General Relativity for Dummies

[Chris Hillman]

Stop worrying about tensor calculus for now!

Let's see if my understanding is correct.

Ricci Tensor: The rate at which the difference in coordinates of two points (t,1,2,3) change in respect to... themselves...

No, that doesn't even make sense.

Since we have agreed (yes?) that you can't try to understand the notion of tensor in a few days, much less explain it to other AP high school students and still have time left to talk about the EFE, you should avoid the word "tensor" entirely! When we pointed you at Baez and Bunn, we meant (or at least I meant) that you should just try to get the physical picture they are suggesting, which doesn't even require mention of the word "tensor". To wit: after explaining world lines and timelike/null/spacelike vectors/paths as per Geroch, you can say something like this:

How did Einstein distinguish between "gravitational force", which by his elevator experiment must be as fictitious as "centrifugal force", and genuine forces which cannot be made to vanish by a change of coordinates? He adopted Minkowski's redefinition of acceleration of a particle as the path curvature of its world line in spacetime. Then he sought to represent the gravitational field entirely in terms of the curvature of spacetime, so that any particle experiencing no genuine forces (in Newtonian terms, no "nongravitational forces") has a world line which is a timelike geodesic. As John Wheeler puts it: "the curvature of spacetime tells freely-falling matter how to move".

To complete the picture, Einstein needed to specify how a given amount of mass-energy creates a specific amount of spacetime curvature. This information is contained in the Einstein field equation (EFE). As Wheeler puts it: "matter tells spacetime how to curve".

Physically speaking, this is what the EFE says: imagine that we have a small sphere of freely falling coffee grounds ("test particles") which are initially comoving with respect to each other. Suppose that there is a bit of fluid inside this sphere, but that the particles are still able to move freely (without experiencing any accelerations). Then all other things being equal, the gravitational attraction of this mass-energy should cause the sphere to contract. The EFE says that that volume changes like
$$\frac{\ddot{V}}{V} = \rho + 3 \, p$$
This formula captures both parts of Wheeler's slogan: "the curvature of spacetime tells freely falling matter (the coffee grounds) how to move, and matter (the blob of fluid inside the sphere of coffee grounds) tells spacetime how to curve".

Compare Newtonian gravitation, which is nonrelativistic, where the pressure term is absent. This means that pressure increases the gravitational attraction; in sufficiently extreme situations, such as a collapsing supernova core, this effect is sufficient to overcome extreme pressure and the result is complete collapse to form a black hole.

Here, we need to use a small sphere of coffee grounds because in curved manifolds, large spheres can have a very complicated relation between volume and surface area. But small spheres are pretty much just like the euclidean ones you know from high school solid geometry. (Well, you probably don't know, because American schools stopped teaching solid geometry about 1920. Physics/math students need to somehow acquire this essential knowledge out of class...)

The explanation on the article explained it in terms of a spherical cluster of points centered on point P (coffee grounds, specifically), each initially comoving with each-other. As this ball continues to move, each individual point is affected differently because it's specific path through the curvature of space-time is slightly different. The article explained that the "Ricci Tensor" is then, the rate at which such a sphere would change volume, though I don't understand what that is in respect to since time itself (to my understanding) would change slightly between the points.

Forget the Ricci tensor and focus on the physical intutition. It's not that different from Newtonian intuition, which is partly the point.

A Ricci Scalar seems like it would be easy to understand once I grasp what a Ricci Tensor is from my basic knowledge of scalars and vectors. Since "Speed" is the magnitude of "Velocity", seems that the Ricci Scalar would than the be magnitude of Ricci Tensor, Using the ensteinian summation technique.

The Ricci tensor is just a trace of the Riemann tensor, and the Ricci scalar is just the trace of the Ricci tensor. Since you haven't studied linear operators, you probably don't know what traces or eigenthings are yet, so I'd put all this aside for now.

For future reference, Baez and Bunn are "secretly" talking about something called the Raychaudhuri equation, which involves something called the kinematical decomposition of the covariant gradient of a vector field and something called the Bel decomposition of the Riemann curvature tensor. But you will need more time and more mathematical/physical background than you currently have to really understand all that!

