Relativity Discussion Forum

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Question by Chuck Davis
I do not have the expertise to dispute the muon experiment which seems to prove time dilation. How would you explain the detection of abundant muons at sea level - in line with the calculations of relativity - when, logically, the half life of a muon would only allow it to travel the 660 meters as accepted by most scientists?

Let me first point out that the impossibility of time dilation is, as indicated on my page regarding the Time Dilation and Twin Paradox, purely an issue of conceptual logical consistency i.e. a conclusion which can be reached independently of experiments, so any experimental results apparently indicating time dilation must be interpreted by other means. The most obvious explanation for the circumstance that cosmic ray muons travel further in their life time than they ought to according to Relativity, would simply be that they have in fact a velocity higher than c. The point is that usually the velocity is not being measured in these experiments but only the muon numbers at different heights (at least I am not aware of any data which would unambiguously prove that the cosmic ray muons have in fact the velocities they are supposed to have according to Relativity).
Alternatively, it is also possible that the lifetime of the muon is affected by travelling through the magnetic field of the earth (this interpretation could also be applied to the lifetime of muons observed in accelerators).

Comment by Rhys Davies
I quote directly from your "Gravitational Lensing" article:
"On the other hand, it is unreasonable to assume that immaterial and massless objects like light can be in any way subject to a gravitational interaction. It is much more likely that the propagation of electromagnetic waves is, by their very nature, only affected by electromagnetic forces."
This statement is blatantly at odds with experimental fact. I refer you to the famous Pound-Rebka experiment: "Apparent Weight of Photons" (R. V. Pound and G. A. Rebka, Jr., Phys. Rev. Lett. 4, 337-341 (1960)),
This is characteristic of a lot of the claims on your site. You also cite 'logic' often, when your statements are far from logically tight. Physics is all about questioning and searching for new, better explanations, but they have to be consistent with experiment, and most of what you claim is not.

Until a few hundred years ago it was also a 'fact' that the sun was revolving around the earth. The point is that observational phenomena are one thing and their theoretical interpretation is quite another. Nobody questions that the Pound-Rebka experiment resulted in an energy shift of the 'photon', but this could in principle also be due to other physical causes than gravity. For the case of the redshift of spectral lines in the solar gravitational field (as well as the bending of starlight by the sun) I have shown for instance that an electric field could explain this (see the page Plasma Theory of 'Gravitational Lensing' of Light on my site I am not necessarily saying that the same effect explains also the Pound-Rebka experiment (I have not looked into this possibility yet), but other explanations are always a theoretical possibility. It could for instance be that the supposed gamma-ray 'photons' were actually no photons but massive particles created in the corresponding nuclear decay and interacting resonantly with the absorber. In this case one would obviously expect the energy of these particle to be affected by gravity.

Note that in this context I did actually not claim that a gravitational effect on light can logically be ruled out, but merely that it is implausible and defies the nature of light on the one hand and gravity on the other. The point is that there are no theoretical grounds to connect the two in any way. General Relativity claims that the connection is due to space-time being curved by gravity. It is this concept of curvature of space which can be ruled out logically however, so effectively the suggested mass-light interaction is theoretically without any basis in my view.

Rhys Davies (2)
By what mechanism do you propose an electric field can affect light? Electrodynamics is linear (ie. light cannot be a source of light), so this idea would require a completely new theory. Considering the many successes of classical as well as quantum electrodynamics, and the fact that no deviation from linearity has ever been observed, this seems - to put it mildly - a little far-fetched.

I also dispute that you have in any way argued that a curved spacetime is illogical. Your "Curved Space" entry contains the following:
"However, a surface is only a mathematical abstraction within the actual (3-dimensional) space and one can in fact connect any two points on the surface of a physical object through a straight line by drilling through it." This seems to indicate a lack of appreciation of the difference between the intrinsic properties of a manifold and those associated with its embedding in a particular space. Obviously we are able only to observe intrinsic properties of the spacetime in which we reside. We can indeed follow your idea and say that "space can only be defined by the distance between two objects". Mathematically, of course, 'distance' is given by a metric. Einstein showed that the metric of space(time) is a dynamical object, and is coupled to energy density.

