General Relativity for Dummies

[Alabran]

Hey there folks,

Within my AP Calc-Physics class everyone is beginning an independent study of (semi)-modern physics, and I, in my ignorance, chose General Relativity because I felt it would be interesting to learn all of this about time dilation and etc. I have till next Thursday to finish a research paper and prepare a lesson plan so I can teach General Relativity (to an extent) to the rest of the class on that day. I have finished the AP Calculus BC course and AP Calc-Physics course, so that is the prior mathmatical knowledge I have available to me. Unfortunately, after scanning through several canonical, jargon-filled books, I seem to be woefully unprepared for a mathmatical understanding of General Relativity. I generally latch onto mathmatical concepts very easily, but even trying to understand what terms I need to understand, to understand General Relativity is very difficult as I cannot seem to find an explanation that uses the same terms I'm trying to understand (If you understand what I'm saying .)
I come here in hopes that I can attain some aid in this matter from those of you who already understand the material. I don't need a very in-depth understanding of General Relativity, as I only have one class period to explain the subject to my peers but I'm hoping to get into something "cool" such as time dilation near a body of mass or other such intriguing phenomena of general relativity. If I can get any help whatsoever, I would very much appreciate it.

Thanks.

 

[cristo]

General Relativity is an advanced theory, which needs a fair bit of advanced mathematics in order to understand it. There is no chance that you will be able to learn GR in a week, let alone learn it and teach it!

I would suggest picking something else. Have you learned Special Relativity? This doesn't need anything like that same amount of mathematics as GR, and you may actually have a chance of getting somewhere with it.

 

[Alabran]

I understand how complex it is, and I by no means expect to understand the entire subject. I also appreciate your concern on my ability to grasp the subject and I'm sure there is quite advanced mathematics involved. However, I and everyone in my class are very strong mathematicians (as much as that can mean at the senior high school level) and I'm sure if I can get some pointers I'll be able to grasp it to at least an introductory level. My class has already been through special relativity to an extent and are able to perform such actions as Lorentz Contractions and Time Dilation within the bounds of Special Relativity. All I want is to take it a step past Special Relativity (if there is such a thing), since I'll only be teaching for an hour anyway I don't have enough time to go so indepth in any aspect of the theory. I'm confidant if I have some help I could understand it, I just need someone willing.

Thanks again.

 

[pervect]

There are a few books I can think of that you might get something out of if you can get a hold of them. However, I don't think that "next thursday" will be a sufficient period of study to get a whole lot out of them even for your own education. Teaching a class by next thursday is wildly ambitious.

What I would recommend are

General Relativity from A to B by Geroch. Especially if you don't have a lot of background with SR, this is probably the place to start.

Exploring Black Holes: Introduction to General Relativity, by Edwin F. Taylor and John Archibald Wheeler. You can download some of the introductory chapters from http://www.eftaylor.com/download.html to see how it works for you.

If you aren't familiar with SR, I'd also recommend "Spacetime Physics" by the same authors. They cover a very small amount of GR related material as well. If "Exploring Black Holes" is too advanced, you may have to read "Spacetime Physics" or another SR book just to get started. There are also some downloads of a few chapters of the earlier editions of this book on the same website as above.

A rather old and perhaps odd choice that leverages on electromagnetism (I am assuming that you've at least seen Maxwell's equations and vector calculus) is Bondi's "The Physical Foundation of General Relativity". It's not terribly technical, but you do need some background in Maxwell's equations to appreciate the analogies Bondi draws to gravity.

I think there have been some past threads on this topic as well.

[add]
You might seriously think of trying to cover relativistic electrodynamics rather than general relativity. There is a lot of stuff that will be very useful to you there in understanding GR, so it would be very helpful to learn it first, and it's also quite interesting and definitely challenging. Books that talk about this would be for instance "Introduction to Electrodynamics" by Griffiths. The famous classic text by Jackson "Classical Electrodynamics" would probably be too advanced, being a graduate level textbook - even the relevant parts of Griffiths are probably best suited to a senior level undergraduate college course.

 

[Alabran]

Thank you for your help Pervect, I'll make sure to look for those books.
I currently have two books I was looking over, but which are quite technical. Those are:

Relativity for Scientists and Engineers by Ray Skinner
and
Gravitation by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler.

I can tell they are great books on the material, but, as I said, very technical.

I am currently perusing http://en.wikipedia.org/wiki/Basic_introduction_to_the_mathematics_of_curved_spacetime
which seems like it might have some merit, though it is rather technical in itself. I'm not sure I quite grasp Einstein's Notation. An example is the representation of the "Transformation of dx" within that article. I would copy the png here, but I haven't been able to find how to do that on this forum yet, so I'll simply explain it here, or if you want a cleaner version, check the article.

X^(Mu') = ((Partial)X^(Mu')/(Partial)X^V)dX^V , which is defined as X^Mu',v dx^v.