I already suggested that you just say that because gtr is local field theory, the Newtonian force law is replaced by a focus on the tidal accelerations (these bend world lines since acceleration is path curvature) then carry out a Newtonian computation of the tidal accelerations in a spherically symmetric gravitational field, which is accessible to AP physics students. That satisfies the requirement that you do some math, and then you can discuss conformal diagrams and the Vaidya thought experiment on the formation of a black hole by a collapsing shell of EM radiation.

I can speculate on the meaning of the meaning of the Weyl Tensor, though I haven't actually checked it yet.

The Riemann tensor decomposes into the sum of a "trace part" built out of the Ricci tensor, plus the Weyl tensor. Since the EFE equates the Ricci tensor to the stress-energy tensor (this is equivalent to the physical picture above, but since you don't know what tensors are you can't expect to really understand this yet), it implies that the Ricci part is the part which is directly proportional to the immediate presence of matter (plus the field energy of any EM field). Since outside a star we have essentially vacuum, the EFE must imply that if the Ricci part of curvature is nonzero, it can generate nonzero Weyl curvature, which is the part which can be nonvanishing in a vacuum.

In particular, the field outside a spherically symmetric object like a star or black hole is a vacuum field, so the curvature there is entirely Weyl curvature. In contrast, in the simplest cosmological models, the FRW models, we imagine that the entire universe is filled with dust (pressureless fluid). Then the curvature is entirely Ricci curvature.

A very important prediction of gtr is that when you wiggle bits of mass-energy in the right way (for example, extending and contracting a pneumatic arm changes the mass distribution in the right way), you create gravitational radiation, ripples of Weyl curvature which propagate outward at the speed of light and which can propagate across a vacuum region.

If you wanted to mention this, you could talk about "Ricci curvature" and "Weyl curvature" without mentioning tensors. The important point is that only Weyl curvature can be nonzero in vacuum regions, and there is a differential equation which relates the two types of curvature.

You would have to preface all this by explaining spacetime, lightcones, worldlines, timelike/null/spacelike vectors, and path curvature, which is alot, so I doubt you'd have time in three periods, given the level. Path curvature is $\frac{d\theta}{ds}$, where in two-dimensional Riemannian geometry, $d\theta$ is the small angle through which the tangent vector to the curve turns as you move small distance $ds$ along the curve. In 4-d Lorentzian geometry its a bit more complicated but you don't have time to fuss with that.

I have the awful feeling I am starting to write your talk for you...

 

[Alabran]

No, that doesn't even make sense.

Shoot.

Since we have agreed (yes?) that you can't try to understand the notion of tensor in a few days, much less explain it to other AP high school students and still have time left to talk about the EFE, you should avoid the word "tensor" entirely! When we pointed you at Baez and Bunn, we meant (or at least I meant) that you should just try to get the physical picture they are suggesting, which doesn't even require mention of the word "tensor". To wit: after explaining world lines and timelike/null/spacelike vectors/paths as per Geroch, you can say something like this:

Well, I've abandonned attempting a mathmatical understanding of them, certaintly. Still, since Baez used the term within the article http://math.ucr.edu/home/baez/gr/gr.html" , I thought I should have an idea about what the Ricci Tensor really was, or at least what it represented. What I found from the description he included was that a Ricci Tensor is "the rate of change of [a small sphere of particles'] volume," though I don't know how that would apply to non-spheroid objects or even if it's applicable. So truly I only understand this term in context, perhaps that's the idea.

Here, we need to use a small sphere of coffee grounds because in curved manifolds, large spheres can have a very complicated relation between volume and surface area. But small spheres are pretty much just like the euclidean ones you know from high school solid geometry. (Well, you probably don't know, because American schools stopped teaching solid geometry about 1920. Physics/math students need to somehow acquire this essential knowledge out of class...)

I don't know what I might be missing. Basic solid geometry was taught all through middle school (Surface Area/Volume of basic solids: Cylinders, Spheres, Cones, Prisms, Pyramids, etc.) and more irregular solids is an emphasis within Calculus, (Surface area\Volume of an object composed of (whatever shape) whose (some characteristic) is modeled by these curves, etc.)

