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### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 11:01 am
Some quick comments on the General Relativity discussion, starting here:

viewtopic.php?p=320824#p320824
Halc wrote:Things don't accelerate in such a field, so of course this can be regarded as an inertial frame, since it is one, no?
Yes. In a small enough volume, in which the gravitational field can be regarded as uniform, the frame is inertial. Hence astronauts in the ISS experience what feels (approximately) like weightlessness even though the force of gravity acting on them is almost the same as it is on us.
This being within a concentric distribution of matter is exactly such a uniform gravitational field, and clocks within one are gravitationally dilated compared to a stationary clock well outside that field. Neither can locally detect which is which, but if you compare the two (a non-local test), they'll read different values and then each will know which it is.
I would have thought a uniform concentric distribtion of matter (are we talking about a hollow spherical shell of matter?) would result in no net gravitational force inside it. This is the gravitational analogue of the Faraday Cage and Gauss's law.
This seems poorly worded if not wrong. I see a lot of commentary on the web taking apart the wording of exactly the part you've quoted here. I think it would be more correct if it talked about being in the gravitational field of exactly one point gravitational source, in which case there would be no exception. Me standing on a planet with the ground exerting a force keeping me stationary in Earth's gravitational field is the same as being in an accelerating ship with the deck exerting the same force keeping me stationary in the ship's accelerated reference frame. That's what the GR rule is, but I thought it was about local forces and physics, not time dilation.
I'm sure you may be right about the poor wording. I was just doing a quick search as I'm not sure about this stuff myself. At University, as I recall, we had one lecture course on Special Relativity but General Relativity was only briefly covered. I presume the details of General Relativity are considered to be beyond the scope of a BSc Physics course. So anything I've learned about GR is only from popular science books and the internet. Not from proper textbooks.
I'm not sure!
If I put a clock in a centrifuge and whir it up to 1000G, will it dilate the same as being on a star with that sort of gravity? I mean more than it would dilate just from the linear speed of being in that centrifuge? I tried to keep that to a minimum in my example, but there would be no ignoring SR dilation in a centrifuge that violent.
Interesting experiment! I guess the SR time dilation effects due to the linear speed could be calculated and factored out.
The wiki talks about an exception within (not necessarily at the center like it says) a concentric distribution of matter, where you also cannot locally detect the difference, but any non-point gravitational source would be an exception. Add more points in different places, and the dilation goes up, but the acceleration might go up or down, depending on the distribution of the points. Gravitationally, clocks run slowest on Earth when the moon is full and we're sandwiched between it and the sun, but our acceleration there is the least since the pull is in opposite directions.
Is that right? Has the increase in gravitational time dilation due to the relative positions of Sun and Moon been experimentally measured? I have read that the difference in gravitational time dilation due to a difference in height above the Earth of as little as 1 foot has been experimentally verified. So it wouldn't be surprising.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 11:07 am
Error: I meant Earth and Moon, not Sun and Moon.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 12:00 pm
Steve3007 wrote:Yes. In a small enough volume, in which the gravitational field can be regarded as uniform, the frame is inertial. Hence astronauts in the ISS experience what feels (approximately) like weightlessness even though the force of gravity acting on them is almost the same as it is on us.
An interesting aside: Tidal forces inside orbiting spacecraft mean that the environment in there is usually referred to as "micro-gravity". Any unsecured object that is not positioned exactly at the centre of mass of the spacecraft will tend to slowly drift, because its own natural orbital period around the Earth is slightly different from that of the spacecraft.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 12:57 pm
Tamminen wrote:
October 3rd, 2018, 10:58 am
I have thought about it this way: acceleration makes the flat spacetime curved in the same way as gravity does.
Well that can't be since it would effect stuff around it. I could accelerate an electron at a million G's and if spacetime bent enough to account for that, it would crush the lab with it. No, the non-gravitational acceleration is due to different forces, not spacetime curvature.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 1:14 pm
Steve3007 wrote:
October 3rd, 2018, 11:01 am
Halc wrote:Things don't accelerate in such a field, so of course this can be regarded as an inertial frame, since it is one, no?
Yes. In a small enough volume, in which the gravitational field can be regarded as uniform, the frame is inertial. Hence astronauts in the ISS experience what feels (approximately) like weightlessness even though the force of gravity acting on them is almost the same as it is on us.
Perhaps I mustunderstood 'uniform field'. Given the wording, I think your interpretation is what they meant. Yes, the ISS guys can't tell they're not in an inertial frame unless they do experiments that take several minutes like put 4 marbles stationary in a tetrahedron shape and see where they go after a while. 4 marbles is enough for a 3-D shape.
