this will take some time but I hope in a couple of days
something nice will be written.
This is by no means a eulogy, but I feel it is fitting to dedicate something to this great man.
It is customary to dub people "great" for their accomplishments and otherwise professional stature.
Avi Schiller was a great man, simply because he was a "mench". One of the few.
He would also say: A boor cannot be sin-fearing, an ignoramus cannot be
pious, a bashful one cannot learn, a short-tempered person cannot teach,
nor does anyone who does much business grow wise. In a place where
there are no men, strive to be a man.
Abstract: It's not my fault I'm lazy, Physics made me this way
dedicated to my sister-in-law and her fresh hubby's mawwiege
Mazal Tov!!!
Recently I attended my sister-in-law's wedding...
After a full month of being forgotten, taken as granted and ultimately being treated as some ungainly babysitter to my own girl, it has arrived...
The event dreaded and anticipated, probably by the bride and groom, but also by yours truly -
THE WEDDING
Awesome and gruesome, with a multitude of family-related potential calamities, just waiting to happen, hanging on the off chance of someone sitting in the wrong seat or uttering the wrong sentence, incurring a death sentence, with the obvious exception of the crazy and otherwise fully-certifiable-bunkers pseudo-relative (She can say whatever the hell she wants, nobody's home thus no-one really listens or cares what she says).
The food was good, the company better, and the music divine. well, not really... the first two statements are correct whereas the third - not so much....
The DJ was poor, probably not in the fiscal sense after the presumed hefty sum he was paid, but poor taste in music, poor know-how of his trade, poor execution, and ultimately, the only reason his music was anywhere near palatable, was this was their wedding and we were so damn happy for them, that everything else vanished, not unlike the 10th order Taylor term for a quadratic potential function.
At some point the music was pounding so hard, I literally saw hordes of vikings working at a huge smithy, pounding away with all their might to mold some unfathomably hard metal into some equally mysterious shape.
Now, if you follow my posts you know full well what happens to me when faced with potential demise - physics comes into play!
I immediately went on to contemplate vikings and physics, and stumbled upon Dichroism, which is basically when light polarized in different directions go through a certain special material, only one polarization survives (the otherwise polarized rays are reduced to a point of vanishing altogether).
Now since sunlight is non-polarized, this normally doesn't mean squat, but when there's heavy cloud cover, the light rays disperse over the clouds so you get a general bright haze, but no obvious direction for the sun.
Enter dichroism: by using dichroic glass the light that passes through the stone is brightest in the direction of the sun, because right in front of the sun (through the clouds) the light is differently polarized.
The trick is you have to find the right direction to hold the dichroic stone in, else you'll basically get nothing.
Anyway, this dichroism trick is how vikings were able to sail across wide stretches of open sea, since they didn't have land in sight for coastal navigation, and either they didn't know their celestial-nav basics, or the skies were constantly cloud-covered.
BOOM!!
One of the loud pounding beats threw me back into consciousnesses, in time to realize how horribly out of shape I was...
You see, I was actually dancing (well, more like contorting) on the dance floor, sweating and panting, and the all too familiar "why am I so lazy?" question kept burning in my mind...
Which again set a chain of events in my mind culminating with physics:
It's not my fault!!! Even physics is lazy!!!
And I will prove this immediately:
First off, remember old Newt's law:
"Every body persists in its state of being at rest or of moving uniformly straightforward, except insofar as it is compelled to change its state by force impressed"
Translated: every body is basically a fat blob. you really have to exert force to change its current state.
