time dilation train experiment

Time dilation, length contraction and the relativity of simultaneity are among the strange conclusions of special relativity. This page uses animations to explain them in more detail. There is a little mathematics: we use Pythagoras' theorem about the sides of a right angled triangle, but nothing beyond that.

Time dilation

Prior to the experiment, Zoe parks her car (a 1962 Holden, unexpectedly capable of relativistic speeds ) next to Jasper's verandah so we can observe her clock, shown in the diagram at right. Coincidentally, the car and the verandah are the same length.

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Now the speed of light c in each of the animations is the same - that's what the principle of relativity is all about. But the distances are different: for Jasper, each tick of Zoe's clock is the time T taken for light to trace the hypotenuse of the right angled triangle. For Zoe, the light pulse covers only w, the width of the car, in each tick. So the constant speed of light means that the light beam clock ticks at different rates for Jasper and for Zoe! From his frame, Jasper observes Zoe's clock running more slowly than Zoe does. (Remember, the red counters give the number of ticks of the clock as measured by each of the observers.) Let's call T' the time that Zoe measures for one tick of the clock, so

Pythagoras meets Einstein

T ' 2 = T 2 - (v/c) 2 T 2     whence

T ' = T(1 - (v/c) 2 ) 1/2 = T/γ,    where γ = 1/(1 - (v/c) 2 ) 1/2 .

What about other clocks? The choice of this rather peculiar clock was made only because it is one that so clearly depends on a simple electromagnetic phenomena. Other clocks (quartz crystals, springs, even Zoe's biological clock) depend on complicated combinations of electromagnetic phenomena such as the forces between atoms and molecules, and on Newton's laws. If they didn't differ from Jasper's clocks by the same factor of γ, then we would conclude that the laws of mechanics and/or electromagnetism are different between the two frames, contrary to the principle of relativity. So yes, time dilation would affect biological clocks as well, and Jasper thinks that Zoe is getting older more slowly than he is.

Symmetry . But Zoe also thinks that Jasper is getting older more slowly than she is. (The situation between Jasper and Zoe is symmetrical: if you believe that a '62 Holden can travel at relativistic speeds, you might as well believe that the verandah can, too.) So Zoe observes that Jasper's clocks are running slow by γ, too.

Hold on , I hear you say, how can both clocks run slow? Surely that's impossible? Puzzling, but not impossible, because the clock rates are measured in different frames. Let's suppose that they synchronise clocks when they pass each other. What happens when they compare time later? Now remember that their relative speed is a substantial fraction of the speed of light. Consequently, when Zoe looks at Jasper's clock, the light will take a while to reach her eyes. So she will see the time that the clock read when the light that she is seeing reflected off Jasper's clock. (Think of astronomers looking at very distant galaxies: they emitted the light we see when they and the universe was much younger than it is now.) So distant observes can't just use the time that they see on the clock, they have to correct this for the time of transmission of light (or radio etc) that is used to send the local time.

This synchronisation effect is important enough to need another page on its own, because it is tempting to try to use this symmetry to create a paradox that would disprove relativity. See the twin paradox for an explanation.

We call the time measured by an observer in his/her own frame the proper time . ('Proper' here means 'belonging to' or 'property of', it doesn't mean 'correct'.) So the time measured by other observers (Jasper in this example) is γ times greater than the proper time (Zoe's time in this example). γ is always greater than or equal to one.

Is time dilation true? How big are the effects?

One question at a time. Yes, clocks do run more slowly. Planes travel about a million times more slowly than c (so γ is about 1.0000000000005), but atomic clocks are very precise and so this tiny effect effect can actually be measured. In 1971, J. Haefele and R. Keating took atomic clocks on airliners travelling both East (with the Earth rotating underneath them in the direction of their motion over the Earth) and West (these planes have the Earth's rotation speed opposite that of their terrestrial motion, so the two tend to cancel). Apart from some complications due to the gravitational field variations and their acceleration (which are dealt with by general relativity), this is like the twin paradox , and it gave results in agreement with the relativistic prediction. (See the original paper by J.C. Hafele and R. E. Keating, Science 177, 166 (1972) for details. Also see the diagrams and discussion about this experiment and its complications on the FAQ in high school physics .)

Do people age more slowly? We don't know whether people age more slowly, because even cosmonauts don't travel fast enough for the effect to be statistically observable on their life spans*. However, people's ages are determined by physical and chemical processes in our bodies. Certainly we expect that people would age more slowly at relativistic speeds. Particles certainly do. Particle accelerators generate some short lived particles (eg muons or pions) that travel within a fraction of a percent of c, and (in the laboratory frame) they survive for much longer than their lifetime when at rest in the lab frame. Muons with a half life of 1.5 microseconds are also created several tens of km above the Earth in the upper atmosphere by cosmic rays. Travelling 50 km at c would take 170 microseconds or 110 half lives, so we should expect their numbers to be reduced by a factor of 2 110 ~ 10 33 (ie effectively none) to reach the surface. In fact they are measured at sea level and at various altitudes, with rates that agree with the relativistic dilation of their half lives. Time dilation happens, however counter-intuitive it may seem at first.