I would have thought a uniform concentric distribtion of matter (are we talking about a hollow spherical shell of matter?) would result in no net gravitational force inside it. This is the gravitational analogue of the Faraday Cage and Gauss's law.
Yes... That is what I thought they meant by uniform gravitational field: Net zero force, not just uniform force everywhere as is an approximation of ISS.
I'm not sure!
OK, so neither of us nor that wiki thought-experiment is going to resolve this question. I think we need a better site.
Interesting experiment! I guess the SR time dilation effects due to the linear speed could be calculated and factored out.
Right. So all they need is a clock sturdy enough to operate under enough G's where it can measure acceleration dilation after linear dilation is factored out. Surely this can be done. It would answer our question. There should also be a way to determine it via thought-experiment. Almost all Einsteins's empirical tests were thought-predictions before they were tested. Alice and Bob may yet teach us something, but only with a description that doesn't get sloppy. I'll look at the site again to attempt to glean exactly what it was trying to demonstrate, even if they did it poorly.
Is that right? Has the increase in gravitational time dilation due to the relative positions of Sun and Moon been experimentally measured?
Not sure how you would measure that. You'd need to compare a pair of clocks not in each other's presence. But the acceleration (in an inertial frame) is definitely less, as is the net gravitational gradient, but not the depth of the well. I know the well depth is a factor in dilation, but I don't know if the gradient (acceleration) is. I'm treating Earth as a point-mass here, so you know.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 4:04 pm
Steve3007 wrote:
October 3rd, 2018, 2:42 am
One of the things I've noticed about your long posts (such as your most recent two) which quote individual parts from my posts and then (ostensibly) reply to those parts is that they don't focus on those parts to which they are ostensibly replying. Hence, the apparent strangeness of your "trashing the laws both of physics and mathematics" criticism which appeared to me to be a reply to an obvious truth uttered by me.
Each of my replies always responds to the quoted part, but may also comment more generally. In the case of the specific exchange at issue here, you were trashing physics and maths by denying that an unaccelerated clock in deep space will tick at a constant rate and claiming that you need to measure it against another clock. Experiments have determined the rules, and the rules are that an unaccelerated clock in deep space will tick at a constant rate (unless something affects it in such a way that it affects everything else to match such that you can't identify any variation, such as you might get with an alien twiddling the control knobs of the universe). These experiments were done using multiple clocks, but they show us that we can rely on an unaccelerated clock sitting in deep space to tick at a constant rate. Rejecting that in an attempt to muddy the waters in this discussion is rejecting physics. In thought experiments, we can absolutely rely on such a clock ticking at a constant rate because in thought experiments we can rule out the alien intervention, so it's just another of your diversions.
Obviously another thing that's noticeable is that you spend a huge amount of your time simply telling me what an idiot I am - how I'm parroting dogma, following a cult, terrible at maths and physics, incapable of logical reasoning etc etc.
That doesn't equate to you being an idiot. Geniuses are capable of idiocy without being idiots (and idiots can sometimes come up with ingenious ideas). If you're making mistakes though in maths, logic or physics, you need to know that, and the bigger the mistake, the more chance there is that you can be helped to recognising the error, which is why a word like "idiocy" can be a useful flag. D>L and E>S and D=S and E=L is a big error in maths, and it applies to physics too. If D>L and D=S and E=L, then S>E. That's the asymmetry. Why would you want to protect SR dogma when it goes directly against this maths? The answer to that is simple - for most people, beliefs override reason so they simply switch off their reasoning abilities in order to defend incorrect positions that they've bought into, and it's very rare for them to budge. They aren't all stupid, but they are merely locked in their thinking on some issues. Take them away from those issues and you find that most of them can actually process the maths correctly, though as most of them have never been taught logic, they can still struggle to accept the rules (which are a fundamental part of mathematics). If this wasn't showing up contradictions in Einstein's original SR model, I'm confident that you would have no trouble agreeing that if D>L and D=S and E=L, then S>E; and that if E>S and D=S and E=L, then L>D. As soon as this is used against SR though, you reject it because you think the authority that you respect trumps any argument that opposes it, and you blind yourself to the reality that you're rejecting mathematics in order to defend the authority's error.
OK, first of all I'd like you to tell me precisely and unambiguously what you mean by the term "constant rate" here. And I'd like you to tell me how you know that this clock you've mentioned here is ticking at this constant rate - I mean the method by which you arrive at that knowledge.
You accept, I assume, that light moves at a constant speed. If so, then you are already accepting that a clock ticks at a constant rate if it doesn't accelerate and if it isn't changing its depth in a gravity well. Just put a light clock in deep space and imagine the light going to and fro at a constant speed of c, and there you have it - a constant ticking rate.

