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Here is a traditional derivation of time dilation:

There's a train with a lamp in the ceiling, moving at velocity v with respect to an observer. In the frame of the observer, the path taken by the light from the lamp straight down to the ground is actually diagonal because the train has moved forwards by the time the light hits the ground. Since the speed of light is constant, the time it took for the light to reach the ground must have been GREATER, because the distance traveled was the hypotenuse of a triangle whose side is the height of the lamp and whose base is the distance traveled by the train in the time it took the light to travel.

That's the essence of it, math not included because it's not relevant to my question:

This derivation works for light, yes, but it's based on the fact that the speed of light is identical in all frames, so applying the same procedure to a ball, say, would not work. In short: We calculated that light travel time has been dilated. How does this argument extend for ALL objects, not just light?

Also: I have heard of answers involving light clocks (devices which measure time based on how long it takes light to move some distance), using the following arguments:

  1. Measuring time with a light clock shows that time clearly dilates.

counter-argument: how do you know that the light clock is accurate then? Maybe other clocks would disagree, and time only dilates for light?

  1. If one uses both a light clock AND a variety of other clocks: The argument is that if you used both clock types and only the light clock went out-of-sync, you could tell that YOU were the one moving, so this violates the postulate of relativity (all inertial frames are equally valid; none are "THE" rest frame).

counter-argument: this is okay with me if the person observing a difference is in the clock frame. But if they are not, relativity seems satisfied with the condition that, if a train observer and a "stationary" observer both have both types of clocks, each person sees the other person's clocks as out-of-sync with the other person's light clocks (nobody looks at their own clocks).

I am aware of the experimental evidence that particle decays follow time dilation. I'd just like some evidence that it applies to all phenomena, rather than just the set which we have experimentally verified. Best would be a theoretical argument from Einstein's postulates.

I am an undergraduate in my senior year, who has not yet taken General Relativity, so I would appreciate it if that were kept in mind in any explanation!

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  • $\begingroup$ How would the light clock be inaccurate? You know that the light traveled a distance $d$, and you know the speed of light is $c$, so the amount of time that passed is $d/c$. Do you mean that your measuring tools are imprecise? It's a thought experiment. $\endgroup$ – dfan Dec 26 '14 at 20:13
  • $\begingroup$ The light clock wouldn't be inaccurate, the scenario I was describing was one in which it went out of sync with the other clocks. $\endgroup$ – doublefelix Dec 26 '14 at 20:15
  • $\begingroup$ “Based on the fact that the speed of light is identical in all frames” — that is one of the central premises of special relativity. Together with the dictum “time is what clocks measure”, and the fact that a light clock is a clock, as dfan points out, time time dilation follows. $\endgroup$ – xebtl Dec 26 '14 at 20:31
  • $\begingroup$ I've always had a problem with the derivation of special relativity...I was introduced to the topic right after learning about simple harmonic motion...shouldn't the derivation have some usage of sin/cos? I'm sure that the difference in equations/results would not differ greatly, but why be less accurate than what is readily available? $\endgroup$ – Dutch Brannigan Dec 26 '14 at 21:00
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    $\begingroup$ @DutchBrannigan Here is a source stating that it is exact. If you take a marker to a whiteboard and move it up and down at a steady rate and then you start moving to the right (at a steady rate), the result is a triangle wave. If you have more questions about this, it is probably best to start a new question rather than adding more comments here. $\endgroup$ – dfan Dec 28 '14 at 1:16
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The empirical answer to the question is simple: radioactive beams have longer half-lives as measured in the lab frame than the same particles have when at rest.

This was first noticed in the context of cosmic-ray muons, and later in the hadronic spray emerging from deep inelastic scattering, and these days we build particle accelerators that run radioactive isotopes up to high energy with malice aforethought.

So, long story short: we measure this stuff all the time.

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  • $\begingroup$ Hey! Aye, I'm aware of the empirical evidence, but I was looking for something which come from Einstein's postulates alone, because the claim seems to be that the two postulates are sufficient to derive everything else. $\endgroup$ – doublefelix Dec 27 '14 at 4:50
  • $\begingroup$ Ah ... because the speed of light is constant for every observer then the motion of a beam of light is a clock; that is the distance traveled by light is proportional to what is meant by the word "time". Think about the kinematics of constant velocity. $\endgroup$ – dmckee Dec 27 '14 at 6:44
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Time dilation applies to light clocks and matter (clocks) equally.

This really arises from the following law...

  1. The speed of light is an absolute speed limit for matter.

If you take a property of matter, say temperature, it is just the average kinetic energy of particles moving relative to each other. If these particles are in a frame of reference moving at close to the speed of light then they will experience time dilation. This means the particles will move slower (relative to each other) from the perspective of a stationary observer.