How big are time dilation effects? Note the shape of the curve above: γ only starts to become large at speeds close to c. At 0.99*c, γ is 7. But in many modern devices, electrons are accelerated to higher speeds than this. In a typical electron accelerator used to treat cancers, the electrons have an energy of 20 MeV (see Module 5 ). The speed of such electrons is 0.9997*c and γ is 40.

Now of course an electron cannot go much faster than this, but it can have a lot more energy. In the Large Electron-Positron collider in Europe's nuclear research lab CERN, electrons (and positrons, or anti-electrons) were accelerated to energies of 100 GeV. For such particles, v = 0.999 999 999 95*c and γ is 200,000. Yes, time is slowed down by that factor. And the momentum is increased by that factor too: something that is rather important in the design of the collider because these electrons must be turned to go in a circle.

Nature can produce even larger particle energies. Some particles striking the Earth's upper atmosphere have energies that exceed 2*10 20  eV. If such particles are protons (with mass of about 1 GeV), their speeds would be 0.999 999 999 999 999 999 999 995 c. For them, γ is 10 11 . Now the age of the universe is about 13 billion years for us, but for such particles, the age of the universe would be about (13 billion  years/10 11 ), ie about a month. Such a particle could cross the visible universe in a matter of months (their time).

From special relativity to general relativity

Simultaneity.

In the next animation, Jasper has set up an apparatus to make simultaneous 'events'. A little while after he presses the switch, sparks jump at the two gaps, and two pulses of light travel towards him. (Note that the sparks are not simultaneous with the switch: the electric field in the circuit cannot travel faster than light.) The spark gaps are equidistant from Jasper, the light pulses arrive simultaneously, so he concludes that the two events (the two spark emissions) occurred simultaneously. As we'll see later, relative simultaneity can only be noticeable for events that are well separated in space, but close in time. For this reason, we asked Jasper to set up the spark gaps a long way apart, and so as to observe the small time effect, we've slowed everything down in this animation, compared to the previous ones, just for clarity.

Zoe also receives the two pulses of light, which she observes to come towards her at the same* speed c . Jasper has timed the pushing of the switch so that the two pulses, Zoe and Jasper all meet at the centre of the verandah at the same time*. But Zoe sees the light pulses emitted from Jasper's moving verandah, and in her frame of reference the two events (the two flashes of light) are not equidistant. For the two pulses to arrive simultaneously, Zoe deduces that the right hand pulse (emitted from the approaching spark gap) must have been emitted before the emission of the left hand pulse (the receding one), because of their relative motion.

Another non-intuitive result: events that are simultaneous to one observer need not be simulatenous to another . Indeed, the time order may be reversed: a traveller going from right to left with respect to Jasper would, by symmetry, observe the left hand pulse to be emitted first. (A question for you to puzzle on: look at when the switch near Jasper closes and work out why the sparks occur simultaneously for Jasper but not for Zoe. Answer below.)

Sometimes one encounters this objection: in this example, Jasper, sends a pulse of voltage down wires to create his two simultaneous events. What if, instead, he switched two distant switches using two very long, rigid rods. Wouldn't they be simultaneous then? The key word here is "rigid". When you push on the end of a rod of length L, the other end does not move instantaneously. It takes a time L/v, where v is either the speed of sound in the object or the speed of a shock wave in the object. The speed of sound in solids is typically a few km/s. (It is the square root of the ratio of an elastic modulus to the density.) Interatomic forces are continuous functions of separation (see Young's modulus, Hooke's law and material properties ), you cannot make an infinitely rigid rod, i.e . you cannot make one with an infinite elastic modulus and thus an infinite v. So, although Jasper might see a mechanical wave travel along each rod at the same speed, Zoe would point out that, to her, the waves have different speeds.

The limits to time order reversals

No. As we show elsewhere, if two events are separated by distance L (according to one observer) and time difference Δt (according to the same observer), then their time order can only be reversed for some other observer if L is greater than cΔt. Let's suppose that a cosmonaut dies 80 years after his birth (Δt = 80 years). For an observer to deduce that he died before he was born, his birth and death would have to be separated in space by more than cΔt, which is 80 light years. But (assuming the cosmonaut is present at both his birth and death — even the busiest people manage to attend both!) to get from the place of his birth to the place of his death, he would have to travel more than 80 light years (L greater than 80 light years) in 80 years. He would have to travel at L/Δt, which is greater than the speed of light, which is in turn impossible. (For quantitative details, see Lorentz transforms, the addition of velocities and spacetime .)