As I've said before, there could be an alien tampering with the controls of the universe, but if there is, then (s)he/it will need to vary the speed of light in order to vary the ticking rate of that clock. In physics, it's normally assumed that the speed of light is not being tampered with in such a way, and that a clock that isn't being accelerated sitting/moving in deep space will tick at a constant rate.
I suspect you will say that this clock is ticking at a constant rate by definition.
Clocks aren't fundamental, so we break them down into what they are and do. They always involve something cycling in one way or another, and that cycling is governed by the speed of light. A constant speed of light results in a constant rate of ticking for the clock unless it's accelerating (continually changing the cycle distance) or is changing its depth in a gravity well where the speed of functionality of the clock is affected by changes in the speed of light.
As I've said many times before, time is the thing that is measured by clocks. There is no other way to define time.
Fundamental things cannot be defined other than by how they interact with or affect other things. Time is fundamental. Light moves a certain distance through space in a set length of time, unless it's slowed by an additional medium, such as the one created by energy/matter (which reduces the speed of light around it, more strongly with greater proximity).
The only answer is to define general abstract concepts in terms of individual physical examples (much as we do with the laws of physics). As with the definitions of many things in life, the way to define a clock is to point to lots of clocks and say:

"An 'ideal' clock is that property which all of those things (those 'real' clocks) have in common, and time is that thing which is measured by this idealized clock.".
An ideal clock measures how long it takes unslowed light to cover a set distance.
So what do all clocks have in common? What is this property that we might call clocky-ness? They keep doing the same action over and over again. In other words, they tick. But the only way that we can decide whether each of those ticks is coming at us with the same period of time between each one is to compare them to the ticks of another clock.
We've done all that, and what we've found is that if we move a clock, it under-records time because it measures light over a greater distance than the set distance it's supposed to be measured over. We've also found that if we put a clock in a gravity well, it runs slow because the speed of light is lower there. Light clocks will also run slow if we measure the speed of light through glass unless we reduce the set distance for the course the light moves along to compensate. We have established from all our experiments that a clock sitting in or moving at a constant speed in deep space will tick at a constant rate (unless that alien is interfering or the expansion of the fabric of space is slowing the speed of light over time, etc., but in any such case, the slowing of the clock will not show up to us in any way that we can detect, other perhaps than by seeing an increase in the rate at which the universe appears to expand).
You may recall that I've explained this to you before by referring to the clock on my wall:
Steve3007 wrote:What do you mean by "ticked at a constant rate"? Relative to what? The only way I can assess whether any clock is ticking at a constant rate is by comparing that clock's ticks with those of another clock which I deem to be ticking at a constant rate. As I sit here and look up at the clock on my wall, I deem the second hand of that clock to be ticking at a constant rate. How? By comparing it to my own internal sense of time - my biological clock. Or by comparing it to some other clock. How else would I make that judgement?
viewtopic.php?p=320538#p320538
(You did reply at the time, but your reply appeared to misunderstand what I was saying here, and I let it drop.)
You don't seem to realise that your question here is much bigger than you imagine - you could equally be asking if the speed of light is constant. It has no relevance here, and nor does your issue about whether clocks tick at constant rates. We can assume that they do if they aren't changing speed or changing height in a gravity well, just as we can assume that the speed of light is constant.
How do we know that the period of time between each pair of ticks is the same?

We can measure it.

What with?

A clock, of course.

What clock shall we use?

