robinpike

Perhaps, but I would prefer not to use a scenario where different observers may measure the travelling clock in different ways. If I keep to when the travelling clock begins next to the stay at home clock, and ends its round trip, again next to the stay at home clock, then at least all observers can agree that the travelling clock lost time compared to the stay at home clock. That may make it easier to find if a logical contradiction has occurred? Also it is the acceleration of those two events that results in the travelling clock making different progress through space-time (shorter / as opposed to back to not shorter).

Yes mentioning the direction of the acceleration was a bit muddled - probably because my thoughts went on to mention the comparison about moving from left to right, and right to left. Of course, the travelling clock can choose to arrive back at the stay at home clock from any direction. The bit that I am thinking about is the consequence of the initial acceleration away from the stay at home clock, and the final acceleration that arrives back, stationary again next to the stay at home clock. The first acceleration puts the travelling clock onto a progress through space-time that is shorter than the stay at home clock's progress through space-time, whereas the final acceleration puts it back onto the same progress through space-time as the stay at home clock. Is it logically consistent that those two accelerations can do that? That is where my thoughts are at the moment.

Yes, thanks. What I've written isn't right. The second declaration isn't a declaration at all - it is a deduction based on this declaration below that is missing... 2. The travelling clock is the clock that undergoes acceleration to cause the two clocks to move apart and return back together. So then it becomes an additional deduction... 1. The time difference between the two clocks at the end of the journey is caused by a loss in time for the travelling clock (as opposed to a gain in time by the stay at home clock). Yes thanks, I am going to have to be careful about things like that so as to avoid false contradictions. The next steps are going to take a bit of thought. The reason for those first steps though, is that I wanted to identify deductions that are valid on their own merit - which means that they should remain valid even when the next steps include deductions provided by relativity. The part that I am currently thinking about, is when the travelling clock undergoes acceleration. When the travelling clock first moves away from the stay at home clock, relativity doesn't care about which direction the travelling clock moves off in. And yet the direction of acceleration does seem to be relevant when the travelling clock applies further acceleration to turn around and finally at the journey's end, when the travelling clock comes to a stop next to the stay at home clock. Someone did already mention there is no contention with that, because it is like moving to the left and then moving right to get back to where you started. Or alternatively moving to the right first, and then to the left. Nothing special about positions in space. But my line of thinking is a bit more subtle than just how the acceleration affects the travelling clock's position in space. The travelling clock ends up on a shorter route through space-time after the first acceleration. And yet the final piece of acceleration at the end of the journey causes the travelling clock to return back to the stay at home clock's (longer) route through space-time. So I am just having a think about whether there is any logical inconsistency in the effects of those two accelerations? Of course the expected outcome is that there is no logical inconsistency - but that is what I am going to think about. Yes thank you Tim for noticing that slip up - your correction is what I meant to write.

True, special relativity does give us that information. However, the starting statements do not mention (special) relativity because the initial deductions need to be independent of any assumptions connected to relativity. Any assumptions / premises that relativity requires / predicts will be included in the upcoming steps in logic, as that is when I think the logical contradiction occurs - and then it can be discussed as to why that logical contradiction is flawed / not flawed etc. Before I put together the details of the argument (and as Swansont suggests, I'll try to think of a way to keep it as simple as possible), To summarize, the focus will be on when the travelling clock changes its progress through space-time at the start of the round trip, and when the travelling clock changes its progress through space-time at the end of the round trip. I think those two steps present a logical contradiction for relativity. But please wait for me to post the logical steps first before explaining why there is no logical issue. =============================================================== To help clarify how it is even possible to have a logical contradiction when relativity agrees absolutely with experiment, here is a pretend example of a logical argument... In the above graphic, the mass of the earth causes space-time to bend. If someone were to explain gravity as an object following the curved lines of space-time, then that explanation has a potential logical contradiction. For if gravity is a result of curved space-time, then what is the force that is distorting the space-time in the first place? Before anyone replies - that is just a made up example to help show what a logical argument is.