I see it's composed of partial derivatives of a vector in respect to another vector, though I'm not sure what the superscript "v" represents. Mu is defined as being the original four-vector of x,y,z and time. V was left undefined though while there is a "v" mentioned in http://en.wikipedia.org/wiki/Einstein_summation_convention" , where it seems to represent a vector of arbitrarily large sums, I don't see what it represents in this context.

[add]
While it would've been convenient to switch topics, this was the topic presented to me and I am unable to switch. I am stuck in General Relativity and I must make the most of it.

 

[cristo]

Thank you for your help Pervect, I'll make sure to look for those books.
I currently have two books I was looking over, but which are quite technical. Those are:

Relativity for Scientists and Engineers by Ray Skinner
and
Gravitation by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler.

I can tell they are great books on the material, but, as I said, very technical.

I'm not, personally, familiar with the first text, but the second is probably the most complete GR text around, and is highly technical. I wouldn't bother trying to understand MTW at this time, especially considering the fact that you have a goal to attain in a week or so.

I am currently perusing http://en.wikipedia.org/wiki/Basic_introduction_to_the_mathematics_of_curved_spacetime
which seems like it might have some merit, though it is rather technical in itself. I'm not sure I quite grasp Einstein's Notation. An example is the representation of the "Transformation of dx" within that article. I would copy the png here, but I haven't been able to find how to do that on this forum yet, so I'll simply explain it here, or if you want a cleaner version, check the article.

X^(Mu') = ((Partial)X^(Mu')/(Partial)X^V)dX^V , which is defined as X^Mu',v dx^v.

I see it's composed of partial derivatives of a vector in respect to another vector, though I'm not sure what the superscript "v" represents. Mu is defined as being the original four-vector of x,y,z and time. V was left undefined though while there is a "v" mentioned in http://en.wikipedia.org/wiki/Einstein_summation_convention" , where it seems to represent a vector of arbitrarily large sums, I don't see what it represents in this context.

The Einstein summation convention is rather simple concept, although it can take a bit of getting used to. On that Wiki page you mention, new coordinates x^{u '} have been introduced, which depend upon the old coordinates x^{u}. Then, the "infinitesimal"dx^{u'} can be calculated, using the chain rule, as
dx^{u'}=\sum_v \frac{\partial x^u'}{\partial x^v}dx^v

Now, the Einstein summation convention simply says that repeated indices are summed over; therefore the explicit summation sign in the above is dropped. It may help if you keep the summation signs in for a while if you get confused, then take them out when you understand fully.

As an aside, it may seem that I'm trying to discourage you from GR. I'm not, since GR is a beautiful theory which is well worth learning, but just don't want to see you trying to take on something more than you can cope with at the moment.

 

[neutrino]

You can check if you find anything useful in here...
http://math.ucr.edu/home/baez/RelWWW/HTML/tutorial.html

The whole site, i.e., http://math.ucr.edu/home/baez/RelWWW, has one of best set of links to resources on relativity on the web. But most of the contents there are suited for someone with at least an undergrad-level grasp of maths and physics.

And here's something just to overwhelm you.
http://physics.syr.edu/research/relativity/RELATIVITY.html

 

[Alabran]

Thank you Cristo, That makes much more sense. I had assumed that was some kind of complicated derivative, where really that's simply a standard vector transformation (if I see it correctly.) That much is familar as my Calculus course is currently undergoing three dimensional vectorial calculus.

So, in reality,

dx^{u'}=\sum_v \frac{\partial x^u'}{\partial x^v}dx^v

is akin to the notation I'm familiar with of:

\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}

correct?

 

[pervect]

I have MTW's "Gravitation", it's one of my favorite books. "Exploring black holes" is by two of the same three authors, and has much of the same material at a less advanced level. (I don't own it, however).

The superscripts and subscripts are a part of tensor notation. Learning tensor notation in a familiar context is an excellent reason to study electrodynamics before studying GR.

I'll try to briefly explain tensor notation, we'll see how well it goes

x^v is a "contravariant" vector, which transforms in the manner outlined in the wikiepdia article.

Contravariant vectors have duals, covariant vectors, which are often called "1-forms", which transform in a different manner which you can look up (it's fairly obvious). They are written with subscripts, rather than superscripts, i.e. x_v

The superscript or subscript, v, is just an index. In relativity, it takes on the values (0,1,2,3).

In a particular coordinate system, both covariant and contravariant vectors can be represented by their four components, i.e x^v has four components

<br /> x^v = (x^0, x^1, x^2, x^3)<br />

usually x^0 represents time (and the other three components represent space)

A covariant vector can be regarded as a map from a contravariant vector to a scalar (an ordinary number). This value of this scalar given the components of both vectors can be written out as:

x^v \cdot y_v = \sum_{i=0..3} x^i y_i

The summation sign is usually omitted in tensor notation - this is called the Einstein summation convention.

You might have seen these vectors called "row" and "column" vectors in introductory versions of linear algebra.

