robinpike

If you are going to use Javascript, it can be used for animation by drawing shapes on the html 'canvas' element, which are then redrawn as a loop using requestAnimationFrame().

This works well if you are going to plot points / basic shapes using an equation.

If you want a 3D animation but for simplicity are going to use canvas in 2D, a trick is to calculate the shapes as 3D into an x y z array, include an adjustment for the size of the shapes with regards to perspective, sort the array on what is furthest away from the point of view, and then draw the shapes as 2D onto the canvas in that order. This produces a 3D animation.  Otherwise WebGL can be used for 3D plotting onto the canvas but I have not used that myself.

Notes from w3school https://www.w3schools.com/html/html5_canvas.asp

The HTML <canvas> element is used to draw graphics, on the fly, via JavaScript.

The <canvas> element is only a container for graphics. You must use JavaScript to actually draw the graphics.

Canvas has several methods for drawing paths, boxes, circles, text, and adding images.

On 27/07/2017 at 0:48 PM, tar said:

....  And the rational mind part of humans, the science and the math, and the technological advances and the Turing machines,  laws and religions, came only recently on this planet...in the last 10,000 years or so, and can rightly be thought of as consciousness, outside that that a Zebra is capable of.  And still, even with the advantages that the structure of the human brain brings humans, over Zebras, and the value of the institutions that humans have built using our natural brains and emotions,  we still are, more than 90 percent Zebra, probably. and the various pheromones and hormones and neurotransmitters and body parts and brain parts found in a human are also there in a Zebra.  The differences are slight, but important, and still we have the 90 percent of "instincts" that the Zebra has.  Yet we probably have our first 90 percent of consciousness in common with the Zebra, as well.

 

Regards, TAR

Our ability to be 'more conscious' than say a zebra, would seem to be a result of our ability to communicate using a complex language, rather than because we have a brain that is constructed 'with a greater consciousness'. Imagine a group of modern day people living on the African plains that happen to have never learnt a language. The communication and behaviour between those people would be basic - possibly even similar to the way a group of zebras behave. And yet for this to be so, the construction of their brains need not be changed from that of a modern day person.

This topic talks about the relationship of philosophy, science and reality.

I will expound it thru questions:

...

4. When can we say that a question become philosophical? Can we say that philosophy is an advance science? If yes then we can conclude that the only task of science is to prove philosophy ( is it correct?).

 

DrP,

 

So I take the opposite task and concentrate on the evidence that there is, that you have a mind, regardless of the fact that I can not know this for sure.

 

But follow me here. For me to think I have a mind, I MUST be capable of developing a theory of mind. The ability to view yourself objectively in this fashion, requires having a mechanism within your brain that is capable of taking "you" and putting that observer in someone else's or something else's "place". It is done in science all the time. The very idea of the models we are talking about requires the ability to have one thing stand for another, to consider what it would "be like" to be the thing under study. This operation requires two minds, one's own, and one's own in the place of the entity under study. Einstein had observers all over the place, that were given hypothetical minds. Science has peer reviewers. Other minds, meant to check reality for the same conditions you found. The whole operation, philosophy, science and reality, requires at it's base a mind that is investigating, recording and manipulating the place, and then only when this mind is assumed to be real, is there any reason to continue the investigation. Then the question of whether someone else has a mind, like yours, can be asked...but the fact that there is science (requiring peer review) and language (requiring a second party, or a internal construct,) to share your thoughts with and philosophy (requiring the consideration of other people's thoughts, by definition), and a reality to share with these other minds, already has the question of whether other minds exist, answered in the affirmative.

 

Regards, TAR

for instance, if one scientist makes some progress, but allows for the fact that he/she stood on the shoulders of giants to make such strides...the fact the other scientists existed, and had not only minds, but exceptional ones, is a given

Perhaps the following example shows how science and philosophy can be different.

