o 2 = c 2 t o 2 and so finally:

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How does the Kennedy-Thorndike experiment test for time-dilation?

From the Wikipedia page (as of 3/6/2023),

The Kennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the Michelson–Morley experimental procedure, testing special relativity.[1] The modification is to make one arm of the classical Michelson–Morley (MM) apparatus shorter than the other one. While the Michelson–Morley experiment showed that the speed of light is independent of the orientation of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of the velocity of the apparatus in different inertial frames. It also served as a test to indirectly verify time dilation – while the negative result of the Michelson–Morley experiment can be explained by length contraction alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why no phase shifts will be detected while the Earth moves around the Sun.

The passage says that the original MM experiment had equal length arms (with respect to the rest frame) and the modification that comes with the KT experiment is that one of the arms is shortened. The passage claims that the KT experiment is able to test for time dilation as well as length contraction whereas the MM experiment only tests for length contraction. I understand how MM tests for length contraction (from the point of view of a relatively moving inertial observer), but how does KT test for time dilation? I would be interested in understanding this in more detail.

• special-relativity
• experimental-physics
• time-dilation

The way the KT experiment tests for time-dilation is similar to how the light-clock thought experiment can reveal time-dilation. They're not exactly the same, but I would argue the principles are very similar.

The reasoning here requires us to first accept Lorentz contraction. The MM experiment alone does not prove Lorentz contraction, because there are a number of possible theories that can account for the results of MM (dragged aether, ballistic/emission light theory), but a combination of late 19th century and early 20th century experiments do manage to narrow down on the fact that objects are Lorentz contracted when moving relative to some fixed inertial reference frame. With this in mind, let us take Lorentz contraction for granted.

To simplify things, I will also make the assumption that the lengths perpendicular to velocity are unaffected. This was apparently a hidden assumption of Kennedy and Thorndike when they claimed they could derive the full Lorentz transformation. See this passage in Wikipedia . (In fact, it is worth pointing out that Voigt transformations are an example that fits the MM and KT experiments but the time-dilation factor is different than the one we derive in this post.)

First, let us consider a simplified warm-up thought experiment. There will be issues concerning conventionality and relativity of simultaneity, so in order to avoid them and sweep them under the rug, I will fix a reference frame $F$ as the so-called stationary inertial reference frame, and I will analyze everything with respect to $F$ . Moreover, I will assume the speed of light is constant and isotropic in $F$ .

Let us consider a variant of the simplified MM/KT setup where we have an interferometer with two arms of lengths $L_{1}$ and $L_{2}$ that are $90^{\circ}$ apart, and the apparatus is moving in the direction of arm $\#1$ at speed $v$ according to $F$ . For simplicity, let us start by analyzing the pure kinematics of the situation. We can analyze this with light-rays or point-particles; we'll choose to consider light-rays.

Let $C_{I}$ (interferometer clock) be a clock placed right at the beam-splitter that moves with the interferometer, and let $C_{S}$ (stationary clock) be a clock that is stationary with respect to $F$ .

Suppose a light-ray is shot. When it meets the beams-splitter it splits into two rays going through the two arms. Assuming Lorentz contraction occurs with respect to $F$ , the arm lengths change: $L_{1}\rightarrow L_{1}/\gamma(v)$ and $L_{2}\rightarrow L_{2}$ . By doing some basic kinematics math, we find that the light-ray in arm $\#1$ takes $t_{1}(v) = 2L_{1}\gamma(v)/c$ time to go from the beam-splitter and back, and light-ray going in arm $\#2$ takes $t_{2}(v) = 2L_{2}\gamma(v)/c$ according to $C_{S}$ . Thus $C_{S}$ measures a time difference of $$ \Delta t_{S}(v) = t_{1}(v) - t_{2}(v) = \frac{2\gamma(v)}{c}(L_{1}-L_{2}). $$ Now what does clock $C_{I}$ measure? The results of the KT experiment indicate (this needs an explanation which is provided below) that the time difference measured by $C_{I}$ is \begin{align}\tag{$*$} \Delta t_{I}(v) = \frac{2}{c}(L_{1}-L_{2}). \end{align} If this is true, then it follows that from the point of view of frame $F$ clock $C_{I}$ runs slower, and $$ \Delta t_{S}(v) = \gamma(v)\Delta t_{I}(v). $$ This is precisely the statement of time-dilation.