Here are the first steps in the logic, now rewritten using better phrasing. First statements... When the travelling clock completes its round trip journey, it has less time on its clock than the stay at home clock. This is caused by a loss in time for the travelling clock (as opposed to a gain in time by the stay at home clock). That the travelling clock lost time is agreed by all observers. All observers agree the amount of time lost with respect to the stay at home clock, but it is possible for an observer to measure a different amount of lost time when measured by their own clock. From the above statements the following deductions can be made... At some point in the travelling clock's round trip, the travelling clock lost time compared to the stay at home clock. It is not necessary to identify at which parts of the travelling clock's journey that the time was lost, or the mechanism as to how that time was lost, for deduction 1 to be valid. It is not necessary to state how much time was lost, for deduction 1 to be valid. What cannot be deduced from these first statements (i.e. in the absence of any further statements) is whether the travelling clock at all points on its journey lost time with respect to the stay at home clock, or whether there were some parts of the journey when the travelling clock was gaining time with respect to the travelling clock. I will continue with the next steps in due course.

Point taken, I can see that using the terms 'apparent' and 'real' are bad choices and confusing for this discussion (and probably for any discussion). I will re-phrase the initial statements and deductions in a different way. Let's see if I can do it based on measurements and point of views according to observers.

Tim88, I am at work at the moment and may not be able to reply for a while. I need to read your post carefully before commenting further. This is important to me to sort out, as I want to remove discussions that are at cross purposes.

Thank you Tim88. I think the issue here is a wording one. When I talk about an apparent time loss (or apparent time gain), I am not excluding the presence of a real time loss (or a real time gain if that is appropriate). So if I understand you correctly, I agree with your point. A further conclusion (which please discuss if you disagree), is that it is not necessary to include an apparent time loss in order to deduce the real time loss - they are separate things. When a real time loss occurs, the presence of the real time loss cannot be 'denied' by any particular observer's point of view. Whereas for an apparent time loss, this can be dependent on a particular observer's point of view. So to use the example of two objects in motion, moving at different speeds. This means that the distance between them is changing and thus the change in distance is a real effect. But as to how much that distance is changing, or which one (or both) are moving, or even what their apparent closing speed (or separation speed) is, can be dependent on an observer's point of view.

Describing the whole argument in one post is making the subsequent discussions too difficult for me to ascertain which step(s) in the logic are being refuted. Here are the first steps in the logic. Even at this stage of the discussion I do not know if any of these deductions are being disputed, so continuing in discrete, sequential steps can only help. First statements... When the travelling clock completes its round trip journey, it loses time compared to the stay at home clock. This is a loss in time for the travelling clock (as opposed to a gain in time by the stay at home clock). This loss in time is not an apparent loss - it is a real loss in time. From the above statements the following deductions can be made... At some point in the travelling clock's round trip, the travelling clock loses real time with respect to the stay at home clock. It is not necessary to identify at which parts of the travelling clock's journey that real time is lost, or the mechanism as to how that real time is lost, for deduction 1 to be valid. It is not necessary to identify by how much that real time is lost, for deduction 1 to be valid. It is not necessary to include any apparent loss in time (or any apparent gain in time) during the travelling clock's journey for deductions 1, 2 and 3 to be valid. What cannot be deduced from the statements 1, 2 and 3, is whether the travelling clock exclusively loses real time, or whether at any point in its round trip, it gains real time with respect to the stay at home clock. All that can be deduced with those statements is that overall the travelling clock loses real time - and therefore at some point during its journey, it loses real time with respect to the stay at home clock, i.e.deduction 1.

Tim88, sorry - it is not always obvious for me to know which posts to reply to. Let me review your replies and I will get back to you. And Swansont - I will reply to you as well of course - I'm at work at the moment - so will reply as soon as I can.

