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(SR) Lorentz t', x' = Intervals

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    (SR) Lorentz t', x' = Intervals                    
     (c) Eleaticus/Oren C. Webster

Subject: 1. Introduction with the obvious debunking of the use of 'just coordinates' in any scientific formula. Defenders of the Special Relativity faith are especially fond of telling opponents of their space-time fairy tales that they do not know the difference between coordinates and magnitudes. That may often be so, but the fault lies ultimately with SR dogma. The Lorentz-Einstein transformations cannot possibly be 'just coordinates', which is the interpre- tation required to support the many sideshow carnival acts with which the SR faithful bedazzle the public, and establish their moral and intellectual superiority. If I get in my car and drive steadily for a few hours at 50 kilometers per hour, is 50t the distance I travel? Of course not. The last time my hours-counting 'just coord- inates' clock was set to zero was when Zeno first reported one of his paradoxes to Parmenides. That was a long time ago, so my t is not useful for such purposes unless you also use my clock to established the starting time, perhaps t0, and use the formula 50(t-t0) to calculate the distance. In any case, my t is even then not 'just a coordinate' because it always represents particular elapsed times that can be used in the (t-t0) form to calculate perfectly good time intervals (elapsed times). Alternatively, I could (re)set my clock to zero at the start of some meaningful time interval, in which case my t shows a scientifically perfect current and/or end time. In which case, the Lorentz-Einstein t'=(t-vx/cc)/g is a function of an elapsed time interval (not 'just a coordinate') and a time interval (-vx/cc; the interval amount the t' clock is being screwed up at time t) and thus cannot be 'just a coordinate' since neither of the independent variables is such a 'just' thing. {Their meaning is shown below, step-by-step.] If it takes me 50 minutes to cross the Interstate highway, was x/50 my velocity crossing it? Of course not. The origin of all my axes is at the very spot where Zeno first presented his first paradox to Parmenides. That makes my x equal a couple of thousands of miles, plus, and is not useful for such purposes unless you establish the starting x value, perhaps x0, and use the formula (x-x0)/50 to calculate my velocity. In any case, even then my x is not 'just a coordinate' because it always repesents particular distance intervals that can always be used in the (x-x0) form for any and every scientific purose. Alternatively, I could move my x-axis origin to the starting (zero) point of some meaningful distance, in which case my x shows a scientifically perfect current and/or end distance. In which case, the Lorentz-Einstein x'=(x-vt)/g is a function of a current/ending distance interval (not 'just a coordinate') and a distance interval (-vt; the interval amount the x' axis is being screwed up at time t) and thus cannot be 'just a coordinate' since neither of the independent variables is such a 'just' thing. {Their meaning is shown below, step-by-step.]
Subject: 2. Table of Contents 1. Introduction with the obvious debunking of the use of 'just coordinates' in any scientific formula. 2. Table of Contents. 3. The Lorentz-Einstein transforms. 4. The 'just coordinates' argument. 5. Single-system, little-purpose ambiguity. 6. Relating two coordinate measures/systems. 7. Distances and moving coordinate axes. 8. Time intervals. 9. Einstein's (1905) derivations. 10. A word about intervals. 11. Intervals versus the Twins Paradox. 12. Summary
Subject: 3. The Lorentz-Einstein transforms Special Relativity's space-time circus is based on the 'transformation' equations by which it is believed one can relate a nominally 'stationary' system's space and time coordinates to those of an inertially (not accelerating) moving other observer. That moving observer's own physical body and coordinate system might have been identical in size to those of the stationary observer before the traveller began moving, but are 'seen' as very different by the stationary observer when the relative velocity of the two is great enough, a high percentage of the velocity of light. Concerning ourselves - as is customary - with just the spatial coordinate axis that lies parallel to the direction of motion, and with time, Einstein arrived at these formulas that relate the moving system measures or coordinates (x' and t') to the stationary system coordinates (x and t): x' = (x - vt)/sqrt(1-vv/cc) (Eq 1x) t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) The v is for the two systems' relative velocity as seen by the stationary observer, and is positive if the dir- ection is toward higher values of x. By concensus, the moving system x'-axis higher values also lie in that direction, and all axes parallel the other system's corresponding axis. We used vv to mean the square of v but might use v^2 for that purpose below. Similarly for c. Because it is believed that no physical object can reach or exceed c, the square-root term in both denominators is presumed always less than one, which means that the formulas say both x' and t' will tend to be greater than x and t, respectively. However, SRians call the x' result 'contraction' - which means shortening - and the t' result 'dilation' - which means increasing.