When we consider quantum mechanics, science is able to model reality with QM maths, matching what is observed with what is calculated by the model. Note that science doesn't have to answer the question: How does the mechanism of quantum mechanics work? Or what is QM?

If we do ask, 'How does quantum mechanics work?', this perhaps leads to philosophical reasoning. For example, on trying to explain QM, we note that it has aspects that are very difficult to explain in physical terms, such as quantum spin, quantum entanglement, etc.

So philosophically, we could use the inexplicable physical aspect of QM to arrive at the conclusion that we exist in a simulation! ...since a simulation could have QM in it without the necessity to have a physical mechanism for that QM (which of course reality must do).

The rock, though, will not believe itself to be human.

 

Believe whatever you want - I'm not going round pages and pages with you repeating the same stuff... it's obvious. It's why I said "oh, never mind" with an implied face palm when I first entered the conversation as I knew you would just stick to your guns and just repeat the same stuff without actually bringing any actual substance to the discussion other than "I don't believe it".

 

I'll ask one more time - "HOW can you prove you are not a simulation"? Just saying "It's obvious, I don't believe that I am" is not proof that you are not.

And to make it clear on the above point, it is not about the likelihood (or belief) of what we are, but what can be proved absolutely.

Wouldn't a programmed simulation not follow identical responses to repeated identical stimuli?

 

I would think that could be one test.

Just to be clear, if you mean day by day the 'person simulator' experiencing the same stimuli, bear in mind that the 'person simulator' would be a learning program, in the same way that when we learn, we may change our response to the same stimuli.

 

We aren't computers.

But in the example given, how would the computer know that it is not a living thing?

what is this insistence that everything could be an hallucination?

 

it can't be

 

There is no way such a situation would work out. And it certainly makes no sense.

Take this example. All the inputs that feed into a computer inside a driver-less car are recorded and these are then input into another computer sitting in a lab somewhere, running the same driver-less car software. Can that computer tell that it is not in a driver-less car, controlling it down the road?

Suppose all the inputs that a person receives during their life were to be recorded and then played back to a computer running a 'person simulator'. Would that 'person simulator' think that they are alive and real? How would the 'person simulator' know that they are a simulator and not a living creature?

When light is outside of a black hole's event horizon and is moving away from the event horizon - I understand that the light is red shifted, but can someone just confirm that the light still moves at the speed of light at all times?

I was reading another post on black holes, and it talked about light 'hovering' near the event horizon and wondered what that meant.

I really think the ant on the balloon is a good analogy. The balloon is expanding under its feet, and if there is another ant along the balloon, the distance the ants have to walk to get to each other along the surface increases, but they are not "pegged" to their position on the balloon. They can move around freely as the balloon continues to expand under their feet. That is essentially what happens with us and space.

Thanks Delta1212. The difficulty with the ant on the expanding balloon analogy and bearing in mind we are in a 3D world, is that it requires our 3D world to expand into the 4th dimension. That introduces the issue of what is dragging us in that direction? In the analogy, the rubber of the balloon is pushing the ant into the third direction. What are we 'standing on' that pushes us into the 4th dimension?

Space is not a substance, just the distance between objects. It turns out that in a universe described by the equations of general relativity (as ours appears to be) that objects will tend to get further apart (*) over time if no force acts on them (and if there is an even distribution of matter). This is slightly counterintuitive, as we tend to a some that things will just stay "where they are". But this is the reason for expansion.

 

Locally, things are held together by gravity and other forces, and so the expansion only occurs over very large scales, where these forces become insignificant.

 

There is no "drag" from space. But it was generally assumed that gravity would gradually slow the expansion and start a process of collapse. But then it was found that the expansion is accelerating. This acceleration is attributed to "dark energy" - something unknown, but that can be modelled as an extra energy throughout the universe.

 

(*) Or get closer together. Depending on energy density, initial conditions, etc.

Thanks Strange, but still don't follow the explanation for the expansion of the universe.