A clock whose ticks come at a constant rate.
If both of those clocks are being affect the same way by a change in the speed of light as the alien twiddles its knob, we will not detect the change in the speed of light or the ticking rates of the clocks. What we can do though, which is what we do in my thought experiment, is compare the two clocks against each other. They, my clock and Einstein's clock in the latest version, can be assumed to be ticking at constant rates, and even if the alien is twiddling its knob, they will still behave from our point of view as if they are ticking at constant rates. We can discuss whether one of these clocks might be ticking faster than the other. We can look at what happens when Lorentz returns to me and when you catch Einstein, and then we can see that two of the clocks actually ran slow. Every day, billions of experiments done with GPS confirm that the rules don't change - they are extremely reliable.
David, to avoid circular definitions that are doomed to remain forever in the realm of the entirely abstract we have to accept that all of our abstract concepts, including the concept of an ideal clock that ticks at a constant rate, are only rendered physically meaningful when seen as general inductively arrived-at abstractions from specific observed examples. That's science.
You are merely arguing about whether the speed of light varies in a manner which experiments can't detect. In science, we should certainly recognise the possibility that it does vary, but we also see that if it does it makes no impression on the experiments that we can do. We shouldn't rule it out, but if we are going to consider it, we need to ask what impact it would have on things like relativity. Changing the speed of light locally would show up as gravity. Changing the speed of light universally wouldn't show up at all, other than perhaps in the expansion rate of the universe. Changing the speed of light universally makes no difference to my thought experiment. Indeed, you can run my simulation at different speeds by changing the frequency of your button presses, but none of them make any difference to what happens at the registered times, or to the relative tick rates of clocks on different paths - this is not affected by your alien input.
That's how the laws of physics are arrived at. That is how the concept of a clock is defined in physics. If there were only one clock in the entire universe and all other clocks were absent (including my own biologically-programmed 'clock') then it would be meaningless to call it a clock. There would be no concept whatsoever of its ticks coming at regular time intervals. There would be no way to know whether they were.
If it's a light clock (and all clocks ultimately are light clocks under the hood), then it will tick regularly as the light covers that set distance over and over again. If that isn't constant ticking, then the speed of light isn't constant.
Okay. It's irrelevant.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 4:27 pm
I noticed this statement: "time is the thing that is measured by clocks". Not quite. what a cock does is measuring the passing of time which isn't quite the same thing. In principle, a clock goes tick tack. And when it does, the only thing it tells you is that some amount of time has passed. My undertanding is that when you try to measure time, it seems to be standing still. I believe it was Paul Davies who made this remark.

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 5:16 pm
David, I'll take it slowly.
David Cooper wrote:You accept, I assume, that light moves at a constant speed. If so, then you are already accepting that a clock ticks at a constant rate if it doesn't accelerate and if it isn't changing its depth in a gravity well.
You've simply replaced the term "constant rate" with "constant speed". So your task now is to give me your definition of constant speed.

The fact that light moves at a constant speed relative to all observers is not an axiomatic truth and it's not a tautology. It comes from empirical measurements, and theories which are based on empirical measurements.

How was it determined that light moves at a constant speed relative to all observers?

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 5:50 pm
David Cooper wrote:You don't seem to realise that your question here is much bigger than you imagine - you could equally be asking if the speed of light is constant. It has no relevance here, and nor does your issue about whether clocks tick at constant rates. We can assume that they do if they aren't changing speed or changing height in a gravity well, just as we can assume that the speed of light is constant.
You use this word "assume" a lot. Do you think that the constancy of the speed of light as measured by all observers is an assumption? If so, what do you mean by that?

Do you understand why it was initially proposed that the speed of light is a universal constant?

### Re: Does Special Relativity contain contradictions?

Posted: October 3rd, 2018, 7:43 pm
Steve3007 wrote:Do you understand why it was initially proposed that the speed of light is a universal constant?
David, since replies always take such a long time I'm going to pre-empt any answer that I might get to this question.

From our previous discussions, and your apparent lack of interest in the fact that Maxwell's Equations are the basis and starting point for Special Relativity and essential to understanding why light speed was proposed to be a universal constant, I have to assume that your most likely answer will be "no" and that you neither know nor care why this was proposed. If this is true, then I think it partially explains your apparent fundamental misunderstanding of precisely what it means to say that the speed of light is a universal constant. Because if you don't properly understand what led to it you won't know what they mean by it.

So if you've now decided not to talk to me again, fair enough. But, if so, my advice is to find a good book about the Theory of Relativity and read it. As I've said before, you could do a lot worse than Bertrand Russell's "ABC of Relativity". Considering that it was written so early, it is very good, and a good reminder of how insightful Russell was.

I know you'll probably say that your superior powers of reasoning mean that you don't need to learn about the subject that you're critiquing. But, sadly, you're wrong.

