By Doug Sweetser | April 11th 2011 11:03 PM | 23 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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For the unpopular cutting edge, there is no book with the answers at the back. This post will need to suffice for the snarky puzzles at the end of my previous posts. I am a playful snark, not a caustic one. I used to teach retarded citizens how to add. [clarification: while paying for 1 year of math grad school in Bloomington, Indiana, I volunteered an hour a week for a year at a local center.] I am that patient with myself, and will be so with you. Physics discussions can devolve into that 6 year-old boy fighting kind of groove (hi Sascha!). Since this is research, we do have a chance to learn something new together.
[click or skip a dramatic reading of this post]

The puzzles are meant to be simple. The answer has four parts. First is the quote of the problem, easy to skip is you recall the question. Second is the back story: why I originally did the calculation. Third is the one or two line mathematical answer. Fourth is a discussion of the implication of the solution.

March 14 - Quaternion Multiplication for a Third Grader

The Snarky Puzzle:
"Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Maxwell." Lord K.

Quaternions have been a one-trick pony, good for doing 3D rotations since the 1850’s. We presume you have received a good enough education to write it down as:
R -> R’ = (cos(a), I Sin(a)) R (cos(a), I Sin(a))*
where I is an arbitrary 3-vector. Show how if hyperbolic sine and hyperbolic cosine functions are swapped in, evil results: while two terms are correct, two terms are missing, and two other terms that don’t belong soil the calculation any finely educated simpleton would try first. Your job: unmix the evil. For this not simply connected group, find another two combinations of these hyperbolic functions, R, and conjugate operators that correct the flaws. That is what “not simply connected” means, that things must be added together, or did they skip the obvious in your group theory training? Should you tire of the strain, the answer is on a preprint server. Note, that is not the preprint server, since those living on the ultra-conservative fringe have to build their own servers out of fairy dust and Drupal. Very old school here, pre-school, grad school, fringe school, whatever.

The Back Story:

Take another look at that quote: Lord Kelvin is bitch slapping James Clerk Maxwell. Given how much respect I have for Max, the historical level of contempt for work on quaternions is extreme.

Ivan, from Parahyangan University, Bandung, Indonesia, was majoring in Physics. For his final project, he decided to do rotations using quaternions. This was not the most original idea since that is what Rodrigues did with them in the middle of the 1800s. It was only an undergraduate thesis after all. The bigger problem was his professor who pointed out this had little to do with physics. Professors!

A new idea was proposed: work with the Maxwell equations using quaternions only. I suspect my detailed work on derivation of the four Maxwell equations from the Lagrange density would be on the first page of a web search. That pesky professor wanted still more, like showing what happened to the field strength tensor under a Lorentz boost. Ivan asked for help on this one.

Being a practical guy, I said there are 2 approaches. One says that since the quaternion terms are exactly like the antisymmetric rank 2 tensor, nothing new means nothing new. A second idea was to use a paper by De Leo on the subject of quaternion boosts written in 1996. He thanked me for my time, and did get his degree. Way to go Ivan.

I had to live with guilt. Yes, I had read the De Leo paper. I kinda got it, but not really. When I really get it, then I can turn it into code, either C or Perl, depending on convenience. That did not happen here. I knew I didn’t know it. I felt bad about my reply, even if Ivan didn’t.

This is what motivated me to look slowly and carefully again at regular old 3D rotations as done with quaternions for a century and a half. The plan was to switch to hyperbolic cosines and sines, repairing as needed.

$b'=h b h^*+\frac{1}{2}\left((h h b)^*-\left(h^*h^*b\right)^*\right) \quad eq. 5$

$=(cosh(2\alpha ) c t - sinh(2\alpha ) x, cosh(2\alpha ) x - sinh(2\alpha ) c t, y, z)}$

$=(\gamma c t - \gamma \beta x, \gamma x - \gamma \beta c t, y, z)}$

where b is a spacetime quaternion being boosted, and h is a quaternion with hyperbolic cosines and hyperbolic sines.

The Discussion:
Quaternions are in a lose-lose situation. So long as this result remains obscure, physicists are right to avoid quaternions as boosts are too critical for work in special relativity. Should the result gain some traction, it will not generate interest since boosts are so early 1900s.

Here is the new win. A compact Lie group can be used for 3D rotations. Using quaternions that is the product of 3 quaternions. The Lie group for the Lorentz group is not compact. Why not? Using quaternions, the reason is reflected in using both multiplication and addition. Of course the Lorentz group is not compact, because multiple parts are added together. It is as simple as that.