Dark energy seems to be the most common answer when searching on Google, but here are three options as far as I understand their merit / issues...

1) If the expanding universe were to be due to space expanding literally, then that would suggest that things are pegged to their position in space while space in between increases, causing them to drift apart. But since we are able to move through space unimpeded and are not pegged to a position, that explanation doesn't work.

2) If the expanding universe were to be due to the things in the universe shrinking in size - then the distances between them would appear to be increasing - so that would be an alternative way to get the above to occur. But if that were the case, then distance objects would not occur any red shift - and that contradicts what is observed.

3) If the expanding universe were to be due to dark energy pushing things apart, then that seems to be a better fit to what is observed. But a couple of questions.

3a) If dark energy is the explanation, wouldn't distant galaxies be constrained to move apart within the speed of light?

So although at this moment, light from the most distant galaxies hasn't had time to reach us, eventually all galaxies would come into view?

3b) If dark energy is the explanation, how is it able to push against things, and yet when those things move due to local forces, those things do not experience a drag through the dark energy?

Because the escalator example was used to illustrate the principle of local speed limits vs separation rate. It's not a good analogy for every aspect of expansion.

 

In this case, the critical point is that space is expanding per unit of distance. So, to put it in manageable terms: After X amount of time, every mile gains an inch.

 

Someone who is one mile away will then recede at a rate of 1 inch per X. Someone two miles away at 2 inches per X. Three miles at 3 inches per X. And so on.

 

The farther away you get from something, the faster you will move away from it. There is more space between you to expand, and so you get a larger amount of distance between you being added due to expansion in the same amount of time.

 

Eventually you get so far away that, as explained above, the distance is increasing faster than the light is traversing that distance. Like a train with track being laid in front of it faster than the train is moving, it will never be able to reach the end of the line.

Thanks, that explains how light may not be able to reach every part of the universe. Is it the case that we are happy with how the accelerating distance between galaxies occurs? Or are there issues?

For example, if the increase in distance between galaxies is due to 'dark energy' pushing the galaxies apart, wouldn't objects generally moving through this dark energy feel a drag - and slow down? Or is 'dark energy' like photons, in that regardless of your own speed and direction, 'dark energy' moves at the same constant speed to you, regardless of the speed and direction of your own reference frame. But even then, wouldn't a Doppler shift in the amount of energy of the 'dark energy' still occur for some speeds / directions?

Or perhaps is it that space has 'substance' - and it is space itself that is increasing between the galaxies - and dragging the galaxies with itself? But then wouldn't the same problem occur for objects generally moving through space, in that they would experience a 'drag' from the space, and slow down?

Just wondering if these are valid issues or not?

The difference is that neither galaxy is actually moving "through space"; the separation is due to increase in the amount of space between them.

 

One way of seeing this is to consider the fact that as the distance increases then the speed of separation increases but there is no acceleration. You can tell this is the case, because acceleration means a force is applied (you can tell when your car accelerates because you are pushed back into your seat) and there is no fore in this case.

A small point, but to be pedantic... The absence of a 'feel' of acceleration isn't an absolute test that acceleration isn't occurring. For example, acceleration by gravity affects all the atoms in your body - free falling in space in a gravity field doesn't produce a feeling of acceleration.

Have you ever been on one of those moving walkways that are basically like flat escalators that they have in some places, especially airports?

 

So let's imagine a top running speed of 30 mph. Absolutely no one can get any faster than that no matter how hard they try to run. Now let's imagine you try sprinting down one of those moving walkways.

 

You still can't run any faster, but as the walkway carries you along with it, you could certainly recede from someone standing at one end of the walkway faster than your top running speed wild take you. The speed at which you can run around on that walkway still has the same limit it always has had. You may now move away from someone at one end of the walkway at 45 mph, but you still can't run at 45 mph. You're still capped at your normal sprinting rate compared to someone riding along next to you.