To help steer the discussion, if I am reading the counter arguments correctly, then my logical argument is being refuted on the basis that relativity itself has no issue with the scenario under discussion… Summary: When the two travelling clocks are on their journey, coasting side-by-side away from the stay at home clock, their rate of time is seen as slow by the stay at home clock. When the first travelling clock decelerates so that it becomes stationary with respect to the stay at home clock, the stay at home clock sees the first travelling clock’s rate of time return to its own rate of time. At the same time, the second travelling clock sees the deceleration of the first travelling clock as an acceleration away from itself, and so sees the rate of time of the first travelling clock slow down with respect to its (i.e. the second travelling clock) own rate of time. The steps in logic that appear to being used to refute my logical argument being... The above scenario is in exact agreement with what relativity predicts and is in exact agreement with what is measured by experiment.Therefore relativity is correct.Relativity is correct, therefore its premises are correct.Using the premises of relativity, robinpike’s steps in logic lead to a logical contradiction.Since relativity is correct and robinpike’s logic leads to a logical contradiction, therefore robinpike’s logic is flawed.The above logic proves that robinpike’s logic is flawed and therefore it is not necessary to further identify the step in his logic that is flawed.========================================================= This does not mean that I am not grateful for the examples of relativity that have been provided – I am – but please can the step in my logic that is being refuted be identified. For convenience, here are the steps… 1. To demonstrate the logical contradiction, I will start with the (most) obvious method of how time is lost: the travelling clock’s time runs at a slower rate (overall) during its round trip. Let’s see how this assumed method causes the issue by using one stay at home clock and two travelling clocks. By the way, if you dislike the examples that I have chosen, by all means post an alternative, describing how the travelling clock loses time and where in its journey that loss in time occurs, and I will show how the logical contradiction applies to your example. 2. The two travelling clocks are initially stationary and synchronized against the stay at home clock. They start the round trip by accelerating side-by-side away from the stay at home clock (i.e. they increase their relative speed with respect to the stay at home clock). Since the assumption (in this example) is that the real loss in time is due to the travelling clock’s time running slow, during the trip to the half-way point, their clocks will now be running slow (with respect to the stay at home clock). 3. At the half-way point, one of the travelling clocks decelerates (i.e. reduces its speed with respect to the stay at home clock) to the point where it becomes stationary with respect to the stay at home clock. This means that the travelling clock’s time now runs at the same rate as the stay at home clock’s rate of time. 4. But when the travelling clock decelerated, it effectively accelerated away from the second travelling clock (i.e. it increased its relative speed with respect to the second travelling clock). But such an acceleration is just a repeat of the scenario when the travelling clocks first accelerated away from the stay at home clock (although now with respect to different reference frames). This means that the decelerated clock’s time has to run slow with respect to the second travelling clock’s rate of time, which already is running slow with respect to the stay at home clock’s rate of time – AND yet the decelerated clock’s rate of time has to be at the same rate as the stay at home clock’s rate of time (because they are stationary with respect to each other). 5. If these two conditions were apparent effects – such a scenario would be possible. But the loss in time is real and so these two conditions cannot occur together – hence a logical contradiction has occurred. 6. You can try to avoid this logical contradiction by explain the travelling clock’s real loss in time by other methods, but I think they will all fail for the same reason. 7. For example, try the loss in time is because the travelling clock’s progress through space-time is shorter than the stay at home clock’s progress through space-time. This fails at the turn around point because the decelerated clock’s progress through space-time has to be shorter with respect to the second travelling clock’s progress through space-time, which already is shorter with respect to the stay at home clock’s progress through space-time – AND yet the decelerated clock’s progress through space-time has to be at the same rate as the stay at home clock’s progress through space-time (because they are stationary with respect to each other). If you require more detail on a particular step, please ask. To avoid the temptation of gamesmanship, please, there is no need to suggest that I do not understand relativity, or that the steps are so flawed that it is impossible to comment. If you genuinely feel that is the case, may I suggest that you explain why the first step (for example) is flawed. For example: J.C.MacSwell commented on step 4. With “In this case you are assuming all frames have the same absolute time.” My reply is... that is not the assumption. The assumption in step 4. is that the travelling clocks lose time with respect to the stay at home clock because the travelling clocks are on a round trip away and back to the stay at home clock. ============================================= The following may be of interest to those trying to understand flaws in logic and why it is always beneficial to identify the flaw. This one is an example in the fallacy of exclusive premises… No cats are dogs. Some dogs are not pets. Therefore, some pets are not cats. (This is the step with the false reasoning.) If you cannot see why the above step is false reasoning, here is a more obvious example… No planets are dogs. Some dogs are not pets. Therefore, some pets are not planets. In the second example, the physical difference between a dog and a planet has no correlation to the domestication of dogs. The two premises are exclusive and the subsequent conclusion is nonsense, as the transpose would imply that some pets are planets. Here is a link to the Wikipedia article on logic… https://en.wikipedia.org/wiki/List_of_fallacies#Formal_syllogistic_fallacies