Length contraction

Both agree that the time between the two tags — the time Zoe takes to go past the verandah — is two ticks of Zoe's clock. This is 2T ' for Zoe, so the length that Zoe measures is 2vT ' . But for Jasper, two ticks of Zoe's clock takes 2T = 2T ' γ. The length that Jasper measures for the verandah is 2vT = 2vT ' γ. Jasper measures the verandah to be γ times longer than Zoe measures it.

Further, the situation is symmetrical: Jasper observes the car to be shrunk with respect to the verandah, while Zoe concludes that the verandah has shrunk with respect to the car. The proper length is always longer than a measure of the length from another frame. But can't one make a paradox from this? See the " pole in the barn " paradox.

Severe simplifications have been made in the animations shown above. Even if cars could travel at relativistic speeds, this is not how they would "look", because of aberrations associated with the finite time of flight. See References and caveats for more information.

FAQ . How is the 1961 EK Holden capable of relativistic speeds? This has never been satisfactorily explained. Zoe's car has neither speed stripes nor spoiler, so that can't be the answer. However, as the fins on this model serve no other purpose, we speculate that they may be involved.

Late news. Reader Kevin Jezorek writes: "I have been reading your article on time dilation, and I've noticed an inconsistency in a particular detail of the example. In the introduction, Zoe is claimed to be operating a 1962 Holden. The footnotes describe a 1961 Holden which is not capable of relativistic speeds. This is correct, the relativistic speed propulsion unit was not available on the Holden until 1962. I have a feeling Zoe was driving a '62 Holden the entire time."

September 22, 2014

Einstein's "Time Dilation" Prediction Verified

Experiments at a particle accelerator have confirmed the "time dilation" effect predicted by Albert Einstein's special theory of relativity

By Alexandra Witze & Nature magazine

Physicists have verified a key prediction of Albert Einstein’s special theory of relativity with unprecedented accuracy. Experiments at a particle accelerator in Germany confirm that time moves slower for a moving clock than for a stationary one.

The work is the most stringent test yet of this ‘time-dilation’ effect, which Einstein predicted. One of the consequences of this effect is that a person travelling in a high-speed rocket would age more slowly than people back on Earth.

Few scientists doubt that Einstein was right. But the mathematics describing the time-dilation effect are “fundamental to all physical theories”, says Thomas Udem, a physicist at the Max Planck Institute for Quantum Optics in Garching, Germany, who was not involved in the research. “It is of utmost importance to verify it with the best possible accuracy.”

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The paper was published on September 16 in  Physical Review Letters . It is the culmination of 15 years of work by an international group of collaborators including Nobel laureate Theodor Hänsch, director of the Max Planck optics institute.

To test the time-dilation effect, physicists need to compare two clocks — one that is stationary and one that moves. To do this, the researchers used the Experimental Storage Ring, where high-speed particles are stored and studied at the GSI Helmholtz Centre for heavy-ion research in Darmstadt, Germany.

The scientists made the moving clock by accelerating lithium ions to one-third the speed of light. Then they measured a set of transitions within the lithium as electrons hopped between various energy levels. The frequency of the transitions served as the ‘ticking’ of the clock. Transitions within lithium ions that were not moving served as the stationary clock.

The researchers measured the time-dilation effect more precisely than in any previous study, including one published in 2007 by the same research group. “It’s nearly five times better than our old result, and 50 to 100 times better than any other method used by other people to measure relativistic time dilation,” says co-author Gerald Gwinner, a physicist at the University of Manitoba in Winnipeg, Canada.

Understanding time dilation has practical implications as well, he notes. Global Positioning System (GPS) satellites are essentially clocks in orbit, and GPS software has to account for tiny time shifts when analysing navigational information. The European Space Agency plans to test time dilation in space when it launches its Atomic Clock Ensemble in Space (ACES) experiment to the International Space Station in 2016.

The speed of fast-moving ions means that accelerator experiments can test time dilation more precisely than experiments in Earth orbit, says Matthew Mewes, a physicist at California Polytechnic State University in San Luis Obispo, who is not part of the team. “It’s important to look wherever we can and push the technology whenever possible,” he says.

But the research group is dismantling its longtime collaboration, as there is no larger accelerator they can go to for more powerful tests. “It's been many hours in basements, in shielded rooms with noisy equipment, and in the end you get one number,” says Gwinner. “We’ve been exchanging a bunch of nostalgic e-mails.”

This article is reproduced with permission and was first published on September 19, 2014.