Covariant and contravariant vectors can be intercoverted from one to an other via the metric tensor. The metric tensor defines a map between covariant and contravariant vectors. Since there is a natural map between a covariant vector and a contravariant vector to a scalar, the metric tensor can be also be regarded as a definition of the scalar product of two vectors, and/or as a definition of the length of a vector (since the scalar product of a vector with itself is just the length^2 of the vector).

In Euclidean geometry, the length^2 of a vector is always positive - in relativity the length^2 can be positive or negative, depending on whether a vector is spacelike or timelike.

If you read MTW, you'll see that they use the transformation of the electric field as an example tensor transformation. This is why it would be very helpful to study relativistic electrodynamics first - you'll be able to get used to dealing with tensors representing familiar objects and concepts, such as the electromagnetic field.

A couple of more online sources that may be of some help:

http://math.ucr.edu/home/baez/gr/gr.html
http://math.ucr.edu/home/baez/einstein/
http://www.math.ucr.edu/home/baez/RelWWW/

In particular http://math.ucr.edu/home/baez/gr/outline2.html talks about the different types of vectors, using the name "tangent" and "cotangent" vectors.

http://math.ucr.edu/home/baez/einstein/ is a good paper about the meaning of Einstein's equation.

Chris Hillman's site http://www.math.ucr.edu/home/baez/RelWWW/ has a lot of links to other websites and resources.

 

[Alabran]

Thank you again for all of your help Pervect. I believe I understand the v notation now. I am curious though about the difference of x^mu and x^v. Within the wikipedia article, mu seems to have been represented the same manner as simply an indexing variable for which vector within spacetime is being referred to.

By

x^i y_i

do you refer to the "Inner Product" mentioned within the wiki article as:

(A,B)=A_u \cdot B^u

Or does the subscript mean something else?

[Add]After looking through the notation article, I see that you mean the row and collumn data for a matrix. Got it.

 

[Chris Hillman]

Good luck

Within my AP Calc-Physics class everyone is beginning an independent study of (semi)-modern physics, and I, in my ignorance, chose General Relativity because I felt it would be interesting to learn all of this about time dilation and etc.

Time dilation is part of special relativity, a much more tractable topic to learn in a few days.

I have till next Thursday to finish a research paper and prepare a lesson plan so I can teach General Relativity (to an extent) to the rest of the class on that day.

I am guessing you have one class period (EDIT: confirmed below). In that amount of time, you can only hope to convey some intuition. My first thought was the same book pervect mentioned, the popular book by Robert Geroch. My second was the well known expository paper http://www.arxiv.org/abs/gr-qc/0103044

From these sources you can extract some ideas to try to convey.

I have finished the AP Calculus BC course and AP Calc-Physics course, so that is the prior mathmatical knowledge I have available to me.

Egads, not even linear algebra?! Well, not to worry, trying to convey some of the geometry of special relativity and then maybe the global geometry of an event horizon as per Geroch's pictures will be plenty challenging and fun for your classmates if you do a good job.

Unfortunately, after scanning through several canonical, jargon-filled books, I seem to be woefully unprepared for a mathmatical understanding of General Relativity. I generally latch onto mathmatical concepts very easily, but even trying to understand what terms I need to understand, to understand General Relativity is very difficult as I cannot seem to find an explanation that uses the same terms I'm trying to understand (If you understand what I'm saying .)

That's because AP calculus doesn't meet the prerequisites of any of the sources you have been using! Few textbook authors spell out prerequisites in great detail, because they can assume readers are students in formal college courses, where the faculty has presumably drawn an intelligent list of prerequisites for each course they teach.

time dilation near a body of mass

That's a bad way of thinking about gravitational red shift; rather the curvature of spacetime causes initially parallel radially outgoing null geodesics to diverge. This spreading results in the red shift effect.

Teaching a class by next thursday is wildly ambitious.

Or even learning some serious relativity. It's actually rather a comical thought.

If you aren't familiar with SR, I'd also recommend "Spacetime Physics" by the same authors.

Ditto.

There is no chance that you will be able to learn GR in a week, let alone learn it and teach it!

Ditto.

I would suggest picking something else. Have you learned Special Relativity? This doesn't need anything like that same amount of mathematics as GR, and you may actually have a chance of getting somewhere with it.

That was my first thought too.

I understand how complex it is, and I by no means expect to understand the entire subject.

You shouldn't expect to understand more than 0.0001% under these time limits and given lack of mathematical preparation.

The problem is not so much complexity as "subtlety". All students of gtr seem to stumble over the very same depressing long list of misconceptions, all of which must be patiently explained. This takes substantial effort. In particular, "local versus global" issues are essential but most students in graduate physics courses probably never adequately appreciate these.

I also appreciate your concern on my ability to grasp the subject and I'm sure there is quite advanced mathematics involved. However, I and everyone in my class are very strong mathematicians (as much as that can mean at the senior high school level) and I'm sure if I can get some pointers I'll be able to grasp it to at least an introductory level.