 

Similarly, the speed at which anything can move through local space is still capped at c. However, the region of Spaxe you are moving through is growing more distance from the region of space on the opposite end of the universe because of the metric expansion of space. As they become more distance, the space may "carry you along" so that you recede from the other end of the universe faster than the speed of light.

 

You still can't move any faster than normal through space, but the space you are moving through is not restricted to the same speed limit, so to speak.

If I am understanding this correctly...

IF space were to be expanding at a constant rate, then the amount of separation between two galaxies is irrelevant, for the rate of separation between the two galaxies would remain the same. [in the escalator example above, when you run, either you can always reach the other person, or you can never reach the other person. The starting distance between you and the other person is irrelevant.]

So, even when the distance between the two galaxies is very large, with a constant rate of separation, and assuming a rate of separation less than the speed of light, then light can always reach any other object. The only impact of two galaxies being a great distance apart, is that the light will take longer to reach the other galaxy.

Please can some one step through the explanation of how expansion (whether at a constant rate or increasing over time) is able to prevent light from reaching a distant galaxy? I do not understand how that is possible.

"Gravity that we experience on earth is due to the slowing of time on earth"

I must say that I have no idea what this is about. I have admiration and respect for Kip Thorne so I assume its a either a poor word choosing or theres some more meaning to this. I dont get it.

I think this refers to the suggestion that if objects with mass were to 'prefer' to be in a state with the least rate of time, then this would cause objects with mass to move towards other objects with mass. Since mass bends space-time, and time runs slower the nearer the objects are to each other, thus the attraction attributed as gravity could be just the objects moving so as to be in a state of lesser time. I don't know if this idea has any merit.

Everything could seem shocking because, in fact, it should be nothing, not even the Universe?

 

In a thoughtful day.

 

Everything as the Universe itself... and all its content.

I read this as meaning that logically it is impossible for the universe to exist - therefore it is amazing (shocking) that it does exist.

The steps in logic being...

i) If starting from a state of true nothingness, then it is not possible for the universe to come into existence.

ii) Therefore the alternative must be true - that the universe started from a state of something.

iii) But that then leads to the question of where did the "something" itself come from? and we are back to i)

Hence a logical paradox that must be flawed - since the universe does exist.

AI is intrinsically limited by the intelligence of its programmers, so I guess its not possible for machines to "modelize" the unknown using

algorithms based on mathematics and physicalism. However, AI can certainly extrapolate models (I prefer the term "theories") of the universe but

could probably not produces hard scientific evidences about its validity.

Evolution of us is an example that intelligence is not needed to create intelligence. There is no intelligence driving what changes should / do occur to a life form.

Since entanglement versus classical pairing predicts different results when the filters are at the same angle, does anyone have a reference to those experimental results? Those results will be easy to analyse.

I don't have any handy. Maybe when I'm back at work.

Thanks, seeing the results for such a straight forward experiment would be really helpful - I've only been able to find results for when the polarizing filters are at angles.

At the moment, I have a mental block on quantum entanglement being real, but on seeing the results, would accept it in preference to 'spooky action at a distance'.

 

Bell's test was devised to show that EPR's idea of local hidden variables was incorrect. Simple tests that show a difference between classical predictions and quantum mechanical measurements would still fall foul of EPR's claim that the results could flow from some non-classical but still locally hidden variable. What was Bell's genius was to realise that a simple statistics test can show the possibility of a set of data ie which distribution of results can theoretically be produced from a single joint distribution.

 

Simplistcally, EPR claimed there could be no spooky action at a distance (this was the bit of entanglement they attacked) and that all the predictions of entanglement worked just as well if you had a simple locally set and valid hidden variable. Bell's Theorem provided a way to test this - but you have to be in a regime in which the distributions will be different for a local hidden variable and for a pair of qm entangled particles; thus the Aspect experiment etc which do leverage this statistical fact

Thanks, this is starting to become clearer as to what local hidden variables means and why something was needed to discount them, i.e. Bell's inequalities.