The Ricci tensor is just a trace of the Riemann tensor, and the Ricci scalar is just the trace of the Ricci tensor. Since you haven't studied linear operators, you probably don't know what traces or eigenthings are yet, so I'd put all this aside for now.

I believe I studied traces back with matrices in my sophomore year, though Eignen"things" seem unfamiliar. Regardless, I'll take your word for it and disregard it.

For future reference, Baez and Bunn are "secretly" talking about something called the Raychaudhuri equation, which involves something called the kinematical decomposition of the covariant gradient of a vector field and something called the Bel decomposition of the Riemann curvature tensor. But you will need more time and more mathematical/physical background than you currently have to really understand all that!

Fine.

I already suggested that you just say that because gtr is local field theory, the Newtonian force law is replaced by a focus on the tidal accelerations (these bend world lines since acceleration is path curvature) then carry out a Newtonian computation of the tidal accelerations in a spherically symmetric gravitational field, which is accessible to AP physics students. That satisfies the requirement that you do some math, and then you can discuss conformal diagrams and the Vaidya thought experiment on the formation of a black hole by a collapsing shell of EM radiation.

I appreciate the suggestion and I think I will attempt to use the tidal relation like you suggested. It doesn't seem to be so very related to general relativity (from a quick scan, the "Tidal Tensor" is one of what composes a "Riemann Tensor," which is a method to express the "Riemann Manifold," which, when applied through the "Lorentzian Metric," results in the "Lorentzian Manifold," which is representative of space-time,) but unfortunately it seems as though it's the only mathmatical involvement we are capable of doing. I'll also consider the black hole experiment" while I piece together my talk.

The Riemann tensor decomposes into the sum of a "trace part" built out of the Ricci tensor, plus the Weyl tensor. Since the EFE equates the Ricci tensor to the stress-energy tensor (this is equivalent to the physical picture above, but since you don't know what tensors are you can't expect to really understand this yet), it implies that the Ricci part is the part which is directly proportional to the immediate presence of matter (plus the field energy of any EM field). Since outside a star we have essentially vacuum, the EFE must imply that if the Ricci part of curvature is nonzero, it can generate nonzero Weyl curvature, which is the part which can be nonvanishing in a vacuum.

In particular, the field outside a spherically symmetric object like a star or black hole is a vacuum field, so the curvature there is entirely Weyl curvature. In contrast, in the simplest cosmological models, the FRW models, we imagine that the entire universe is filled with dust (pressureless fluid). Then the curvature is entirely Ricci curvature.

A very important prediction of gtr is that when you wiggle bits of mass-energy in the right way (for example, extending and contracting a pneumatic arm changes the mass distribution in the right way), you create gravitational radiation, ripples of Weyl curvature which propagate outward at the speed of light and which can propagate across a vacuum region.

If you wanted to mention this, you could talk about "Ricci curvature" and "Weyl curvature" without mentioning tensors. The important point is that only Weyl curvature can be nonzero in vacuum regions, and there is a differential equation which relates the two types of curvature.

You would have to preface all this by explaining spacetime, lightcones, worldlines, timelike/null/spacelike vectors, and path curvature, which is alot, so I doubt you'd have time in three periods, given the level. Path curvature is $\frac{d\theta}{ds}$, where in two-dimensional Riemannian geometry, $d\theta$ is the small angle through which the tangent vector to the curve turns as you move small distance $ds$ along the curve. In 4-d Lorentzian geometry its a bit more complicated but you don't have time to fuss with that.

Thank you, I believe I understood most of that. I really wish I could have more time to more fully understand and teach this subject since it all seems so very interesting. That does not seem to be the case, though I look forward to learning more about it this summer if you'd still have me.

I have the awful feeling I am starting to write your talk for you...

I'm sure you've already noticed I'm far too stubbornly independent for that to happen.

 

[Chris Hillman]

Suggestions for talk

I appreciate the suggestion and I think I will attempt to use the tidal relation like you suggested. It doesn't seem to be so very related to general relativity (from a quick scan, the "Tidal Tensor" is one of what composes a "Riemann Tensor," which is a method to express the "Riemann Manifold," which, when applied through the "Lorentzian Metric," results in the "Lorentzian Manifold," which is representative of space-time,) but unfortunately it seems as though it's the only mathmatical involvement we are capable of doing. I'll also consider the black hole experiment" while I piece together my talk.