March 21, 2011 - Lunchbox for a Theoretical Physicist

The Snarky puzzle:
This is Gauss’s Law:
These are two gauges:
1. The Coulomb gauge
2. The Lorenz gauge
Apply these gauges to Gauss’s law. Prove that the resulting equations are not the same. Do not try and impress your friends with this obvious result.
This is the GEM version of Gauss’s Law:
Apply the gauges from above if you can.
“Toto, I have a feeling we're not in [Normandy] any more” Dorothy, misquoted to reference Pierre-Simone Laplace’s home state.
“Only bad witches are ugly” Good looking Good Witch of the North
First and last lines spoken here: http://www.youtube.com/watch?v=EPWenQxryr4

The Back Story:

I was made to feel like an incompetent dweeb by a professor in a rushed discussion of spin angular momentum projection operators for a current coupling term in a Lagrangian. I had never had a discussion with anyone about a spin angular momentum projection operator before. I bet few people have. He has won awards for his work, and is a classically frantic fellow. The message was go study Feynman, don’t bother me again, ever. The message was received.

After getting a beat down, it is easy to feel paranoid. I was writing up my results to submit to a real publication. As a gambler, I thought the odds of getting published were not worth the ante cost, the first chips on the table, chump change. If I cannot read and understand the titles and abstracts of a single paper in “Classical and Quantum Gravity”, why should I submit a paper to that very journal? It was out of respect for the process of science. As I am doing the draft, I am worried about the professor - how would he trash the paper? We are not talking about helpful suggestions, a note of encouragement. I imagined a shredder snark, one that might get some blood.

I thought he would ask me about how my unified standard model proposal would look under a gauge transformation. If I didn’t know, I would have to sit in the corner and sulk. That didn’t sound like fun, fear of failure never is. I lived with that feeling for about a month. Now I know the answer.

Gauss’s law under the Coulomb gauge
$\rho = -\nabla^2 \phi$
Gauss’s law under the Lorenz gauge
$\rho = \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2}- \nabla^2 \phi$
GEM Gauss’s law under the Coulomb or Lorenz gauge
$0=\frac{G \hbar}{c^3} \nabla^2 \phi$

The Discussion:

Crank a Lagrange density through the Euler-Lagrange equations and out comes the field equations. All the discussions of gauge symmetries I have seen appear at the Lagrangian level, not for the fields. If I am just ignorant on this issue, please politely educate me. What I saw going on was people thinking carefully about their gauge choice so the resulting field equations could be solved for the problem at hand.

The GEM unified standard model field equations look every bit as invariant under a gauge choice as does the EM Lagrangian, but not the EM field equations. That feels like I rung a deep bell. It is only a feeling as I don’t know what fancy sounding jargon to cloak the observation.

March 18 - Time in Bed with Space

The Snarky Puzzle:

Consider an event in spacetime (t, x, y, z) in Cartesian coordinates so you do not have to think too hard. First show how this member of the Lorentz group:
diag{-1, 1, 1, 1}
would take the 4-vector into another 4-vector where only time had flipped its sign, no matter if time was positive or negative. Take that new vector and apply the same member of the Lorentz group to it. Repeat 1729 times. Admit you took the short cut of induction and did not complete the letter of the assignment. Nature lacks the power of induction, so does the long form.
Define a 4-vector R. I now bestow upon you the power to multiply 2 4-vectors together, a power you have always had Dororthy, via the rules of quaternion multiplication. Find R such that:
(t, x, y, z) R = (-t, x, y, z)
There can be no doubt such an R exists since quaternions are a mathematical field. The results should look so complicated the weak will leave the room. Let them go. Show how if the scalar is 10 orders of magnitude larger than the 3-vector as happens in classical physics, then to a wonderful approximation, R = (-1, 0, 0, 0). Maybe the weak return, since -1 is easy enough for them to remember. Notice that R is not a member of the Lorentz group. If the 3-vector is tiny, ignore the 3-vector, and flip the sign on time with a -1. That is an approximation whose flaw becomes obvious to systems using 1023 atoms. The arrow of spacetime is obvious to any child, so it should be to any physicist too.
[correction: the t, x, y, and z's should be deltas, dt, dx/c, dy/c, and dz/c. Events in classical physics have tiny relativistic velocities. For example, walking speed translates into relativistic velocity of 5x10-9, indicating the change in space is tiny compared to changes in time.]
The Back Story:

I was lucky to have one of the world’s best teachers on the subject of special relativity, Edwin F. Taylor. One of his classes was on the problem of time. He recommended the book by Price on the subject. Given the fundamental nature of the asymmetry of time reversal, this was an obvious thing to try early in my quaternion adventures (1998?).