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More Relativity: The Train and The Twins

     Michael Fowler, UVa

Einstein’s Definition of Common Sense

As you can see from the lectures so far, although Einstein’s Theory of Special Relativity solves the problem posed by the Michelson-Morley experiment—the nonexistence of an ether—it is at a price.  The simple assertion that the speed of a flash of light is always c in any inertial frame leads to consequences that defy common sense.  When this was pointed out somewhat forcefully to Einstein, his response was that common sense is the layer of prejudices put down before the age of eighteen.  All our intuition about space, time and motion is based on childhood observation of a world in which no objects move at speeds comparable to that of light.  Perhaps if we had been raised in a civilization zipping around the universe in spaceships moving at relativistic speeds, Einstein’s assertions about space and time would just seem to be common sense.  The real question, from a scientific point of view, is not whether Special Relativity defies common sense, but whether it can be shown to lead to a contradiction .  If that is so, common sense wins.  Ever since the theory was published, people have been writing papers claiming it does lead to contradictions.  The previous lecture, the worked example on time dilation, shows how careful analysis of an apparent contradiction leads to the conclusion that in fact there was no contradiction after all.  In this lecture, we shall consider other apparent contradictions and think about how to resolve them.  This is the best way to build up an understanding of Relativity. 

Trapping a Train in a Tunnel

The Tunnel Doors are Closed Simultaneously

The key to understanding what is happening here is that we said the bandits closed the two doors at the ends of the tunnel at the same time .  How could they arrange to do that, since the doors are far apart?  They could use walkie-talkies, which transmit radio waves, or just flash a light down the tunnel, since it’s long and straight.  Remember, though, that the train is itself going at a speed close to that of light, so they have to be quite precise about this timing!  The simplest way to imagine them synchronizing the closings of the two doors is to assume they know the train’s timetable, and at a prearranged appropriate time, a light is flashed halfway down the tunnel, and the end doors are closed when the flash of light reaches the ends of the tunnel.  Assuming the light was positioned correctly in the middle of the tunnel, that should ensure that the two doors close simultaneously. 

Or are They?

Now consider this door-closing operation from the point of view of someone on the train.  Assume he’s in an observation car and has incredible eyesight, and there’s a little mist, so he actually sees the light flash, and the two flashes traveling down the tunnels towards the two end doors.  Of course , the train is a perfectly good inertial frame , so he sees these two flashes to be traveling in opposite directions, but both at c, relative to the train .  Meanwhile, he sees the tunnel itself to be moving rapidly relative to the train.  Let us say the train enters the mountain through the “front” door.  The observer will see the door at the other end of the tunnel, the “back” door, to be rushing towards him, and rushing to meet the flash of light.  Meanwhile, once he’s in the tunnel, the front door is receding rapidly behind him, so the flash of light making its way to that door has to travel further to catch it.  So the two flashes of light going down the tunnel in opposite directions do not reach the two doors simultaneously as seen from the train. 

The concept of simultaneity, events happening at the same time, is not invariant as we move from one inertial frame to another .  The man on the train sees the back door close first, and, if it is not quickly reopened, the front of the train will pile into it before the front door is closed behind the train. 

Does the Fitzgerald Contraction Work Sideways?

The above discussion is based on Einstein’s prediction that objects moving at relativistic speed appear shrunken in their direction of motion.  How do we know that they’re not shrunken in all three directions, i.e.  moving objects maybe keep the same shape, but just get smaller? This can be seen not to be the case through a symmetry argument, also due to Einstein.  Suppose two trains traveling at equal and opposite relativistic speeds, one north, one south, pass on parallel tracks.  Suppose two passengers of equal height, one on each train, are standing leaning slightly out of open windows so that their noses should very lightly touch as they pass each other.  Now, if N (the northbound passenger) sees S as shrunken in height, N ’s nose will brush against S ’s forehead, say, and N will feel S ’s nose brush his chin.  Afterwards, then, N will have a bruised chin (plus nose), S a bruised forehead (plus nose).  But this is a perfectly symmetric problem, so S would say N had the bruised forehead, etc.  They can both get off their trains at the next stations and get together to check out bruises.  They must certainly be symmetrical!  The only consistent symmetrical solution is given by asserting that neither sees the other to shrink in height (i.e. in the direction perpendicular to their relative motion), so that their noses touch each other.  Therefore, the Lorentz contraction only operates in the direction of motion, objects get squashed but not shrunken. 

How to Give Twins Very Different Birthdays

Perhaps the most famous of the paradoxes of special relativity, which was still being hotly debated in national journals in the fifties, is the twin paradox.  The scenario is as follows.  One of two twins—the sister—is an astronaut.  (Flouting tradition, we will take fraternal rather than identical twins, so that we can use “he” and “she” to make clear which twin we mean).  She sets off in a relativistic spaceship to alpha-centauri, four light-years away, at a speed of, say, 0.6 c .  When she gets there, she immediately turns around and comes back.  As seen by her brother on earth, her clocks ran slowly by the time dilation factor , so although the round trip took 8/0.6 years = 160 months by earth time, she has only aged by 4/5 of that, or 128 months.  So as she steps down out of the spaceship, she is 32 months younger than her twin brother. 