The loss in time refers to after the instantaneous acceleration/deceleration, when the first travelling clock is now stationary with respect to the stay at home clock, and now coasting away from the second travelling clock. At this stage, the first travelling clock is now losing time with respect to the second travelling clock but not losing time to the stay at home clock. For the logical issue to be discussed, it is not necessary to continue with the scenario. Of course if the scenario is completed, then the travelling clock's round trip is continued by further accelerating/decelerating so that it can return to the stay at home clock. But also - and relevant - the first travelling clock could just as easily accelerate back to the second travelling clock to complete a round trip with respect to that clock. Whichever scenario is completed, the 'round trip' clock will have a loss in time to the clock that it returns to. When an observer sees a clock ticking faster, it is an apparent increase not a real increase in the clock's tick rate.

In what way do you mean I am assuming all frames have the same 'absolute time'? I note that the two travelling clocks lose time with respect to the stay at home clock's time. When the first travelling clock de-accelerates, it loses time with respect to the second travelling clock, and by the same action, returns to not losing any further time with respect to the stay at home clock. Since the second travelling clock is losing time with respect to the stay at home clock, is there anyway that both can be achieved? I am suggesting that it is not possible (within the remit of relativity). What everyone agrees on is that the travelling clock loses time on returning to the stay at home clock. With reference to the travelling clock and the stay at home clock, if the travelling clock does not lose time during the outward portion of the trip (and on the return portion of the trip) - then when does it lose time? Is your concern that different observers can see that loss in time in different parts of the travelling clock's journey? Note that the argument only needs to consider the real loss in time of the travelling clock - not the apparent gains / loss in time as seen by different observers. In my discussion I chose the point of view of the first travelling clock when it de-accelerated because it is this clock and at this point in its journey that the logical contradiction can be seen. Of course, please think up ways to test and challenge this logical argument and I will try to rebuff them.

Tim88 - sure your clarifications are most helpful. The key thing to keep in mind when discussing the steps in the travelling clock's round trip, is that the loss in time for the travelling clock is a real change. Of course, there are apparent changes going on as well - and clarity is required to not inadvertently merge these two things together. By the way bvr, thank you for taking time to understand the points in my post - much appreciated.

By choosing to measure only whether my brother's height has changed after his jog - and not by how much - removes the necessity to compare rulers. As long as nothing happens to the rulers (that is they do not move in the context of relativity), then each ruler can be used to detect whether my brother's height has changed or not changed.

Can you give details of the reality - that will enable me to comment - thanks.