$R=(-t^2+x^2/c^2+y^2/c^2+z^2/c^2, 2 \, t \, x, 2 \, t \, y, 2 \, t \, z) / (t^2+x^2+y^2+z^2)$
If t >>> x, y, z:
$R\approx (-1, \vec{0})$

The Discussion:

Any issue that has been around since Boltzman’s day must be tricky. This could be a removing the thorn from the paw of a lion story. Other professionals think the issue has been resolved, ignoring the modern analysis by Price. I have too much experience with frenetic physicists too busy to do a quaternion calculation, so I don’t worry. Spacetime has 3 arrows, more than enough to solve the arrow of time problem.

April 4, 2011 - Relativistic Rocket Science for Astrophysics

The Snarky Puzzle:

Here again is the relativistic rocket science technical speculation:
With the bold stroke of a pen, eliminate Newton. Isolate the small m and its differential to one side of the equation. Integrate both sides. If you are rusty, ask someone like alpha for help:
Integrate 1/m
Integrate 1/R2
Take the exponential of both sides. Only the second one is not exceedingly trivial:
exponential of the integral of 1/R2 series expansion for large R
Notice the series solution for R as R goes to infinity delivers what was promised.

The Back story:

I found a way to go from a force law for gravity to the exponential metric. It was a long, odd path. The first draft was shown to be wrong using Mathematica. I found a reasonable alternative that got the green light from my impartial, strict judge. At one step, the product rule came into play. I chose mA for sentimental reasons. My experience with shotgun DNA sequencing said I must also work the other road. Being complete is good. That led to a derivation of the relativistic rocket science term, not some inspired guess.

$\sqrt{\frac{G}{\hbar c}} m = \left(1 - \frac{G M}{V c R} + …\right)$

The Discussion:

In the biology world, there is something called DNA recombination. It is a way that the DNA cards one is dealt from ones parents get shuffled. In the cells that make eggs or sperm, the amount of shuffling is impressive. Animals do not pass along genes from mom or dad, but a collage from both.

The relativistic rocket science effect could be viewed as the recombination of an equation. All the same letters are used, but in a new combination. I hope someone with the numerical integration chops steals this one liner and puts it through tests.
$\frac{G}{c^4} m \frac{d V}{d t} \hat{R}\rightarrow \frac{G}{c^3} V \frac{d m}{d R} \hat{V}$
The part I am most proud of is the idea of a new direction for an effect of gravity, the direction of motion, v-hat. It sounds novel without cheating.

Next late Monday: We don't need no stinkin' Higgs!

Please elaborate as to why t is "10 orders of magnitude larger than" x,y,z in classical physics.
Colin:

I added the following to the time reversal puzzle:

[correction: the t, x, y, and z's should be deltas, dt, dx/c, dy/c, and dz/c. Events in classical physics have tiny relativistic velocities. For example, walking speed translates into relativistic velocity of 5x10-9, indicating the change in space is tiny compared to changes in space.]

Hope that is elaborate enough.

Doug
Thanks, and you probably want to make the trivial correction to your correction, changing the last word to "time" because "the change in space is tiny compared to changes in space" isn't what you're trying to say.

In the interest of having the patience to teach the retarded to add, or just good technical writing, please spell out what approximating time reversal by multiplying by the quaternion -1+0i+0j+0k at classical velocities has to do with making the one-way direction of time obvious to physicists.
Trivial correction corrected.
Will spell out the issue here.

There is the exact reversal quaternion, call it Q for now, which is complicated.
$(dt, d\vec{R}/c)\frac{(-dt^2 + dR^2/c^2, 2 \, dt \, d\vec{R}/c)}{(dt^2 + dR^2/c^2)} = (-dt, d\vec{R}/c)$

Here is my gut reaction: that quaternion Q looks to complicated to be useful to anyone ever.