I will now delve into a bit of Lagrangian physics to prove a point:
<Warning: Physics ahead> This is a simplification of deriving Lagrangian Mechanics but here goes: Consider Newton's 2nd Law:
\[F=ma\]
And if you're a physicist you know this is a simplification of:
\[\frac{dP}{dt}=F\] So, now, applying D'Alambert pricnciple we can constrict ourselves to "external" forces, and taking F to always mean a conservative force (i.e. arising from some scalar potential function) we can write: \[F=-\nabla \left(U\right)\] where U is just the potential function otherwise known by the name "Potential Energy". If we constrict ourselves to one dimension we can see: \[F=-\frac{\partial U}{\partial x}\] Looking at the left hand side of the above equation where we treat \(P=mv\) let's explore the relations between the kinetic energy term - \(\frac{mv^2}{2}\) and the above term. It's fairly easy to see that: \[ mv=\frac{\partial}{\partial v}\left(\frac{mv^2}{2}\right) \] Let's call the kinetic energy term T and the potential energy term U, so we now get: \[\frac{d}{dt}\left(\frac{\partial T}{\partial v}\right)=\frac{\partial (-U)}{\partial x}\] Understanding kinetic energy to be non-dependant on location, and potential energy to be solely location dependant we can add expressions to both sides that vanish in derivation thus defining: \[\mathcal{L}=T-U\] to get the Euler-Lagrange equations \[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial \dot{q}}\right)=\frac{\partial\mathcal{L}}{\partial q}\] Where here I've also switched to some generalized coordinates \(q,\dot{q}\). Now, defining the action \(\mathcal{A}\) as the time integral of the Lagrangian \(\mathcal{L}\), or in other words: \[\mathcal{A}=\int\mathcal{L}dt\] We have now successfully exchanged Newton's rules formulation by the "Principal of least action", since it follows that demanding minimal Action, actually yields Euler-Lagrange equations which are the equations of motion. A word of caution though: First off this derivation is a sketch, the true process is a lot more rigorous, but I assure you, it works. Secondly, Lagrangian Mech is a heavy canon, all you need to do is describe the problem in some sensible coordinates, find the kinetic energy and potential energy, perform some derivations... and BAM!!! you get equations of motion, as well as conserved quantities and symmetries of the system like magic!! But, since we use a couple of assumptions in deriving the Lagrangian, this actually covers LESS general physical situations. In reality there's ALWAYS friction, which is by no means a conserving force, thus adaptation to Lagrangian Mech are needed and they are not always as simple. <Warning: Physics ahead/>
The obvious conclusion of all this mess, is that Newton laws of motion are equivalent (under some basic assumptions) to a first principle that states the following:
"Every physical system aspires to minimize the action taken"
for instance, water will flow in the path of least resistance, and light will travel the path of least (optical) distance, and I, ladies and gentlemen will aspire to do practically NOTHING if I can possibly get away with it.
This means I will absolutely abhor every single time I really have to get up, for instance to go to the fridge and get some grub, I'd much rather my lovely wife make lunch for me and serve it while I leisurely sit back and enjoy the latest mind-numbing episode ofsome stupid comedy show.
By the way, this is EXACTLY why light bends in the presence of gravitation. Not to watch stupid comedy shows, the other thing - being lazy.
See, gravitation can be thought of as a property of space(-time), rather than some force or field that occupies said space. by comrade Einstein's equivalency principle you can not tell whether you are being pulled by gravity, or being accelerated in the opposite direction.
And by that virtue, even light itself (having no mass is irrelevant here) bends to accommodate for this equivalency.
One might even say (and be wrong) that light is the submissive partner in the energy-gravitation relationship... But as I said, this is not correct, since light is a form of energy, and gravitation is a product of energy concentrates. So really, it's a marriage of energy and gravity, or in oriental philosophy terms, marriage of heaven and earth, or ying and yang, or horse and carriage (err... that's from a different philosophy methinks).
To conclude, marriage is such a blessing in one hell of a disguise, since left to our own devices we will do absolutely nothing until some external force propels us to action. When in a relatively successful marriage, the same is true, only the external force is called a wife, and hopefully, most of the times she kicks you in the gonads, it's for your own good.
This also enables two people to become a system of coupled equations, thus redefining the least action to actually produce some interesting dynamics, at least until a third little equation comes into play to dominate the hell out of their lives... if you're a parent, you know what I mean, right?
My own sweet third coupled equation
Alternatively, if your marriage sucks, well, this explains why people stay in dysfunctional relationships for too long - it simply is to much ACTION to get out of one, better (by physical standards, not mine), to become mentally dislodged and otherwise apathetic to a fault.
But we know it's the former rather than latter. we really do.
To those who had the (mis)fortune of having me as a part of their lives, weight gain, and weight loss are all too familiar. if not by first experience, they have probably witnessed my own physique change, all across the range spanning from wirey-thin to brown dwarf ( a so called Failed-Star, or UltraCool Dwarf, which is a stellar body of roughly 13-85 Jupiter masses), and hopefully, one day, back again.