Well, define "introductory". Try the paper by Baez and Bunn; if you can convey that you will have done a good job. Don't expect to teach any of the mathematical techniques, the meaning of tensor equations, etc.--- if your class hasn't even studied linear algebra yet, much less tensor algebra, much less calculus on manifolds, your audience can't possibly have the mathematical maturity to really understand the EFE at an advanced undergraduate level.

My class has already been through special relativity to an extent and are able to perform such actions as Lorentz Contractions and Time Dilation within the bounds of Special Relativity. All I want is to take it a step past Special Relativity (if there is such a thing)

That sounds much more reasonable. The book by Geroch and Baez/Bunn should provide some good ideas. Have fun!

 

[Alabran]

Thank you for your response Mr. Hillman.

While initially I was unsure what exactly what you meant by "Linear Algebra", after searching it I've found that we all have had at least the basics of it. We haven't however, under my understanding of tensors, investigated that quite yet. Though we may be actually be working on it to a very small extent through the three-dimensional unit of Calculus at the moment.

I'll be sure to look for this text by Geroch. Meanwhile, I'm going to continue through the wiki article and ask questions here if have any. Thanks again.

[Add]What's more, I'll continue posting what new understanding I receive here. If I misunderstand something any correction would be appreciated.

 

[Alabran]

Co and Contravariant Vectors

Ok, so from Pervect, under einstein tensorial notation, a superscript represents a contravariant tensor whereas a subscript represents a covariant tensor.

A "contravariant" tensor is represented as:

A^u&#039;=x^u&#039;_{,v} A^v=\frac{\partial x^u&#039;}{\partial x^v}A^v

under which u' is the translation into the new coordinate system of vector u and v is the original vector of {0,1,2,3} representing time and the three positional coordinates. The partial derivative of each new coordinate is taken in respect to their original coordinates and then multiplied by said coordinate, (I believe).

"The squared length of the vector" (A, A) is equal to the sum of all but one of the individual coordinates squared and subtracted by A^0^2 which represents time. This makes sense as the relation to the Pythagorean therum. Thus, the extended version of the contravariant tensor is as follows:

\frac{\partial x^0&#039;}{\partial x^1}X^0 + \frac{\partial x^1&#039;}{\partial x^1}x^1 + \frac{\partial x^2&#039;}{\partial x^2}x^2 + \frac{\partial x^3&#039;}{\partial x^3}x^3

 

[Chris Hillman]

Thank you for your response Mr. Hillman.

Nobody calls me that; I go by "Chris".

While initially I was unsure what exactly what you meant by "Linear Algebra", after searching it I've found that we all have had at least the basics of it. We haven't however, under my understanding of tensors, investigated that quite yet. Though we may be actually be working on it to a very small extent through the three-dimensional unit of Calculus at the moment.

Trust me, if you knew linear algebra, you'd know it. I don't see why you are fighting so hard to reject what several posters have gently told you, that high school students (even AP students) simply don't possesses the mathematical prerequisites for a book like MTW. The point is that once you recognize this you can set more reasoanble goals for your talk, both in terms of what you can hope to learn in a few days and in what you can hope to explain to your classmates in a one period.

Meanwhile, I'm going to continue through the wiki article

No, No, NOOO! That's the worst thing you can possibly do at this point if you want to give a good talk which minimizes unintentional disinformation, which I certainly hope you wish to avoid! First, WP is too unreliable for students to use as an information resource Second, that particular article is just about the worst source for an AP high school student which I can imagine, since it focuses on math which everyone is telling you to avoid trying to learn and talk about on Thursday (and in any case, doesn't even try to explain said math, much less the physical meaning of the EFE, which is the heart of gtr).

I'll continue posting what new understanding I receive here. If I misunderstand something any correction would be appreciated.

Sounds to me like you are still avoiding doing any reading. There's really nothing more to be said here--- you've already received from several knowledgeable posters the advice to reassess what you can hope to accomplish and go read some good books focusing on achieving your new and hopefully more reasonable goals. So stop posting, print out the Baez/Bunn article right away and go try to find the book by Geroch right now! It might be in a local bookstore, or your public library.

I like your enthusiasm and hope you will continue to study gtr at a higher level after you give a nontechnical talk on Thursday, but first things first, eh?

 

[Chris Hillman]

You are still going about this assignment all wrong

Ok, so from Pervect, under einstein tensorial notation, a superscript represents a contravariant tensor whereas a subscript represents a covariant tensor.

A "contravariant" tensor is represented as:

A^u&#039;=x^u&#039;_{,v} A^v=\frac{\partial x^u&#039;}{\partial x^v}A^v

This is sufficiently misleading to be wrong. Stop trying to learn tensor calculus and start reading Geroch or Baez/Bunn. Several have told you this and its good advice. (For the duration of the next week, anyway.)