Subject: 4. The 'just coordinates' argument The 'just coordinates' argument is so patently ridiculous that even opponents have a hard time accepting just how simple and obvious its debunking can be, as shown in this section. However, further sections take a more arithmet- ical approach that you'll maybe find more professorial. The 'just coordinates' argument is that t is mot a duration, not a time interval; it's just an arbitrary clock reading. But what if the moving system observer comes speeding by while you make your annual 'spring forward' or 'fall back' change? The formula says that the moving system clock's 'just coordinate' reading can be calculated from yours: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Imagine the moving system oberver's confusion if his clock changes its reading while he's looking at it! If his clock doesn't change when yours does, the formula is wrong; if it is truly a 'just coordinates' formula. And then what happens if you realize you were a day early and put your clock back to what it had said previously? And suppose you are in NYC and your twin in LA and both are watching the moving observer. You'll both be using the same v because you are at rest wrt (with respect to) each other. You're on Eastern Standard Time and your twin is on Pacific Standard Time maybe. You have three hours more on your clock than does your twin. On which 'just coordinate' clock will the Lorentz transforms base the 'just coordinate' time the moving system clock says? The formula applies to both of your t-times: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Sure, the idea that you can change someone else's clock with no connection of any kind is really ridiculous, but Eqs 1x and 1t aren't MY equations. Are they yours? And we aren't the ones to say x, t, x', and t' are just coordinates. If the t' formula is actually either an elapsed time formula, or the basis of a t'/t ratio, then there is no implication that one clock's reading has anything to do with the other's. It can only be rates of clock ticking, or how one time INTERVAL compares to the other that the formula is about.
Subject: 5. Single-system, little-purpose ambiguity. Since we're going to be comparing measurements on two coordinate systems in the next section, let's go to our supply cabinet and get our yard-stick (which we use to measure things in inches) and our meter-stick (which we use to measure things in centimeters). Here, I'm getting mine. Oh! Oh! There's an ant on mine, and he ... she ... sure is hanging on, right at the 3.5 inch mark of the yard- stick. Let's see if I can wave the stick around enough that she'll let go. Nope. However, before I gave up I waved the stick and the ant 'all over the place". Always, however, the ant was at the 3.5" mark on the yard-stick, and always 3.5" away from the end of the stick, however far and wide I have transported her. Neither of those 3.5" facts means very much. Of the two, the distance aspect meant almost nothing. So the distance was 3.5" from the end. So what? That length, distance, was not in use. And only maybe the ant might have been concerned with just what location, 'just coordinate', on the stick she was at. Just so with x and t. So, is the 3.5" reading just a coordinate? Or a distance/length? It's ambiguous in and of itself, and really makes no difference what you say until you try to make use of the number. Hey, my address is 5047 Newton Street. If you are looking for me and you're at 4120 Newton, it is helpful information, because it tells you which direction to go. Is that 'just coordinate'? Where it really becomes useful, perhaps, is in telling you how far away I am. That's not just a coordinate value, that's a distance, length, interval. However, it is subtracting 4120 from 5047 that tells you which direction and how far. It is only because both 5047 and 4120 are distances from the same point - ANY same point - that the result means anything. My x - my yardstick reading - is always a distance or length; it is impossible to be otherwise with an honest, competently designed yardstick. Whether or not its reading is of good use in some particular scientific formula depends on whether I put the zero end of the yardstick at some useful place. As in the introduction, we should either put it at the starting location/end, or use two readings from it: (x-x0).