Since my travels through mass-space have been oscillatory in nature (for the most part), it is with pain-laced mirth I give you the following analysis and analogy of myself, and the Cosmos.
I know, I know, this seems a bit megalomanic on my part, but I assure you... I am.
Well really, since my current field of study is Cosmology, and specifically inflation models, I found it might be funny to recount my thoughts and findings and relate it to the human condition, or rather, the fatso condition...
To any and all cosmologists out there, who might be reading this right now, be gentle, I am a mere neophyte to this field, and so I appeal to your sense of humor, and sympathy (enter violins please).
In order to first pique your interest, please listen to this number by Monty Python:
Which, by the way have some major faults in cosmological terms, for instance, the universe (acording to leading theories) actually does NOT expand at the speed of light. our EVENT-HORIZON (or simply horizon) does. It's meaningless to ascribe velocity to the expansion of the universe, since very close to us, the universe recedes from us in small velocities and far from us the universe seems to recede with greater velocity, as reflected by Hubble's law - velocity is proprtional to distance from us... \(\left(v=H_{0}D\right)\) .
By the way, this also implies that Galaxies which are far enough from us, recede at velocities greater than the speed of light (yes it IS possible), and thus they "drop off" our horizon, since they fade away faster than their light travels to us.
This also foretells a dark and lonely eventual demise, for our universe.
At any rate, I would like at this juncture, to acquaint the reader with Friedmann's equation, which I will first write down and derive LATER (in two ways by the way...)
\[\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
Where \(R\) is the radius of an arbitrary sphere in space. this equation describes the evolution of \(R\) due to gravitational forces alone.
In layman terms:
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.
For those who have some interest of keeping their sanity, you are invited to skip these derivations.
For those of brave soul and not so sound mind, please take special interest in the general relativity case...
(Important conclusions, in green)
<derivation - Newton style>
from Newt's 2nd law: \[F=ma\\
\Rightarrow F=m\ddot{R}\]
Gravitational force due to enclosed mass, on the perimeter of a sphere:
\[F=-\frac{GMm}{R^2}\]
Equating these yields:
\[\ddot{R}=-\frac{GM}{R^2} \Rightarrow \dot{R}\ddot{R}=-\frac{GM\dot{R}}{R^2}\]
Integrating, we get:
\[\dot{R}^2=\frac{2GM}{R}-\kappa\]
So, now, for matter in a sphere with a radius of \(R\), \(M=\frac{4\pi R^3 \rho}{3}\), thus:
\[\dot{R}^2=\frac{8\pi G \rho R^2}{3}-\kappa \Rightarrow \left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]
taking \(R=R_0\cdot a(t)\) we get:
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G \rho}{3}-\frac{\tilde{\kappa}}{a^2}\] </derivation - Newton style> It's fairly easy to see that, according to Newton, the term \(\frac{\kappa}{R^2}\), is simply a term that is a result of integration, i.e. an integration constant. And so according to this derivation \(\kappa \) is simply given by initial conditions, and given no other incentive, we can gauge it away.
Let's see what old Einei have to say about this... <derivation - Einstein style>
I'll try to be brief yet informative:
The metric for a spherical symmetric curved space is given by:
\[ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-\kappa r^2}+r^2d\Omega\right]\]
Or in matrix form:
\[g_{\mu\nu}=\left(\begin{array}{cccc}
-1&&&\\
&\frac{a^2}{1-\kappa r^2}&&\\
&&r^2&\\
&&&r^2\sin^2(\theta)
\end{array}\right)\]
From here it's relatively easy to find the Christoffel connections:
\[\Gamma^{0}_{11}=\frac{a\dot{a}}{1-\kappa r^2}\;;\;\Gamma^{0}_{22}=a\dot{a}r^2\\
\Gamma^{0}_{33}=a\dot{a}r^2\sin^2(\theta)\;;\;\Gamma^{i}_{0j}=\delta_{ij}\frac{\dot{a}}{a}\\
\Gamma^{1}_{11}=\frac{\kappa r}{1-\kappa r^2}\;;\;\Gamma^{1}_{22}=-r(1-\kappa r^2)\\
\Gamma^{1}_{33}-r(1-\kappa r^2)\sin^2(\theta)\;;\; \Gamma^{2}_{12}=\Gamma^{3}_{13}=\frac{1}{r}\\
\Gamma^{2}_{33}=-\sin(\theta)\cos(\theta)\;;\; \Gamma^{3}_{23}=\cot(\theta)\]
all other symbols vanish.
