A miscellany of physics

[doc benway]

 

Well first, what is a tangent plane at a point, intrinsically speaking? It is the space of all the "velocity vectors" an ant can have at that point (!). (A vector is a quantity with both magnitude and direction, a velocity vector is therefore a direction and a speed, a "bearing" if you will).

 

Is it more accurate to say that, it is the space of all the "velocity vectors" an ant can have at that point that are tangent to the sphere at that point?

 

 

It is the connection that tells you what the analog of a straight line will be in an arbitrarily curved space. If the ant "follow it's nose" so to speak, without turning left or right, it's velocity vector is the parallel transport of itself (!). A curve whose velocity vectors are parallel transports of themselves is called a geodesic. In flat space, these are the straight lines (!). On a sphere, they are the circles of maximal diameter.

 

What about circles of lesser diameter? Are they not geodesics because the vector is changing in multiple planes?

 

Thanks - I've never been good at geometry

[Creation]

Is it more accurate to say that, it is the space of all the "velocity vectors" an ant can have at that point that are tangent to the sphere at that point?

Good question!

 

One thing I didn't make clear: When I say sphere, I mean the two dimensional surface of the sphere. Similarly, if I say circle I mean the one dimensional boundary. A mathematician calls the 3 dimensional region bounded by a sphere a ball, and the 2 dimensional region bounded by a circle a disc.

 

If that was not the confusion, here is some more about tangent vectors: The possible velocity vectors for an ant constrained to the surface of the sphere will be 2 dimensional. Thinking of this sphere as embedded in 3 dimensional space, this space of velocity vectors will embed as a 2 dimensional subspace of the 3 dimensional space. The point here is that since the ant is constrained to the surface of the sphere, any velocity vector it can have will automatically be tangent to the sphere. So the two dimensional subspace of velocity vectors for an ant will be precisely the tangent plane. But if you want to think intrinsically, you can use this fact to safely say tangent vector = velocity vector.

 

 

What about circles of lesser diameter? Are they not geodesics because the vector is changing in multiple planes?

Right, to get a smaller circle the ant would have to feel like it is turning. If it "follows it's nose", it will always trace out a circle of maximal diameter.

 

--

 

If anyone else has any feedback on my attempt to explain differential geometry, let me know. It took me years to understand this stuff. But it has been so long that some of the things I take for granted might be totally unfamiliar to you guys. So let me know.

 

If it might help, try the one dimensional case: A smooth curve will have a tangent line at each point. For example, if a circle is "sitting" on a line, that line only touches the circle at the bottom, so it is the tangent line to the circle at the bottom point.

 

If an ant is crawling on the circle, it's possible velocities will just specify if it is going forward or backward and at what speed, so they will form a line. And if you think about what a tangent line is a bit, it will make sense that thinking of the circle embedded in the plane, the line of velocities at a point on the circle will be the tangent line to the circle at that point.

 

You know a curve is curved (kind of bad terminology, I know, but it is standard) if the tangent lines are not parallel. If you think about it, the tangents will only be parallel if the "curve" is actually not curved, i.e. a line.

 

But intrinsically speaking, since there is no room to turn on a one dimensional space, it is impossible for a one dimensional space to be intrinsically curved! But even thought, say, a circle can't have intrinsic curvature in the sense I've defined, surely there is some way in which a circle is intrinsically curved because you can walk without reversing direction and get back to where you started, which is impossible on something un-curved like a line. The type of curvature I defined above is a "local" form of intrinsic curvature, because you can measure such curvature in a small region by comparing parallel transport along paths only in that small region. But to detect this other kind of intrinsic curvature that a circle does have you need to look at the whole circle; if you even remove one point on the circle you can't go all the way around. So one might say a one dimensional space cannot have local intrinsic curvature, but it can have global intrinsic curvature. The of study global properties of spaces is called topology.

Edited April 4, 2012 by Creation

[doc benway]

That helps clear things up, thanks!

 

Not to get too far afield but in thinking about curved surfaces, here is an interesting link.

It probably touches on things you will discuss more in the future but if you have comments about it, I'd love to here them.