Feel free to come back and ask about tensor calculus later, after you've studied linear algebra (we can recommend books, but only after you've gotten past your current assignment).

 

[robphy]

In my opinion,...
whether you are mathematically inclined or not, tensors are the wrong place to start in trying to understand relativity. You need some physical intuition to support the mathematics. This intuition can be developed by operational definitions that are carried out in idealized thought experiments.

So, I cast another vote for Geroch's book... especially if your plan is convey some understanding in one lesson plan, with a short time to prepare. (A novel feature of Geroch's book is his definition of the [square-]interval between two nearby events using three clock readings from a radar experiment. [I believe that this formulation is originally due to AA Robb (1911).] )

 

[Alabran]

Just saw these last three posts...

I understand that the mathematics of General Relativity is incredibly advanced for my level. Unfortunately, some grasp of the mathematics is required as par the project. This may be irrational (and my professor is certaintly notorious for that) but that's how it is. I certaintly don't intend to be filling the entire presentation with mathmatical teachings. Some conceptual understanding of the concept is also required, as is a historical knowledge of the subject. I plan to find Geroch's book and read that for the conceptual understanding, but now is an inconveniant time for me to rush off to a bookstore or library and I'm able to work on the math now so that's the most productive use of my time at the moment as, from what I've read on this thread, that is going to be the most demanding part of this project. Clearly, though, you are all becoming exasperated with this "stubbornness" so I'll spare you my ramblings and reserve this thread for conceptual questions once I pick up the book. Thanks again for all your help.

 

[Chris Hillman]

Well, the paper by Baez and Bunn is on-line and you can grab it in about three seconds. You are clearly not going to understand tensor calculus this week, so I really urge you to stop struggling with that until you are better prepared, and to try Baez and Bunn instead http://www.arxiv.org/abs/gr-qc/0103044 (see also http://math.ucr.edu/home/baez/gr/gr.html and maybe other suggestions in http://math.ucr.edu/home/baez/RelWWW/HTML/tutorial.html ) After you read it once through, we can probably help you render stuff like dV/V for AP high school students.

Hmmm... another idea might be to use high school trig and bit of elementary calculus to compute the tidal force in the Schwarzschild vacuum, which you can compare with the Newtonian tidal force. They agree, which ought to be surprising! Then you can talk a bit about how this can be understood. See http://en.wikipedia.org/w/index.php?title=Tidal_tensor&oldid=28781955
and see if you can figure out the derivation I have in mind. Don't try to understand the Newtonian tidal tensor, obviously--- look for the two arguments I gave for the components. One is an AP calculus level argument for why the radial tidal force is a tensile stress scaling like 2 m/r^3. The other is a high school geometry argument (with appeal to the small angle approximation, which says \sin(\alpha) \approx \alpha for small angles) for why the orthogonal tidal force is a compressive stress scaling like -m/r^3.

Another idea: you can explain in words and a bit of high school geometry type math how to interpret Carter-Penrose conformal diagrams, which are a convenient way to exhibit the global structure of the the full Schwarzschild vacuum solution (and other solutions). You can compare with the diagram for a black hole formed by gravitational collapse, and then you can discuss a thought experiment in which a hollow spherical shell collapses. The shell could be made of matter, e.g. dust particles, but it's more fun to consider a collapsing shell made of radially infalling EM radiation. You can sketch the event horizon in the conformal diagram and point out that this shows that you can be inside a black hole before you know it, i.e. in the case of a collapsing shell of EM radiation (the field energy of the EM field contributes to the curvature of spacetime, as per Einstein's field equation), the shell is approaching at the speed of light, so you don't know its comin g until it passes your location. After it passes, you find you are falling toward what a concentration of mass-energy (the shrinking shell of radiation). If you are very unlucky, you are in fact inside the horizon of a newly formed black hole. In the conformal diagram you can see that the EH has expanded past your location even BEFORE the shell arrives. The beautifully illustrates the global nature of horizons. This is very exciting and subtle idea which can however be explained using diagrams.

(For a pedantic citation, I offer a picture in the monograph by Frolov and Novikov cited here http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#advanced )

Any talk on theory X should end with mention of some outstanding open problems concerning theory X. I'd suggest saying something like this:

You probably know that finding a quantum theory of gravitation is a very hard open problem. But even in classical gtr many important problems are still unsolved. For example, the solution space of the Einstein field equation is not yet very well understood in many respects. The EFE is a kind of generalized wave equation, but it is nonlinear. One can obtain much insight into the solution space of linear wave equations by methods such as Green's functions, but these integral transform methods mostly break down for nonlinear equations. So for example, showing that all vacuum solutions which are "nearby" flat spacetime are in some sense "almost flat" was fabulously difficult, and still has hardly been generalized at all. We'd like to have a similar result describing nonlinear perturbations of the Kerr vacuum used to describe the exterior field of a rotating black hole, for example.