I will try to be as clear as possible so that my use of the terms “apparent” and “real” are not misunderstood. By way of example, the following context shows how I intend to use these terms… If I measure the height of my brother, who say is standing on the far side of an athletics stadium, by sighting him against a ruler held at arm’s length, his height perhaps comes out as 1 inch. Although that is a real measurement, it is only an apparent height. To determine his proper height, I can do some calculations based on distance and perspective and from those infer that his height is 6 feet, but this would still be an apparent height (this apparent height could be the same as the real height, but it doesn’t have to be). To be sure of getting his real height, I need to walk up to my brother and measure his height while standing next to him, which say comes out as 6 feet. This measurement of 6 feet then is his real height (although you might note that this value is only real with respect to the feet and inches marked on my ruler). If my brother then does a lap around the track and on returning to me, I measure his height again when he is standing next to me and find that his height is now 5 feet 11 inches (his shoes fell off while running) – that measurement is now the new real value. We know that it is a new real value because the ruler has not changed - so therefore it is his height that has changed. But his running coach disagrees with my measurement – using his ruler the coach stands next to my brother and measures my brother’s height as 5 feet 10.5 inches. So which measurement is to be taken as my brother’s real height? To side-step this dilemma, the coach and I use our respective measurements before and after my brother ran around the track, to determine only whether my brother’s height got shorter, or got taller, or did not change. So now the coach and I can agree: my brother’s height got shorter and that this change in his height was a real change. This avoids any arguments as to the amount that his height changed, and who has the real measure of feet and inches on their ruler. =============================================================== I thank people for the diagrams posted, but if I may, I would like to continue with a verbal description of the issue (bearing in mind the above example of how I want to use the terms apparent and real). To briefly summarise the travelling clock scenario… The initial conditions are that the travelling clock and the stay at home clock are stationary next to each other, both ticking at the same rate, and both showing the same time. The final position is that the travelling clock has completed its round trip and is again stationary next to the stay at home clock, both ticking at the same rate, but now the travelling clock shows less time than the stay at home clock. When the travelling clock goes away and comes back to the stay at home clock, the one thing that can be stated with certainty as being real, is that the travelling clock lost time with respect to the stay at home clock. When the travelling clock starts its journey and accelerates away from the stay at home clock, somewhere along its journey the travelling clock must lose time compared to the stay at home clock (rather than the stay at home clock gaining time), because the loss in time is real and nothing changes for the stay at home clock. From the point of view of both clocks, during the first part of its journey, the travelling clock is moving away from the stay at home clock. But once its acceleration stops, who can say which clock is moving and which clock is stationary? In fact, while the clocks are moving away from each other, both clocks see the other clock as running slow. After the travelling clock has turned around to return back to the stay at home clock, both clocks now see the other clock as running fast, although there is a period at the beginning of the return trip when the travelling clock sees the stay at home clock running fast while the stay at home clock still sees the travelling clock running slow. =============================================================== Note that there is no issue with the calculations of relativity, as these agree with the amount of time lost by the travelling clock, with respect to the stay at home clock. The issue is a logical contradiction that appears to be caused by relativity, revealed when a method is chosen as to how the travelling clock loses the time. Working through the chosen method reveals a contradiction in logic – which at first simply suggests that the chosen method is invalid. So another method is chosen, and again the same contradiction in logic is found, and so on. This leads to the general conclusion that any chosen method will result in the logical contradiction. But the loss in time is real so there must be a method by which the travelling clock loses the time. This suggests that the fault lies somewhere else – and the obvious candidate for that is relativity itself. To demonstrate the logical contradiction, I will start with the (most) obvious method of how time is lost: the travelling clock’s time runs at a slower rate (overall) during its round trip. Let’s see how this assumed method causes the issue by using one stay at home clock and two travelling clocks. By the way, if you dislike the examples that I have chosen, by all means post an alternative, describing how the travelling clock loses time and where in its journey that loss in time occurs, and I will show how the logical contradiction applies to your example. The two travelling clocks are initially stationary and synchronized against the stay at home clock. They start the round trip by accelerating side-by-side away from the stay at home clock (i.e. they increase their relative speed with respect to the stay at home clock). Since the assumption (in this example) is that the real loss in time is due to the travelling clock’s time running slow, during the trip to the half-way point, their clocks will now be running slow (with respect to the stay at home clock). At the half-way point, one of the travelling clocks de-accelerates (i.e. reduces its speed with respect to the stay at home clock) to the point where it becomes stationary with respect to the stay at home clock. This means that the travelling clock’s time now runs at the same rate as the stay at home clock’s rate of time. But when the travelling clock de-accelerated, it effectively accelerated away from the second travelling clock (i.e. it increased its relative speed with respect to the second travelling clock). But such an acceleration is just a repeat of the scenario when the travelling clocks first accelerated away from the stay at home clock (although now with respect to different reference frames). This means that the de-accelerated clock’s time has to run slow with respect to the second travelling clock’s rate of time, which already is running slow with respect to the stay at home clock’s rate of time – AND yet the de-accelerated clock’s rate of time has to be at the same rate as the stay at home clock’s rate of time (because they are stationary with respect to each other). If these two conditions were apparent effects – such a scenario would be possible. But the loss in time is real and so these two conditions cannot occur together – hence a logical contradiction has occurred. You can try to avoid this logical contradiction by explain the travelling clock’s real loss in time by other methods, but I think they will all fail for the same reason. For example, try the loss in time is because the travelling clock’s progress through space-time is shorter than the stay at home clock’s progress through space-time. This fails at the turn around point because the de-accelerated clock’s progress through space-time has to be shorter with respect to the second travelling clock’s progress through space-time, which already is shorter with respect to the stay at home clock’s progress through space-time – AND yet the de-accelerated clock’s progress through space-time has to be at the same rate as the stay at home clock’s progress through space-time (because they are stationary with respect to each other). .