Ok, but Kennedy and Thorndike didn't have access to atomic clocks in 1932, so how could you do this without atomic clocks? They may not have had access to atomic clocks, but just like Michelson and Morelay, they could take advantage of light interference to tease out timing differences.

We will relate light interference and clock timings to each other in the following way. Let $C_{I}$ be a clock placed right at the beam-splitter that moves with the interferometer, let $C_{I}'$ be a clock placed right at the light source that moves with the interferometer, and let $C_{S}$ be a clock that is stationary with respect to $F$ . Suppose the setup is moving at speed $v$ in direction of arm $\#1$ wrt $F$ .

Let $C_{I}'$ measure the time period $\Delta T_{I}(v)$ of each wave-cycle of the light leaving the light source. Let $C_{I}$ measure the difference $\Delta t_{I}(v)$ between the two times it takes for light to make a round-trip in each of the arms. We also let $\Delta T_{S}(v)$ and $\Delta t_{S}(v)$ be the respective times as measured by clock $C_{S}$ .

We make the assumption that clock $C_{I}'$ and the light source, if they are slowed or sped up at all wrt $F$ , are slowed or sped up by the same factor wrt $F$ . Thus, \begin{align}\tag{1} \Delta T_{I}(v) = \Delta T_{I}(0) \end{align} for all $v$ . Further, we make the assumption that clocks $C_{I}$ and $C_{I}'$ are slowed or sped up by the same factor wrt $F$ , which implies that both are slower or faster than $C_{S}$ by the same factor. Consequently, we have \begin{align}\tag{2} \frac{\Delta T_{I}(v)}{\Delta T_{S}(v)} = \frac{\Delta t_{I}(v)}{\Delta t_{S}(v)} \end{align} for all $v$ . Also, we point out that at $v=0$ , clocks $C_{I}$ , $C_{I}'$ , and $C_{S}$ all tick at the same rate, so \begin{align}\tag{3} \Delta T_{S}(0) = \Delta T_{I}(0). \end{align} By the same reasoning as in Part 1, $$ \Delta t_{S}(v) = \frac{2\gamma(v)}{c}(L_{1}-L_{2}) $$ for all $v$ . In particular, \begin{align}\tag{4} \Delta t_{S}(v) = \gamma(v)\Delta t_{S}(0) \end{align} for all $v$ .

As the light splits at the beam-splitter, makes its round-trips, and returns to the beam-splitter, the resulting phase difference is $$ \Delta\phi(v) = \frac{\Delta t_{S}(v)}{T_{S}(v)}. $$ The result of the KT experiment is that $\Delta\phi(v)$ does not depend on $v$ . In particular, we have $\Delta\phi(v) = \Delta\phi(0)$ , so then \begin{align}\tag{5} \frac{\Delta t_{S}(v)}{\Delta T_{S}(v)} = \frac{\Delta t_{S}(0)}{\Delta T_{S}(0)} \end{align}

Now we will put all the equations $(1)$ - $(5)$ together. By $(5)$ and $(4)$ , $$ \gamma(v) = \frac{\Delta T_{S}(v)}{\Delta T_{S}(0)}. $$ By $(3)$ , this is $$ \gamma(v) = \frac{\Delta T_{S}(v)}{\Delta T_{I}(0)}. $$ By $(2)$ , this is $$ \gamma(v) = \frac{\Delta T_{I}(v)\cdot \Delta t_{S}(v)/\Delta t_{I}(v)}{\Delta T_{I}(0)} = \frac{\Delta T_{I}(v)}{\Delta T_{I}(0)}\cdot\frac{\Delta t_{S}(v)}{\Delta t_{I}(v)}. $$ By $(1)$ , this is $$ \gamma(v) = \frac{\Delta t_{S}(v)}{\Delta t_{I}(v)}, $$ and we arrive at the equation $$ \Delta t_{S}(v) = \gamma(v)\Delta t_{I}(v), $$ which is the statement of time-dilation.