Subject: 6. Relating two coordinate measures/systems. Taking care to not damage our brave little ant, I place my yard-stick onto the table, zero end to the left, 36" end to the right. Now I place the 'just coordinate' meter-stick on the table in the same orientation, in a random location, and find that the ant's coordinate on the meter-stick is 51. The formula relating centimeters to inches is cm=i*2.54 but we want a formula similar to x'=(x-vt)/sqrt(1-vv/cc). That would be c=i/.03937 approximately, but let's use x' for the meter-stick reading, and x for the inch reading: x'=x/.3937. 3.5/.3937 = 8.89 Wait a minute. It's not just science but definition that says c=i/.3937=8.89, so something is wrong. 8.89 is not 51. We already knew that 51 cm was just an arbitrary coordinate. Arbitrary not because that point isn't 51 cm from the zero end of the meter-stick, but because the zero point was in an arbitrary position. Let's put the meter-stick in a position where it's zero point is at the yard-stick zero point. What is the centimeter coordinate now? Hey. 8.89, just like the formula says. The only way for a 'transform' like x'=x/g to work, whatever g might be, is for both coordinate systems to have their zero points aligned, in which case saying the two measures are not intervals is pure idiocy. Noe that with both zero points at the same position both x' and x are great measures for scientific purposes, in any and every case where we were smart enough to put those zero points at a useful location. There is one extension of x'=x/g that will let us use the meter-stick in arbitrary position. When the cm reading was 51, the zero point of the yard-stick read (51-8.89=) 42.11 cm. If we call that point x.z' we get x' = x.z' + x/.3937. = 42.11 + 3.5/.3937 = 42.11 + 8.89 = 51. Obviously, in this formula x/.3937 is the distance from the x' coordinate of the location where x=0. An interval. Just as obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x' coordinate, does not mean that x' is anything more than nonsense for use in any scientific formula. Unless we were smart enough to put the x zero point in a useful location, and use (x'-x.z') in the scientific formula. (x'-x.z') equals the useful, Ratio Scale value x/.3937. So, we have discovered a basic fact: a transformation formula like x'=x/g works only if the two zero points of the coordinate systems coincide. That makes it non- sense to say the two coodinates are only coordinates and not intervals. Both must be values that represent distances from their respective zero points unless you take the proper steps to adjust for the discrepancy. Make sure you understand that although the inclusion of x.z' made it possible to correctly calculate x', the result is nonsense when it comes to use of x' for general length/distance purposes; it is x'-x.z' that is a useful number in such cases. It could be that we're measuring a sheet of paper with one end at x=0 and the other at x=3.5; x'=51 is nonsense as a centimeter measure of the paper. But, you say, the Lorentz transform contain a -vt term.
Subject: 7. Distances and moving coordinate axes. We discovered x'=x.z' + x/g as the correct formula for relating one coordinate to another system's. But the Lorentz transform contains another term, -vt/sqrt(1-vv/cc). What is it? Let's start with our x'=51 cm, x=3.5", x.z'=42.11 example. Every minute, let's move the meter-stick one inch to our right. At minute 0, the cm reading was 51 cm. At minute 1, the cm reading is now 50 cm. At minute 2, the cm reading is now 49 cm. In this instance, v=1 inch/minute. And t was 0, 1, 2. What has happened is that we have made our x.z' a lie, and increasingly so. -vt/.3937 is the change in x.z'. x' = (x.z - vt/.3937) + x/.3937. Obviously, vt/.3937 is not a coordinate; even most SRians wouldn't imagine it was. It is an interval, the distance over which the moving system has moved since t=0. And, of course, x/.3937 is the distance of our brave little ant from the point where x=0 and the centimeter reading is x.z'-vt/.3937. Yes, every minute the meter- stick moves to the right and the meter-stick coordinate of the spot where x=0 gets less and less - and eventually negative. Make sure you understand that every minute the x' coordinate, because of -vt/g, becomes a better measure of, say, the 3.5" paper we might be measuring with the yard-stick, given that 51 was too big a number and -vt is negative. That is, until the two origins coincide at x'=x=0, and then it gets worse and worse. With -vt positive (because v<0) the situation is different. With 51 and -vt positive, x' just gets worse and worse over time. Quite obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x' coordinate even when the x' axis is moving, does not mean that x'is anything more than nonsense for use in any scientific formula. Unless we were smart enough to put the x zero point in a useful location, and use (x'-x.z'+vt/.3937) in the scientific formula. (x'-x.z'+vt/.3937) equals the useful, Ratio Scale value x/.3937.