Reply (2)
The mechanism would not have to be non-linear. As follows from my order of magnitude estimates on my page Plasma Theory of 'Gravitational Lensing' of Light, the effect is simply so weak that it is virtually impossible to reproduce in the lab, as one would need electric field strengths comparable to the inner-atomic field to produce an observable effect over such short distances. Also, note that an effect of magnetic fields on light for instance is well known in terms of the Faraday rotation (rotation of the polarization axis of light), so one could say that the suggested redshift and bending of light by electric fields in a way complements this. I wouldn't push this comparison too far though as it is merely a plausibility argument. Basically you could say that it is a new theory as it is not implied by existing theories. The important point is that it is not ruled out by existing theories either, and to my knowledge neither by experimental or observational evidence (in fact, the redshift and bending of light would be evidence for it as suggested).

Regarding the 'curved space': yes, the distance between two points depends on the metric used, but the Euclidean metric will always provide the shortest distance between them (which is by definition a straight line).
Also, General Relativity claims that gravity is due to the curvature of space-time. This would actually contradict the locality of the interaction, because if you assume that the acceleration a(s) at location s depends on the gradient of some function f(s), then this would imply that a point mass at location s would also be affected by the location s+ds as the gradient is defined as (f(s+ds)-f(s))/ds (with ds being infinitesimally small but different from zero). This is clearly logically not acceptable.

Comment by Mark Mattice
Kudos to your catch of the algebraic error in Einstein's 1905 paper , such a sophomoric mistake mathematically yet such a huge mistake conceptually. Is the Lorentz transformation incorrect? So it seems...

From your website: "The only reason behind the existence of the Special Theory of Relativity is the experimental fact that the speed of light is independent of any motion of the source or receiver."
In your thought experiment on the speed of light web-page you demonstrate that when the source of light remains stationary and 2 observers, one in a moving car and one stationary, measure the speed of light they both get the same value: without rescaling x and t in the 2 different reference frames. And the speed of light is independent of the motion of the receiver. A very intuitive example, showing that the speed of light is invariant to the observers reference frame, when the light source is stationary. However there is another example in question: when the source of light is not stationary. In the case when the light source is in motion, why don't the velocities of the source of light add or subtract to the velocity of the light when measured by a stationary observer? In the Galilean sense whenever 2 velocities are in the same reference plane the velocities always add or subtract. A classical thought experiment is a train where a gun is fired. The velocity of the bullet adds or subtracts from the velocity of the train. So, why doesn't light add or subtract from the velocity of the train? Trying to answer this question is what led to special relativity. But the answer seems so much more intuitive to me than rescaling x and t in 2 different reference frames. The reason the speed of light does not add or subtract from the motion of the source is that light cannot be accelerated or diminished, because light has no mass. You cannot push or pull light faster or slower, because there is no mass to push or pull.

In the example where a train is traveling with a source of light, the source is only capable of generating or creating light at a specific moment in time. At the moment of creation, the light begins moving at the speed of light. The light does not begin moving at the speed of the source of light plus the speed of light, because the light did not exist on the train until the moment it was created. If you were able to contain a photon of light in a container, put the container on a train, then release the photon, then the velocities could add or subtract. But you can't contain a photon in a container, because light has no mass. All you can do is put a light source on a train and create light at the moment you wish, then measure the velocity. The velocity will always be the distance from the source (at the moment of creation) to the observer. As you have already demonstrated the observer can be in motion, and the distance when the light is observed divided by the time it takes the light to travel will always equal c.