Some other notes:

• The result of Part 2 can be used to deduce equation $(*)$ in Part 1. This is the way the KT experiment implies $(*)$ .
• The real experiment took place not only at different speeds, but also at different orientations.
• As I said in my post, the real experiments didn't involve precision timing with atomic clocks. So then what exactly was time-dilated? The answer is the light source. If you trace the logic carefully, you will find that the time-dilation only pertains to the light source. I invoked clocks explicitly only for the sake of exposition, but they are of course not needed here.

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How a failed experiment led to Einstein’s first big revolution

• Even though the speed of light is fast, at nearly 300,000 km/s, Earth moves fairly quickly around the Sun as well, with an average speed of 30 km/s in orbit.
• The Michelson-Morley experiment sought to leverage this fact to measure how light appeared to shift when you moved with (or against) the Earth’s motion, instead of perpendicular to its motion.
• Yet it saw no effect: giving a null result for the experiment. This was one of the most important measurements in physics history, culminating in Einstein’s original, special theory of relativity.

Imagine being alive in the late 1800s, and thinking about one of the most important physical phenomena in the Universe: light. A number of things that we take for granted today were already known about it. We knew that light:

• moved at the speed of light, around 300,000 km/s,
• exhibited wave-like behaviors such as interference and diffraction,
• and was electromagnetic in nature, with oscillating in-phase electric and magnetic fields.

We did make an underlying assumption about light, however, that wasn’t necessarily true: that, like all known waves, it required a medium to travel through. Just like water waves required the water, seismic waves required the Earth, and sound waves required the air to travel through, light was assumed to have a medium as well, known as the luminiferous aether .

Since light was known to propagate through a vacuum — such as the vacuum of space that separated the Earth from the Sun — it never occurred to most that light didn’t need a medium to propagate through; it was assumed that such a medium must exist, and simply had properties that were much more difficult to detect than the medium transporting other phenomena. In the 1880s, however, Albert Abraham Michelson , better known as A.A. Michelson, finally devised a method to detect and measure the effects of this aether itself. The experiment’s failure led to arguably the biggest revolution in the history of science.

We had already known, since the time of Galileo, how velocities worked for normal, low-speed, conventional objects. If you were a fish swimming through the waters of a river, for example, it would take you a different amount of time to:

• swim upstream and downstream by a certain distance, returning to your starting point,
• compared to swimming that same distance across the stream and back again, returning to your starting point,

because of the effects of the current of the river on you. The faster the current, the longer not only the time would be for each of these two scenarios, but the greater the time difference.

Similarly, if the river didn’t flow at all, but were rather composed of completely still water, those two scenarios would lead to not only a faster time than if the river possessed a strong current, but those times — swimming upstream and downstream versus swimming across the stream and back again — would then be identical. As long as the fish swims at a specific speed, and the water that the fish swims in is stationary, it doesn’t matter which direction the fish swims in at all. It will traverse the same amount of distance in a given amount of time.

As a matter of fact, the only “pathological” scenario, where the fish would fail to complete its journey, would be if the current in the river were faster than the fish’s maximum swim speed.

Michelson decided to use a very similar method to test out the behavior of light as it traveled through its alleged medium: the luminiferous aether. We don’t have a “river” to act as the medium that light travels through, as light simply travels through space. However, we do have something like a raft atop that river: planet Earth, which not only rotates about its axis, but which revolves around the Sun. In terms of relative speed, Earth’s orbital speed dwarfs its rotational speed by far, as it speeds around the Sun at an average speed of ~30 km/s, compared to ~1.67 km/s for its maximum rotational speed, which is achieved at Earth’s equator.