[dwai]

Sure is

 

http://en.wikipedia.org/wiki/Gravitational_constant

 

Yet G is just a fudge created to mke the equation work...is there evidence of ths gravitational constant beyond the realm of balancing out the equation?

[Apech]

Yet G is just a fudge created to mke the equation work...is there evidence of ths gravitational constant beyond the realm of balancing out the equation?

 

Its not a fudge its a mathematical way of showing a direct proportionality between the force exerted by gravity, the mass of the objects and the reciprocal of the square of the distance between them. Compare it to ...'Pi' is a universal constant relating the area and the square of radius of any circle.

[Zhongyongdaoist]

Sure is

 

http://en.wikipedia.org/wiki/Gravitational_constant

 

 

Yet G is just a fudge created to mke the equation work (emphasis mine, ZYD)...is there evidence of ths gravitational constant beyond the realm of balancing out the equation?

 

dwai,

 

I hate to point out that, had you even bothered to follow the link which joeblast provided you might have save yourself some embarrassment and discovered the distinction between a mathematical constant, such as Apech describes and a physical constant, which is derived from actual observation, preferably under experimental conditions. As a physical constant, G is the worst example which you might have chosen and thus indicates your basic ignorance of the subject which you wish to criticize. I might have chosen renormalization in quantum mechanics as a possible example, but to be honest I don't understand it enough to be able to be able to do anything other than be a little suspicious of it. Maybe at some point Creation can address it.

 

I have pointed out one of the most famous examples of a scientist introducing a constant into the mathematics elsewhere ( http://www.thetaobums.com/index.php?/topic/7654-chi-in-nature-taoism-and-mak-tin-si/page__view__findpost__p__307052__hl__Einstein__fromsearch__1 ), and that is Einstein's fooling with the equations of General Relativity to support his belief in a "steady state" universe rather than the expanding universe which the pure mathematics of General Relativity entailed. Had Einstein kept faith with the mathematics he would have predicted the expansion of the universe more than a decade before it was observed.

 

The relationship between observation and mathematics in physics is one of the most interesting aspects of the scientific endeavor. The famous physicist Eugene Wigner wrote a paper examining it in 1960 called The Unreasonable Effectiveness of Mathematics in the Natural Sciences, http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences .

 

You are an intelligent person with interesting things to say, but please, in the future avoid embarrassment, follow the link before posting. While I have a good understanding of mathematics and physics, the passage of time has made it vague enough that I will always review my understanding by looking on the internet before posting on these subjects. By doing so I have often refined my understanding and learned new things, both valuable on their own.

[doc benway]

A little off topic, but here is a cool link that shows the relative size of things.

Scroll in and out slowly.

I enjoyed it.

 

Now back to our regular programming

[dwai]

dwai,

 

I hate to point out that, had you even bothered to follow the link which joeblast provided you might have save yourself some embarrassment and discovered the distinction between a mathematical constant, such as Apech describes and a physical constant, which is derived from actual observation, preferably under experimental conditions. As a physical constant, G is the worst example which you might have chosen and thus indicates your basic ignorance of the subject which you wish to criticize. I might have chosen renormalization in quantum mechanics as a possible example, but to be honest I don't understand it enough to be able to be able to do anything other than be a little suspicious of it. Maybe at some point Creation can address it.

 

I have pointed out one of the most famous examples of a scientist introducing a constant into the mathematics elsewhere ( http://www.thetaobums.com/index.php?/topic/7654-chi-in-nature-taoism-and-mak-tin-si/page__view__findpost__p__307052__hl__Einstein__fromsearch__1 ), and that is Einstein's fooling with the equations of General Relativity to support his belief in a "steady state" universe rather than the expanding universe which the pure mathematics of General Relativity entailed. Had Einstein kept faith with the mathematics he would have predicted the expansion of the universe more than a decade before it was observed.

 

The relationship between observation and mathematics in physics is one of the most interesting aspects of the scientific endeavor. The famous physicist Eugene Wigner wrote a paper examining it in 1960 called The Unreasonable Effectiveness of Mathematics in the Natural Sciences, http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences .

 

You are an intelligent person with interesting things to say, but please, in the future avoid embarrassment, follow the link before posting. While I have a good understanding of mathematics and physics, the passage of time has made it vague enough that I will always review my understanding by looking on the internet before posting on these subjects. By doing so I have often refined my understanding and learned new things, both valuable on their own.

 

http://www.npl.washington.edu/eotwash/experiments/bigG/bigG.html

[Creation]

Interesting link dwai, but that only points out the difficulty in measuring G.

 

G is not a fudge to make the equation work. As Apech pointed out, constants of proportionality are common in mathematics. And as Zhongyongdaoist pointed out, G is not just a mathematical constant of proportionality, but a physical one. An observation I would like to add is that if you take G out of the equation, the units don't even match. So G serves as a unit conversion factor, whose value will depend on the units you are using. In other words, it can't not be in the equation, although you could choose units such that it's value would be 1.

 

 

steve,

 

That was the coolest flash animation I have ever seen. The absolutely awe inspiring nature just how big and how small things in the universe are has never been made so explicit to me. We have come a long way since Archimedes' Sand Reckoner.

 

 

Zhongyongdaoist,

 

Sure, I can say a bit about renormalization. I'll add it to the queue.

[Birch]

Pizza.

Yep!

 

Still, I'm curious. Taomeow, where does the pic come from?

Possible new thread about strange-looking objects in paintings awaits! Just so one can never say "look, all that symbolism was far too obscure for me, I hadn't realised you actually meant XYZ'. Because consent is always required

[Taomeow]

Yep!

 

Still, I'm curious. Taomeow, where does the pic come from?

Possible new thread about strange-looking objects in paintings awaits! Just so one can never say "look, all that symbolism was far too obscure for me, I hadn't realised you actually meant XYZ'. Because consent is always required

 

• Title: God as Architect/Builder/Geometer/Craftsman
• From: The Frontispiece of Bible Moralisee
• Style: Gothic
• Date: mid-13th C.
• Location: France
• Codex Vindobonensis 2554 (French, ca. 1250), in the Österreichische Nationalbibliothek.

Famously used as the first color illustration to Benoit B. Mandelbrot's The Fractal Geometry of Nature...

 

...but of course "God is a geometer" was proclaimed much earlier, by Plato, in a dialog with Aristotle.

[Creation]

Wow Taomeow, I thought for sure that was a modern piece done in Gothic style because of the fractal. Very cool.

[Creation]

OK, a bit about renormalization before getting back to geometry.

 

Renormalization is something that you use to get a finite answer when your theory keeps spiting out infinity as the answer. These infinities can arise in several ways. Here are a couple.

 

1. Statistical field theory.

 

Statistical mechanics is the study of systems with extremely large numbers of particles that behave effectively randomly, like gas in a room has an enormous number of atoms that you cannot possibly keep track of, so you just use probability and statistics to calculate large scale properties like average energy.

 

But what if you are considering turbulent fluid, moving for all intents and purposes random? If you model the fluid as a continuous flow, and try to calculate thing like average energy, this is an example of statistical field theory. But this causes some mathematical difficulties.

 

If you rescale a figure in the plane by a factor of two, it's area gets multiplied by 4. If you rescale a figure in 3-space by a factor of 2, it's volume gets multiplied by a factor of 8. Generally, if you rescale an n-dimensional figure by a factor of c, it's appropriate measure of n-dimensional volume will be multiplied by c^n. Now when you want to calculate the average of something, you will need to know it's n-volume, n being the number of parameters needed to specify a point a particular thing in your collection. To specify the positions and velocities of all atoms in a gas cloud, you will need a huge but finite number of parameters. So OK, no problem. Take the average.

 

But a continuous flow of fluid requires an infinite number of components to specify: at the very least it's velocity at ever point in space. So if you want to calculate averages over all fluid flows, you will need a notion of volume on an infinite dimensional space. But if such a thing existed, it would have to have the property that if you scaled up any region, however small, by any factor k however close to 1 (say 1.0000000001), the factor of increase in ∞-volume would be k^∞ which is ∞.

 

So when you try to take averages over all configurations of a continuous thing like a fluid or the electromagnetic field, you run into infinity at every turn.