As indicated on my page regarding the Speed of Light, what led to Special Relativity is the attempt to apply the usual concept of 'velocity' (which implies that velocities add or subtract) to light, but still to have a constant velocity in all reference frames (as required by observations). Einstein tried to solve this logical contradiction by re-scaling the space and time units in a velocity dependent way such that formally a constant speed of light in all reference frame is maintained, but this means that the relationship between the units x and t on the one hand and the velocity v on the other becomes circular as x and t depend on v and, by the usual definition, v=x/t. All quantities are thus actually undefined as they mutually depend on each other (my page regarding the Lorentz Transformation makes this inconsistency also clear from a purely mathematical point of view).
The error Einstein made was thus to consider the velocity v as a more basic unit as the the coordinates x and t, which of course means turning things on its head as the velocity (in its usual sense) is indeed merely a derived quantity as v=x/t. So the correct way to go about this is to change the concept of 'velocity' for light rather than the concept of space and time. As explained on my 'Speed of Light' page, the only way to have a constant 'velocity' (i.e. one that is independent of any velocity of source and receiver) is to define it by the travel time of the light signal and the distance when the signal is emitted (not when it is observed as you stated in your last sentence). As I have shown by the 'moving car' thought experiment on that page, both the stationary and moving observer agree about the time it takes a light signal to reach the car, i.e. the 'velocity' of light is constant for both. Relativity on the other hand would yield inconsistent results here as it could only achieve an identical velocity in both frames by making the time and distance units velocity dependent, which however would contradict the physical symmetry of the thought experiment.

Comment by Stan Byers
In reference to the article Speed of Light and Theory of Relativity on this web site, which says "The only reason behind the existence of the Special Theory of Relativity is the experimental fact that the speed of light is independent of any motion of the source or receiver. This is because the usual concept of speed is inconsistent with the invariance of c unless one redefines length and time units accordingly":
The Jupiter / Io demonstration of light speed charted in relation to the relative velocity between the Earth and Jupiter shows that the speed of light is not independent of any motion of the source or receiver (
The given facts in this analysis are:
* The light from Jupiter takes 1003 seconds to cross from the near point of Earth's orbit to the far point at the speed of 300,000 km/sec, (c) in relation to Jupiter.
* It is known that Earth takes about 200 days to make this same trip. The maximum retreating speed in relation to Jupiter is approximately 29.79 km/sec, which is Earth's sidereal orbital velocity.
* When two things race between two points at the same time at differing speeds...they have a relative speed that is equal to the difference between the highest and lowest speed. Therefore the slowest relative speed is approximately 300,000 minus 29.79 km/sec.
Conclusion:...This relative velocity between the retreating Earth and the Jupiter/Io light train is not constant at 300,000 km/sec, and therefore light does not maintain a constant speed relative to all observers, as postulated by Special Relativity.

The observed delay is due to the varying distance between Jupiter and Earth, and does thus not imply a direct dependence on the velocity (in principle, one should observe the same delay if you could briefly stop Jupiter and the Earth in their orbits when the light signals are emitted and received). If there would be a direct velocity dependence of the speed of light, then this would merely correspond to the distance change during the 1003 sec the signal travels. This would thus only amount to a factor v/c = 30/300000 =10-4 of the delay due to the varying positions, i.e. the delay due to a non-constant speed of light would only be 0.1 sec for the 113 eclipse periods between opposition and conjunction of Jupiter and Earth, and a mere 10-3 sec for one eclipse cycle. So this would be pretty difficult if not impossible to verify in this case.

You are of course intuitively right to claim that the speed of light can not be constant if the source and observer are moving relatively to each other, but this is only if you adopt the usual concept of 'speed'. If however you assume that the time it takes a light signal to travel from A to B does only depend on the distance of both at the moment of emission but not on the relative velocity of the two, then this issue does not even arise and the 'speed' of light is trivially constant without involving Relativity (see my page Speed of Light and Theory of Relativity for more).

Comment by Thomas Spellman
I think your page /lightspeed2.htm, is flawed. In it, you give an illustration with a ground based observer (GBO), a car, and a light signal. You claim that at t2, the light signal emitted by the GBO reaches the car, but that is false. At t2, the light reaches the position (x1) that the car was at when the light was emitted. But the car has continued on, as you say in your illustration, to x2, but you don't account for the additional time (t3-t2) it takes for the light to get to x2 at t3. But again, the car is at position x3, and so on, in infinitely smaller amounts of time. The answer can be found with a simple system of equations, that of the position of the car from the moment the light is emitted (vt+x1) and that of the light itself (ct). The equation is ct = vt + x1, and solving for t gives you t = x1/(c-v). If the car was stationary at x1, then your answer would be correct, t = x1/c, but it's not stationary, and that makes all the difference.