The Ricci scalar is then given by:
\[R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]\]
The Einstein equation:
\[G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}\]
or, equivalently :
\[R_{\mu\nu}=8\pi G \left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\]
Taking the 00 term we get:
\[3\left[\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]=8\pi G\rho\Rightarrow \left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G\rho}{3}-\frac{\kappa}{a^2}\]
And, if we use the other notation also we get:
\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}
(\rho+3p)\]
</derivation - Einstein style>
For now, we'll leave the second equation be, and look at the first. We'll notice a couple of things first - Although it SEEMS as though the equations (Newt's, and Einei's) are the same, really the real Friedmann equation (derived from General Relativity) is the more general case. that is due to the former derivation relying on dynamics of MATTER only, and the latter does not, though, a careful massage of the former with reletavistic ideas, might give the correct equation for radiation, and other cosmic stuff... The second thing I wish to emphasize is the existence of \(\kappa\) and it's meaning:
In the Newtonian analysis, it was simply an integration constant, but in the Einsteinian analysis this is an intrinsic factor to the very fabric of the universe, this is the CURVATURE signature of the metric, that governs whether the cosmos is positively, negatively or null curved (i.e. flat) like in this picture:
Curvature types: positive, negative, and null.
<derivation - Gangnam style>
And this is just for fun:
</derivation - Gangnam style>
Admission of Guilt, errr... Ignorance
But wait!! What does all of this have to do with inflation?
Well, in fact, suppose space is sufficiently dillute, the matter density is sufficiently close to zero, and if it's sufficiently cold, radiation density drops to zero (almost), with an (almost) flat curvature, we are then left with some quantity we'll call \(\rho_0\), ("Rho naught"), and we get a dynamic equation, with an inflationary/deflationary solution:
\[\left(\frac{\dot{a}}{a}\right)=\pm\sqrt{\frac{8\pi G\rho_0}{3}} \Rightarrow a=A\exp\left(\sqrt{\frac{8\pi G\rho_0}{3}}t\right)+B\exp\left(-\sqrt{\frac{8\pi G\rho_0}{3}}t\right)\]
The second part drops off rapidely, and so we are left with an inflationarry solution.
"What is this witchcraft?!?!" you ask - where does this \(\rho_0\) come form?
well, suppose you're a fat guy, and you go into a fasting mode, note that you are actually GAINING weight (at least in the short run)... and by the time you've lost the battle against hunger, and went on a burger binge, guess what? you've now underwent inflation.
Well, actually this has nothing to do with \(\rho_0\), here's the REAL explanation for this:
WE DON'T KNOW!!!!
Oh my god, I can't believe I said that!!! this is the absolute NO NO for physicists!!!
To actually state that I don't know something? to recognize that everything we *THINK* we know is simply an approximate modeling of the awsome and complex reality we live in???!?
I should stop. NOW!!! I hear them knocking already... don't let them take me! NOOOOOOOOOOOOOOOOO!!!!
Ok, done with that gag, are we?
Moving on, there are several possible explanations for this, most are problematic to say the least, but hey, this is what it means to be in the front lines of science. you either get promoted or get diced....
At any rate, Inflation is a widely accepted theory of the adolescent universe, whatever mechanism manifests it.
What about oscillations?
In a nutshell - at the onset of the early universe, after inflation, some major ocillations occured in matter density, affected by ordinary (barionic) matter as well as dark matter, these oscillations are known as BARIONIC ACOUSTIC OSCILLATIONS (or modes) , and they could be seen quite nicely in analysis of the cosmic background radiation.
Moreover, when the universe was matter dominated, the same dynamics might have happened on a cosmic scale - Suppose matter is the dominant part of the universe, the dominant force then is gravitation, thus the universe itself, aspires to CONTRACT, offset only by radiation pressure, there MIGHT have been an epoch of slight contraction on the universe's part.
Sadly I didn't find a sufficiently fascinating animation to show, of the acoustic oscillations, but maybe some other time...
In conclusion, much like myself, the universe was "born", and began inflating.
Undergoing some oscillations, and at a certain point (just about.... now!) the universe moved from matter driven dynamics into moderate inflationary epoch.
In layman terms:
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.