Similarly, we aren’t “fish” swimming through this river, but we can test out the behavior of light as it moves through space. After all, light isn’t just a beam or a ray, but has wave-like properties as well. Light of any variety comes in a particular wavelength, with that wavelength also defining light’s energy and color. When we take light of the same wavelength from two distinct sources — such as light that’s passed through a double slit — we observe the classic characteristic behavior associated with waves: both constructive and destructive interference. Where the “crest” of one wave lines up with the “crest” of another, we get constructive interference, resulting in brighter light, while when a “crest” lines up with an “anti-crest,” we get destructive interference, resulting in no light at all.

Michelson’s big idea was to take two beams of light and race them the same way two fish sent in perpendicular directions to one another would race across a river: one going upstream and then downstream, the other going across the river and then back across in the other direction. He couldn’t, of course, set up any apparatus that was fast enough to make this measurement directly, as the speed of light was remarkably, incredibly fast, at around 300,000 km/s, and the differences in arrival time would be tiny owing to the relatively slow speed — at least, compared to the speed of light — of Earth’s orbit around the Sun of just 30 km/s, or about 0.01% the speed of light.

Instead, Michelson realized that if he made the path length of both light paths identical, then he could send one down a direction that was oriented at 90° relative to the other one. As they reached the end of the line, he could set up a mirror at the end of the apparatus, and that mirror would reflect the light back toward their original location. When those two separate beams of light arrived at their point of origin, they should arrive simultaneously, assuming they covered the same amount of distance in the same amount of time. However, when the two light paths came back together, he could recombine them to see if there was even a tiny shift, which would produce an interference pattern.

The big idea of Michelson was this. By seeing how the interference pattern shifted (or didn’t shift) when he then rotated his apparatus through a variety of angles:

• some oriented along with Earth’s motion around the Sun,
• some oriented perpendicular to Earth’s motion around the Sun,
• some oriented in an anti-aligned fashion to Earth’s motion around the Sun,
• and some at all the angles in between,

he would see whether the effect of Earth’s motion around the Sun did indeed change the amount of time it took that light to traverse those two mutually perpendicular paths.

As Michelson and many others imagined, if the apparatus were completely stationary relative to the luminiferous aether, then the light from both beams would traverse the distance traveled in the same amount of time. When you brought those two beams back together, you would observe that there was no shift in the observed interference pattern (sometimes called a fringe-shift) so long as they traveled the same distance in the same time.

However, if the apparatus were in motion relative to the aether because of Earth’s motion around the Sun, similar to how a raft floating atop a river moves because of the current beneath it, then it would take that light a longer amount of time to traverse the direction that co-moved with the aether compared to the direction that moved perpendicular to it. This, to Michelson, would be the critical test.

Michelson performed his first sets of experiments using precisely this type of apparatus back in 1881, and was immediately disappointed. He had built a large interferometer — a device that’s now known as a Michelson interferometer — with an arm length of 1.2 meters for each of the arms. His detector was sensitive down to a precision of 0.02 fringes, and with an expected shift of 0.04 fringes, he was chagrined to have not detected any such effect at all. Despite the fact that he performed the experiment multiple times throughout the day and night, even as the Earth rotated (which changed his position and his apparatus’s orientation relative to Earth’s orbit), he could not detect any such effect.

What Michelson did next was remarkable. Rather than claim a null detection, he went back to the drawing board to build a bigger, better, and more precise apparatus. With a new collaborator on board, Edward W. Morley , the two of them set up and executed many further refinements to this interferometer-type experiment, improving their techniques and the precision of their results many times. By 1887-1888, they had built a Michelson interferometer that was ten times the size of the original, and that could be rotated on a platform to any angle they desired at any time during the day. By the time they were ready to publish, their experimental precision was down to 1/40 th of the expected, theoretical fringe shift predicted by aether theories.