Your argument is based on the usual concept of a frame(observer)-dependent velocity. The speed of light is however not frame dependent: if one has a light source at distance x1 at the moment the light flash is emitted, then it is irrelevant whether the light source is moving relatively to the observer or resting; the flash will always reach the observer after a time x1/c. This is the whole point of the 'speed' of light being independent of the relative motion of source and observer.

Comment by Ray Staines
It seems to me that if a theory generates a paradox either that theory is logically inconsistent or the paradox is only apparent. It follows that any apparent paradoxes generated by SRT must be explained for the theory to remain intact. My critical appraisal of SRT has therefore recently run along the lines of looking at the various apparent paradoxes it generates and then examining the explanations put forward to resolve them. Having exhausted the supply of paradoxes I could find on the internet (and finding the explanations at least plausible to a non-mathematician) I moved on to inventing some of my own, the simplest of which I present below in the form of a thought experiment. I have been unable to resolve this (apparent?) paradox myself and I invite solutions or comments from your readers.

An observer on a space ship far out in space builds the following apparatus: Two identical rods of spring steel are arranged at right angles to each other. The rods are fixed at one end and the free ends are fitted with identical weights such that when set in motion they will act as spring pendulums that posses a simple harmonic motion. These pendulums (P1 and P2) being identical in every respect will both vibrate at the same frequency. The free end of each pendulum is fitted with a small electrical conductor (brush), which makes contact with a corresponding fixed electrical contact when the free end of each pendulum is midway along its trajectory. A lamp and battery are wired to the fixed contacts in such a way that when both contacts are closed a current will flow down one pendulum and up the other thus lighting the lamp. It can be seen that provided the pendulums are identical and set in motion together the lamp will flash at the same frequency as the periodic motion of the pendulums. The observer now places the apparatus on board another spaceship and blasts it into space along the axis of P1 (P 2 is at right angles to the direction of motion) When the rocket reaches c/2 the motor is cut and the rocket settles down to a steady state of motion with respect to the observer back on the mother ship. Now the spring pendulums are set in motion by an observer on the remote space ship.

SRT makes the following predictions:
Prediction 1: The observer on the moving space ship will obtain the same result whatever his motion w.r.t. the mother ship and the lamp will flash in sympathy with the harmonic motion of his two identical pendulums.
Prediction 2: According to the observer on the mother ship the pendulum P1, which has its axis along the direction of travel, will be shorter than pendulum P2, which has its axis at right angles to the direction of travel. The frequency of the simple harmonic motion of the two pendulums will be different and the lamp will not flash.

SRT thus makes two predictions that contradict each other. Both observers can see the lamp. Either it flashes or it does not. They can't both be right.

The resolution of your paradox is simply that there is no length contraction. The latter is, like the time dilation effect, the result of a logically and mathematically inconsistent interpretation of the invariance of the 'speed' of light (see my page Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation).

Question by David DeFoe
If object A is moving toward object B at half the speed of light and object B is moving toward object A at three quarters of the speed of light, what will the outcome be? Will they reach each other before they see each other?

Yes, they will (although Relativists would obviously disagree about this).

Question by Chip Hogg
On your page Twin Paradox Debunked, do the rods have the same length in A's frame, or the same rest length? They are started and stopped simultaneously, but simultaneously in which frame?

The clocks are started and stopped simultaneously by definition in both frames: the first contact starts each clock from zero, the second stops them after a time corresponding to the relative speed of the rods and the travel time of the contact signal in the rod (which is the same given fact that both rods move with the same speed relative to each other, and if both rods are assumed as identical geometrically and physically).
(I have added now a corresponding clarification to the page).

Chip Hogg (2)
Your goal with the Twin Paradox Debunked page is to demonstrate that SRT is not self-consistent, right? That under its own assumptions it leads to contradictions which are more than just apparent? I'm asking because if so, I can analyze the situation under the framework of SRT, which I'm familiar with, and not under your framework which I haven't learned.