Subject: 8. Time intervals. Instead of using our sticks, let's get out two clocks. Mind you, we're not going to deal with different clock rates here, just establish the same basics as for distance. Your clock says 9:00 Eastern Standard Time (EST) and we note that t=540 minutes when we put down the clock. Blindly, let's turn the setting knob of your twin's Pacific Standard Time clock and put it down before us. According to what we see, EST's 540 minutes (9:00) corre- sponds to PST's 14:30; t'=870. We know the formula relating PST to EST is t' (pacific) = t (eastern) - 180 (minutes). Thus, it is not correct that the second clock can have an arbitrary setting, because 870 <> 540-180. We know that the two clocks are related by t' = t/1 since both are using the same second, hour, etc units. But 870 (14:30 in minutes) is not 540/1-180, so once again we know something is wrong. However, t'=t.z' + t/1 works. EST midnight equals PST 0.0 (midnite) - 180, so t.z' = -180, and t' = -180 + 540/1 = 360. Since EST-180=PST, 9:00 EST is 6:00 PST = 360 minutes. We see thus that like distance measures/coordinates, time axis origins (zero points) must either be 'lined up' or adjusted for. So, the Lorentz/Einstein t'=t/sqrt(1-vv/cc) must be the moving system elapsed time interval since the time axes were both at a common zero. There is no t.z' adjustment: t' = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Make sure you understand that in the clock case, if the EST is showing a good number for elapsed time since the travelling observer passed NYC, then the PST clock is silliness. t.z' must be zero or must be taken out of time lapse calculations for the PST clock to be used intelligently, just as was true for x.z'. What is lacking as yet for Lorentz t' is the -vx/cc term that corresponds to the x' formula -vt term. Break it up into two parts: v/c and x/c. v/c is a scaling factor that changes velocity from whatever kind of unit you are using over to fractions of c. x/c is distance divided by velocity, which is time. x/c is thus the time interval since the two time axes had a common zero point - which they have to have in the Lorentz transforms which do not have the t.z' term we learned to use above. Thus, (-vx/cc)/sqrt(1-vv/cc) is the interval amount the moving system clock has been changed - since the common/ adjusted time - over and beyond the elapsed time interval represented by x/sqrt(1-vv/cc). We have discovered that the only way for t' to be t/g is for t' and t to have a common zero point, just as for x' and x. It would be otherwise if the t' formula contained an adjustment t.z' under some name or other, but the necessity to include such a term correlates 100% with t' numbers that aren't directly usable. As for x and x', our knowledge of how to setup a proper formula relating t and t' is of no use unless we use the knowledge in scientific formulas; (t'-t.z'+xv/gcc) gives us the only directly useful value: t/g.
Subject: 9. Einstein's (1905) derivations. When we return to Einstein's derivations of the transform formulas with a well-focused eye, we find he was a wee bit confused - or at least self-contradictory. When he set up his (at first unknown) tau=moving system time formulas, he created three particular instances of tau. Tau.0 is the time at which light is emitted at the moving origin toward a mirror to the right that is moving at rest wrt that moving origin and at a constant distance from that origin. He lets the stationary time slot have the value t, a constant, the stationary system starting time. Tau.1 is the time at which the light is reflected. He lets the stationary time be t+x'/(c-v); t is still a constant and x'/(c-v) is the time interval since t. Tau.2 is the time at which the light gets (back) to the moving origin. The stationary time value is put as t + x'/(c-v) + x'/(c+v); t is still a constant and x'/(c-v) + x'/(c+v) is the time interval since t. On the thesis that the moving observer sees the time to the mirror as the same as the time back to the origin, he sets .5[ tau.0 + tau.2 ] = tau.1. Tau.0 completely drops out of the analysis and leaves no trace, and has no effect. Further, the t you see in tau.0, tau.1, and tau.2 also completely drops out with no trace and no effect, leaving us with exactly what you'd get if you had explicilty said t' is an interval and so is t. What doesn't drop out in the stationary time values is x'/(c-v) and x'/(c+v), the time interval it takes for light to get to the fleeing mirror, and the time interval it takes for light to get back to the approaching origin. Thus, his resultant t' formula is strictly based on time intervals in the stationary system. Time intervals since some starting time, yes, but time intervals. There is absolutely nothing in the derived formulas that depends on arbitrary coordinates like the constant t in the stationary time arguments. Let's look at the x dimension; it is x'=x-vt [as x increases by vt, the effect over time is x'=(x+vt)-vt)], which Einstein explicitly sets up as a constant stationary distance. He uses that x' not just in the time interval parts of the stationary time arguments, but also in the x (distance) stationary system argument for the tau at the time light is reflected. x' can't be the stationary system coordinate of the mirror at that time. That value is x'+vt. x' is explicitly an interval, distance. Thus, the whole tau derivation of the t' formula is fully and explicitly based on x' - a spatial length/distance/interval - and the two time interals x'/(c-v) and x'/(c+v). While we're at it, if the starting t is not zero, his x'=x-vt formula is complete nonsense also. Given that there was some L that was the mirror x-location and length when the light is emitted, if t was already, say, 500, then x'=L-vt could have been a very negative length.