The first question though, is to ask if it is right. Multiply on the left by the inverse of (dt, dR/c). If one is familiar/reminded of the rules of complex numbers, the inverse of a quaternion is the same. Take the conjugate (toss a minus sign in front of the 3-vector) and divide by the norm (the scalar squared plus the 3-vector squared). So the question becomes:

inverse of (dt, dR/c)...(dt, -dR/c)/(dt^2 +dR)^2 multiply that by (-dt, dR), will that be the time reversal quaternion Q, the big ugly fraction?  Make that look prettier:
$\frac{(dt, -d\vec{R}/c)}{(dt^2 + dR^2/c^2)} (-dt, d\vec{R}/c) = ?$

The denominator is right. The scalar has the a dt(-dt), so make as -dt2, check. There is also a dot product of the 3-vectors. A minus times a plus is minus. This is were one must learn that quaternions are 3 complex numbers that share the same real value. The dR has 3 imaginary vectors, so squaring the imaginary numbers tosses in one more minus sign. minus plus minus make the dR.dR dot product positive. The 2 dt dR are formed by multiplying the scalar and a corresponding 3-vector, either both positive or both negative, so they add up. Since dR points in exactly the same direction as dR, there is no cross product to include.

The complicated term is exactly right. It is so complicated, there must be limits where this is more useful. The question was about the classical limit, with dt >>> dR. The denominator gets simpler, leading to this approximation:
$Q=\lim_{dt->big} \frac{(-dt^2 + dR^2/c^2, 2 \, dt \, d\vec{R}/c)}{dt^2} \approx(-1,\vec{0})$

This is a most excellent approximation for classical physics for a small sample set. For stupidly large numbers of atoms, a moles worth, this breaks down, or one could say the second law of thermodynamics shows up. The quaternion Q provides a reason for the asymmetry in thermodynamics because spacetime has 3 arrows.

A fun bonus problem is to look at a different limit. For light, changes in time are the same size as changes in space. This limit produces an exact result:
$Q=\lim_{dt = |dR/c|} \frac{(-dt^2 + dR^2/c^2, 2 \, dt \, d\vec{R}/c)}{dt^2 + dR^2/c^2} = (0,\vec{1})$

I decided to check this, using the rules of complex products:
$(t, \vec{R}/c)(0, \vec{1}) = (-R/c, t \, \vec{1})$

This works only if the 3-vector 1 above points in exactly the same direction as the 3-vector R.

"Obvious" is probably a bad word choice. One needs to kick this quaternion time reversal around a little, see how it works in a few limits.

For standard physics, time reversal using the global Lorentz transformation remains a problem.

Doug
Somehow I missed this good detailed response to my question until now.

What I get from your explanation is that at classical speeds, your exact time-reversal quaternion shows the 3 arrows of space only become significant when there's a thermodynamically relevant number of particles, which shows thermodynamically it's not surprising there's a difference between the huge number of particles and just a couple when looking at time reversal.  I have a hard time directly relating the ugly time-reversal quaternion with the probability of time reversal, but I guess I agree it feels right to say having to account for all the 3 space directions of all the particles in doing a time reversal would make it less likely the more particles there are.
Yup, you have the core idea. While this is suggestive, a concrete connection to equations that appear in thermodynamics has yet to be made. What I did in my first research efforts was take a class on special relativity. The experiment was that I had to solve every single problem set question posed using the standard way, and using quaternions exclusively. If there was one problem that could not be solved with quaternions, the proposal would be wrong. 53 questions later, quaternions passed the test.
The same standards apply for thermo. I really enjoyed my statistical thermodynamics class back in 1982. It does not look like I have the time now to tackle a set of problems in thermodynamics. So it goes.
Why should we believe the quaternion operations that change the quaternion scalar from plus to minus corresponds to an exact relativistic solution to reversing time in Minkowski space-time?

In other places in these comments, you say the 4-vector quaternions transform as 4-vectors under Lorentz transforms, but that seems to be because you put your 4-vectors into Minkowski space-time to do the transform, and don't use the properties of quaternion space-time at all.

In your inversion of time Snarky  Puzzle, you are keeping the 4-vector in quaternion space-time and just using quaternion multiplication and division to get your answer.  Why should we believe that has anything to do with how time would be inverted in relativity?
"I used to teach retarded citizens how to add. I am that patient ... and will be so with you."
Readers can see ever more clearly how you actually feel about them. Nobody successfully hides their true nature for long. Some gave you a chance up to now - after all, belonging on the couch does not necessarily imply bad science. But you cannot even refrain the immature urge of putting a personal insult against a commentator ["6 year-old boy fighting kind of groove (hi Sascha!)"] straight into the head of your article? Extremely bad style, inferior taste, six year old attitude, quite correct. Did it hurt so much, giving the boy a taste of some grown up science?

