Today is National Pi Day

[Mathman]

Quick, give me a number between 1 and 10. If you said Pi, you are ready to celebrate National Pi Day, today, March 14.

http://www.piday.org/

In honor of the occasion, Stephanie Godden, a sixteen-year-old student from Sterling Heights Michigan, recited from memory the first 2000 diigits of this famous transendental number: Pi = 3.14159265358...

The coolest thing about Pi (defined to be the ratio of the circumference of a circle to its diameter), is that we can measure the curvature of space by calculating how much experimental measurements of Pi vary from the expected Euclidean value just given.

For instance, if we measure Pi by computing the ratio of the circumference of the earth to its diameter, we get

Pi = 3.14159265286...

This is because of the distortion of the geometry of space due to the earth's gravitational field (the diameter is about 3 millimeters longer than our Euclidean geometry textbookssay it should be.)

(You're welcome. )

 

[dorispulaski]

arty:arty:

Happy Pi Day!

 

[beep_beep]

Thanks, for the reminder MM. :agree:

Pi is everywhere! Let's raise our glasses and celebrate ! arty:

And since we are on the geek notes (it's a joke, please take no offense):
Did you know that 2009 is the International Year of Astronomy ?

http://www.astronomy2009.org/

So take a look at the sky at night (unless it's cloudy or you live in Las Vegas )

 

[Particle Man]

I love it when people say "sound waves need air to travel through, but light needs no medium to travel through." Then later they talk about curvature of spacetime. Uh, ok. Then obviously spacetime is then the MEDIUM for light. It HAS a medium, the medium is spacetime. How can you curve something that doesn't exist? :sheesh:

Curved space is hard to comprehend though. Helps to think of a 2D example, like stretching a balloon material flat over a ring, then putting a weight on it to distort it. Traveling through the center (diameter) will take longer than it would if it were flat. Or if you put 2 ball bearings on this stretched material, they would not only distort the surface but also tend to come together and stay together, because of the depression each creates in the surface.

In this 2D scenario, the 3D gravity creates both the curvature of the 2D surface and the apparent attraction of objects, each of which could be perceived by any "2D creatures" as gravitational effects existing entirely inside their 2D plane. So, maybe gravity is a 4D (or higher dimensional) force, perhaps acting toward a central point in hyperspace, creating both the curvature effect (from mass) on our 3D space and apparent attraction of objects due to curvature of that space? Also, maybe gravity is weak because the hypersphere universe has expanded really far from the center? (well this could be tested by testing whether the gravitational constant appears to be decreasing )

 

[Tonichelle]

my favorite is pumpkin.... I want some now.

 

[Mathman]

So, maybe gravity is a 4D (or higher dimensional) force, perhaps acting toward a central point in hyperspace, creating both the curvature effect (from mass) on our 3D space and apparent attraction of objects due to curvature of that space?

In my opinion, this is a misleading view of curvature. The model of the spatial universe as a three dimensional object (or of space-time as a four dimensional object) embedded a higher dimensional "hyperspace" gives rise to what is called "extrinsic" curvature. Such considerations have led people fallaciously to speculate that this higher dimension must actually exist, because how can an object have room to "curve" except in extra dmensions?

What Einstein actually studied (following Gauss and Riemann), and incorporated into his field equations in general relativity, is "intrinsic" curvature, or metric curvature. Space does not twist, bend or warp -- if you think about it, what could this possibly mean? -- rather, the yardstick-plus-clock that we measure distances with changes from point to point. The result is that when we measure the distance from here to there, the answer is different from what a "flat" Euclidean metric predicts.

(But despite years of trying, I have not been able to convince my phyicicist friends to stop using the "bowling ball on a rubber sheet" model to explain the physical interpretation of the Schwarzschild solution. )

 

[Particle Man]

If only objects like yardsticks and clocks and protons and electrons are affected, why is massless light also affected by the "appearance of curvature" which is not actually curvature? Light exists in a pure state but only our appearance or measuring of it is messed up? What about black holes then?

Also, if space or spacetime has no fabric or "real" quality, how can it exhibit quantum vacuum effects? Virtual particles and production of particles etc.

 

[Mathman]

If only objects like yardsticks and clocks and protons and electrons are affected, why is massless light also affected by the "appearance of curvature" which is not actually curvature?