Swansont - I agree. Let me think about how to state the questions so that they are not misunderstood / discussed at cross purposes. I don't want this to just peter out - I want to understand that there is no (logical) issue with relativity, or if there is, then what is the argument that reveals that logical issue. PS Studiot - I haven't been ignoring you! A lot of cross purpose replies were happening (inadvertently) and I couldn't delve into everything that was being suggested, while I was trying to figure out why.

Dictionary definition: Apparent - seeming real or true, but not necessarily so. Dictionary definition: real - actually existing as a thing or occurring in fact; not imagined or supposed. An apparent value is different to a real value by definition. Swansont are you being deliberately obtuse? Yes I know the travelling clock ran slower. I asked how (not what). What you have done is replied a 'what' reply to a 'how' question!? For example: How does gravity work? And the reply: It pulls you towards the ground. Does not answer the How question! Relativity equations do not distinguish between apparent or real - relativity is a set of equations that equate to what is measured.

Well, maybe it leads to questions rather than misunderstandings. An effect that is classified as real and an effect that is classified as apparent are not artificial classifications - apparent and real mean different things. I think a better way to state this would be to say that apparent and real effects are both in the classification relativistic?

Swansont - I do not understand how the equations distinguish between an apparent value and a real value? When you say the 'answer', to what question? I can understand for the question: How much time did the travelling clock lose? But what if the question is: How did the travelling clock lose time? What is the answer (with certainty) to that question?

I think I have just realized why this discussion has become so difficult to resolve. When I used the description ‘unfortunate observer’, the responses just used ‘observer’. I wondered why? Could this be because the equations of relativity don’t care about the history of the travelling clock and don't care about which part of the observed value is apparent / real? The equations describe what is being observed now (by an observer), but they do not break the observed value into an apparent value and a real value. (Of course, real effects are not denied by the equations, such as when the travelling clock is compared to the stay at home clock at the end of its round trip.) So when I asked “how does the travelling clock lose time?” how can the equations of relativity reveal how? Sure an explanation could be inferred from the equations, such as shorter route through space-time, or the travelling clock ticks at a slower rate, or its length contracts, or a combination of all of those perhaps – but the equations cannot be used to deduce with certainty the answer? Does the above make sense?

So to the next step that I want to understand. (Apologies if I my phrasing is not always as it should be - I am trying my best to use the correct terms.) When the travelling clock is at rest and next to a stationary clock, the two clocks can synchronize with each other. At this point, we / all observers can note that their routes through space-time must be the same, that is they have the same spatial co-ordinates and their progress along their timelines stay abreast with each other. The travelling clock then goes on its round trip, eventually returning back to the stationary clock and halting alongside that clock. At this point, we / all observers can note that their routes through space-time must again be the same, that is they have the same spatial co-ordinates and their progress along their timelines stay abreast with each other. From experiment, we also know that the travelling clock will have lost time compared to the stay at home clock. What I would like confirmed is how the travelling clock lost time. My naive deduction is that during its journey, the travelling clock was on a shorter route through space-time than the stay at home clock? So first point, is that the correct explanation? And thinking ahead, assuming that is the correct explanation, then I note this point... For an observer, who by misfortune did not see the travelling clock start its journey, and only sees the travelling clock coasting through space away from the stationary clock (or towards it if it is on the return leg of its round trip), that observer is therefore unable to determine if it is the 'travelling' clock on a shorter route through space-time, or whether it is the 'stationary' clock that is on the shorter route through space-time. Despite this unfortunate observer not being able to deduce (or measure) which is which, it does not alter that it is the travelling clock that is on the shorter route through space-time.

