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Time and space are joined at the hip. I mean this in a completely naked, having an amazing animal experience together kind of way. A review of the history of physics will support this scandalous claim.
[click or skip, the video is a reading of contents contained herein]

Sir Isaac Newton developed the mathematics of change, the calculus.
In looking at the logical structure of the formalism, he concluded that the math treated time as absolute, and space as absolute. He was not happy with this result, but logic is logic.

As a person, Newton was not known for being a nice guy even though he had great hair. He and God were on the same level, far above the commoner and counterfeiter. One rumor that has lasted centuries is that he died a virgin. At MIT at 18, I had never kissed a girl. Should I follow in Newton’s footsteps? I chose to become a biologist because at least they study sex.

In 1905, Einstein found a link between time and space.
Not being diligent in math class, it took his professor Minkowski to recognize that the Lorentz transformation was a rotation of time into space, and space into time. The party between time and space was beginning. Like an exclusive nightclub, the fun was restricted to jet setters who travelled at high speeds in straight lines.

After ten years of trial, error, and hard work, time was allowed to mix it up with space in any way using the math of the Riemann curvature tensor.  Toiling on the Russian front, Karl Schwarzschild solved the comparatively easy case of a spherically symmetric, uncharged, non-rotating gravitational source.  Good going Karl, but he died on the front due to a nasty autoimmune disease.  Finding new solutions is rare.  The math is too hard, even today with supercomputers.

Einstein had skills with the ladies. At ETH Zurich, there was one woman in his physics classes. She got pregnant with his child, then, well, wasn’t. They did marry and have two children. He got a great job in Berlin, but she stayed in Zurich. Long distance relationships didn’t work then, often don’t today. He remarried a first cousin but not to worry, no kids. They were together for his superstar years.The 1919 NY Times article “Lights all askew in the heavens. Men of science more or less agog over results” which I hope others also have reproduced and framed in their living rooms. The Nobel prize finally was awarded in 1921. Had to be a scary time in Germany for someone of his ansestry. He had a nice time out at Cal Tech, then set up shop at Princeton.

Again, it is just a rumor,
but I heard after the misses passed away, he had a room for entertaining female friends. The Russians sent a lady spy after the old man. While she got in “the room”, I don’t know how many secrets about the Manhattan project she obtained.

I am a radical anti-Newtonianist. I never allow time to separate from space. Likewise, I never allow energy to be separated from momentum. Surely I must be joking, Mr. Stand-Up Physicist. Take Newton’s second law. If you want to figure out the physics of bowling, nothing more is needed. How do you know this is classical physics? Say you were in a bar and pretty drunk, like, probably-shouldn’t-try-to-stand-up drunk. Someone slaps down two equations written on a napkin, and asks you to identify which one is classical physics, which one is relativistic quantum mechanics:
1. $(0, \frac{G}{c^4} \vec{F}) = (\sqrt{\frac{G \hbar}{c^5}}\frac{d}{d t},\vec{0})(\sqrt{\frac{G}{\hbar c}} m, \vec{0})(0, \frac{1}{c} \frac{d \vec{R}}{d t})$
2. $\left(\frac{\hbar }{m c^2}\right)^2\left(\frac{\partial^2}{\partial t^2}-c^2\nabla^2,2c\frac{\partial }{\partial t}\vec{\nabla }\right)\psi =\left(1,2 \vec{\beta} \gamma^2\right)\psi$
People write weird things on napkins in this bar. Look through all the tens of thousands of books that reference Newton’s second law, none will write things out this way. Really being dimensionless is like smoking much better dope than you are used to: thought you knew what high was, but it is not so. Sorry, loss of f o c u s, a common problem when drinking. The classical equation is the one with all the Easter Eggs, zeroes for everyone. The absolute separation of time and space is apparent when all the zeroes are written out explicitly. Newton took the correct first step. It is relativistic quantum field theory that has tried to fill in all slots. The scalar operator in equation 2 is known as the Klein-Gordon equation. Anyone reading this blog know the name of the lady vector equation curled up beside KG? [Celtics fans want to know].

Time and space get kinky in relativistic quantum field theory, something only a sliver of the folks reading this blog have studied. The Dirac equation introduces the dance of the 16 Dirac gamma matrices. There are a bunch of relationships between these gammas that make for good tests at one’s skill with abstract algebra. The high priesthood can talk about different types of representations and the implications, not that anyone else understands.

Let’s ask a much simpler, working class question: what the fuck does one of these Dirac gamma matrices do? Look at gamma 1:$\gamma^1 = \begin{pmatrix}0&0&0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$
This shows up playing “spin the spinor” games inside the Dirac equation. That is too complicated, so for simplicity, take Joe event in spacetime, (t, x, y, z), see what happens when it plays with this matrix:
$\gamma^1 (t, x, y, z) = (z, y, -x, -t)$
Now the value of the z position is literally time. Let that idea bother you as it should. The value of time is looking into its past so it can then point along the z axis. The gender roles for x and y have swapped, leaving x negative on the experience.

There are 16 Dirac gamma matrices, so information is systematically scrambled up. This is how Feynman’s “sum over all possible histories” happens. Go gamma go girl.

There are now two extremes. Newton’s pure classical world with a zero for every young man to guarantee he remains chaste. KG knows how to throw a fully relativistic field theory party so that no one cares who they end up with or what role they take.

