An addendum concerning the last question I previously asked, and the first answer of Dr Percival to my comments.


* * * * *


At the end of the previous answer to Percival's proposal, I asked the following question:


* "are we in front of something which could be said just conventional (namely, depending from the "appearance", from the conventions we use in our measurements), or not, that is to say an objective FACT of Nature?"


I replied to * saying that: "today, I would rather say that this second case is more correct...". I still believe that, but I wish now to expand more about this opinion, showing that well-known experiments of the kind of the increasing of the mean-life of travelling muons SEEM indeed in favour of relativity as a NON-CONVENTIONAL PHYSICAL THEORY. If this is so, any criticism of SR MUST find the "other possible explanations" for such phenomenena, to which I generically alluded before. For this reason, I want to give furthermore in this addendum an HINT about a possible "mechanism" for the increasing of this mean-life from the point of view of an aether theory.


* * * * *


i - First of all, let me remark that in the case I discussed above, A' is in front of A etc., there is indeed the presence of a clock-synchronization between the clocks of A and B, which would make the whole argument apparently depending on it. But we can try to "avoid" this synchronization in many ways. In the P.S. I suggested to avoid it supposing that B starts to light his own candle just when he directly sees the candle of A' lighting on, but as I already told, it is easy to see that in this case it is impossible, from the point of view of SR, that the candle of A' is still lighting on in front of B, while the candle of B, surely lighted on AFTER the candle of A', is not. This is indeed true, but one can imagine other ways to be sure that the phenomenon under examination started just when A' was in front of A, no earlier and no later, no matter at which time it happened. From this point of view, the only relevant information for B are the following ones:


a - B knows the speed k of A' just by a "local" measurement, and can have "good reasons" in order to suppose that this speed did not change all along the path L;


b- So B has good reason to say that the elapsed time of the travel of A', from A to B, is L/k, independently from any value marked by the clock of A when A' was in front of A;


c - B MUST then be surprised if a phenomenon, which he knows did start exactly at a distance L from him, and whose duration was less than L/k in HIS laboratory, is still going on when A' is in front of him.


Having sincerely recognized that such a phenomenon would be in favour of relativity (it is impossible to believe that any classical physicist would have foreseen that!), one can nevertheless add the following remark: in the famous muon experiment, which is the real counter-part of the "gedanken experiment" described above, there is no evidence for a SYMMETRY of the observed phenomenon. Only this symmetry would be the key-point for claiming that the relativistic interpretation of this "time dilation" is the unique correct one. In other words, until NEW experimental evidence will be available, one is allowed to suppose that the increasing of the mean-life of the muons is a real physical fact, which would yet possibly depend on the "absolute speed" k of the muons. I assert that in the hypothesis, which I like better, that the aether is, more or less, at rest with the Earth, and so that the speed of the muon with respect to the atmosphere would indeed be even its absolute speed.


ii - A possible aether-based explanation for the experiment under discussion could come from an ANALOGY.


Let us imagine an ordinary cartesian orthogonal reference frame, with an origin O and three axes Ox, Oy, Oz , and a rain falling down along z-axis with intensity s (for instance millimeters of rain for time unity: s has the physical dimension of a VELOCITY). This means that, if we take a container (a tea-cup!) in form of a paralleliped (suppose that it has a vertex in the origin, and the three sides coming out from this vertex along the three axes of the given reference frame, in such a way that the length of the side along x-axis is L, M along y-axis, and H along z-axis), of volume LMH, after a period of time T we will find a volume of water inside the container which is equal to LMsT. Suppose now to turn the cup along the y-axis of an angle th (theta), less than 90, and then to ask the same question as before: what will be the volume of water inside the "cup" after a time T? It is clear that the answer will be Lcos(th)*MsT , and so far so good. Suppose at last that the cup is moving along x-axis with a velocity v = (v,0,0), and then to ask once again the same question: after the same time T, what will be the volume of water inside the cup? It is rather clear that, from the point of view of an observer moving with the cup at the same velocity v , the rain will not fall anymore along the perpendicular to the "base" of the cup, since one must take in account a kind of "composition of velocities", and the rain will fall along the direction (v,0,s), "aberrated" of an angle th, with respect to z-axis, such that: cos(th) = s/sqr(v^2+s^2) . This obviously means that the case we are studying now is exactly the same as before, as if the cup was simply rotated along y-axis with the same angle th, and then the answer will be once again:

Lcos(th)*MsT = LMTs^2/sqr(v^2+s^2) = LMsT/sqr(1+v^2/s^2) .