When we speak of the metric tensor we are not talking about physical yardsticks and physical clocks. We are talking about the metric sturcutre of space-time itself. Light, like anything else in space-time, must "follow the rules."

This is made explicit by the Einstein field equations. The formal definition of a space-time is a four dimensional manifold (continuum) together with a Lorenzian metric g satisfying (in local co-ordinates) the system of second order differential equations

Rij - (1/2)Rgij +Lgij = 8 pi G/c^4 Tij

(Hey look, there's Pi! )

All "four-dimensional continua" are indentical locally. What distinguishes one universe from another, then, is the metric g. In Einstein's equations, the left-hand side describes the geometry of the universe and the right-hand side (the stress-energy tensor) describes its physical properties, at least as far as gravitational phenomena are concerned.

But notice what "the geometry of the universe" consists of. gij are the local components of the metric tensor (the God-given magic giant yardstick-clock-protractor in the sky that determines the geometric stucture of the universe.) Rij and R are respectively the Ricci curvature and the scalar curvature of the metric (locally they are various combinations of first and second derivatives of the components of the metric tensor.) But the main point is, when we say "curvature" in general relativity, it is the curvature (second order rate of change) of the metric that we are talking about, not any property of the underlying topological space.

L = lambda, by the way, is the infamous "cosmological constant." As far as we can tell at the moment, this correlates with "dark energy" (discovered in 1998 by astronomers studying the inhomogeneities of the cosmic microwave background radiation) which in this era apparently drives the expansion of the universe.

..."appearance of curvature" which is not actually curvature

In this I fault Einstein. In my opinion Einstein made two terrible blunders when he and his friends tried to explain relativity to non-scientists. The first was his use of the term "relative to the observer." This made it seem like events somehow weren't real unless they were "observed," perhaps by some sort of giant eyeball floating around in space. Nothing could be farther from Einstein's view of physics (he was what philosphers of science call a "naive realist," which is why he never truly accepted quantum theory.) Einstein should have said, "relative to a choice of local space-time co-ordinates" instead.

Einstein's second "public relations mistake" was his use of the word curvature (following Gauss and Riemann). If he had just called it something else instead, say, "flarb," he could have spared several generations of graduate students in physics a ton of grief. Instead of getting all confused trying to picture space as somehow "curving" -- in the ordinary sense of the word, the sense in which objects in the universe can be flat or curved -- we could instead concetrate on the scientific investigation of the flarb of the universe, first having given a formal mathematical definition of this term. (BTW, the flarb of the universe as a whole is either 0 or so close to 0 that we have not been able to detect it despite decades of intense observation and experimentation.)

Also, if space or spacetime has no fabric or "real" quality...

No real quality? From the point of view of general realativity, space-time and the metric tensor are the only things that are "real."

...how can it exhibit quantum vacuum effects? Virtual particles and production of particles etc.

I don't know. It is my firm impression, having attended many scholarly conferences on this topic (at some of which I discussed my own work) that no one else knows either.

At one time it was hoped that string theory would give some insight into such phenomena. But now that string theory has petered out without accomplishing anything of note, it looks like we are back to square one. hwell:

 

[Tonichelle]

anyone else have that urge to fall asleep and drool on a text book, or is it just me?

 

[Mathman]

Hey Tony, they have just invited me to guest star on the TV show "The Big Bang Theory." My role is to be so nerdy that Sheldon looks cool! (Rock-paper-scissors-lizard-Spock!)

(Did you know that rock-paper-scissors-lizard-Spock is actually based on the existence of an Euler circuit in the complete graph on five points (It looks like a star inscribed in a pentagon)? )

 

[beep_beep]

Wow, that's a lot (even for me)!

I have a B.A. in Physics.
I took the mandatory basic astronomy/astrophysics, relativity and quantum mechanics 1 courses and went on the way of condensed matter and experimental physics (ion beam analysis).
My knowledge on the more advanced topics of cosmology and other more contemporary topics of theoretical stuff is pretty minimal.

This discussion forced me to dig in my books and the internet for things I haven't heard of in years. This geek says "thanks".

My experience as a high school teacher told me that more transcedental topics like cosmology, or even basics quantum mechanics, are pretty hard to explain without resorting to some higher Math.
These concepts have no match in the "real" (newtonian) world. So, in an attempt to help, people come up with ideas like bowling balls in trampolines and the Schrödinger's cat experiment (grrrr...hate it).

Oh, look at that: pi has made me just a little less dumb today...