There must be something between these two extremes. There is: classical quantum mechanics. It looks much like Klein-Gordon, but changes in time are not nearly so crazy, so only one time derivative is necessary. One full 4-derivative is an exact replica of the two found in Klein-Gordon. The other operator has the constant 1 in the place of the partial derivative with respect to time. To make the spatial derivative dimensionless, it needs a factor of hbar over 2 m. The mass shows up as a number instead of an energy-3-momentum quartet because the second time is a constant. Drop that operator in, and Schrödinger pops out:
3. $(1,\frac{\hbar}{2 m c} \vec{\nabla})\left( \sqrt{ \frac{G \hbar}{c^5}} \frac{\partial}{\partial t}, \sqrt{ \frac{G \hbar}{c^3}} \vec{\nabla}\right)^* \psi = (\sqrt{ \frac{G }{\hbar c^5}} V, \sqrt{ \frac{G }{\hbar c^3}} \vec{P}) \psi$

$\sqrt{ \frac{G}{\hbar c^5}}\left(\hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2 m} \nabla^2, \hbar c \vec{\nabla} - \frac{\hbar^2}{2 m c} \frac{\partial}{\partial t} \vec{\nabla} - \frac{\hbar^2}{2 m} \vec{\nabla} \times \vec{\nabla}\right) \psi = \sqrt{ \frac{G }{\hbar c^5}} (V, c\vec{P}) \psi$

Some will say it is silly to have the same collection of constants on both sides of the equation. I am playing handball against Nature, on her court, it is her home rules. I recognize the importance of the work of al-jabr, and I have a great imagination, but I cannot see Nature using anything but naked numbers. I will openly ignore the conventions chosen by Schrödinger.

[art sidebar: I have an idea for a poster. It features four dimensionless equations: the GEM unified standard model equations, the certainty and uncertainty principles, Schrödinger and Klein-Gordon. The alliteration, “Nature Nurtures Naked Numbers” adorns the poster. I need one other element - an image of a naked woman. As I understand it, there are a few on the Internet somewhere. I hope to obtain an image of someone who is familiar with one or two of those equations so there is a tangible connection. I don’t care if it takes me years to find such a well-educated model since I work on long time scales.]

Others might say I should use natural units. It is so easy to get lost playing in the world of physics equations, I cling to my physical constants. I like seeing one hbar next to the time derivative, but two next to the spatial one. I like seeing no c’s in the scalar term, yet some are in the 3-vector.

Again I ask the question: who is that equation in a 3-bed next to Schrödinger? It has the same operators in different combinations. If someone can name this dragon, it will be appreciated.

Now move to one of the great remaining mysteries in physics:  the source of asymmetry in the second law of thermodynamics, the proverbial arrow of time. It may have been Ludwig Boltzmann that pointed to the asymmetry being put in “by hand”. Once there, everything follows. I did struggle through Huw Price’s “Time's Arrow and Archimedes' Point: New Directions for the Physics of Time”. His presents an updated version of this issue, how there really is no reason for the asymmetry. He goes after others who have claimed to have resolved it, and puts up a good fight, saying, not so fast sailor (Price is an idea-smith, so his critiques are far more sophisticated than I can do with my sock-puppet prose).

The Foundational Questions Institute, FQXi, sponsored a contest called “The Nature of Time”, which was won by Julian Barbour’s well-titled submission “The Nature of Time”, the only contestant with that much moxy. These good FQXi folks apparently ignored my own contest held in March 1996, posted in the newsgroup sci.physics research. Here were the rules:

>of time.  There are two constraints:  it must be based on mathematics
>or physics - not philosophy - and it must be two sentences or less.

The results were reported on April 23 after a month of submissions (http://bit.ly/define-time). It was this very contest a decade a half ago that got me to reject time by itself, and give spacetime a full bear hug using quaternions. Always. Consistently. Like shut up already you bore me.

For the issue at hand, it means that the question about the arrow of time is not a well formed question. I am borrowing the notion of a well-formed question by reading and rereading “Godel’s Proof” by Nagel and Newman. What does one plus equal? Since it is not a well-formed question when mapped to math, the inquiry can never be answered. Should anyone provide me with a definition of time, even set up an experiment to show the work in practice, I will watch while sitting on a cosmic ray. All the data on time looks to me like data on time and space. The definition of time looks to the jet-setters like a definition of spacetime. Hello, basic vanilla special relativity. I know the august group that submitted essays longer than two sentences to FQXi knows special relativity. They are following a long, cultural tradition, and will be stuck with Boltzmann on the arrow of time issue. The essays do bring up other vital topics, like how you measure anything, let alone time, so are worth reading. My critique is targeted.

Does spacetime have an arrow? The time part is a scalar. Scalars have no pointiness to them, never have, never will. Space is pointy. If you wish to get technical, solve the snarky puzzle. The two most important laws in physics today, the standard model and general relativity, are local. Might as well make time reversal local too. Thank you for watching to this show. Please deposit the global Lorentz transformations in the waste receptacles on your way out.

As for my own sexuality, it is something I know very little about. I do take as many adult adult ed classes as I can. Such information can be used soon after the class to positive effect.
I have received instruction from both Carol Queen and Tristan Taormino. Ms. Taormino is the only teacher I’ve known to show up in a short, form-fitting, bright yellow latex dress. They both have books and videos worthy of study. Two examples I have enjoyed: “Exhibitionism for the Shy” by Carol Queen, and “Pucker Up: A Hands-on Guide to Ecstatic Sex” by Tristan Taormino. While I have not met Ian Kerner, Ph.D., the title of his book “She Comes First: The Thinking Man’s Guide to Pleasuring a Woman” will get a thinking man thinking, and perhaps take the correct form of action.

Doug

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 muliplication. 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 [correction: division algebra]. 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

Next Monday Night: Relativistic Rocket Science for Astrophysicists