 

2. Field singularities.

 

The electromagnetic field associated to a point particle is infinite at the location of the particle, because assuming the charge is concentrated at a point means the charge density there is infinite. Now, electromagnetic fields carry energy, so to have a self-consistent theory of charged point particles, you must take into account this energy attached to the particle when calculating the mass of the particle (that old E=mc^2 thing). But upon calculating the total energy, it is infinite because of the infinite value of the field at the location of the particle. So the mass of the particle should be infinite! Since we can model many things by point particles, and these things we are modeling don't have infinite mass, we would like to have a way out of this dilemma.

 

What you can do is to only calculate the energy of the electromagnetic field more than some distance r from the particle and get a finite answer. Then you add this to the "bare mass" that the particle has without taking the field into account. This is your "total mass". Now, you demand that the total mass is equal to the mass that you actually observe the particle having. Then, finally you take the limit as r goes to zero, demanding that the total mass stays equal to the observed mass. For this to happen, the bare mass will have to become negative infinity. But to the extent that you are a pragmatist, you will not care because the final answer for the total mass gives you the right answer. After all, only the total mass is observable. The "bare mass" is unobservable so why do we care if it is negative infinity? This is called "renormalization".

 

In the case of the statistical field theory example you might choose to look at a finite number of points at which to consider the electromagnetic field, so it only takes a finite number of parameters to specify. Then you once again find some extra parameters in the theory to be considered "bare quantities", and let them absorb the infinities that arise when you take the limit of considering all of spacetime so that the observable quantities come out right.

 

In the case of quantum field theory you have both of these issues and more besides causing infinities to crop up. So given a quantum field theory a big question you want answered is are there enough terms with parameters that you can shuffle around to absorb all the infinities. In quantum electrodynamics, this is possible (Feynman, Schwinger, and Tomonaga all proved this independently of each other, and shared a Nobel Prize for doing so). In the first theory of the weak interaction this was not possible. For a long time didn't think it was possible to find renormalizable quantum field theories of the weak and strong forces. But then some people figured out how to do it in each case, and got Nobel prizes for it.

 

When you try to make general relativity into a quantum theory you find that it is also non-renormalizable. This is the "quantum gravity" problem that everyone gets so hyped up about. Should we modify general relativity or quantum field theory to make them fit together? Both? Neither, but we just need to rethink how we are trying to make a quantum field theory out of gravity? String falls into the "modify general relativity" camp, and loop quantum gravity falls into the "neither, but rethink how we are doing it" camp.

 

Now, even if you have a renormalizable theory, you can't help but think, "wait a second, what we are doing here is crazy." For example, how do you know that renormalization is even consistent, i.e. that following different procedures to renormalize a theory give the same answers? Some things have made the procedure more respectable, though. There is some mathematical work towards proving that renormalization is in fact a mathematically consistent procedure. Also, people using renormalizaion in statistical field theory realized that perhaps non-renormalizability isn't so bad. For example, if you are using statistical field theory to study fluid turbulence, if you can't take the limit from your discretization to the continuum and absorb all the infinities, so what? Fluids are not continuous at the microscopic level anyway, but are made of molecules. This is called "effective field theory": if there is some scale at which you know new physics will emerge anyway, why bother trying to take the limit beyond this scale?

Edited April 5, 2012 by Creation

[doc benway]

To specify the positions and velocities of all atoms in a gas cloud, you will need a huge but finite number of parameters. So OK, no problem. Take the average.

 

But a continuous flow of fluid requires an infinite number of components to specify: at the very least it's velocity at ever point in space. So if you want to calculate averages over all fluid flows, you will need a notion of volume on an infinite dimensional space. But if such a thing existed, it would have to have the property that if you scaled up any region, however small, by any factor k however close to 1 (say 1.0000000001), the factor of increase in ∞-volume would be k^∞ which is ∞.

Why can you get away with a finite number of parameters for a gas cloud but an infinite number of compoonents are required for a continuous flow of fluid?

Does it have to do with making more general predictions about the gas behavior vs trying to make more specific predictions with respect to the fluid or field behavior?