Well, I would say a lot more readers gone with this distasteful rubbish - bye bye "snarky".

PS: I have to admit you are really funny sometimes:
"If I cannot read and understand the titles and abstracts of a single paper in “Classical and Quantum Gravity”, why should I submit a paper to that very journal? It was out of respect for the process of science."
Sascha, Sascha, Sascha, darling (oops, probably the wrong salutation)

I play a game of generous tit-for-tat. If you recall your opening salvo, in that other post, it would call into question all I had written ever. You decided to select the most distasteful thing, a mischaracterization of the dark energy hypothesis.
I learned a few things from the exchange. First was a style of correction. I keep the text as is, then add the comment [correction: whatever I need to say]. I used that same style to clarify an issue in this post. I did try to correct an impression that the phrase "dark arts" does not mean any overlap between dark matter and dark energy.

Being a tit-for-tat guy, I did fire back. It was not aimed at you. It was aimed at the way dark matter as a hypothesis is commonly described. I also learned that you can really anger some folks by questioning the content of their blog.

If you want to see that I can have a mature exchange, look at the dialogue at the end of the time in bed with space post. I happen to be using my math lingo incorrectly, and David is helping me say things right. Quaternions are a division algebra (knew that), but I sometimes refer to it as a mathematical field, which is wrong since quaternions do not commute. Looking back on equations in my paper, it is now clear I was always working with Lagrangians, not actions per se. I think I should also use an explicit symbol for the quaternion products as well as the hypercomplex ones. That exchange has had a positive benefit to the technical aspects of the work.

I will make you a few promises. One is to never type "Sascha" in a blog again. Ever. Simple to do.

I promise I don't hate you. I don't care about you because none of your comments has technical content about my work (I don't work on the dark matter or dark energy hypotheses). My guess is that will angry you more, but of course, that does not matter to me.

My final promise is not to reply should you decide of your own free will to reply to this reply. You proved the point of the sentence by going after the bait, no more proof is needed.
I for one look forward to reading your posts because they put me in the right mood to study math I want to understand but fell asleep trying to read the first time. I don't really see why the scaler part of quaternions should be identified with time, but you motivated me to understand quaternions better which lead to understanding a lot of other math better too.

You are also hilarious at times esp. in some of your comments and I have laughed out loud while walking my dogs and remembering one in particular "The odds of a non-physicist guy getting a spin-g factor right whatever-the-donkey-kong that is are low, like don't bother me low"
As long as I have one dedicated reader, that is enough for me. Science is not a popularity contest as we all know. The biggest benefit of writing is that it clarifies my own thinking.
How do you think about time? Do you consider time to be a real number? It turns out that the real numbers are a subgroup of quaternions. It is that subgroup, the scalar part of a quaternion, that I am giving the role of time. The 3-vector then is space, so the 4 numbers together is an event in spacetime, like a fingersnap, done at a particular time and place.

This really is a port of what goes on in special relativity. There are many 4-vectors in special relativity, and I treat each one as a quaternion. That means energy takes the pole position, with the 3-momentum following the pace car. I do the same with derivatives. I do the same with 4-potentials. Consistency is good, if a bit dull.

Note: quaternions cannot do it all. I also use a number that has similar rules, except that everything is positive. I needed to rediscover those to do gravity work.
My problem with interpreting quaternions t+xi+yj+zk as 4-vectors in special relativity has to do with calculating the interval between two 4-vectors.  I assume matrices are not allowed since they aren't quaternions. So, we need to find another quaternion t'+x'i+y'j+z'k such that
(t+xi+yj+zk)(t'+x'i+y'j+z'k)=-tt+xx+yy+zz (don't know how to type in superscript 2 for squared)

I can remember when first learning the interval in 2 dimensions was -tt+xx, why not use complex numbers?  If t always has an i associated with it, then t squared would be negative.  Of course the problem is that (ti+x)(ti+x)=-tt+2xti+xx.  What are we supposed to do with the 2xti?

You found R such that:
(t, x, y, z) R = (-t, x, y, z) which seems in the right direction so might as well repeat your result:

[1/(tt+xx+yy+zz)](-tt+xx/cc+yy/cc+zz/cc+2txi+2tyj+2tzk)

We want a result like xx-aa.  When I first learned to factor polynomials, I loved the fact that xx-aa had a very easy solution: (x+a)(x-a).  So it is very natural to try

(t+xi+yj+zk)(t-xi-yj-zk) and as we all know, that is the usual way to measure the distance between quaternions, giving the unfortunate result
tt+xx+yy+zz with no minus sign to be found anywhere because ii=jj=kk=-1 changes the hoped for minus signs into plus signs.