The purpose of using a framework of clocks, static within that framework, is to help me understand that there is no issue when a clock travels from one point to any other point in that framework, or from any point in that framework back to its starting point. Yes, the synchronization of the travelling clock to a stationary clock is when it halts next to that stationary clock. To avoid complications as to how separated clocks synchronize, I only consider loss in time for the travelling clock when it starts and ends its journey with the same stationary clock. It is then certain that the two clocks are at the same point in time and at the same location in space as each other, at the start of the journey and at the end of the journey. This means that the loss in time of the travelling clock must be a real physical loss in time (rather than an apparent loss in time). Having a second framework that is moving compared to the first framework, helps me to understand that there is no issue when a travelling clock moves from any point to any other point within any single framework. And so there is nothing special about which single framework (frame of reference) is chosen. The part that I do not understand is when a clock travels from one framework to a different framework (i.e. the frameworks are moving through space at different speeds). Ideally I would like this to be explained one step at a time. So the first step (I think) is for me understand how clocks move through space-time, when moving at different speeds to each other through space. Taking the given answer: clocks moving through space at different speeds, see the other as moving slow through space-time, this being mutual. Perhaps my choice of phrasing is not very good though. So perhaps it is clearer to say that each clock sees the other clock as taking a shorter route through space-time than itself? Sure, that wasn't a very clear phrase to use. I was trying to describe the passage though space-time and the 'rate' at which the travelling clock was making progress through its route through space-time. Does this simply equate to saying that the route through space-time is shorter or longer?

Thank you all for your help so far. Here is a very basic example of the issue that I am struggling to understand. If a series of locations in space all have a clock permanently based at their locations (like an array of dots positioned throughout space) and these locations are all stationary with respect to each other, then a travelling clock can travel between these locations, and each time it halts at one, it will find that its rate of time is the same as the clock stationed at that location. If during its travels, the travelling clock synchronizes the time on its clock with a stationary clock, then on returning to that location, the travelling clock will have less time on its clock than the stationary clock. This loss in time is attributed to the travelling clock taking a shorter route through space-time on its round trip away and back to the stationary clock. Let's say in this example, the stationary clocks are all moving through space-time at a 'rate of 1' and when the travelling clock journeys from one location the another, it moves at a speed of ‘8’ with respect to the stationary clocks speed of ‘0’ and its route through space-time is at the 'rate of 0.8' with respect to the stationary clocks ‘rate of 1’. So far so good. Since there is nothing special about this array of stationary clocks in space, another array of clocks in space can be considered, again all not moving with respect to each other, but nonetheless they are moving with respect to the first array of clocks. Let’s say they are moving at speed ‘8’ with respect to the first array of clocks. This means that their route through space-time is also at the ‘rate of 0.8’ with respect to the first array of clocks route through space-time, which is at the ‘rate of 1’. However, there is nothing special about the first array of clocks, and so it is equally valid to consider the second array of clocks as being stationary, and the first array of clocks to be moving. In which case, it is equally as valid to say that the second array of clocks are moving through space-time at the ‘rate of 1’ and the first array of clocks are moving through space-time at the ‘rate of 0.8’ with respect to the second array of clocks route through space-time, which is at the ‘rate of 1’. Because travelling clocks lose time when they perform a round trip back to a stationary clock, the assumption is that the change in rate through space-time is a real change. In which case, the first question… Although no preference can be made between the first or second array of clocks, and although their routes through space-time are different (because the clocks are in physical different locations in space?) is it correct to say that their rates through space-time are at the same rate as each other?