 

 

Yes, that's pretty much exactly the experiment that has been done, many times. Since entanglement is still part of mainstream physics, what do you think the result was?

I've only seen results that support entanglement... but those experiments do not have the polarizing filters at the same angle. Those experiments typically have the polarizing filters at a difference of 120 degrees and Bell's inequalities are used to assess if the results agree with entanglement or classical pairing. The analysis of those experimental results are complex to follow.

Since entanglement versus classical pairing predicts different results when the filters are at the same angle, does anyone have a reference to those experimental results? Those results will be easy to analyse.

Indeed, why is it even necessary to devise a complicated experiment involving Bell's inequalities?

(Note: orthogonal, or "opposite" polarizations are at 90º, not 180º)

 

If you have an entangled pair, you know the correlation of the polarizations, but not the polarization of a particular photon until it's measured. So saying 50% make it through the filter is inot the whole story. That's true if you have a known polarization passing through a filter that's at 45º to that polarization. But that also applies to the second filter as well — 50% chance of making it through. But that means 25% of the time you get both photons, 25% of the time you get no photons, and half the time you get one photon at one detector but not the other. The photon polarizations are not correlated.

 

If the photons are entangled, you always get the second photon, if you detect the first one, and never get the second photon if you don't get the first one.

Thanks Swansont. Is it possible to perform the experiment such that the photon pairs can be detected as individual hits on their respective photon detectors and therefore the hits correlated as being from the same photon pair? If so, then entanglement will produce pairs of hits being detected at the same time (for a perfect experiment, entanglement will produce detection at the same time or both detectors nothing). Whereas classical pairing will, as you mention, produce 25% of the time you get both photons, 25% of the time you get no photons, and half the time you get one photon at one detector but not the other.

Assuming the above to be correct, has this type of experiment be performed? If so, did it demonstrate entanglement or classical pairing?

I find it difficult to follow the analysis on experiments that investigate quantum entanglement, there seems to be something fundamental about quantum entanglement that I am missing.

Perhaps if I step through a simple example, it will show up what it is that I am misunderstanding.

Say a source of entangled photon pairs are used in an experiment, such that the photons pass through two vertically orientated polarizing filters, positioned either side of the photon source, with one filter slightly further away from the source than the other. For the sake of simplicity, it is assumed that the polarizing filters are perfect - that any photons that make it through are polarized to the orientation of the polarizing filter.

For the photon of an entangled pair that reaches its filter first, its wave function collapses and its orientation is at some, random angle to the filter. At that same moment, the other photon collapses its wave function too - and its orientation is at 180 degrees to the other photon's orientation.

The probability of a photon passing through a filter is dependent on its angle of orientation, and with an equal probability of any angle occurring, 50% of the photons will make it through the filters, and 50% of photons will fail to make it through the filters.

Except for the special cases when the photons are at 0 degrees to the filters, or at 90 degrees to the filters - where either both photons pass through the filters, or both photons fail to make it through the filters, there is a probability of whether a photon is absorbed or not, based on the angle of the photon's orientation to the filter. In these cases, there is no guarantee that if one photon of the pair makes it through its filter, then the other photon will too, and vice versa.

So how is this different to the classical view, where the photon pairs are emitted with an orientation that is at some random angle to the filters, with the orientation of the two photons being at 180 degrees to each other? Or is it, that for this particular set up, there is no difference between the two explanations?

 

I have no strong opinion if there is indeterminism in the world. Generally physics appears to function with cause and effect. However some processes look truly random. As a result I simply don't know if we have a free will or not.

Can such a thing be proven?

Tim, even if some processes have random outputs, I don't think that will give us free will as we have no control over that output?