I didn't work anything out before typing this response and was mistakenly thinking I could combine your time flipping result with the usual way of doing things to somehow get the minus sign back in front of time, but now realize I wasn't thinking very clearly.

Anyway, have you worked out what quaternion multiplies t+xi+yj+zk to give the scalar result -tt+xx+yy+zz (or with all the signs flipped)?  Doesn't seem very easy, which is why I don't see why the scalar should be identified with time in special relativity.

On the other hand, I do appreciate special relativity can be deduced from EM and you have quaternion equations for EM so that in those equations, the scalar must in fact be time.  So, what is the quaternion that gives the correct interval?
So, what is the quaternion that gives the correct interval?
The answer to this question is why I own quaternions.com :-)

First a note on notation. Brains are great at handling pairs of numbers. This is how the folks that multiply 2 8 digit numbers do the task, splitting each number into 4 pairs. Brains can handle 3, worse for 4, and on down it goes. Knowing this, I like to represent quaternions with 2 letters, a real (hopefully lowercase) and a 3-vector (uppercase), so (t, R). In LaTeX, I put an arrow on a 3-vector in a 3-vector slot. The R in tR would get the arrow, but the scalar R.R does not. These are my conventions, designed for how the brain works most efficiently.

Square the delta time and delta space between 2 events in spacetime:
$\frac{c^5}{G \hbar}(dt,d\vec{R}/c)^2= \frac{c^5}{G \hbar}(dt^2 - dR \cdot dR/c^2, 2 \, dt \, d\vec{R}/c)$
There is the Lorentz invariant interval in the first position. Nice. There is also a well-formed Lorentz covariant 3-vector. Lorentz covariant is just a fancy way to say we know how it transforms (like gamma2 beta if you must know).

This was the first calculation I ever saw with a quaternion, a plain old square. I recognized the interval instantly. It rang a bell in my mind - this is not chance. I had been taught of the importance of this interval in special relativity. The simplest darn thing one can do with a quaternion is a key tool. This was my starting point.

The norm uses a mirror:
$\frac{c^5}{G \hbar}(dt,d\vec{R}/c)(dt,d\vec{R}/c)^*=\frac{c^5}{G \hbar}(dt,d\vec{R}/c)(dt,-d\vec{R}/c)= \frac{c^5}{G \hbar}(dt^2 + dR \cdot dR/c^2, \vec{0})$

Both of the interval and norm are used extensively.
Those 2 results are what I already noticed, but can you get
(dt, dR/c)(simple quaternion based on dt, dR/c) = (dt2-dR.dR/c2, 0)?
Quaternions are a division algebra. That means there is necessarily a way to go from one point to another no matter what that other is:
$\frac{c^5}{G \hbar}((dt,d\vec{R}/c)^2+((dt,d\vec{R}/c)^2)^*)/2=\frac{c^5}{G \hbar}(t^2 - dR^2/c^2,\vec{0})$

That works by blasting away the 3-vector, so it may not be a satisfying answer. What is being wasted? The 3-vector-with-no-name, 2 dt dR. In a different thread, I pointed out that stretching the 3-vector dR by the amount dt is not rocket science. What is stunning to me is that this thing doesn't have a name as far as I know. Maybe I should know more. Yet it sits right next to one of the most famous invariants in physics. "Lorentz invariant interval, I'd like you to meet ... whatever". Physics is so over-studied, I want to hang out with the new girl.

The Lorentz invariant is interpreted as the time the object being measured experiences in it's own rest frame.  Why should the time experienced in an object's rest frame be in a vector that also has a physical displacement term?

Just as much to the point, why should the norm be interpreted as some sort of time?  We have to keep it in the 4-vector in the scalar slot because it's not invariant, which you want everyone to interpret as time, but what is the physical interpretation of the quaternion norm in terms of time?  I think you understand now my confusion in interpreting the scalar as time.  Besides, doesn't the word "scalar" usually imply invariance?  If so, time is certainly not a scalar because it is a component of a vector whose value depends on choice of basis vectors, i.e. choice of inertial frame.  Is there a natural way to change the quaternion basis vectors such that the real component transforms the way time does in Relativity?