Comparing this value with the value LMsT found at the very beginning, we find, as it was obvious to expect, that the water inside the moving cup is LESS than the water inside the standing cup, and, pushing further our analogy, if


(1) T = V/LMs


is the time required for getting a given volume V of water in the standing cup, this same time will be equal to


(2)...T(v) = (V/LMs)*sqr(1+v^2/s^2) = T*sqr(1+v^2/s^2)


in the case of a moving cup, a time which is bigger than (1).

I claim that we could possibly be in front of a similar phenomenon in the case of the increasing of the mean-life of a speedy muon. Let us call T the mean-life of a muon in a terrestrial laboratory: then this T will be, presumably, a function of the given particle and of its interaction with the "surrounding aether". Then we know that a speedy particle will "appear" to have, from the point of view of the terrestrial laboratory, a BIGGER mean-life, equal to


(3) T(v) = T/sqr(1-v^2/c^2) ,


which is the value predicted by SR. As a matter of fact, let us now compare (2) and (3), putting in the first, as it would be "natural", s = c. Up to second order in beta = v/c , (2) can be written as


(2') T(v) @ T*(1+beta^2/2) ,


while (3) can be written as


(3') T(v) @ T/(1-beta^2/2) @ T*(1+beta^2/2) ,


and so we get a value which is EXACTLY EQUAL to (2')! Could we be quite sure that the true physical explanation of the famous muon experiment is not of the kind of the "moving cup" model, at least in absence of any evidence of symmetry, which would be essential for the relativistic explanation?


iii - This relativistic interpretation would have a real good point in its favour not only if one could check the asserted relativistic symmetry of the phenomenon, but even if ALL phenomena would show exactly the same slowing down: in such a case one should indeed acknowledge that SR is correct (at least about this matter). For instance, going back to the previous case, A' travelling from A to B, we should study the behaviour of the mean-life not only of muon, but even of a candle, of a match, of a sandglass, and so on... Suppose that A and B are at the same equipotential level in a gravitational field, for instance at the Earth's surface, and suppose to have a pendulum P travelling from A to B with a uniform speed k. Then, even according to SR, the length L of P would not change (no transversal length contraction), either with respect to the reference frame in which the pendulum is moving, or to the (proper) reference frame in which the pendulum is standing still, so one should think that the (classical) pendulum's rate, 2p sqr(L/g), would not change (now we would have indeed a difference between Lorentz hypothesis of longitudinal length contraction, and FitzGerald hypothesis of transversal length dilation, which are equivalent with respect to the Michelson-Morley experiment! - see for instance the papers in the point 15 and 16 in this same page, which are unfortunately written in Italian). So we could think of a pendulum* which starts its ticking exactly when it is in front of A, and then count these tickings. If they are less than L/k, when the pendulum is in front of B, if an experiment like that would really give an outcome of the same kind predicted by relativity, questions of symmetry apart, well, I would begin to change my opinion about the validity of the theory, and I would stop fighting against it...


* [NO atomic clocks, please, because their frequence could perhaps change according to the "absolute speed", exactly in the same way as we remarked in point ii]


* * * * *


Dear Prof. Bartocci:


Thanks very much for taking so much time and making such a major effort to respond to my comments in your last two emails. Since we are on opposite sides of a debate, let me first emphasize that should any of my comments below appear to be arrogant, rude, obnoxious, etc., please assume that that is NOT what is in my mind, but rather it is due to:

1) our "arguing", 2) the nature of ad hoc emails to be read with a different tone than intended, 3) something being "lost in translation", 4) my being too passionate about my favorite topic etc., etc.

I will explicitly comment on this at the beginning, but will not do so throughout the letter as that would make it too long. I hold you in the highest regard and have been impressed by your knowledge of Special Relativity and related topics.


I believe that our area of disagreement is quite small - but very important. Perhaps, further communication will reveal that our differences are really smaller still.