Why is this?

Well the particles can't move at the speed of light, BUT if you imagined they were then you would see that they can't have any movement relative to each other due to #1 above. E.g. they're already travelling at the speed of light in one direction so they can't have speed in any other direction (other than to decelerate) otherwise they would be travelling faster than the speed of light. E.g. for them time is "frozen". More realistically if you imagine they have a speed close to light speed then you can see that they can't have much speed in other directions, or again they would be travelling faster than the speed of light. So now they are moving slower relative to each other1.

So by looking at temperature, we can infer that the movement of particles relative to each other also slows down (at relativistic speeds). If we think of our body (it has a temperature) and all the movement within it (e.g. blood cells circulating the body, etc) we can see that again due to #1 above the movement of cells is limited by the speed of the frame of reference they are in. So all movement in our body will also slow down at the same rate. E.g. we will age slower.

Essentially anything comprised of matter will move/age slower at relativistic speeds for the same reason. Even within the atom there is movement and so sub-atomic processes are also effected by time dilation.

So it doesn't matter what type of clock you choose, whether it is a light clock or atomic clock or mechanical clock, they all slow down equally. And because everything slows down at the same rate you don't notice anything different, e.g. it is from the perspective of someone in a stationary frame of reference which notices time slowing down in the spaceship moving past. On the spaceship time appears to move at it's normal rate. The best way to think of this is that your thinking also slows down so its running at the same slowed down speed as everything around it.

EDIT:

From Wikipedia (and in response to the comment below):

  1. First postulate (principle of relativity)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion. OR: The laws of physics are the same in all inertial frames of reference.

  1. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. OR: The speed of light in free space has the same value c in all inertial frames of reference.

Consider a spaceship travelling at .95c and this spaceship fires a projectile forward (in the direction of travel) at .25c (as measured by someone on the spaceship) and at the same time fires a light beam in the same direction.

From #2 above both a stationary frame of reference and the frame of reference of the spaceship will measure the light beam to be travelling at c.

From #1 above this scenario is the same if performed in the spaceships frame of reference as if it were performed in a stationary frame of reference. E.g. If performed in a stationary frame of reference the projectile travels out behind the light beam. Therefore this is also the case from the spaceships frame of reference.

As such a stationary observer sees the light beam travel out from the spaceship at speed c with the projectile behind it at a slower speed (somewhere between .95c and c, e.g. always sub light speed). This will always be the case, the projectile will always be slower than the speed of light. If it wasn't (and the projectile could travel faster than the light) then #1 above couldn't be true as the projectile would overtake the light in one frame of reference but not in another (the frames of reference would not be equivalent).

So from #1 and #2 above you cannot add the velocities of a frame of reference and the projectile as you might intuitively think. Even if the frame of reference is going at .99c and fires a projectile forwards at .99c the resultant speed (as measured by a stationary observer) will still be less than c. So if the speed of matter starts off below c then it can never be accelerated above c.

The logical conclusion to this is that time dilation must occur at high speeds (e.g. for someone on the spaceship). A stationary observer notes the projectile moving forwards (relative to the spaceship) quite slowly (< 0.01c). They also note that time has slowed down for the person on the spaceship so from their perspective (if you speed up time for the person) they will see the projectile leave them at .99c.

1. It is interesting to consider that a gas travelling at relativistic speeds (from the perspective of a stationary frame of reference) is similar to a Bose-Einstein condensate.

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  • $\begingroup$ I like this answer a lot, and it helped me intuit SR a bit more, so I upvoted. But I was hoping to be able to figure this out without making the assumption of light being the universal speed limit, because SR apparently follows only from Einstein's postulates of the constancy of c and equivalence of inertial frames. Can you get the universal speed limit-ness from those? $\endgroup$ – doublefelix Dec 27 '14 at 0:30
  • $\begingroup$ @user31415926535897932384626433 The speed of light as a limit for matter has been confirmed empirically, e.g. accelerating an electron to .99c, then increasing the energy to accelerate further doesn't result in a linear increase in speed (it tends to the limit of c). I've also edited my answer to try to answer in regards to the postulates of SR. $\endgroup$ – Quantumplate Dec 27 '14 at 22:14
  • $\begingroup$ Why from wikipedia? :( $\endgroup$ – Dutch Brannigan Dec 28 '14 at 5:41
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To add to Dmckee's answer, violations of Lorentz invariance are actively sought as a major part of modern physics because such violations are consistent with certain, beyond-standard-model and beyond-GTR theories, and so far none have been found. See the Modern Searches for Lorentz Violation Wikipedia Page for a summary. Dmckee is not joking when he said that this stuff is tested all the time independently of the electromagnetic field, making Lorentz covariance one of the most experimentally verified results in physics.

