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- Subject: (SR) Lorentz t', x' = Intervals
- From: Thnktank@concentric.net (Eleaticus)
- Followup-To: poster
- Organization: Think Tank Eleataic
- Approved: news-answers-request@MIT.EDU
- Newsgroups: sci.physics,sci.physics.relativity,alt.physics,sci.math,sci.answers,alt.answers,news.answers
- Summary: The Lorentz transforms themselves are proof t' and x'
- cannot possibly be just coordinates. Examination
- of their derivation verifies their identity as
- intervals.
- X-Disclaimer: approval for *.answers is based on form, not content.
- Originator: faqserv@penguin-lust.MIT.EDU
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-
-
- (SR) Lorentz t', x' = Intervals
- (c) Eleaticus/Oren C. Webster
- Thnktank@concentric.net
-
-
- ------------------------------
-
- 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 ThnkTank@concentric.net ?
- ! "Anything and everything that requires or encourages systematic ?
- ! examination of premises, logic, and conclusions" ?
- !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?---!---?
-
-
-