For nearly a century, some of the brightest scientists who have walked the earth have debated BOTH SIDES of the Twin Paradox. While your colleague calls it a "dead issue" that is only because he is acting as "prosecutor, judge and jury". I doubt that he would agree with you regarding the solution to the Twin Paradox - particularly at the detail level - and there are many others who share his contempt for the topic that hold mutually exclusive views with him and with you. Furthermore, neither Dingle nor anyone else of note on his side ever found merit with the rebuttals. Chang of Harvard did extensive research on the Twin Paradox and concluded that that the debate could be characterized as the two sides talking at cross purposes (i.e., Dingle's points were never addressed).


It is, therefore, no surprise that our first two rounds can be characterized as "talking at cross purposes". This is NOT your fault. This is NOT my fault. This is the nature of the Twin Paradox - it is a very complex topic (even if Special Relativity itself is very simple) - it is very sophisticated and even perverse in its subtlety - it goes well beyond SRT.


In addition to spending 35 years studying the Twin Paradox, I have spent the last six years trying to develop a more effective way of communicating about the Twin Paradox and I believe that I have significant progress. I believe that if we follow my methodology that we can avoid the qualitative, vague, semantic, ambiguous, marshmellowy exchanges that are "business of usual" in a Twin Paradox debate and be rigorously precise and quantitative - speaking a minimum of prose and a maximum of physics.


However, before I embark on this latter [f]act (Part II), I do have a few remarks that fit into the former category. I do this only in hopes of planting some seeds of doubt in your mind and hoping to avoid the reverse effect. Part II, however, will be where the progress is made.


Part I: "Qualitative, Vague, Semantic, Ambiguous, Marshmellowy" Remarks


I agree with virtually all that you said except for your detailed discussion of how the net time difference accumulates and even there further clarification may bring us closer together.

Please also note that unlike Dingle and most others on his side of the debate I do NOT hold that the Twin Paradox shows that SRT is logically inconsistent or has flaws. Rather I say that the net time difference cannot be explained in terms of SRT per se. However, the "real" cause is not only compatible with SRT per se, but also extends SRT.


You say (in your email of May 6) , "You seem to me wrong in the first assertion, the effect is a GLOBAL effect, and it is not possible to "split" it "for the constant velocity part".


I have two comments on that:


1) I did NOT assert anything. I asked a question. That is the methodology that I use in Part II. You do NOT have to rebut my ideas. You merely have to state your views unambiguously and in detail and let logic/physics dictate whether they are consistent. The question I asked remains a very good question. Its essence is Dingle's question. To elaborate a little more, but not exhaustively, it says:


"Given classic Twin Paradox scenarios in general, we have a net time difference that is a function of the parameters of the constant velocity parts of the round trip. The duration of the periods of acceleration can be made arbitrarily small with respect to the constant velocity parts of the round trip. Hence, if we are to explain in detail how the net time difference accumulates, we must do so in terms of the constant velocity parts of the trip. Yet, for the constant velocity parts of the trip, both twins are inertial observers and must be treated equally and symmetrically within the context of SRT. Hence, the paradox. If the net time difference must be explained in terms of the constant velocity parts of the trip - even if the acceleration is IMPORTANT, even if it's a GLOBAL effect - then it must be that some effect outside of special relativistic effects needs be invoked to give a detailed understanding of what's occurring during the constant velocity parts of the trip."


2) I've spent a lot of time analyzing the Twin Paradox and, in particular, I've spent a lot of time debating the Twin Paradox with many people. When I hear the phrase "GLOBAL" applied, I can guess where you are on the road of analyzing the Twin Paradox. I believe that you have analyzed a variety of detailed explanations of the net time difference (e.g., due to relative velocity, due to the time dilation effect, due to an artificial gravitational field caused by the turnaround acceleration, etc., etc.) and that you correctly found problems with each and, hence, you logically rejected these logically flawed explanations. And, finally, you felt that you gained great insight in "seeing" that it was a GLOBAL effect. It makes perfect sense to you.


Let me recount how my discussions go when I encounter such a response. I ask if that means that one is unable to tell me what time the travelling and stay-at-home clocks record at various well defined events. "NO", one answer and gives me specific answers which tend to imply one of the specific causes discussed above and only when confronted with associated logical contradictions does one retreat back to the safety of the vagueness of "it's a GLOBAL effect ".