You say time is a scalar, but I think you mean proper time is a scalar.  It's certainly true that all observers can calculate the proper time of an event and all will agree.  So, yes, time is a scalar if you only talk about the time you are personally experiencing, and only look at the interval, which is the proper time, when looking at other events.  In general, time is not invariant, so does not justify being treated as a scalar in math.

I'm no expert in Relativity, having only studied it 25 years ago and never having worked many problems (being just as lazy then as I am now), but I'm pretty sure the issue I'm bringing up is the reason time was not identified with the quaternion scalar after Relativity was developed.
Looks like my answer failed, so it goes.

The way I use the word scalar and 3-vector was the way Hamilton did. For the sake of clarity in this reply, I will prepend those words with quaternion. A quaternion scalar and quaternion 3-vector have not a thing to do with how things transform under a Lorentz transformation. So the dt in (dt, dR/c) is a quaternion scalar, dt2 - dR2/c2 is a quaternion scalar in  (dt2 - dR2/c2, 2 dt dR/c), and dt2 + dR2/c2 is a quaternion scalar in  (dt2 + dR2/c2, 0).

When one collects data about events in spacetime, that goes into the quaternion (dt,dR/c), with dt being the difference in the time measured by whoever. We know how this transforms under a Lorentz transformation. It goes like so:$(dt, \mathbf{dR}/c)\rightarrow (dt', \mathbf{dR'}/c) = (\gamma dt, \gamma \beta \mathbf{dR}/c) \quad eq. 1$

Square an event quaternion, or Lorentz 4-vector:
$(dt, \mathbf{dR}/c)^2= (dt^2 - dR^2/c^2, 2 dt \,\mathbf{dR}/c ) \quad eq.\,2$

The quaternion scalar is a Lorentz invariant scalar. The way the quaternion 3-vector transforms is really much messier than I had initially thought. How it transforms can be calculated, but I will skip that.

Take the Lorentz 4-vector, reflect it in a mirror, and form the product:
$(dt, \mathbf{dR}/c)(dt, -\mathbf{dR}/c) = (dt^2 + dR^2/c^2, \mathbf{0} ) \quad eq.\,3$

This time it is the quaternion 3-vector that is Lorentz invariant. The quaternion scalar will be complicated under a boost.

These look like 3 different animals to me, both algebraically and how they transform. I don't come with any expectations of how they should behave. Turtle, meet duck and rock.  All are distinct.

In 1910 and 1911, two different guys tried to use a quaternion triple product to do a Lorentz boost. That does not work. They both figured out how to do it by tossing in an extra factor of I, using biquaternions. Biquaternions are not a division algebra, so I don't work with them. It took a century of non-effort, but the answer to how to do a boost with real-valued quaternions is right here in the original blog.

That is a better answer, although I still don't see how to do a Lorentz boost using just the properties of quaternions.  You seem to simply put your (t,R) vector in Minkowski space, do the change of inertial frame there without making use of ijk, and then say your result is still a quaternion because it has the same form, (t,R).  I would like you to answer a different question though.

You identify the quaternion scalar with time, and the ijk parts of the quaternion with vectors in space, so that for you, quaternions provide a description in space-time.  You then use the ability to multiply and divide these vectors in quaternion space-time to invert time in a 4-vector in quaternion space-time.

My latest question then is, why should we believe inverting time in a 4-vector in quaternion space-time using quaternion multiplication has anything to do with inverting time of a 4-vector in Minkowski space-time?
Being a good skeptic, I will not ask you to believe. Instead, I request you look through my proof. Oops, I don't have one at this time. All I have is something suggestive, a technical speculation (defined in a different post as something so specific it can be captured in a line of algebra). There is a well established problem with the arrow of time that someone like that book by Huw Price goes into detail. This is an old problem. I might have the kernel of an idea to resolve it. To construct a proof, I would need to take my tiny observation and connect it mathematically to an established law of thermodynamics. That has not been done, so remain skeptical. As has been learned time and time again, reasonable ideas do fail. It would be silly for me to claim that mine was different from others.