 

In my opinion when indeterminism would be true, it would mean we have no (or less) free will. I really hope that I am determining my actions. If there would be no causal relationship between my thoughts, plans, desires etc on one side, and my actions on the other,, there would not be a way to be free at all.

Eise, evolution means that our brains are constructed so as to produce rational thoughts but nonetheless we are constructed from atoms. So yes, atoms can produce rational reasoning but I don't see how atoms can produce free will?

A culture's ethics are subjectively derived, thus, it follows that it's citizens ethics are subjectively derived, generally.

Something that is lawful, or something that is accepted by a majority, does not make that something ethical. Ethical and lawful are separate things - although hopefully the two overlap.

Could not possibly disagree with this more vehemently or passionately if I tried. Morality is NOT relative, especially when it comes to life/death. You cannot even get a moral code up and running without this basic, fundamental understanding that things like reciprocity and concern for fellow man leads to propagation of the species, and a better experience doing so, and that killing (even for retribution) does not benefit this, but damages it. By your position, the governments of medieval Europe "had their own morality," so they were okay. I guess North Korea is also okay in the way they treat their people? And the various theocracies of the middle east. They've got their own morality, who are we to judge? I cannot understand this way of thinking. No, morality is not relative. Even if people can form their own opinions on what's moral and what's not, morality still isn't relative. Abu Bakr Al Baghdadi definitely thinks what he's doing is moral, but he's objectively wrong. What he does leads to suffering, pain, death, and detriment to human life, and a threat to the human species, as do the others mentioned above and others like them.

As Tampitump says, ethical / moral has a predefined intention - it is not relative. Ethics deals with such concepts as: upholding rights that are self-evident, protecting those that need protecting, ensuring fairness, etc .. Legalities deal with such concepts as what is allowed / not allowed by a society, punishments etc...

.

That's not language we use on this forum. You came up with subject to discuss, and you have to deal with opponents with different view than yours.

 

 

That's very simple if we start with laser and polarization filter.

Laser is sending beam of photons with random polarizations, or random circular polarizations.

Then they interact with polarization filter.

50% of beam of photons is reflected,

50% of beam of photons pass through.

Polarization Filter.png

If you have just single photon, which way it will go?

 

Suppose so that after reflecting there is photodiode, or other optoelectronics, which is triggering nuclear explosion (similar case like in Schrödinger's cat)

 

You're sending single photon with random polarization.

There will be explosion or not?

It could be that it only seems to be indeterminate because we do not have enough knowledge of the system.

Take a double pendulum for example. If all you can see is the tip of the pendulum swinging under the glare of a strobe light, you could be forgiven for thinking that you are watching random movement that over time obeys a probability wave function.

https://en.wikipedia.org/wiki/Double_pendulum#Lagrangian

I think my original question was incorrect - as my concerns are probably with explanations of relativity (such as space-time etc) and not relativity itself.

Since it would be incorrect to now re-direct this thread along those lines (and there are a couple of space-time threads currently on-going anyway), I would just like to conclude by thanking everyone for their help on this discussion.

 

You got it correct, except for the last sentence: when d catches up with b, it doesn't show anything about a.

After being synchronized with a, d lost time wrt b.

 

a[0] ->

<- b[0]

----------------------------------------

<- b[1] a[1] ->

c[1] ->>

<<- d[1]

----------------------------------------

<- b[3] a[3] ->

<<- d[2] c[2] ->>

 

Note that, according to b, a will also lose time, but b will have to calculate that, or send EM signals to verify. And according to a, b will lose time.

 

You will never find a contradiction, but I understand your search. Those concepts are frustrating, and SR doesn't explain. It's more: "Don't think about. It works, so accept it."

 

Thanks for spotting that. Even though relativity isn't an explanation (nor does it have to be in order for its equations to work and to be useful), I am still thinking about whether there is a logical contradiction in there somewhere. There is something that doesn't seem to be quite right when trying to apply an explanation of what happens to a clock when it changes from one reference frame to another. But since relativity is not an explanation per se, it could well be that any such issues would be irrelevant to relativity anyway.