But wait a minute—how does this look from her point of view?  She sees the earth to be moving at 0.6 c , first away from her then towards her.  So she must see her brother’s clock on earth to be running slow!  So doesn’t she expect her brother on earth to be the younger one after this trip?

The key to this paradox is that this situation is not as symmetrical as it looks.  The two twins have quite different experiences.  The one on the spaceship is not in an inertial frame during the initial acceleration and the turnaround and braking periods. (To get an idea of the speeds involved, to get to 0.6 c at the acceleration of a falling stone would take over six months.)  Our analysis of how a clock in one inertial frame looks as viewed from another doesn’t work during times when one of the frames isn’t inertial—in other words, when one is accelerating. 

The Twins Stay in Touch

To try to see just how the difference in ages might develop, let us imagine that the twins stay in touch with each other throughout the trip.  Each twin flashes a powerful light once a month, according to their calendars and clocks, so that by counting the flashes, each one can monitor how fast the other one is aging. 

The questions we must resolve are:

If the brother, on earth, flashes a light once a month, how frequently, as measured by her clock, does the sister see his light to be flashing as she moves away from earth at speed 0.6 c ?

How frequently does she see the flashes as she is returning at 0.6 c ?

How frequently does the brother on earth see the flashes from the spaceship?

Once we have answered these questions, it will be a matter of simple bookkeeping to find how much each twin has aged. 

Figuring the Observed Time between Flashes

To figure out how frequently each twin observes the other’s flashes to be, we will use some results from the previous lecture , on time dilation.  In some ways, that was a very small scale version of the present problem.  Recall that we had two “ground” clocks only one million miles apart.  As the astronaut, conveniently moving at 0.6 c , passed the first ground clock, both that clock and her own clock read zero.  As she passed the second ground clock, her own clock read 8 seconds and the first ground clock, which she photographed at that instant, she observed to read 4 seconds. 

That is to say, after 8 seconds had elapsed on her own clock, constant observation of the first ground clock would have revealed it to have registered only 4 seconds. (This effect is compounded of time dilation and the fact that as she moves away, the light from the clock is taking longer and longer to reach her.)

Our twin problem is the same thing, at the same speed, but over a longer time - we conclude that observation of any earth clock from the receding spacecraft will reveal it to be running at half speed , so the brother’s flashes will be seen at the spacecraft to arrive every two months, by spacecraft time. 

Symmetrically, as long as the brother on earth observes his sister’s spacecraft to be moving away at 0.6c, he will see light from her flashes to be arriving at the earth every two months by earth time. 

To figure the frequency of her brother’s flashes observed as she returns towards earth, we have to go back to our previous example and find how the astronaut traveling at 0.6 c observes time to be registered by the second ground clock, the one she’s approaching. 

We know that as she passes that clock, it reads 10 seconds and her own clock reads 8 seconds.  We must figure out what she would have seen that second ground clock to read had she glanced at it through a telescope as she passed the first ground clock, at which point both her own clock and the first ground clock read zero.  But at that instant, the reading she would see on the second ground clock must be the same as would be seen by an observer on the ground, standing by the first ground clock and observing the second ground clock through a telescope.  Since the ground observer knows both ground clocks are synchronized, and the first ground clock reads zero, and the second is 6 light seconds distant, it must read -6 seconds if observed at that instant. 

Hence the astronaut will observe the second ground clock to progress from -6 seconds to +10 seconds during the period that her own clock goes from 0 to 8 seconds.  In other words, she sees the clock she is approaching at 0.6 c to be running at double speed . 

Finally, back to the twins.  During her journey back to earth, the sister will see the brother’s light flashing twice a month.  (Evidently, the time dilation effect does not fully compensate for the fact that each succeeding flash has less far to go to reach her.)

We are now ready to do the bookkeeping, first, from the sister’s point of view. 

What does she see?

Her return trip will also take 64 months, during which time she will see 128 flashes, so over the whole trip she will see 128 + 32 = 160 flashes, so she will have seen her brother to age by 160 months or 13 years 4 months. 

What does he see?

As he watches for flashes through his telescope, the stay-at-home brother will see his sister to be aging at half his own rate of aging as long as he sees her to be moving away from him, then aging at twice his rate as he sees her coming back.  At first glance, this sounds the same as what she sees—but it isn’t!  The important question to ask is when does he see her turn around?  To him, her outward journey of 4 light years’ distance at a speed of 0.6 c takes her 4/0.6 years, or 80 months.  BUT he doesn’t see her turn around until 4 years later, because of the time light takes to get back to earth from alpha-centauri! In other words, he will actually see her aging at half his rate for 80 + 48 = 128 months, during which time he will see 64 flashes. 

When he sees his sister turn around, she is already more than half way back! Remember, in his frame the whole trip takes 160 months (8 light years at 0.6 c ) so he will only see her aging at twice his rate during the last 160 - 128 = 32 months, during which period he will see all 64 flashes she sent out on her return trip. 