This time, they confidently could show there was a null result, and that the predicted shift did not occur at all. In fact, the only “shifts” seen at all were consistent with the experimental error: at 2.5% of the predicted value or less, consistent with absolutely nothing.

Did this lead to a rejection of the aether?

Not immediately. Some contended that the aether was being “dragged” by planet Earth, and that since the Earth was in motion around the Sun, the aether must be being dragged along with it, meaning that the Earth must be co-moving with the aether. However, that was swiftly shown to be wrong , as the observed phenomenon of stellar aberration and the previously-conducted Fizeau experiment ruled out the aether-drag hypothesis.

The next attempt to explain Michelson’s and Morley’s observations came from George FitzGerald and Hendrik Lorentz , who suggested that when objects move close to the speed of light, they experience lengths in the direction-of-motion to be contracted. By contracting the length that the light co-moving along with Earth’s orbit experienced, a null result would again be predicted, thereby saving the aether hypothesis if one insists on it.

For the next 18 years, the series of Michelson-Morley experiments were the key puzzling result that physicists were attempting to make sense out of. It was in 1905 that a series of brilliant papers by a young physicist named Albert Einstein finally solved the puzzle, eliminating the need for the aether entirely.

Although it’s not quite true that the Michelson-Morley experiment was the key motivator for special relativity , with Einstein himself instead crediting an earlier realization about induction by Michael Faraday , the resolution to the puzzle didn’t rely on an aether at all, but instead Einstein took a wholly new approach: by considering the possibility that the true invariant quantity in nature was actually the speed of light.

In other words, the question of whether the aether was:

• stationary with respect to the Sun,
• dragged by the Earth,
• or whether a more sophisticated “Lorentz ether drag” was at play,

could all be done away with if one instead assumed that there was no medium necessary for light to travel through; that simply having empty space would be enough. The only catch was that neither space, nor time, nor motion was absolute, but rather everyone would agree that the speed of light, to all observers, was the one true invariant.

It then became easy to show that no matter how you oriented the Michelson interferometer, and no matter how you put that interferometer in motion, there will never be a shift observed due to the orientation of the two arms. It further became clear that both the phenomenon of length contraction and the related phenomenon of time dilation could both be derived from that Einsteinian principle: that the speed of light was a constant to all observers in all reference frames. The aether suddenly became extraneous and unnecessary, and it was apparent that physics would be no worse off if the concept were discarded entirely.

Remarkably, however, the Michelson interferometer would come back in a remarkable way in the context of general relativity, as propagating gravitational waves could alternately expand and contract space itself in mutually perpendicular directions. As the path length increased in one direction (while shrinking in the perpendicular direction), there would be a periodic difference in how the fringes would appear to shift, in proportion to the frequency and amplitude of the passing gravitational waves. Today, observatories — such as LIGO, Virgo, and KAGRA — leverage the most precise Michelson interferometer setup ever to measure, directly, the effects of these gravitational waves.

The Michelson-Morley experiment remains, even today, arguably the most scientifically impactful null result in the history of science, although other null results are far more precise today. This was reflected in 1907, when one of the earliest Nobel Prizes in Physics was awarded to A.A. Michelson , for “his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid.” For 18 years, physicists strove to reconcile the nature of light with the concept of the aether in light of Michelson’s results, and it wasn’t until Einstein came along that a satisfactory answer was achieved.

Even two years after Einstein’s great achievement, it was clear that it was a high-quality experiment that drove our understanding of nature forward more than any single realization. Today, it’s arguable that we find ourselves in much the same boat, as the overwhelming cosmological evidence for dark matter and dark energy demonstrates an enormous gap between how our Universe is behaving as compared to how we’d expect it to behave from what’s known and understood about physics and the behavior of matter. Perhaps it will take a 21st century genius of even greater caliber than Einstein to solve these modern day puzzles, but it will be the results of experiments, observations, and precise, high-quality measurements that motivate the next great scientific revolution, whatever it turns out to be.