 

After a hasty read I see that you made a quantitative beginner mistake, which you can easily spot if you plug in 0.5c for "0.5s" in your symmetrical point of view, and then transform back to your first view, using your calculus: you will find that clocks are supposed to move at the speed of light! Transformations of relative velocities are more complex in SR than in classical mechanics.

If you won't use the exact numbers then it's an irrelevant glitch, but else it needs correcting.

I thought it would be easier to follow if I plugged some numbers in! Never mind, in that case best to just ignore those example speeds.

To reiterate what Tim88 stated, the measurements are relative, so you can only state that one clock ran slow as compared to another, which means that the other ran fast in comparison. There is no absolute time that you can use as a basis for the analysis.

 

 

There is no contradiction here. Again, the issue is that clock A runs slow relative to clock B in clock B's frame, and B runs slow relative to A is A's frame.

 

Your error is ignoring that clock rates are not invariant values. Your invalid assumption in your logic is that they should be. A simpler example:

 

In A's frame it is is rest and has no kinetic energy. B is moving, and has KEB. KE_B > KE_A (=0)

 

In B's frame it is is rest and has no kinetic energy. A is moving, and has KEA. (0=) KE_B < KE_A

 

Is there a contradiction in those statements? You can argue that they can't be true simultaneously, but you can't be in two frames at once, so that's not a problem.

Putting together the above and all the other points people have made to help me, I believe the following shows how two clocks moving relative to each other can both lose time with respect to each other...

So that we do not have to worry about clocks accelerating / decelerating, as someone mentioned previously, passing clocks can be used to communicate / compare times.

So the stay at home / travelling clock example can be stated like this...

The stay at home clock (a) is at rest and a travelling clock (b) flies past the stay at home clock, say from right to left at speed '1s'. As the travelling clock passes the stay at home clock, the two clocks are synchronized. The travelling clock continues on its way to the left at speed '1s'.

A second travelling clock ( c ) that is moving from left to right at speed '1s', passes the first travelling clock, and at the moment of passing synchronizes its clock with the first travelling clock. This second travelling clock continues left to right at speed '1s' until it passes the stay at home clock, at which point its time is compared to the stay at home clock's time. It is found that the travelling clock ( c ) has lost time compared to the stay at home clock (a).

But who is to say which clock is the 'travelling' clock and which clock is the 'stay at home' clock? To demonstrate that point, here is a slightly extended version of the above scenario...

This time, instead of describing the stay at home clock's reference frame as stationary (i.e. speed '0s'), it will be considered to be moving from left to right at speed '0.5s' and the travelling clock (b) moving at speed '0.5s' from right to left. This is still the same scenario as before, as the relative speed between the two clocks is still '1s'. [Any reference frame could be used to state the relative speeds of the clocks, but this one conveniently emphasizes the symmetry between the stay at home clock and the travelling clock.] As before, when the travelling clock (b) passes the stay at home clock (a), the two clocks are synchronized.

The second travelling clock ( c ) that is moving from left to right at speed '1.5s', passes the first travelling clock, and at the moment of passing synchronizes its clock with the first travelling clock. The second travelling clock continues left to right at speed '1.5s' until it passes the stay at home clock (a), at which point its time is compared to the stay at home clock's time. It is found that the travelling clock ( c ) has lost time compared to the stay at home clock (a).

And to show that it is equally valid to consider that it is the stay at home clock (a) that loses time compared to the travelling clock (b), a fourth clock (d) travelling from right to left at speed '1.5s' can pass the stay at home clock (a) and synchronize its time with the stay at home clock.

When this travelling clock catches up with the travelling clock (b) and their times compared, it is found that the stay at home clock (a) has lost time compared to the travelling clock (b).

So if I have got the above correct, then what I need to demonstrate is does the above lead to a logical contradiction?