"Bill Owen" wrote in message news:kd2djc$ods$1@news.jpl.nasa.gov…

Lord Androcles, Zeroth Earl of Medway wrote:

- — -

> "Bill Owen" wrote in message news:kd1hm6$fid$1@news.jpl.nasa.gov…

> Please be patient … I didn’t get a chance to check the newsgroup over

> the weekend. More below.

> Lord Androcles, Zeroth Earl of Medway wrote:

>> "Bill Owen" wrote in message news:kcnq4b$pua$1@news.jpl.nasa.gov…

>> [snip]

>>> Why is Savard multiplying xi by beta instead of dividing?

>> Because the expression on the right-hand side uses tau, not t. It’s

>> certainly true that x = xi/beta + vt.

>> Here’s the trivial derivation.

>> 1) Solve tau = beta(t-vx/c^2) for t:

>> t = (tau/beta) + vx/c^2.

>> 2) Multiply both sides by v:

>> vt = (v tau)/beta + (v^2c^2) x.

>> 3) Solve xi = beta(x-vt) for (vt):

>> vt = x – xi/beta.

>> 4) Equate the right-hand sides of (2) and (3):

>> (v tau)/beta + (v^2/c^2) x = x – xi/beta.

>> 5) Solve for x:

>> (v tau)/beta + xi/beta = x – (v^2/c^2) x

>> = (1 – v^2/c^2) x

>> = x / beta^2

>> 6) Multiply both sides by beta^2:

>> beta(v tau + xi) = x

>> which is half of the inverse transformation. Solve for t in terms of xi

>> and tau similarly, by solving the equations for xi(x,t) and tau(x,t) for

>> x, equating the two expressions for x, and solving for t. The result is

>> t = beta (tau + v xi/c^2).

>> ================================================================

>> v = x/t, so

>> t = beta (tau + x/t xi/c^2).

>> Incomplete, you still have 1/t on the RHS, Compo.

>> "Because the expression on the right-hand side uses tau, not t. " — Bill

>> Owen, above.

>> v = dx/dt (which can be integrated to x plus a constant) and since v is a

>> constant too we have v = x/t anywhere on the x-axis.

>> This the velocity of the train/ship/plane/car in the Earth’s Euclidean

>> coordinate system.

>> Unfortunately for you the velocity of Earth anywhere on the xi-axis of

>> the

>> vehicle’s coordinate system is upsilon = dxi/dtau = xi/tau and upsilon

>> does

>> not equal v.

>> To demonstrate, let v = 0.866c so that beta = 1/sqrt(1-0.866^2) = 2

>> Choosing x = 0.866, we have t = 1 for simplicity.

>> Hence tau = t * sqrt(1-0.866^2) = 0.5

>> xi = beta(0.866 – 0.866*1) = 0, the Earth is at the origin of the

>> vehicle’s

>> coordinate system at tau = 0.5, so it must have been at beta(0-vt) at tau

>> =

>> 0 and travelled a distance dxi = 0-upsilon.tau in time dtau = 0.5-0 with

>> velocity upsilon = xi/tau = 1.732/0.5 = 3.464c

>> Perhaps you can explain how the Earth exceeds the speed of light in the

>> frame of the vehicle, but I’m certainly not going to accept yours or

>> Savard’s ASSUMPTION that upsilon = v when length x<xi and time t> tau,

>> that way lies insanity.

>> So why is Savard multiplying xi by beta instead of dividing?

>> Answer: To deliberately obfuscate and pretend he’s slick, the

>> contemptible

>> arrogant bastard.

>> — This message is brought to you from the keyboard of

>> Lord Androcles, Zeroth Earl of Medway.

>> When I get my O.B.E. I’ll be an earlobe.

>> P.S. Kelleher, you’re welcome.

> I’ll have to think about this a bit more, but I *suspect* that you may

> be confusing v — the relative velocity of the two coordinate systems —

> with the velocity of an object as measured in one or the other of the

> coordinate systems. The latter is indeed dx/dt or dxi/dtau, but that’s

> not necessarily the same as v.

> BTW, my earlier post was pure algebra, showing how one can take two

> equations tau(x,t) and xi(x,t) and invert them to get t(xi,tau) and

> x(xi,tau). That’s straightforward, and the math holds regardless of the

> meaning behind the symbols.

> =======================================================

> You’ll have to think about this a bit more, but I *suspect* that you may

> be

> confusing v, the velocity of a point travelling a distance from zero to x

> in

> duration t with upsilon, the velocity of a point travelling a distance

> from

> (omega - xi) to omega in duration tau, which is definitely not the same

> as

> |v| in relativistic drivel but is the same in the real world.

> zero(0)——————–>x velocity v = x/t

> xi<—————-omega(0) velocity upsilon = xi/tau

> I refer you to

> http://www.fourmilab.ch/etexts/einstein/specrel/www/

> There is no tau(x,t) or xi(x,t) in Einstein’s inequalities, perhaps you

> mean

> tau(x’,t) and xi(x’,t) where x’ = x-vt.

> In particular, I refer to the inequality

> http://www.fourmilab.ch/etexts/einstein/specrel/www/figures/img22.gif

> You’ll have to think about this a bit more, without the "but".

> — This message is brought to you from the keyboard of

> Lord Androcles, Zeroth Earl of Medway.

> When I get my O.B.E. I’ll be an earlobe.

> — Bill

> P.S. I see that I accidentally omitted a slash in step 2 of my earlier

> post. Obviously the last term should have been (v^2/c^2) x.

> ============================================

> Fair enough.

Wholeheartedly agreed that I’ll have to do some more thinking. It’s

been over 20 years since I’ve done any coursework in relativity (either

special or general) and as an astrometrist I don’t often need to use it.

For my JPL colleagues who do radio navigation it’s an entirely

different story.

So again I will ask for your patience while I try to find the time to

get the dust off the textbooks, not to mention the neurons.

– Bill

=====================================================

It’s been over 40 years for me, Bill, and over twenty five years (1987)

since I discovered why stars vary in intensity, so I have a great deal of

patience for those with intelligence and an open mind but little patience

for bigots like Savard who think they know it all but can’t do the math.

My mathematical model reproduces this curve as well as that of eclipsing

variables and cepheids, all of which have a common cause.

http://www.britastro.org/vss/gifc/00918-ck.gif

You could call it a follow up to Swan Leavitt’s work.

The point is, if the star is a constant emitter and moves in an elliptical

orbit as per Kepler, and if space has no properties, then the star varies in

intensity with the period of its orbit.

– This message is brought to you from the keyboard of

Lord Androcles, Zeroth Earl of Medway.

When I get my O.B.E. I’ll be an earlobe.