If you want to look at this stuff a little differently, then you can think of Galileo's relativity, which follows from the first postulate of relativity and was essentially the assertion that no experiment can be done to detect inertial motion of a frame without looking outside that frame. He describes this postulate in his Allegory of Slaviati's Ship.

If you assume an absolute time (i.e. that the time between two events is the same for all inertial observers), then the transformation laws are uniquely defined by this postulate as well as a homogeneity of space and time postulate and are the group of Euclidean isometries and Galilean boosts, as I discuss in my answer here.

The homogeneity of space postulate implies the transformations act linearly on spacetime co-ordinates, as discussed by Joshphysic's answer to the Physics SE question "Homogeneity of space implies linearity of Lorentz transformations". Another beautiful writeup of the fact of linearity's following from homogeneity assumptions is Mark H's answer to the Physics SE question "Why do we write the lengths in the following way? Question about Lorentz transformation".

So now we have our familiar Galilean transformations, uniquely defined by homogeneity, the first relativity postulate and an assumption of absolute time (that the time between two events is the same for all inertial observers).

If, however, you relax the assumption of absolute time and derive the most general group of transformations consistent with homogeneity and the "Salviati" postulate (first relativity postulate), you get the Poincaré group, but with an unspecified $c$. That is, a whole family of transformation groups is possible, which family includes Galilean relativity (with $c=\infty$) and it now becomes an experimental question of what the value of $c$ is. You can thus think of Galileo as one of the main originators of the relativity concept, and Einstein as the guy who was bold enough to broaden it by relaxing the absolute time assumption. I say more about Einstein's generalisation of Galilean relativity in my answer to the Physics SE question "What's so special about the speed of light?"

Notice we have not yet invoked the second postulate of the constancy of the speed of light. What we notice, when we derive the Poincaré group as the generalisation of Galileo's relativity, is that if there is anything moving at a velocity given by this weird $c$ parameter, then its velocity is $c$ in all reference frames. The derivation of the Poincaré group does not show that there has to be anything moving at $c$, but if it were, then its velocity would have this unwonted invariance property. But it would be a reasonable hunch that there might be something moving at this velocity.

It then becomes an experimental question of whether we can find something with a velocity that transforms in this way. If we can find such a thing, then we know our Universe is characterised by a finite $c$. Of course we know, through the Michelson Morley experiment, that there is and this thing is light. It also follows from dynamical considerations of Lorentz boosts that anything travelling at $c$ must have zero rest mass.

So the Michelson-Morley experiment, from this standpoint shows the following:

Given spacetime homogeneity and Salviati-ship-style independence of physical laws on inertial frames (first relativity postulate), then (1) our Universe is characterised by a finite $c$ rather than the $c=\infty$ of Galileo, (2) the speed of light is the same as this $c$, to within experimental error and (3) light is mediated by a zero rest mass particle.

Einstein himself didn't proceed as above and instead used the constancy of the speed of light postulate and an assumption that Maxwell's equations retained their form to derive the Lorentz boost. He was, however, the first to clearly understand that one must forgo an assumption of absolute time to make all this work.

Hopefully you can see this approach of generalising Galileo's relativity under a relaxation of the absolute time assumption is independent of light. The first person to clearly understand that this approach could be taken seems to have been Ignatowski in 1911. Even Einstein's 1905 paper mentions the group postulates, but more as an afterthought, perhaps even after discussions with his mathematician wife Mileva Marić, as Einstein does not seem to be thinking of the group theoretical ideas as central in the paper.

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I would say you just need to look at the two fundamental postulates of special relativity (the light clock derivation assumes they both hold, obviously it's an experimental question whether they do in reality, but so far all the evidence supports them). The second postulate says that light should be measured to have the same speed of c in every inertial frame, and that's the basis for the conclusion that the light clock shows time dilation as seen in a frame where it's not at rest. The first postulate says that the laws of physics must work the same way in every inertial frame--this is sometimes called the "principle of relativity" which was already part of classical mechanics, although without the additional postulate that all inertial frames measure the same speed of light. A simple way of thinking about this principle is that if you have a windowless sealed laboratory moving inertially in flat spacetime (no gravity), with no external references to observe outside the lab, then the principle of relativity says that any experiment you do inside the lab should give the same result regardless of the lab's velocity relative to any particular choice of frame (like the Earth's rest frame). So say you set up an experiment to measure the ratio between ticks of two clocks A and B, where A is a light clock with mirrors one meter apart, and B is some other clock, like a mechanical clock or an atomic clock based on cesium oscillations or even a clock based on the half-life of some radioactive element. The first postulate then says that the experimentally observed value of this ratio should be the same regardless of the velocity of the lab relative the chosen reference frame--and if your other clock B ticks at a fixed rate relative to the light clock A regardless of the velocity of both clocks, this means if a frame in which A is moving sees A display time dilation, then B which is moving along with it must display the same time dilation.