Again, no one is discouraging you from studying linear algebra, tensor calculus, exterior calculus, differential equations, the theory of manifolds, and other background for gtr. To the contrary, this is such wonderful stuff that it is well worth learning and I wish everyone were as eager as you are to do just that! What we are saying, though, is that something well worth doing is worth doing well, but you can't possibly master the math, much less the physics behind said math, in a few days, given where you are starting from.

Giving a talk is a wonderful experience and if all goes well can be very rewarding. But I think you owe it to your classmates to do the best job you can, which requires having clear and achievable goals for what points you want to get across. I don't know what those could be--- you can decide what you most want to convey after reading Baez and Bunn.

Don't forget to cite your sources

 

[Alabran]

Very well then, I'll look over this Baez & Bunn article and Baez's "general relativity tutorial" looks amusing. Do you recommend any place to start within your own list of tutorials?

 

[Alabran]

"Let me describe the Ricci scalar, R, in 2d. This is positive at a given point if the surface looks locally like a sphere or ellipsoid there, and negative if it looks like a hyperboloid - or "saddle". If the R is positive at a point, the angles of a small triangle there made out of geodesics add up to a bit more than 180 degrees. If R is negative, they add up to a bit less." - http://math.ucr.edu/home/baez/gr/oz1.html

This is mind blowing.

 

[pervect]

I'm afraid you're rather missing the point on the vectors. The vectors being talked about are what one would call "4-vectors" in special relativity, i.e for example

(u^0, u^1, u^2, u^3 ) = (t,x,y,z)

The square-norm of a 4-vector, what I have been informally calling length^2, is the actually the square of the invariant Lorentz interval.

GR is built on special relativity. The geometry of space time in GR s constructed from the geometry of the Lorentz interval. With a metric tensor of
\left| \begin{array}{cccc} -c^2&amp;0&amp;0&amp;0\\<br /> 0&amp;1&amp;0&amp;0\\<br /> 0&amp;0&amp;1&amp;0\\<br /> 0&amp;0&amp;0&amp;1\\<br /> \end{array}<br /> \right|<br />

the standard Minkowski tensor for a flat SR space-time, using a more-or-less standard sign convention, the square-norm of a contravariant vector with components (t,x,y,z) would be

-c^2 t^2 + x^2 + y^2 + z^2

or in the usual notation. the square norm of u^i would be g_{ij}x^i x^j

You can thus see that the square-norm of a unit timelike vector (1,0,0,0) is -c^2, while the square-norm of a unit spacelike vector, for example (0,1,0,0) or (0,0,1,0) or even (0,0,3/5,4/5) is +1.

You'll find much more about 4-vectors in Space-time physics. You will notice a few small differences in approach, which will perhaps make things easier. For instance, you'll probably say "hello" to ict.Note that you can grab the first few chapters of an older edition of space-time physics online from Taylor's website.

There is very little sense in struggling with tensors until you learn the basics of 4-vectors. It sounds like the SR treatment you had didn't cover them :-(.

 

[Chris Hillman]

Very well then, I'll look over this Baez & Bunn article and Baez's "general relativity tutorial" looks amusing. Do you recommend any place to start within your own list of tutorials?

Yes, the Baez & Bunn article

 

[Chris Hillman]

Curvature

"Let me describe the Ricci scalar, R, in 2d. This is positive at a given point if the surface looks locally like a sphere or ellipsoid there, and negative if it looks like a hyperboloid - or "saddle". If the R is positive at a point, the angles of a small triangle there made out of geodesics add up to a bit more than 180 degrees. If R is negative, they add up to a bit less." - http://math.ucr.edu/home/baez/gr/oz1.html

This is mind blowing.

Yup, and you can explain it with a good diagram too. That's what we are all trying to say-- look for cool stuff you can explain well with diagrams and perhaps a few lines of math your classmates are already comfortable with, like trig and a bit of differential calculus.

BTW, the fact about angles has nothing to do with gtr per se; this is one of the most basic results in the theory of surfaces, which was created by Gauss c. 1830 and later picked up by Riemann to create what we now call Riemannian geometry. Riemann himself speculated that curved manifolds could be useful for describing physics, but died before he could pursue this. This fact about angles is in fact integral for one definition of the Gaussian curvature, which in gtr appears as sectional curvatures (the components of the Riemann curvature tensor taken wrt a frame field).

 

[pervect]

"Let me describe the Ricci scalar, R, in 2d. This is positive at a given point if the surface looks locally like a sphere or ellipsoid there, and negative if it looks like a hyperboloid - or "saddle". If the R is positive at a point, the angles of a small triangle there made out of geodesics add up to a bit more than 180 degrees. If R is negative, they add up to a bit less." - http://math.ucr.edu/home/baez/gr/oz1.html

This is mind blowing.

And you can draw triangles on a sphere to illustrate this, using geodesics (great circles) for straight lines.

It's easy enough to draw a triangle with three right angles on a sphere. Try doing that on a plane :-).