Sometimes its easier to see flaws in others' arguments so let me recount an analogous discussion. My "opponent" claimed that the cause of the net time difference was "the nature of space-time" and it was obtuse of me to ask for anything more specific.

I made it clear that I agreed that it was correct to cite the "nature of space-time", but that that explanation left out 99.9999% of the relevant physics. I gave the analogy that if there were twins in a gravitational field and one twin moved (at non-relativistic speeds) along the field's gradient and stayed a specific amount of time at some different point and then returned, we could correctly say that the "net time difference" between the two clocks was due to the "nature of space-time", however, we would not be telling the whole story. In fact, if someone claimed that no further detailed description of how the net time difference accumulated, then that person would not know the detailed physics of the situation.


What's good SRT for the stay-at-home is good for the traveler. I objected to one's trying to explain the net time difference in terms of special relativity by applying special relativity in the stay-at-home frame for the constant velocity parts of the round trip, but saying that the same application of special relativity could NOT be made for the travelling frames (separately) for the constant velocity parts of the round trip. I pointed out that if one consistently used the given criterion for not applying special relativity using the travelling frames, one would rule out using special relativity in all real world situations.


You gave a rather general reply. Part of your answer was to say, "But to understand WHEN it can be applied in order to give good results, or not, it does require a deep understanding of the theory, which does not seem very common. But to understand WHEN it can be applied in order to give good results, or not, it does require a deep understanding of the theory, which does not seem very common."


I do not doubt that you have a deep understanding of special relativity (deeper in many areas than I and broader as well). However, I am quite competent at understanding a logical explanation of such things. In the past, I have debated with people who quite explicitly said, that applied one or another construct of SRT using the stay-at-home frame and then claimed that that same SRT construct could not be used for the traveling twin for the constant velocity legs "because the traveling twin accelerated". This was said with great assurance as though "everyone knew this". However, when I probed and asked how this could be derived or even logically justified, no one even got off the ground. I cannot even imagine what a logical first step would be - not SRT's postulates, not the time dilation equation, not length contraction, not E=moc2 . On the one hand, I would guess that one could find 100 references echoing his claim. On the other hand, do you know of any credible derivation of that assertion?


Clearly, there are ways to mis-apply SRT - no argument. Clearly, there are illegal operations one could do involving the traveling twin's two different frames of reference. However, that does is not the issue here. This is not a case of my making some naïve oversight. Dingle, Prokhovnik, Ives and many others who had a deep understanding of SRT didn't even flinch when the argument in question was put forward.


SRT and acceleration

This is looking at the above question from a slightly different perspective.


In SRT, special relativistic effects are, by and large, "symmetric". By that I mean, if we have A and B at rest in different inertial frames, then whatever A observes about B, B observes about A. If A observes the time dilation/length contraction factor for B to be 0.675823, then B observes the time dilation/length contraction factor for A to be 0.675823. If A observes the kinetic energy per unit rest mass for B to be 67.5823, then B observes the kinetic energy per unit rest mass for A to be 67.5823. Etc., etc.


Furthermore, if I wish to compute the special relativistic effects of a period of acceleration, that is straightforward and also symmetric. Even though acceleration is "absolute", SRT effects are a function of relative velocity so if we treat uniform acceleration as "a uniform changing of relative velocity", we can break the acceleration period into, say, a million segments of (increasing/decreasing) constant velocity and readily compute the instantaneous and cumulative effects for known special relativistic effects - these effects will obviously be symmetric.


Hence, when you discuss deriving asymmetric results (i.e., the results apply to the traveling twin, but not the stay-at-home twin) what are you saying in the context of SRT?


Let me repeat the point made above. When you have an equation that contains "g" (acceleration), one might tend to think, "Well, only the traveling twin accelerates, therefore, the equation only applies to the traveling twin." However, within the context of SRT "g" represents the change in relative velocity and, hence, the equation can be viewed, if consistent with SRT per se, as applying to both twins.