Under that assumption, I've done the algebra in detail, and found no contradiction under special relativity. When you say that 'both rods are assumed as identical geometrically and physically', I'm assuming that to mean that they have the same rest length. (Obviously, this would give the situation the maximum amount of symmetry, which is a central point of your argument.) This means that, if we pick A's frame to analyze the situation from, rod B will actually appear shorter by a factor of gamma (henceforth abbreviated G).
Thus, if a lightspeed signal is sent to each clock from the contact points, clock B will start before clock A (in A's frame). (Actually, this is due not only to the length contraction, but to the fact that B is moving towards the signal source as well.) So, the clocks are not started simultaneously in A's frame, and it's not clear to me how we can simply define them to be simultaneous unless we also assume that the signal propagation is different along each rod, which would wreck the symmetry.
Doing the algebra, I find that the time which clock A measures, from start to finish, is tA = [L(G+1)/(G*v)], where L is the rest length of either rod, G is gamma, and v is the relative speed. The symmetry of the situation absolutely requires that B's clock read this same value once the clocks are compared: here, I fully agree with this central point of yours. But special relativity also predicts that the time between those two events (starting and stopping of B's clock), as measured in A's frame, must be longer by a factor of G.
Checking this claim, we find that B's clock starts at a time tB0 = L/[2G(c+v)], and it stops at a time tB1 = L(G+1)/(G*v) + L/[2G(c-v)]. The time between starting and stopping B's clock, as measured in A's frame, is therefore tB = (tB1 - tB0) = L(G+1)/v = G*tA, exactly as predicted. (I confess to some amazement at seeing how perfectly it worked out.)

Reply (2)
If you agree that the clocks in frames A and B should show the same time when compared (due to the symmetry of the situation), then obviously the prediction that special relativity makes (and which you used here) must be wrong, as it would be a logical contradiction if A concludes that the clock in B runs slower, but B concludes that the clock in A runs slower. So this logically proves the impossibility of time dilation and length contraction and the invalidity of the Lorentz transformation as such (which as shown on my pages regarding Einstein's original 1905 derivation and his later algebraic derivation has in fact been derived mathematically inconsistently).

Chip Hogg (3)
The great difficulty in resolving any SR paradox is the need to be completely explicit about details which are irrelevant in classical theory. For example, before relativity, it would have been irrelevant whether we're talking about whether the time for B's clock was measured in B's frame or A's, since time was considered to pass at the same rate in all frames of reference. Now, though, it is a crucial distinction.

We're considering the time between two events: the starting and stopping of B's clock. In A's frame, this time will be longer by a factor of G. But B's clock doesn't measure the amount of time which passes in A's frame; it of course measures the time passed in its own frame. That will be shorter. (By the way, as you probably already know, the time passing in its own frame is called the 'proper time', and it is always the shortest time that can be measured between two events. The 'proper time' is defined as the time which has passed between two intervals in a reference frame where they take place at the same physical location. This frame can be uniquely determined, of course, up to some irrelevant arbitrary rotation. As I said, you probably already know this but I just wanted to make sure.)

Now doesn't this seem to contradict other experimental results said to verify time dilation, such as the atomic clocks on the supersonic jets? I think not, because there is an essential difference. In your thought experiment, the clocks are not at all accelerated while they are running, whereas the atomic clocks are accelerated while running. If one is to observe the effects of time dilation experimentally in terms of getting different readings on the clocks, one has to accelerate them to bring them together at rest, and this acceleration would break the symmetry.

Hmm... a thought just occurred to me. What about a version of your paradox where you have two clocks moving relative to one another, only now these clocks are each attached to a time bomb? Let's say that contact simultaneously starts both clocks, and each is programmed to self-destruct after some arbitrary amount of time, t, has passed: the same for both clocks. We still have no need to introduce symmetry-breaking accelerations into the picture, and we have the added bonus that the clocks don't have any length along the boost direction, so we don't need to worry about length contraction or signal propagation either!
In this case, in A's frame, it explodes after a time t has passed, and B is observed to explode after time G*t, i.e. (G-1)t after A has already exploded. This is a way to observe the effect of time dilation: since B is moving, the rate at which time passes is dilated compared to A; hence, it exploded later.
The symmetrical argument still applies for B's frame, of course, and would seem to cause a contradiction: which one 'really' explodes first?! In this case it's the relativity of simultaneity which solves the contradiction. The explosion events are separated by a spacelike interval -- in other words, the light from one explosion always reaches the other after it's already happened. Thus, their time ordering depends on the reference frame, since they can't be causally related.