Subject: 10. A word about intervals. There are intervals, and there are intervals. If we put our yard stick zero point at one end of a piece of paper and read off the coordinate at the other end of the paper, we have a good measure of the paper's length, a Ratio Scale measure. [Absolute temperature scales are ratio scale.] If instead we put the one end of the paper at the one inch mark (or the zero end of the stick one inch 'into' the length of the paper) we get measures that are one inch off the true, ratio scale length. The two messed up measures are still intervals, but they are Interval Scale measures. [Household temperature scales are interval scale, which is why your physics and chemistry professors won't let you use them without first converting to the ratio scale absolute temperatures.) t'=t/g and x'=x/g represent ratio scale measures, given that t and x were ratio scalae to start with. t'=t.z'+t/g and t'=t/g-vx/gcc are both interval scale measures, even given a good ratio scale t and a good ratio scale x. x'=x.z'+x/g and x'=x/g-vt/g are both interval scale measures, even given a good ratio scale x and a good ratio scale t. Look for the "(SR) Lorentz t', x' = degraded measures" document soon at a newsgroup near you.
Subject: 11. Intervals versus the Twins Paradox. t'=(t-vx/cc)/g shows t' being greater than t. The reason Special Relativity will not allow the use of its basic time equation in determining what SR has to say about the twins' ages, is that t' and x' are supposedly just coordinates, and they say you have to take the coordinate pairs (t',x') and (x,t) into consideration in both the time and place the twins' separation started and the time and place the twins reunited. Since t' and x' are actually both intervals, not just coordinates, the 'excuse' is spurious, and is so even without use of the obvious (x_b-x_a) and (t_b-t_a) usages. However, SR is right to be embarrassed by their transformation formulas. Look for the "(SR) Lorentz t', x' = degraded measures" document at a newsgroup near you.
Subject: 12. Summary A. t'=t/g and x'=x/g can be almost 'just coordinates' in the sense that the values obtained may not be of much use except in the most primal and useless way: how long and how far since/from the time/ place they were zero. Even here, however, the zero points within each of the two scale pairs (t',t) and (x'.x) must have been lined up. If the zero points have been intelligently selected (such as at the starting point and time of a trip) they can be rationally used 'as is' in any valid sci- entific equation. B. Even the interval scale t'=t.z' - xv/gcc + t/g and x'=x.z' - vt/g + x/g are not 'just coordinates'. They can be used to good effect by establishing the relevant starting times/points and using (t'-t.z'+xv/gcc) and (x'-x.z'+vt/g), as the situation may require. C. When you see vx/gcc or vt/g in use in any guise with non-zero values, you know the resultant t' or x' is a degraded, interval scale value. E-X: Anytime you do not see what amounts to t.z' and xv/gcc in the time case, or x.z' and vt/g in the distance case, you know that the t' and/or x' in use are intervals. Period. Y: Either set your clock to zero at the start of the relevant time interval, or use (t-t0), with both being readings on the same clock. Either move your x-axis origin to the starting end or point, or use (x-x0), with both being readings on the same axis. Z: In _(SR) Lorentz t', x' = Degraded (Interval) Scales_ we see that t' and x' satisfy the mathematical tests for/of interval scales when -vt and -vx/cc are not zero; thus, they must be intervals. When -vt and -vx/cc are zero, t' and x' satisfy the much better mathematical definition of ratio scales, and are thus not just mere intervals, but (rescaled) good ones. Eleaticus !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ! Eleaticus Oren C. Webster ? ! "Anything and everything that requires or encourages systematic ? ! examination of premises, logic, and conclusions" ? !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?

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