Now I have nothing against adding an asymmetric construct to SRT. I do that myself. One must do it to explain an asymmetric result, namely, the net time difference in the Twin Paradox. But you need to explicitly acknowledge that you are not only adding a new construct to SRT but an asymmetric construct that is fundamentally different from the constructs of SRT per se (i.e., asymmetric vs symmetric).


Now, like all the topics above, I have "peeled the onion" endless number of times going deeper and deeper (Hopefully, you have spent your life in a broader range of more productive pursuits.). I'll stop here and merely say that when one gives an equation, there are many possible physical interpretations of that string of symbols. You give a lot of equations and very little explicit physics. However, how you use that equation implies a particular physical meaning for that equation and, for the area under discussion, implies a particular space-time model. You feel that your equations are right and you know that SRT is correct, so it's a small jump for you to feel that your equations are classic SRT. Are you so sure? Have you thought about this topic in depth?


"Hiding" Acceleration

I said "I contend that when you modify the classic Twin Paradox scenario by using a "uniformly accelerated normalized" observer (instead of having the main portions of the round trip done at constant relative velocity), you add an unnecessary element of complexity and confusion..."


To which you replied, "Of course I do not agree. The main question is: since there must be an acceleration as a physical reason for the twin coming back, where do you want to HIDE this acceleration?"


I guess we must be talking at cross purposes here or my words "complexity" and "confusion" mean different things to you and me.


Let me just say that I will be analyzing the classic Twin Paradox and analyzing the classic issue of the Twin Paradox. I will not be "hiding" acceleration, but rather I will be defining it in terms of specific events and giving it specific numeric parameters. It may well be that some variation on the classic Twin Paradox has helped give you great insight into this topic. If that insight is general and not just applicable to that special case, then it should be applicable to the classic Twin Paradox that people have been debating for the past century.




You say that "If the twin comes back, then THERE MUST BE an acceleration somewhere." As stated that is true. But there is the well known "acceleration-less" scenario with "triplets" in which there are NO accelerations and the consensus is that the net time difference is the same (making arbitrarily small corrections for the fact that there are no accelerations). The "acceleration-less" scenario is used equally by proponents of SRT (myself included) as well as SRT opponents.

There is also the converse side of the above. Suppose that, in the classic Twin Paradox, "simultaneous" with the "traveling" twin doing his turnaround acceleration, the "stay-at-hand" twin does the same of amount of accelerates and then decelerates and returns to his initial position in his initial rest frame. The period of the stay-at-home's acceleration is equal to or greater than the "traveling" twin, however, the stay-at-home's round trip can be made arbitrarily small with respect to the traveling twin's.




The turnaround acceleration is indeed an asymmetry. Everyone agrees with that. However, just because there is a bona fide asymmetry does not give us carte blanche to do any kind of asymmetric treatment of SRT equations we want to. To know the complete answer, we must be able to say here is an asymmetric effect, here is the physics of that asymmetric effect: the details, the physics and the calculations, need to be consistent with accepted theory. That's what was missing with the standard approach of "Acceleration introduces and asymmetry and, therefore, one can use SRT constructs which in any other context would not be justified." That problem remains.




You say, "Let me add that the "smartest minds" you refer to, have given exactly the same answer to the paradox I did before: I was in no sense ORIGINAL, all "good" people gave the same reply all "good" people gave the same reply " This is an area that I know very well and your statement bears little resemblance to what I've read. You may have done very selective reading in this area. I know that, by (your) definition, you decree that Langevin, Dingle, Sachs, Ives, Prokhovnik and a very long list of others are not of the "smartest minds" nor are they "good" people. However, of those who feel that the net time difference in the Twin Paradox can be explained using SRT per se, there are a large number of different and mutually exclusive causes and explanations put forth. While this is not shouted from the roof tops, I believe it to be well acknowledged by both sides. The time dilation equation (including equivalent Minkowski diagrams and the Minkowski integral), relative simultaneity, the turnaround acceleration as inducing an artificial gravitational field (and other possibly equivalent GRT based explanations), lines of simultaneity, lines of synchronicity, the Kerr metric, the "nature of space-time", various geometric theorems, the parallax distance, the Eifso effect, the "Sachs" interpretation, Bondi's k factor have been put forward as reconciliation arguments in the topjournals. Do you and Max Born see eye-to-eye or is he not that smart or not that good? How about Moeller? Etc.