My favourite version of the twin paradox, since it's the most physically realistic, is the constantly-accelerated spaceship version. It was a problem in one of my E&M homeworks a few years ago. We accelerate a spaceship at a constant proper acceleration of g, for five years, as measured in the spaceship's frame. Then we decelerate for five years. Going home, we repeat the same procedure. This lets us make the effects of the acceleration quantitative, and I found that after the total of 20 years passed on the spaceship, on Earth 338 years would have passed, according to SR. Pretty neat!

Reply (3)
But the paradox we are dealing with here is introduced by SR, not resolved by it. Without the incorrect derivation of the Lorentz transformation and its implication of a time dilation, the issue would not arise in the first place. What the observer A thinks the clock in system B shows is completely irrelevant here. The clock times that are being compared at the end depend solely on the physical properties in the corresponding system:   the contact triggers a signal (not necessarily a light signal) which propagates with a known velocity in that frame, starts the clock, and after a set time (which is identical in both frames by assumption here), a second contact stops the clock. Everything is fully defined in each system separately, and with the physical conditions in each system being identical, the clock times, when compared later, must be identical as well. There is no way that the relative velocity of the two systems as such can produce any changes in the clock readings in the sense that one clock should show a different time. If there is any difference, then it must have something to do with physical differences between the system (I have for instance suggested that the corresponding experiments with clocks travelling on aircraft could be explained by the fact that the motion through the magnetic field of the earth causes the clocks to run slower because of the associated Lorentz force (see this entry in the discussion forum)). It is only in this sense (namely that forces are being applied) that an acceleration could lead to different clock readings, but this is obviously not a relativistic effect then. According to relativity, clock differences should arise solely due to a constant relative velocity, but as this leads to the twin paradox, a hand-waving argument is being made that one system has to turn around (i.e. accelerate) to compare the clock (which of course doesn't resolve the paradox, as a) you could in principle make the period of acceleration arbitrarily short, and b) it would violate causality as the clock difference would have to build up already before the acceleration occurs). This was the reason why I suggested this particular thought experiment which does not involve any acceleration whilst the clocks are running, and which thus, due to its symmetry, would not enable to resolve the twin paradox (thus proving a time dilation is logically impossible.

Regarding your alternative 'time-bomb' experiment:   first of all, this would actually be somewhat inconsequential, as you would still need the contact to start the clocks, and then you could anyway simply stop the clocks after a certain time (and trivially of course they will then show the same time when compared).
But again, you shouldn't make the mistake here to take time dilation as a given fact. In the context of these thought experiments you can only take it as a hypothesis, and then check whether this leads to logical inconsistencies or not (and here it is clear as well that because the situation is completely symmetric (i.e you must be able to exchange A and B without changing the result), a time dilation for either reference frame would be a logical contradiction).

Chip Hogg (4)
Again, for clarity, I emphasize that all I'm trying to do is to explore whether special relativity is self-consistent, with respect to time dilation, in the case of this paradox. Whether the Lorentz transformations were derived properly or not is not our concern; I am taking the formulae for time dilation (and length contraction) as given, and seeing whether they force a contradiction.

Let me also say that we all agree that the clocks will show the same time. This is perfectly clear by the symmetry of the situation.
Now, if I'm reading you right, you seem to be saying that if time dilation is real, it should show up in the reading of B's (or A's) clock, right? Otherwise, how can we hope to measure it?
The key point is that the dilated time is the time for B, as measured in A's frame, not as measured in B's frame. We pick a reference frame -- A's frame -- and set up synchronized clocks at suitable locations. All these clocks tick away at the same rate, so we can unambiguously talk about 'time' in A's frame. In this reference frame, we note the time (on any clock) for first contact, and also the time for second contact. Subtracting the readings on the two clocks will give a value larger than the reading on B's clock by the factor gamma. This is the time dilation effect.
You might object that this is fundamentally different from the usual twin paradox, since both clocks remain the same, whereas one twin is aged and the other is not. You would be right: when you removed the source of acceleration, i.e. removing what distinguishes one object from the other, you've changed that part of the paradox so that neither can differ from the other, since there is nothing to break the symmetry.
Let me clarify it by putting it this way: in order to show a true time dilation paradox, you need to have
- two initially identical objects undergoing relative motion and being brought back together, such that one is predicted to be different from the other, and
- nothing asymmetrical about how they were treated.
In altering the twin paradox to achieve the second point, you've left the first point out.