### Re: Does Special Relativity contain contradictions?

Posted: October 4th, 2018, 3:27 pm
Steve3007 wrote:
October 3rd, 2018, 5:16 pm
You've simply replaced the term "constant rate" with "constant speed". So your task now is to give me your definition of constant speed.
We're up against fundamentals and circular definitions, as you know full well.
The fact that light moves at a constant speed relative to all observers is not an axiomatic truth and it's not a tautology. It comes from empirical measurements, and theories which are based on empirical measurements.
You don't know that it moves at a constant speed at all. It could vary wildly, but if that happens, that is kept hidden from us. I told you last time that if you run the simulation on my page you can change your clicking rate on the buttons to make the action run through at different speeds, but none of that makes any difference to the results or relative ticking rates for clocks within the simulated universe. You are trying to take things on a diversion again instead of addressing the issue.
How was it determined that light moves at a constant speed relative to all observers?
It wasn't. It was determined by measurements that it appears to move at a constant speed relative to observers if they built into their measurements an assumption that light is moving relative to them they at c from the start. They then read back out the same assumption at the end, getting back exactly what they put in.
Steve3007 wrote:
October 3rd, 2018, 5:50 pm
You use this word "assume" a lot. Do you think that the constancy of the speed of light as measured by all observers is an assumption? If so, what do you mean by that?
There is no guarantee whatsoever that the speed of light is constant, but we can treat it as if it is - if an alien is twiddling the knobs that control it, it has no impact on events here. We look out into space and don't see the optical distortions that would show up if there was localised tampering by the alien, which means that whatever (s)he/it might be doing is being done universally. This means that when we talk about the speed of light being constant, that needn't mean constant in absolute terms, but it is effectively constant for our purpose here as everything adjusts to hide the difference (if such a difference and such a ridiculous alien exist).
Do you understand why it was initially proposed that the speed of light is a universal constant?
It pops out of equations, but that isn't automatically a speed of light. In 4D models and set zero models there is no speed of light, so it's a mere illusion of a speed in those models.
Steve3007 wrote:
October 3rd, 2018, 7:43 pm
From our previous discussions, and your apparent lack of interest in the fact that Maxwell's Equations are the basis and starting point for Special Relativity and essential to understanding why light speed was proposed to be a universal constant, I have to assume that your most likely answer will be "no" and that you neither know nor care why this was proposed. If this is true, then I think it partially explains your apparent fundamental misunderstanding of precisely what it means to say that the speed of light is a universal constant. Because if you don't properly understand what led to it you won't know what they mean by it.
Maxwell's Equations are not fundamental to this at all - they just follow on from the actual fundamentals which are that time runs and light moves through the fabric of space at c. They are also a simplification which depends on relativity for producing right answers in all frames, failing to consider the underlying reality. That makes them highly misleading if you imagine that they provide any insight into this.
So if you've now decided not to talk to me again, fair enough.
Where did that come from? You're the one who decided not to speak to me for a while earlier.
But, if so, my advice is to find a good book about the Theory of Relativity and read it. As I've said before, you could do a lot worse than Bertrand Russell's "ABC of Relativity". Considering that it was written so early, it is very good, and a good reminder of how insightful Russell was.
No amount of reading on my part will fix the faults in your understanding. You are running away from the important issue here, and that is that you approve of this shocking piece of maths:-

D > L and E > S and D = S and L = E.

I don't approve of it, and nor does any competent mathematician on or off the planet. The correct maths is D = S > L = E.
I know you'll probably say that your superior powers of reasoning mean that you don't need to learn about the subject that you're critiquing. But, sadly, you're wrong.
I looked into it and found it to by irrelevant. I did ask you before to show me how Maxwell's Equations produce a speed of light relative to an experiment without using any distance terms loaded with biased values based on the experiment being at rest, but you failed to do so. You won't find anything in Maxwell's Equations or your physics books to justify D > L and E > S and D = S and L = E. That is the thing you should be addressing here, but you don't want to touch it because it shows you to be plain wrong, so you're using diversion tactics instead in order to avoid commenting on this crunch point. I have deliberately not asked you a direct question about this up to now just to see how long you go on avoiding it, but it's now time to push you.

Do you approve of the maths of the following (these values are tick rates of clocks relative to each other, all applying to the same moment in time):-

D = S
L = E
D > L
E > S

I say it's impossible as it contains a contradiction.