Therefore, by counting the flashes of light she transmitted once a month, he will conclude she has aged 128 months on the trip, which by his clock and calendar took 160 months.  So when she steps off the spacecraft 32 months younger than her twin brother, neither of them will be surprised!

The Doppler Effect

The above analysis hinges on the fact that a traveler approaching a flashing light at 0.6 c will see it flashing at double its “natural” rate—the rate observed by someone standing still with the light—and a traveler receding at 0.6 c from a flashing light will see it to flash at only half its natural rate. 

This is a particular example of the Doppler Effect , first discussed in 1842 by the German physicist Christian Doppler.  There is a Doppler Effect for sound waves too.  Sound is generated by a vibrating object sending a succession of pressure pulses through the air.  These pressure waves are analogous to the flashes of light.  If you are approaching a sound source you will encounter the pressure waves more frequently than if you stand still.  This means you will hear a higher frequency sound.  If the distance between you and the source of sound is increasing, you will hear a lower frequency.  This is why the note of a jet plane or a siren goes lower as it passes you.  The details of the Doppler Effect for sound are a little different than those for light, because the speed of sound is not the same for all observers—it’s 330 meters per second relative to the air. 

An important astronomical application of the Doppler Effect is the red shift .  The light from very distant galaxies is redder than the light from similar galaxies nearer to us.  This is because the further away a galaxy is, the faster it is moving away from us, as the Universe expands.  The light is redder because red light is low frequency light (blue is high) and we see low frequency light for the same reason that the astronaut receding from earth sees flashes less frequently.  In fact, the farthest away galaxies we can see are receding faster than the 0.6 c of our astronaut!

In the next lecture, we shall brush up on the pre-relativistic concepts of momentum, work and energy to be ready for their relativistic generalizations. 

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To understand relativity in time with time dilation experiment

The time dilation experiment involves two frames in relative motion, let one at ground and other at train with velocity V. The light clock runs faster in rest frame, as seen by an observer A at rest in train ( just beside clock ) than that observed by an observer B in ground frame which observes moving clock. But here the discussion in both conditions are just about clocks everytime in Train's frame observed by both. So, how do I conclude that the time goes faster in ground frame while slower in moving frame,  that is the man at ground ages faster than the man in train. And, I know about the reciprocity and symmetricity for Twin's paradox. So, let's take ground to be at rest and train to be moving only.          One thing that I think that I can do is that the light clock, which is at rest in A's frame, can also be kept in ground frame and I also think that the time of rest-clock in ground frame will be same as that of rest-clock in train frame as the light covers same distance D with same speed of light, c.  Then, the time can be compared for the two light-clock set-ups after looking at both : one kept on the ground and the other at train, both observed from ground by observer B. And, phenomenon of Ages of observers A and B can be explained. But here is another problem that then the equation  

 ∆t'(observer:B) =   (gamma)*∆t(proper/observer:A) 

will not even be concluded because as I mentioned earlier that this one is for the same event placed at train's frame as seen by both observers A and B. Please tell me if my reasoning is wrong or right 😬 and help to sort out this issue i.e. Explain me the relativity of Time with this time dilation experiment.

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

• 3 $\begingroup$ See What is time dilation really? $\endgroup$ –  John Rennie Commented Apr 9 at 18:00

2 Answers 2

You are right it is symmetric, So in the moving train you age the same as in the nit moving. It is just the other observer who thinks there is a difference. So if there were twins, they just age the same. Its only if one of them "returns" that you see an age difference.

• $\begingroup$ the problem is not symmetric. the age difference comes from the fact that in order to return to their initial position, one of the twins has to change the direction of their velocity. change of velocity means that they are no longer in an inertial frame, and all observed age difference happens during this acceleration. the comment on the original post links to some very well written exposition on the subject $\endgroup$ –  paulina Commented Apr 9 at 19:35
• $\begingroup$ Thats what I said in my answer, symmetry only if no return. $\endgroup$ –  trula Commented Apr 10 at 17:01

The point people often struggle to appreciate in SR is that time dilation is symmetric between two inertial reference frames. What that means is that if you are time dilated in my frame, then I am time dilated to exactly the same extent in yours.

To see this, imagine there is a light-clock with the people on the train and another light-clock with the people on the platform...

To the people on the train, the light in their clock travels straight up and down, but to the people on the platform it follows a longer diagonal path. Therefore it must take longer for the train light-clock to tick in the platform frame than it does in the train frame- ie it appears time dilated to the people on the platform.

However, to the people on the platform, the light in their clock travels straight up and down, but to the people on the train it follows a longer diagonal path. That means the platform clock takes longer to tick in the frame of the train than it does in the frame of the platform.