My experience with quaternions has been amazing though. That I can document. All the published books out there use biquaternions to do the Lorentz boost or derive the Maxwell equations. I have been skilled enough in this odd craft to do both. Remain skeptical, but keep reading the new posts :-)
Of course it's not a satisfying answer because you got your answer by adding instead of multiplying.  But, what's really bothering me is that the norm is not invariant - it depends on choice of observer's speed relative to the 2 events being measured.  The whole point of special relativity is that it's the interval that's invariant in all inertial reference frames.  So, if the norm used by quaternions isn't invariant, then how can you say it's a natural choice for Special Relativity?  Using Quaternions, you would have to always keep the norm in the scalar slot of the vector to remember it's not really a number that's the same for all observers, but the interval is the same and should be taken out of the vector.

You've gotten me studying again, and that's good.  The only math/physics book I've got at the moment is The Road To Reality, so that's what I'm working through.  I had started it before you started blogging here, but had stopped soon after the section on quaternions.  I was excited by quaternions, but then he quickly went on to Clifford algebras and Grassman algebras which I fell asleep on.  With your help, I became excited by the math again and this time understood how Clifford algebras follow from quaternions and also the geometrical spinorial-object models given in the book which are a natural physical interpretation of quaternions and Clifford algebras.  Just now I peaked way ahead at how he treats Special Relativity and he actually uses complex values for all four coordinates.  Then he points out he can slice through the space to get all reals for Euclidean, or take a slice where, as with quaternions, the space parts are imaginary and that gives Minkowski space.  Can also take the slice where time is imaginary and space real for Minkowski space.  Of course, he is not using quaternions, but it's interesting that modern physics isn't too far off, maybe, from your quaternion approach.
Your previous 2 posts sound like we have an issue of language. You are using a lexicon that is relevant to special relativity. Mine is rooted in math. Here is the definition I try to use consistently:

Definition 1). A scalar is the first term of a quaternion that has no "pointiness". No transformation properties are ever implied.

The common use by physicists is:

Definition 2). A scalar is a Lorentz invariant quantity.

There are many situation where the same darn word is used in different ways. I could claim that Hamilton coined the words scalar and vector, but that would be both true and silly. Let's use definition 1 on our limited cast of characters.

(dt, dR/c)
The scalar dt and the 3-vector dR/c together transform as a 4-vector under a Lorentz transformation.

(dt, dR/c)2 = (dt2 -dR2/c2, 2 dt dR/c)
The scalar dt2 -dR2/c2 is invariant under a Lorentz transformation. The 3-vector 2 dt dR/c transforms under a Lorentz transformation as gamma^2 beta.

(dt, dR/c)* (dt, dR/c) = (dt2 + dR2/c2, 0)
The 3-vector 0 is invariant under a Lorentz transformation (way to go zero). The wrong one was invariant :-) The scalar norm dt2 + dR2/c2 transforms like so:
$dt^2 + dR^2/c^2 \rightarrow (dt^2 + dR^2/c^2)' = (dt^2 + dR^2/c^2) \frac{1 + \beta^2}{1 - \beta^2}$
Does that ratio of betas have a name?

Folks who work in special relativity are justified in using their own lingo so communication is efficient. It does mean I have to be more verbose. I can always figure out how something transforms since I start out with quaternions that transform like 4-vectors.

Of course it's not a satisfying answer because you got your answer by adding instead of multiplying.  But, what's really bothering me is that the norm is not invariant - it depends on choice of observer's speed relative to the 2 events being measured.  The whole point of special relativity is that it's the interval that's invariant in all inertial reference frames.  So, if the norm used by quaternions isn't invariant, then how can you say it's a natural choice for Special Relativity?
I have seen this argument before. This is the "calling a chicken a stapler" problem. A chicken is not a stapler, so calling a live chicken a stapler will only generate confusion. The norm of a quaternion is not the Lorentz invariant interval. If one wants the norm, just square a quaternion. That was easy. I have an aversion to anything being called natural since nature does too many odd things. Natural really implies the bias of the viewer. I take what Nature or math provides. I enjoy when my hopes get dashed. If I find something that works, I keep it. As I may have said here recently, I can use quaternions to solve problems in special relativity. Quaternions do not make physics easy, but it can be done, I have the problem set answers to prove it (on the web, off of the quaternions.com page).

I think I love this blog.

The drama that one time with douchie felt like an Olympian level of magnificent discord.

I hope you are well, Douglas, and sorry I didn't not reply to your last very patient response to my ridiculously inadequate understanding of math and physics. However I do love asking questions and getting answers that explain the shape of the landscape of my lack of understanding. This you have consistently done. Thanks.

Martin

[snark on] Look you jerk, you keep pumping up my ego like that and I might consider answering another one of your questions [snark off]. This keeps me up late for a good reason.