Edited April 5, 2012 by steve
trying to answer my own question...

[Creation]

Why can you get away with a finite number of parameters for a gas cloud but an infinite number of compoonents are required for a continuous flow of fluid?

Does it have to do with making more general predictions about the gas behavior vs trying to make more specific predictions with respect to the fluid or field behavior?

In both cases you are only going for general predictions.

 

If you have a one particle system, you need 3 numbers to specify it's position and 3 to specify it's velocity, 6 total. This completely characterizes the "configuration" of the system, meaning you can then apply Newton's laws of motion and predict where it will be. If you have a two particle system you need 6 for each particle, so 12. If you have an N particle system, however large N may be, you will need 6N parameters to specify it's configuration. So finite but huge number of particles = finite but huge dimension of configuration space.

 

When you model a field or a fluid as a continuous thing, it's configurations will involve functions from space to some other space. For example, the electromagnetic field assigns each point in space electric field strength vector and a magnetic field strength vector, so you need 6 parameters at every point of space. So instead of needing 6 parameters for each of a huge but finite number of particles, you need 6 parameters for every point in space, of which there an infinite number. A finite number of parameters could only specify the values of the field at a finite number of points, or some other approximation to the exact configuration.

 

Granted, we don't care about exact configurations in the end, because we will be taking averages, but to calculate the average in the first place you need to know in principle what the possible exact configurations you will be averaging over are.

 

Thanks for the question!

Edited April 5, 2012 by Creation

[Zhongyongdaoist]

OK, a bit about renormalization before getting back to geometry.

 

Renormalization is something that you use to get a finite answer when your theory keeps spiting out infinity as the answer. These infinities can arise in several ways. Here are a couple.

 

Thanks Creation, when you said that you would put it in the queue, I didn't think that you meant at the top of the stack.

 

Anyone who follows the discussion can see why I would nominate renormalization for a fudge, but whether it is or not, or just a very creative way of working around an infinity of difficulties is another matter.

 

I am pressed for time, but wished to express my thanks in a timely fashion. There are many interesting aspects to the matters raised here and If I have some time I may come back to what you said about units in the equation for G. It raises some interesting points which many people don't understand because such equations are written without the units, but the units are an integral part of such equations and how they interact and are an important part of manipulating equations which cannot be ignored. The part which units play is one of the fascinating aspects of that "unreasonable effectiveness of mathematics" which I have mentioned.

[Creation]

Thanks Creation, when you said that you would put it in the queue, I didn't think that you meant at the top of the stack.

Well I'm still mulling over how to proceed with the geometry-physics connection. I understand that better than renormalization, which makes it somewhat harder to write an exposition that I am satisfied with.

[Birch]

[doc benway]

That makes perfect sense.

Making predictions about discreet "things" vs every point in a field or fluid.

thanks for the clarification.

I'm really enjoying the thread.

I was a chemistry major way back in the '70's and have always had a fascination with physics but never studied it seriously other than some "pop" physics books over the last few decades.

Steve

 

In both cases you are only going for general predictions.

 

If you have a one particle system, you need 3 numbers to specify it's position and 3 to specify it's velocity, 6 total. This completely characterizes the "configuration" of the system, meaning you can then apply Newton's laws of motion and predict where it will be. If you have a two particle system you need 6 for each particle, so 12. If you have an N particle system, however large N may be, you will need 6N parameters to specify it's configuration. So finite but huge number of particles = finite but huge dimension of configuration space.

 

When you model a field or a fluid as a continuous thing, it's configurations will involve functions from space to some other space. For example, the electromagnetic field assigns each point in space electric field strength vector and a magnetic field strength vector, so you need 6 parameters at every point of space. So instead of needing 6 parameters for each of a huge but finite number of particles, you need 6 parameters for every point in space, of which there an infinite number. A finite number of parameters could only specify the values of the field at a finite number of points, or some other approximation to the exact configuration.

 

Granted, we don't care about exact configurations in the end, because we will be taking averages, but to calculate the average in the first place you need to know in principle what the possible exact configurations you will be averaging over are.

 

Thanks for the question!