What I have described shows that the light-clock on the platform is time-dilated in the frame of the train, and the light-clock on the train is time-dilated in the frame of platform. The effect is entirely symmetrical.

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Physics Notes - Herong's Tutorial Notes - v3.24, by Herong Yang

Physics Notes - Herong's Tutorial Notes

∟ Time Dilation in Special Relativity

∟ Demonstration of Time Dilation - Amy on the Train

This section provides a thought experiment to demonstrate time dilation by starting with Amy synchronizing a clock with a bouncing light pulse on a moving train.

The most common way to demonstrate time dilation is to follow a thought experiment using a clock on a train moving relative the ground.

Part 1 - Amy on the Train : The first part of the thought experiment is to synchronize the moving clock and a laser light pulse bouncing perpendicular to the moving direction of the clock. This part consists of the following:

• Amy stays on a train that is moving at a speed of v relative the ground.
• Amy carries a clock which is also moving along with the train at the speed of v meters per second.
• Amy installs a laser meter on the floor of the carriage on the train and a mirror on the ceiling of the carriage. The distance between the meter and mirror is L meters.
• Amy releases laser light pulses from the meter, and waits for pulses to be reflected back from the mirror.
• Amy observes that a light pulse always takes T seconds on her clock to finish a round trip from the meter, to the mirror, and back to the meter.
• Amy uses the train as her frame of reference x.
• Assuming the distance from the carriage floor to the ceiling is L meters, Amy observes that the light pulse travels twice the distance of L in her frame: one time going up from the meter to the mirror and one time going down from the mirror to the meter.
• Amy establishes a relation between the elapsed time of T seconds on her clock with the distance L meters using the speed of light c: 2*L = c*T.
• Amy declares that her clock is synchronized with the light pulse bouncing between the laser meter and the mirror. Each time her clock moves T seconds, the light pulse completes a single round trip of bouncing.

Based on Amy's observations in her frame, we can derive a formula to express time T in terms of distance L:

Notice that T also represents the elapsed time observed by Amy between two events: event A when the light pulse is leaving the meter, and event B when the same light pulse is returning to the meter reflected back from the mirror.

With Amy's clock synchronized with the bouncing light pulse, we can say that the bouncing light pulse itself is also a moving clock. It moves T seconds per click.

Now we are ready to measure the speed of time on the moving frame from a stationary frame by observing the elapsed time of between two clicks of the bouncing light pulse clock. Continue with the second part of the thought experiment in the next section.

Table of Contents

  About This Book

  Introduction of Space

  Introduction of Frame of Reference

  Introduction of Time

  Introduction of Speed

  Newton's Laws of Motion

  Introduction of Special Relativity

► Time Dilation in Special Relativity

  Time Dilation - Moving Clock Is Slower

► Demonstration of Time Dilation - Amy on the Train

  Demonstration of Time Dilation - Bob on the Ground

  Demonstration of Time Dilation - Formula

  What Is Lorentz Factor

  Reciprocity of Time Dilation

  Elapsed Time between Distant Events

  Length Contraction in Special Relativity

  The Relativity of Simultaneity

  Introduction of Spacetime

  Minkowski Spacetime and Diagrams

  Introduction of Hamiltonian

  Introduction of Lagrangian

  Introduction of Generalized Coordinates

  Phase Space and Phase Portrait

  References

  Full Version in PDF/ePUB

Demonstration of Time Dilation - Amy on the Train - Updated in 2022, by Herong Yang

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A question about Relativity of Time using Time dilation experiment

• Thread starter rahaverhma
• Start date Apr 9, 2024
• Tags Frames Observer Time dilation
• Apr 9, 2024

A PF Molecule

• Researchers discover new material for optically-controlled magnetic memory
• PhAI—an AI system that figures out the phase of x-rays that crystals have diffracted
• Achieving quantum memory in the notoriously difficult X-ray range

A PF SuperCluster

rahaverhma said: The time dilation experiment involves two frames in relative motion, let one at ground and other at train with velocity V. The light clock runs faster in rest frame, as seen by an observer A at rest in train ( just beside clock ) than that observed by an observer B in ground frame which observes moving clock. But here the discussion in both conditions are just about clocks in Train's frame observed by both. So, how do I conclude that the time goes faster in ground frame while slower in moving frame, that is the man at ground ages faster than the man in train.

rahaverhma said: One thing that I think that I can do is that the light clock, which is at rest in A's frame, can also be kept in ground frame and I also think that the time of rest-clock in ground frame will be same as that of rest-clock in train frame as the light covers same distance D with same speed of light, c. Then, the time can be compared for the two light-clock set-ups after looking at both : one kept on the ground and the other at train, both observed from ground by observer B. And, phenomenon of Ages of observers A and B can be explained. But here is another problem that then the equation ∆t'(observer:B) = (gamma)*∆t(proper/observer:A) will not even be concluded because as I mentioned earlier that this one is for the same event placed at train's frame as seen by both observers A and B. Please tell me if my reasoning is wrong or right and help to sort out this issue i.e. Explain me the relativity of Time with this time dilation experiment.