As for my time bomb re-imagining of the paradox, it's more than just a gedanken experiment (which is why I picked it). Muons (in fact, all unstable particles) are found to have mean lifetimes which depend on their velocity in exactly the way predicted by special relativity. A muon decay event is conceptually the same as a time bomb going off, after being started by contact.
If you have a bunch of muons at rest, every 2.2 usec you will have 1/e as many. If, however, you have a beam of muons moving with speed v, then you have to wait a time [G * (2.2 usec)] for there to be 1/e as many, where G = 1/√(1-(v/c)2) is the usual gamma factor. This happens at particle accelerators all the time.

Returning to the time bomb formulation, I disagree with your final statement that a time dilation for either case leads to logical contradiction. Let's examine it in each frame.
In A's frame, it is standing still, and B comes whizzing by. After a time t has passed, A explodes (or gives off a light signal), and a time (G-1)*t later, B explodes.
In B's frame, it is B that is standing still, and A comes along in the opposite direction. B explodes after a time t, and a time (G-1)*t later, A explodes.

The reason there is no logical contradiction is that SR predicts that, in some cases, the time ordering of two events can depend on the frame. Thus, the question of which one 'really' exploded first isn't meaningful. It's perfectly consistent for A to believe that it exploded first, while B believes that B exploded first, and have each be right in their own frame. When we demand, "but which one *really* exploded first?" we are trying to impose the idea of a frame-independent time which is the same for all observers, something that SR says doesn't exist. SR may well be wrong, of course, but if we're trying to show that it's not self-consistent, we can't use this concept of a frame-independent time to do so.

Reply (4)
Not seeing a logical contradiction and one not being there is not necessarily the same thing. And in view of what has been said before, I don't actually quite understand how you fail to recognize it. Even SR itself does, as it goes to great lengths to discover some kind of asymmetries in various suggested 'twin-paradoxes'. The simple fact is that a perfectly symmetric situation, as suggested by me, logically rules out that the clocks show different time when compared, i.e. that the time in either of the moving systems passes slower than in the other (your 'key point' that the dilated time is the time for B, as measured in A's frame, not as measured in B's frame is actually a bogus argument; there is no such time; the time in system A is defined by clock A, and the time of system B by clock B, and if you want to tell whether both systems have measured the same time, you have to compare clock A with clock B (so contrary to what you said above, a 'frame-independent' time does not come into it)).

So this then logically rules out time dilation (in the sense of special relativity) as an explanation for corresponding experiments like those involving muons for instance (with regard to 'cosmic ray' muons, I have suggested already right at the top of this page that the explanation could actually be a speed higher than c; the accelerator experiments could be explained if the muon lifetime depends on the magnetic field for some reason).

So it is all about having logically and physically valid explanations of experiments. SR does not provide either of this (and as mentioned already, it isn't mathematically valid either).

Chip Hogg (5)
The perfectly symmetrical situation you describe is not a 'twin paradox' type experiment, since SR itself does not predict that the clocks will show different times.

Reply (5)
Then SR should for instance not predict a time dilation for the case of the 'cosmic ray' muons:   if you assume that the traversal of the muon through two successive detectors starts and stops a clock on the earth respectively, and imagine that the same events also start and stop a clock travelling with the muon, then this is exactly the same situation with regard to the symmetry as in the thought experiment I have been suggesting.
Anyway, either of the clocks could not care less about whether the situation is symmetric or not. Each clock just takes its time and doesn't know anything about the other clock and how it moves.

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