Did I ever I imagined that I would be having this type of discussion on GS? Never.

C'mon Toni, let us geeks have fun...

 

[Tonichelle]

*cough*TonI*cough*

and I still haven't seen an episode of that show though their version of rock paper scissors did peak my interest

 

[Joesitz (RIP)]

Does Pi go beyond circles? and are sphere's involved?

There are huge amounts of angles in life that can be observed. Are any of them affected by Pi? Think of a hexiconical cone.

 

[Mathman]

My experience as a high school teacher told me that more transcedental topics like cosmology, or even basics quantum mechanics, are pretty hard to explain without resorting to some higher Math.

These concepts have no match in the "real" (newtonian) world. So, in an attempt to help, people come up with ideas like bowling balls in trampolines and the Schrödinger's cat experiment (grrrr...hate it).

Last summer I wrote an article explaining how our pedagogical techniques for teaching general relativity are all backward. I submitted it the journal, The Physics Teacher.

The editor kindly thanked me for thinking of his journal, but "due to the large number of submissions and limited journal space," go soak your head.

As for Schrödinger's cat, all we can hope for is that by then the students are too confused or bored to pipe up and say, "hey, wait a minute!?"

Does Pi go beyond circles? and are sphere's involved?

Yes. In Euclidean space (what we study in high school geometry), the surface area of a sphere of radius r is given by the formula A = 4 pi r^2.

In a negatively curved 3-dimensional space, A > 4 pi r^2, and in a positively curved space A < 4 pi r^2.

There have been quite a few astronomical studies to try to decide which of these three formulas holds in our actual universe. The "radius" of a big shere in space centered at the earth is determined by red shift, and the "surface area" is estimated by a count of galaxy clusters at that distance.

(In pracfice, there are many obstacles for carrying out this experiment. We do not know as much as we would like about the formationand evolution of galaxies, and furthermore the effect of dark energy tends to mimic negative curvature.)

There are huge amounts of angles in life that can be observed. Are any of them affected by Pi?

Yes! Here is another way that we could, in principle, measure the curvature of the universe.

In Euclidean geometry, if you measure the angles of a triangle in radians, the three angles always add up to...pi radians (same as 180 degrees.)

In a positively curved space the three angles always add up to more than pi radians, and in a negatively curved space they add up to less than pi radians.

So if we could draw a big triangle in space and measure its angles, by comparing the total number of degrees to 180 we could find out which type of universe ours is.

Homework. We own a 1053-acre dude ranch. It is triangular in shape. One day we decided to measure the angles of this triangle. Imagine our surprise when, in defiance of Euclid's Proposition 32, the angles turned out to be 90.000003°, 60.000002°, and 30.000001°!

Find the raius of the earth.

Hint: The three angles add up to 180.000006 degrees, or 3.14159275831 radians (a tiny bit more than pi.) Now use the formula for sperical triangles (this is called the Gauss-Bonet Theorem in fancier settings) that says that for a triangle drawn on a sphere of radius r, the three angles of a triangle add up to pi radians + (1/r^2) times the area of the triangle. (One acre = the amount of land a team of oxen can plow in a day. There are 640 acres in a square mile.)

See how practical all this is? "Geo-metry" is Greek for "measuring the farm" (true.)

 

[Dee4707]

anyone else have that urge to fall asleep and drool on a text book, or is it just me?

Hey Toni, doesn't Math sound like he would make a great date??? You know I mean in his unmarried days....I bet Mrs. Math married him for his money.

Did someone say this is Wine day??? I'll have a glass.

 

[Mathman]

Hey Toni, doesn't Math sound like he would make a great date??? You know I mean in his unmarried days....I bet Mrs. Math married him for his money.

Did someone say this is Wine day??? I'll have a glass.

^Ah, how you take me back. We would park along lovers lane and look at the stars.

"Now, sweetie-pie, if you look over there at inclination 15 degrees 32 minutes you will see Betelgeuse, at 135 lumens the ninth brighest star in the sky. Do you know that it's diameter is 950 times that of the sun and its surface temperature is estimated to be 3500 degrees Kelvin?"

 

[Joesitz (RIP)]

MM - I grew up in the Great War era and we had to learn Spherical Trigonometry with all those Logarithms as well as Sines Cosines, Tangents and the lot. Did Pi have a hand in those tables?

 

[beep_beep]

Did someone say this is Wine day??? I'll have a glass.