### Re: Does Special Relativity contain contradictions?

Posted: October 4th, 2018, 3:36 pm
For anyone who hasn't read the discussion up to this point, I'll spell it out more clearly:-

D = S because D and S are co-moving (and are side by side) throughout the first leg of the thought experiment.

L = E because L and E are co-moving (and are side by side) throughout the first leg of the thought experiment having set off from the place where D and S are.

D > L is an assumption which we might make at the end of the experiment when L returns to D and is found to have ticked more slowly than D.

E > S is an assumption which we might make at the end of the experiment when S catches up with E and is found to have ticked more slowly than E.

It is clear from this that if D > L, E must be less than S.

It is also clear that if E > S, D must be less than > L.

It is also clear that if D = L, then S = E. Remember though that these values only apply to the first leg of the trip, so in this case, we can see that L2 = L and that S2 < S.

Whichever way you try to cut the cake, you get an asymmetry. It is unavoidable.

### Re: Does Special Relativity contain contradictions?

Posted: October 4th, 2018, 3:38 pm
How about a quick recap of this...

viewtopic.php?p=320629#p320629

Just out of interest, I'm re-iterating the above, emphasizing what each observer observes and making clear the distinction between what is directly observed (the raw sensory data) and what might be calculated from those observations, from the point of view of different inertial reference frames. In this way, I hope it demonstrates how the interpretation of raw observations with respect to different reference frames works, and the precise sense in which each observer "sees" the other's clock running slowly.

All numbers rounded to 2 decimal places.

The Situation

Let person A stay stationary in an inertial reference frame.
Let person B travel from A, at a speed of 0.866c relative to A, to a point that is 8.66 light-years away as measured by A. We'll call that point C.
Assume no nearby large masses.

The Lorentz factor is:
1 / sqrt(1 - v2/c2)
For a speed of 0.866c that is 2.

So, while in transit, person B measures (WRT his reference frame) the distance traveled to be half of what person A measures it to be: 4.33 light-years. And person A measures (WRT his reference frame) person B to be length contracted - squashed in the direction of travel - by a factor of 2.

What A and B Directly Observe

The relativistic Doppler Effect describes the relative tick rates that are directly observed, and is given by:

sqrt( (1 + v/c) / (1 - v/c) )

With v/c = 0.866, this results in a value of 3.73.

If they both zero their clocks when B sets off, then according to B's clock, when B experiences the event of reaching C, he sees that his own clock reads 5 years. That's how long the journey takes him, as measured against his reference frame travelling at 0.866c. 4.33/0.866 = 5. When he gets there, if he momentarily stops and looks back at A, he sees A's clock reading (10-8.66) = 1.34 years. If he wants to, he may calculate from this, and from his knowledge of the distance to point C and the speed of light that:

1. measured against his (B's) outward moving reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 2.5 yrs.
2. measured against his (B's) momentarily stopped reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 10 yrs.
3. measured against his (B's) inward moving reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 17.5 yrs.

But what B actually sees (from a distance) is A's clock reading 1.34 years. And, as Halc says, the A-ticks that he actually sees don't suddenly jump forward by 15 years. But B's judgement as to which A-ticks are simultaneous with which B-ticks does, due to the frame change. This can be seen clearly in this Minkowski diagram that illustrates the lines of simulteneity WRT the two reference frames in this particular case, in the reference frame of A:
MD_Small.png (31.55 KiB) Viewed 729 times
It can be seen from this type of diagram that the discontinuity at the point of inertial reference frame switch causes B to calculate this jump based on both his observations and the definition of simultaneity.

Likewise, when A sees B reaching C he sees his own clock reading (10+8.66) = 18.66 years, and sees B's clock reading 5 years. So, if he wants to, he may calculate from this, and from his knowledge of the distance to point C, measured against his (A's) single reference frame, that the "Clock A tick event" which is simultaneous with the "B reaching C" event is 10 years.

What A sees (from a distance) is B's clock reading 5 years.

Person B then, as measured by his reference frame, takes another 5 years to return from C to A. So at the end of the journey, A and B can both see that B's clock reads 10 years and A's clock reads 20 years.