A PF Planet

rahaverhma said: , how do I conclude that the time goes faster in ground frame while slower in moving frame, that is the man at ground ages faster than the man in train.

A PF Singularity

Another thing to note is that there is also no such thing as ”a moving” inertial frame or ”a stationary” imertial frame. They are moving/stationary relative to something. Neither frame can be universally acclaimed to be ”the” stationary frame. This is a basic cornerstone of relativity and something many people have difficulty grasping.  

A PF Asteroid

rahaverhma said: One thing that I think that I can do is that the light clock, which is at rest in A's frame, can also be kept in ground frame and I also think that the time of rest-clock in ground frame will be same as that of rest-clock in train frame as the light covers same distance D with same speed of light, c.

• Apr 11, 2024

The time dilation result derived is a bit strange, no doubt, but there doesn’t seem to be anything downright incorrect about it until we look at the situation from A’s point of view. A sees B flying by at a speed v in the other direction. The ground is no more fundamental than a train, so the same reasoning applies. The time dilation factor, γ , doesn’t depend on the sign of v, so A sees the same time dilation factor that B sees. That is, A sees B’s clock running slow. But how can this be? Are we claiming that A’s clock is slower than B’s, and also that B’s clock is slower than A’s? Well ... yes and no. Remember that the above time-dilation reasoning applies only to a situation where something is motionless in the appropriate frame. In the second situation (where A sees B flying by), the statement tA = γ tB holds only when the two events (say, two ticks on B’s clock) happen at the same place in B’s frame. But for two such events, they are certainly not in the same place in A’s frame, so the tB = γ tA result in Eq. (11.9) does not hold. The conditions of being motionless in each frame never both hold for a given setup (unless v = 0, in which case γ = 1 and tA = tB). So, the answer to the question at the end of the previous paragraph is “yes” if you ask the questions in the appropriate frames, and “no” if you think the answer should be frame independent. A passage on Time Dilation from David Morin's book on Classical Mechanics. Help me understand what the 2nd para in the passage means to say.  

rahaverhma said: A passage on Time Dilation from David Morin's book on Classical Mechanics. Help me understand what the 2nd para in the passage means to say.

Ibix said: It says that A determines that B's clock is ticking slowly and B determines that A's clock is ticking slowly. And that this is not a contradiction since A derives the result based on an analysis of two events that are at the same place in B's rest frame, which are therefore not in the same place in A's rest frame. So B cannot be using the same pair of events as A, since B needs a pair of events that are in the same location in A's frame. Note that this doesn't guarantee that you don't have a contradiction, just that you don't have a contradiction yet . It will turn out that you don't have a contradiction at all once you derive the Lorentz transforms and come to understand relativity of simultaneity, but at this point all Morin is doing is explaining that A and B are following the same process, but cannot be applying it to the same events, so one shouldn't look at the symmetry of time dilation and immediately conclude it's crazy.

rahaverhma said: I understood it. One thing more though, what is meaning of an "appropriate frame" as mentioned in the passage ?

• May 22, 2024

ESponge2000 said: am I making sense?

The OP question appears to have already been answered, so this thread is now closed.  

ESponge2000 said: There can’t be smoothing of forces across a stick ?

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JSmol Viewer

A multi-dimensional reverse auction mechanism for volatile federated learning in the mobile edge computing systems.

1. Introduction

2. related work, 3. system model and problem definition, 3.1. system model, 3.2. auction framework, 3.3. reputation calculation, 3.4. problem definition, 4. mratr auction mechanism, 4.1. problem transformation, 4.2. selecting client training data based on mratr.

4.3. Payment

4.4. properties, 5. simulation results, 5.1. experiment settings, 5.2. experiment analysis, 6. conclusions, author contributions, data availability statement, conflicts of interest.

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Hong, Y.; Zheng, Z.; Wang, Z. A Multi-Dimensional Reverse Auction Mechanism for Volatile Federated Learning in the Mobile Edge Computing Systems. Electronics 2024 , 13 , 3154. https://doi.org/10.3390/electronics13163154

Hong Y, Zheng Z, Wang Z. A Multi-Dimensional Reverse Auction Mechanism for Volatile Federated Learning in the Mobile Edge Computing Systems. Electronics . 2024; 13(16):3154. https://doi.org/10.3390/electronics13163154

Hong, Yiming, Zhaohua Zheng, and Zizheng Wang. 2024. "A Multi-Dimensional Reverse Auction Mechanism for Volatile Federated Learning in the Mobile Edge Computing Systems" Electronics 13, no. 16: 3154. https://doi.org/10.3390/electronics13163154

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