If you have a saddle, I suppose you could draw triangles on a saddle-surface to illustrate the case where R is negative, but a sphere is easier to deal with.

 

[Chris Hillman]

You can also make a cardboard model of geodesics on surface with negative and positive curvature concentrated at points, to show that it isn't necessary that spacetime be curved where you are to have measureable effects "in the large". This actually edges toward discussing the Aharonov-Bohm effect http://en.wikipedia.org/wiki/Aharonov-Bohm_effect

I am thinking of a model of certain "Sturmian tilings" (these are generalizations of Penrose tilings, due to de Bruijn) which can be obtained by looking from the right angle and an array of stacked cubes in E^3. There are three types of vertices: ones where three squares meet (angular deficit \pi/2; positive curvature concentration) and ones where six squares meet (angular deficit -\pi; i.e. an angular excess; negative curvature concentration).

(I should stress that we are discussing two dimensional Riemannian manifolds here. Spacetime models are four-dimensional Lorentzian manifolds, which are significantly different both mathematically (noncompact symmetry group, geodesic paths maximize proper time under small endpoint-fixing perturbations, etc.) and conceptually (timelike geodesics can't "turn around in time", etc.) from Riemannian manifolds. Again, you should avoid trying to actually learn much about manifolds for purposes of preparing for your talk.)

Alabran, one of the great results of Gauss in what we now call Riemannian geometry is the Gauss-Bonnet theorem, which states that in such models of a surface which is topologically spherical, the angular deficits must add up to 4 \pi, the area of a unit sphere. Try this with a cube: it has eight corners, each having deficit \pi/2, since three squares meet at each corner. Now put a small cube on one face. That adds four new vertices with angular excess of \pi/2 and four new verticles with angular deficit of \pi/2, so the sum is unchanged. With a lot of work, you can add a square pyramid (say) to another face and verify that the sum is still unchanged.

Want a proof of the Gauss-Bonnet theorem? Good for you! Well, one nifty approach is based upon Sperner's lemma, which is of independent interest. Ask after you've given your talk!

I once had the pleasure of pointing out to Penrose that you can interpret a rhombic Penrose tiling as a tiling by squares thought of as an abstract manifold with curvature concentrated at the vertices. Naturally it has net curvature near zero over any large disk. Even better, you can declare that each square has metric tensor ds^2 = -dt^2 + dx^2! Then, the null geodesics are extremely intricate and interesting! Superficially they resemble something studied by Conway but they are not the same.

 

[Thrice]

Any chance you could switch topics to special relativity? You might have a prayer there.

 

[Chris Hillman]

Any chance you could switch topics to special relativity? You might have a prayer there.

Ditto the thought, but that was the first thing at least two posters suggested. He's already rejected the notion of changing topics, however.

 

[Alabran]

Thank you all for your help, this has been very illuminating.

While we're all familiar with gaussian surfaces through it's use in finding electric and magnetic flux, I doubt any of us understood it at this level.

I just returned from dinner in consideration on how to make a 270 degree triangle. The concept of drawing triangles on the surface of a sphere, treating it as a plane, as Pervect mentioned is revolutionary in my mind. I didn't really understand this "The Parable of The Apple" when I tried to get into the MTW Gravitation book, I think I do now. Never-the-less, I think I'll have to perform it myself.

"You can also make a cardboard model of geodesics on surface with negative and positive curvature concentrated at points, to show that it isn't necessary that spacetime be curved where you are to have measureable effects "in the large". This actually edges toward discussing the Aharonov-Bohm effect http://en.wikipedia.org/wiki/Aharonov-Bohm_effect"

I'm familiar with this effect, it seems to be quite famous as far quantum mechanics go.

I am thinking of a model of certain "Sturmian tilings" (these are generalizations of Penrose tilings, due to de Bruijn) which can be obtained by looking from the right angle and an array of stacked cubes in E^3. There are three types of vertices: ones where three squares meet (angular deficit ; positive curvature concentration) and ones where six squares meet (angular deficit ; i.e. an angular excess; negative curvature concentration).

I'm a little curious about how to go about doing this. This is the "cardboard model" of geodesics you mentioned. By E^3 do you mean stacked cubes along the direction Ek? Or something else. If it's feasible to make such a model for class, I'd consider doing it, though I don't think I could construct one from this description.

Also, (assuming a "Unit Sphere" to be a sphere of radius r=1) wouldn't the area of a Unit Sphere be (4/3) \pi using the equation for the volume of a sphere: (4/3) \pi r^3 ?

I don't quite understand why where 3 squares meet there is an "angular deficit" or what that implies. You mean there are \pi/2 less radians within a corner of a cube than within a quarter of a sphere?

[Add] I just noticed you said "area" and you probably did not mean volume. Area of a sphere is, of course, 4pir^2.

 

[Alabran]

I just made a 270 degree triangle on a basketball. Exciting!