So, in summary, this is a table of the raw observations of A and B of their own and each other's clocks:

Code: Select all

``````                      A as seen by A       B as seen by B       A as seen by B       B as seen by A
Start                       0                    0                    0                    0
Observed mid-point        18.66                  5                   1.34                  5
End                         20                  10                   20                   10``````
So you can see from this that, just in terms of their raw uninterpreted senses, they each see the other's clock running 3.73 times as slow during the first leg and they each see the other's clock running 3.73 times as fast during the second leg, in agreement with the relativistic Doppler effect calculation. And at the end they will both agree that B has aged half as much as A.

What A and B Calculate from their Observations about Simultaneity

As I said, above is the situation simply as observed by the two people. If A and B decide to do some calculations based on that raw sensory data then the results will depend on the reference frame that they use to do those calculations against. This is what Halc was talking about. This is the sense in which they both "see" each other's clocks running slow on both legs of the journey, and it's where the concept of simultaneity comes in.

If B wants to work out which of A's "clock tick events" he ought to regard as being simultaneous with the "B arriving at C" event, he has to consider what he actually sees (the raw sensory data) and then account for the distance from which he sees it and the speed of light. The way that he does this depends on the reference frame against which he does it. But, at C, he switches between two different inertial reference frames.

WRT the first reference frame (that he is stationary WRT during leg 1) he will judge that the "A's clock tick event" that is simultaneous with the "B arrives at C" event is 2.5 years. Which he will judge to be simultaneous with the "B's clock tick event" that is 5 years.

WRT the second reference frame (that he is stationary WRT during leg 2) he will judge that the "A's clock tick event" that is simultaneous with the "B arrives at C" event is 17.5 years. Which he will judge to be simultaneous with the "B's clock tick event" that is 5 years.

On the other hand, B remains in the same inertial reference frame throughout. So he will make just one judgement. He will judge that the "B's clock tick event" that is simultaneous with the "B arrives at C" event is 5 years and that it is simultaneous with the "A's clock tick event" which is 10 years.

Here's a new table to show all this. In this table the columns show the above calculations for A (in his one reference frame) and B (in his two reference frames). The two numbers in each column show the A and B clock events that each observer calculates to be simultaneous.

Code: Select all

``````               A          B1         B2
Start          0,0        0,0        0,0
Mid-point      10,5       2.5,5      17.5,5
End            20,10      20,10      20,10``````
So you can see from this that both A and B calculate, based on their direct observations, that the other is travelling 2 times more slowly than themselves on both legs of the journey. And the final asymmetry is a result of B's reference frame change. This reference frame change also highlights that the question of whether two events can be regarded as simultaneous depends on the movements of the observer.

General Relativity

You can also see from this that another way of looking at it is that the act of B turning around (accelerating) causes (in B's judgment) A's clock to go forward 15 years. This is where the alternative treatment of this situation, invoking General Relativity and the Equivalence Principle (between gravity and acceleration), comes in. We can either treat this as an idealized instantaneous (zero time, infinite acceleration) transition between two inertial reference frames (SR) or as a transition between two inertial reference frames via a non-inertial reference frame of constant finite acceleration for finite time (GR).

The defining feature of a "special case", as opposed to the more general case, in physics, is that in the special case simplifying assumptions - idealizations - are made. In this instance, the special case is Special Relativity (hence the name). And the idealized simplifying assumption is that the acceleration during the turnaround can be allowed to go arbitrarily high so that the time taken to turn around (the time spent in a non-inertial frame) can be allowed to get arbitrarily close to zero. Hence the non-inertial frame is removed.

This is why, when we consider the problem using Special Relativity, we get the sudden, discontinuous jump in the Minkowski diagram, from 2.5 to 17.5 years. It reflects the discontinuity of the sudden change in inertial reference frame. If we allow ourselves to use General Relativity then we can replace that sharp discontinuity in the diagram with a curve. The rate of curvature would show the size of the acceleration. And the jump from 2.5 to 17.5 years would not be all in one go.

### Re: Does Special Relativity contain contradictions?

Posted: October 4th, 2018, 3:39 pm
Editing error in: "It is also clear that if E > S, D must be less than > L."

The second ">" sign shouldn't be there.

### Re: Does Special Relativity contain contradictions?

Posted: October 4th, 2018, 3:45 pm
David Cooper wrote:Maxwell's Equations are not fundamental to this at all...
They, and the experimental evidence that gave rise to them, are the reason for the existence of Special Relativity. There's a little discussion of it here:

viewtopic.php?p=320976#p320976

but a textbook on Relativity (or a general undergraduate level physics textbook) would be much better.