You seem to bring up this argument yourself, but then you say:

'counter-argument: this is okay with me if the person observing a difference is in the clock frame. But if they are not, relativity seems satisfied with the condition that, if a train observer and a "stationary" observer both have both types of clocks, each person sees the other person's clocks as out-of-sync with the other person's light clocks (nobody looks at their own clocks).'

It's a basic principle of relativity (and all theories that assign position and time coordinates to events) that all frames must agree on which pairs of events coincide "locally" at the same position and time. If this wasn't the case, different frames could disagree on physical questions like whether a person was right next to an exploding bomb when their stopwatch read 100 seconds, and thus would disagree in their predictions about whether the person lived on past 100 seconds! So since the two clocks A and B can be right next to each other and at rest relative to one another, if one frame says that a given pair of ticks of A and B happened at the same position and time, then all other frames must agree.

It's common in relativity problems to idealize various objects as being pointlike (or infinitesimal in size), that's why many objects in relativity problems are assigned worldlines (where the object has a single position coordinate at each time) as opposed to world-tubes. In this case, the worldlines of two objects can coincide at a single point in spacetime, so my phrase "which pair of events coincide locally at the same position and time" would have an unambiguous meaning. I would say that it's simplest to think of the light clock derivation as dealing with this sort of idealization, where we can treat both the bottom mirror of the light clock and the other non-light clock as pointlike objects whose worldlines are described by identical coordinates.

If you want to deal with more realistic models of extended objects which always have some finite distance between them, we can use the Lorentz transformation to derive the following conclusion: if two clocks are synchronized in their own rest frame (i.e. they both show the same reading at a given time-coordinate in that frame), and they are a distance D apart in that frame, then in another frame which measures the clocks to be moving together at speed v, at any given time-coordinate in this second frame their readings will differ by $vD/c^2$. So as the relative velocity between the frames approaches c, the clocks approach being out-of-sync by a maximum of $cD/c^2 = D/c$, meaning for example that if two clocks synchronized and are 0.0000001 light-seconds apart in their rest frame (or 29.9792458 meters), then no other inertial frame will see their times differing by more than 0.0000001 seconds at any given instant. So for example if you're rounding off times to the nearest microsecond, then for clocks with 30 meters or less between them, it's safe to say that if their times match at any given moment in their own rest frame, their times will still match in any other frame. So a 30 meter separation between the light clock and the other clock would be "local enough" to derive the conclusion that if they are synchronized in their rest frame, they still appear synchronized in other frames, and thus that the second clock must dilate by the same factor (to the nearest microsecond) as the light clock.

Of course, this last argument depends on already knowing the Lorentz transformation, and if you know the Lorentz transformation it's simple to derive time dilation directly from that, so the light clock is no longer necessary. But this calculation at least shows in retrospect that the approximation of just treating the light clock and the other clock as pointlike was a reasonable one that didn't lead us astray in our conclusions about time dilation.

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  • $\begingroup$ "It's a basic principle of relativity (and all theories that assign position and time coordinates to events) that all frames must agree on which pairs of events coincide "locally" at the same position and time." Can you expand on that? How local must things be, etc? And how do we know? $\endgroup$ – doublefelix Dec 27 '14 at 7:05
  • $\begingroup$ I edited the end to add more on the notion of events coinciding "locally", let me know if it helps or if there's still some aspect that you would like elaborated. $\endgroup$ – Hypnosifl Dec 27 '14 at 19:28
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They are called Lorenz transformations because Lorenz applied them to light long before Einstein thought to harness them to matter also. They are inherent in classical electrodynamics.

The inconsistency of Newtonian mechanics with Maxwell’s equations of electromagnetism and the inability to discover Earth's motion through a luminiferous aether led to the development of special relativity, which corrects mechanics to handle situations involving motions nearing the speed of light.

Notice the "corrects mechanics". Mechanics is trains and clocks.

It was the experimental results that led to special relativity. Thought experiments with clocks and trains came much later as intellectual toys.

As dmckee says in his answer there have been innumerable verifications of the validity of special relativity in all particle interactions.

You comment

because the claim seems to be that the two postulates are sufficient to derive everything else.

plus using Lorenz transformations, and the panoply of Maxwell's equations, very mathematical .

The postulates are a general framework for interpretation of the physics results of what the mathematical models model. You could never get Lorenz transformations out of the two postulates alone.

The postulates in physics theories are not axioms, as in mathematical theories, where everything with a lot of manipulation falls out of the axioms. The postulates define the correspondence of the mathematical model to the measured quantities in experiments.

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