Everyday is wine day.

 

[Particle Man]

The following is a hypothesis which uses an expanding hypersphere to explain observed effects of gravity & relativity. If a hypersphere confuses you, just visualize a normal sphere (that's what I do.)

Universe from time=0 to time=whenever --- a "solid" hypersphere
Universe at any instant --- a "hollow" hyperspherical surface.

IF you could move toward the inside of the hypersphere, you would be moving backward in time. Moving outside the hypersphere would be moving forward in time.

1.) Mass somehow distorts the hyperspherical shell, pulling it inward radially. More mass = more distortion of the hyperspherical shell, i.e. the "bowling ball on the rubber sheet" analogy. Another object in the vicinity (say mass m) would not only be pulled into the bowling ball's distortion of the surface (force of gravity of bowling ball on m) but also create its own distortion (force of gravity of m on bowling ball) - mutual attractive force of gravity.

In addition, this effect of mass on the topology of the hyperspherical shell should logically have the following effects:

- Local surface area (really 3D volume) increases beyond the "normal" euclidean amount - as you say, A > pi*r^2.
- Any stretching distorts the "perfect" spherical shape, thus distorting the surface in the radial (time) dimension.

2.) Light's "speed limit" could be theorized to be a geometric property of travelling the hyperspherical shell. The fastest possible speed (light speed) is a geometric property of the curvature of the hyperspherical shell you are currently in. Greater radius = less curvature = faster maximum (light) speed. Smaller radius = more curvature = slower maximum (light) speed.

I believe there are some working theories that light may have travelled slower in the past?

3.) Relativistic effects - Mass (or anything else) in motion is traversing the hyperspherical shell (as it expands.) The faster it travels the curved surface, the more it attempts to push against the curvature of the surface.

Relativity demonstrates that, to external observers, objects travelling near lightspeed (such as muons entering Earth's atmosphere) appear to experience time more slowly. The aforementioned muons to appear to decay much more slowly than they would at rest.

Objects at rest are moving radially outward, passively travelling along with normal hyperspherical expansion.

Objects in motion (in any direction on the expanding hyperspherical surface) are not moving entirely radially outward, but also have a tangential component to motion. Their motion vector, which is less "outward" than the rest mass, makes them travel somewhat against the normal outward-radial expansion, thus making them appear to age more slowly.

Other thoughts:

- The properties of the graph of KE vs speed (basically the graph of Lorentz factor, which is pretty straight up until about 2/3 lightspeed then starts curving sharply) may be instructive as to the fundamental nature of the properties of spacetime and how resistive it is to distortion. Or, to the properties of the hyperdimensional geometry of spacetime.

- Scientists have always been puzzled why an object's mass affects not only inertia, but also gravity, and why gravity is pretty much indistinguishable from physical acceleration. Perhaps the local distortion of the hyperspherical surface (perceived as gravity) is a simple result of the inertia of the mass resisting the full effect of the hyperspherical expansion. Mass has inertia, inertia resists "pushing". This creates surface distortion in spacetime and all the effects of gravity.

Is the hyperspherical expansion also accelerating? If so, no mysterious 4D force is necessary to explain the effects of gravity, in fact nothing is needed other than inertia itself. The mass simply doesn't want to be accelerated outward (due to inertia) and that alone explains the distortion, and what we observe as gravitational effects. In short: inertia + acceleration of universe expansion = gravity.

- In the "expanding hypersphere" hypothesis, both mass and motion cause similar time-distortion effects as a simple consequence of geometry. Fast motion and presence of mass both cause a slowing effect, since both resist the "normal" full-speed expansion in the +time dimension.

In relativity, it is proven that fast motion and presence of mass both cause a slowing effect.

Any comments?

 

[Mathman]

MM - I grew up in the Great War era and we had to learn Spherical Trigonometry with all those Logarithms as well as Sines Cosines, Tangents and the lot. Did Pi have a hand in those tables?

Pi is all over trigonometry!

Factoid: The beginnings of trigonometry took place in India (although the Indian mathematicians did not appreciate the role of ratio as thoroghly as the Greeks.) Anyway, the word "sine" comes from a Sanskrit word meaning "bowstring."

The idea is that if you have a curved bow (as in bow and arrow), the bowstring represents the chord of the arc. (Later, for various reasons, they decided to define the sine to be "half a bowstring" instead of the entire chord.