Many of your rebuttals are not relevant to the point I raised. They are non sequitors addressing a straw man. Perhaps, I am not writing clearly, but it makes me worry that, due to time pressure plus your conviction that you are right and I am wrong, you are not really reading or analyzing what I am saying. I am sure I do the same - I think we al suffer from the Twin Paradoxitis disease. However, the next section should help.



Part II - The Good Stuff


Refreshingly, the prose takes a back seat now. Just a few rounds of (simple) questions.


In the classic Twin Paradox scenario, A and B start together with clocks set to zero and they end the scenario back together and they compare the readings on their clocks. They are comparing their proper times - the number of ticks that each clock recorded between the start and end events. A and B do NOT compare A's observations of B's clock during the trip to B's observations of A's clock during the trip and compute net differences in observed number of ticks. "Explaining" the net time differences in terms of A observing B or B observing A was at the heart of Dingle's problem with the Twin Paradox.


So I would like to focus just on the accumulation of proper time for the twins and, hence, indirectly the accumulation of the net time difference.


I am going to define a classic, standard Twin Paradox scenario in terms of events that mark the beginning or end of accelerations. And, I will ask you to tell me the number of ticks on the each twin's clock between each set of events. (I'll initially fill in the numbers as a way of helping to describe what I'm doing, but you're free to rearrange the numbers any way you want as long as they add up to the "right" answer.)


All the events will be defined "in terms of the travelling twin" (e.g., the travelling twin begins accelerating, the travelling twin ends accelerating). Hence, for the stay-at-home twin, we will really be giving the number of ticks that occurred on his clock between events that he says are simultaneous with our defined events.


We'll say the stay-at-home twin observes the constant velocity in bound and outbound leg to each take 5 Million ticks on his clock. The constant velocity is 0.866c. The stay-at-home twin observes that it takes 30 ticks to accelerate from 0 to 0.866c or from 0.866c to 0.


Below, I will describe each section of the Twin Paradox scenario and I will give what seems to be the "current" consensus SRT view of how the net time difference accumulates. I will leave a space where you can fill in your numbers if different.


E0 to E1: E0 is defined as the travelling twin begins the initial acceleration that takes him from being at rest in the stay-at-home twin's frame to going at constant velocity 0.866c with respect to the stay-at-home twin's frame.


E1 is defined as the travelling twin ending the initial acceleration


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24


E1 to E2: E1 is defined above. E2 is defined as the travelling twin begins the initial deceleration that takes him from going at constant velocity 0.866c with respect to the stay-at-home twin's frame to being at rest in the stay-at-home twin's frame. In other words, this is the constant velocity part of the outbound leg.


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million


E2 to E3: E3 is defined as the travelling twin ending the initial deceleration that takes him from going at constant velocity 0.866c with respect to the stay-at-home twin's frame to being at rest in the stay-at-home twin's frame. In other words, this is the 1st half of the turn around acceleration.


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24


E3 to E4: E4 is defined as the travelling twin ends the re-acceleration that takes him from being at rest in the stay-at-home twin's frame to going at constant velocity 0.866c with respect to the stay-at-home twin's frame back toward the stay-at-home twin. In other words, this is the 2nd half of the turn around acceleration.


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24


E4 to E5: E5 is defined as the travelling twin begins the initial deceleration that takes him from being at rest in the stay-at-home twin's frame back with the stay-at-home twin.

In other words, this is the constant velocity part of the inbound leg.


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million


E5 to E6: E6 is defined as the travelling twin ending the final deceleration that takes him from going at constant velocity 0.866c with respect to the stay-at-home twin's frame to being at rest in the stay-at-home twin's frame. In other words, this is the final deceleration.


Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24



Total Ticks Accumulated Between the Start and End Events E0-E6

Stay-At-Home Clock: Approximately 10 Million

Travelling Clock: Approximately 5 Million

Just cut and paste Part II and fill in your answers and send me the result and then I'll ask you round two and we'll be almost done. Feel free to answer my "once over lightly" points in Part I, but you needn't spend an inordinate amount of time on them unless you think it's a killer point.


Sincerely and gratefully,

Nick Percival