 

[Chris Hillman]

Clarification

While we're all familiar with gaussian surfaces through it's use in finding electric and magnetic flux, I doubt any of us understood it at this level.

Anyone with graduate training in math or physics knows the Gauss-Bonnet theorem, even good undergraduate students know it, but yes, this would surely be new to most high school students.

I just returned from dinner in consideration on how to make a 270 degree triangle. The concept of drawing triangles on the surface of a sphere, treating it as a plane, as Pervect mentioned is revolutionary in my mind.

This was indeed revolutionary in 1830 or so, when the idea was independently invented by Gauss, Lobachevski, and Bolyai. But by 1870 there were so many noneuclidean geometries that it was becoming difficult to keep track of them all! After your talk, see recent This Weeks Finds by John Baez, in which he has been discussing Kleinian geometry, one of the best ways to organize homogeneous geometries like the hyperbolic plane, the sphere (or a "round" metric version of RP^2, a certain nonorientable boundaryless surface, the "quotient" of the sphere obtained by identifying antipodal points), plus the less structured notions of geometry such as affine or conformal geometry (the former naturally belongs on the plane, the latter on the toplogical sphere).

I didn't really understand this "The Parable of The Apple" when I tried to get into the MTW Gravitation book, I think I do now. Never-the-less, I think I'll have to perform it myself.

After you've studied Taylor and Wheeler, since for spacetime you need the Lorentzian version whereas we are talking about the Riemannian version.

"You can also make a cardboard model of geodesics on surface with negative and positive curvature concentrated at points, to show that it isn't necessary that spacetime be curved where you are to have measureable effects "in the large". ...

I am thinking of a model of certain "Sturmian tilings" (these are generalizations of Penrose tilings, due to de Bruijn) which can be obtained by looking from the right angle and an array of stacked cubes in E^3.

I'm a little curious about how to go about doing this. This is the "cardboard model" of geodesics you mentioned. By E^3 do you mean stacked cubes along the direction Ek? Or something else. If it's feasible to make such a model for class, I'd consider doing it, though I don't think I could construct one from this description.

Well, Quasitiler with D=3 is what you want, but that seems to be broken.

Well, for a surface made of square faces which is naturally embedded in five dimensional euclidean space (!) try http://www2.spsu.edu/math/tile/aperiodic/penrose/josleys.htm Can you see three rhombs meeting at a common vertex as three faces of a cube? This particular picture won't "pop into" E^3 but rather into E^5, but this surface made of squares in E^5 does remain close to a hyperplane. When you have given your talk we can talk about decompositions of rotations in E^n; the hyperplane is an eigenspace (linear algebra) of the rotation which cycles the five basis vectors and that is why Penrose tilings---but not most other Sturmian tilings from D=5--- appear so symmetrical! This example is more complicated, so we have vertices where 7,6,5 4, or 3 squares meet instead of just 3 or 6. Can you spot some?

Also, (assuming a "Unit Sphere" to be a sphere of radius r=1)

Correct.

wouldn't the area of a Unit Sphere be 4}/3\pi using the equation for the volume of a sphere: 4/3(\pi)r^2 ?

No, but you can figure this out by differentiating 4/3 \pi r^3 wrt r!

I don't quite understand why where 3 squares meet there is an "angular deficit" or what that implies.

In the plane, four squares meet at a vertex. I said that in the surface I have in mind, sometimes three squares meet at a vertex (missing one right angle, so a deficit of one right ange) and sometimes six (two extra right angles, so an excess of two right angles). I couldn't find a three-dimensional picture on-line, so try to figure it out from the five-dimensional one.

BTW, a phenomenon discovered by Conway, "empires" is a lovely demonstration of how surprising global order can arise from local rules. This is the same "local to global" issue I keep mentioning. See http://www2.spsu.edu/math/tile/aperiodic/empires/jackempire.gif for an empire in another variant of Penrose tilings (combinatorially equivalent but drawn with nonrhomboidal tiles). Whenever you see the blue tiles, you must also see the green tiles, in those precise positions relative to the blue tiles, but the other tiles are not determined by the blue tiles. A rather fascinating idea which I and others have bandied about is the notion that this suggests a geometric approach to mathematical logic. After all, we know that in mathematics combining lemmas can sometimes enforce a surprising conclusion.

An example much more important for science/statistics as a whole is the Szemeredi lemma. This ties together combinatorics, number theory, ergodic theory, Ramsey theory, hard analysis, and many wonderful things. It basically says that complete combinatorial disorder is impossible. That's far more profound than noneuclidean geometry, one could argue! http://www.arxiv.org/abs/math/0702396

Having mentioned number theory, I should point out that Sturmian tilings are in a sense merely a way of realizing geometrically certain phenomena in simultaneous rational approximation, a classical topic which goes back to Lagrange.

If you ever get a chance to give another talk, you should give a talk on random graphs. Because this is such a wonderful and amazing subject, yet so accessible, even at high school level.