Good, good.

General relativity reduces to Newtonian gravity for weak fields and slow speeds (wrt c). That's terribly important. So you can say, as I suggested, that to get rid of the force law (which can't be fixed to work with relativity), focus attention on tidal accelerations, which cannot be removed by coordinate changes, so they are physically/geometrically signficant. To get an idea of what to expect, compute these in Newtonian theory for the gravitational field of the Sun. You can do that using only AP calculus and high school trig. Then you can say that we should get a similar answer in gtr, and we do--- in fact, in a sense we happen to get exactly the same answer. You can say that gtr represents gravitation by spacetime curvature, and the thing that is used to represent/measure curvature, the Riemann tensor, breaks up into pieces which are important in their own right. One such piece is the tidal tensor of gtr. What I just said is that for the Schwarzschild vacuum solution, the gtr tidal tensor happens to agree with the Newtonian tidal tensor.

So if gtr gives the same result at Newtonian gravitation, who needs it? Well, it doesn't give the same result for most things! If you compute the motion of the planets orbiting the Sun, or light rays passing near the Sun, you find slight discrepancies between gtr and Newtonian theory. These have been tested and results agree with gtr, not Newtonian gravitation.

Sometimes gtr gives DRASTICALLY different results from Newtonian gravitation. The best known example is the black hole, where the very tidal accelerations which on small scales agree perfectly with Newtonian theory, turn out to give a dramatically different conclusion on large scales.

(Then you can start talking about light cones in black holes and then the Vaidya thought experiment.)

Thank you, I believe I understood most of that. I really wish I could have more time to more fully understand and teach this subject since it all seems so very interesting. That does not seem to be the case, though I look forward to learning more about it this summer if you'd still have me.

Good, good, I'm sure many here will be happy to help you learn more about gtr. It really is beautiful stuff, although certainly not the only beautiful theory out there.

 

[Chris Hillman]

Vaidya Thought Experiment

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 )

Oh dear, I didn't look hard enough for an on-line graphic, but Jacques Distler has provided exactly the picture I wanted (in the context of discussing a bizarre fringe claim):
http://golem.ph.utexas.edu/~distler/blog/archives/000530.html
This is exactly the same as the picture in Frolov and Novikov, only bigger and nicer.

In this Carter-Penrose conformal diagram, each point represents an entire sphere (two-dimensional round sphere) of events. The diagram correctly represents "angles" but distorts distances in such a way as to make an infinite spacetime fit in a finite diagram.

A "timelike" curve in the diagram represents the "world sheet" of a sphere of observers (concentric with the place where the hole will form). A "timelike geodesic", the same, for a sphere of inertial observers. An "infalling null geodesics" represents a sphere which contracts at the speed of light.

In the picture, you can see the shell of infalling EM radiation as the blue diagonal band running from lower right to upper left; as bits of shell collapse to a point they encounter a strong spacelike curvature singularity represented by the green horizontal line at top. The Ricci curvature of this Vaidya model (which is an exact spherically symmetric but nonstatic null dust solution of the EFE) is confined to the location of the shell (the blue band). Below this band, the spacetime is indistinguishable from Minkowski spacetime. Above it, from a Schwarzschild vacuum with mass parameter corresponding to the mass-energy contained in the shell. So: our spacetime model has a flat vacuum region below the blue band, a nonvacuum null dust region in the blue band (with nonzero Ricci and Weyl curvature), and a curved vacuum region (Weyl curvature only) above the blue band.

The event horizon is the dotted black diagonal line running from lower left to upper right. The events at the right (like the right half of a "diamond", well outside the EH) represents events very distant from the newly formed BH.

And now the point of this figure: the stick figure man represents an inertial observer who is coasting along in a locally flat region of spacetime and who believes himself to be in Minkowski spacetime. Unbeknownst to this hapless victim, he is actually momentarily on the event horizon of a black hole! He isn't falling yet because curvature is flat in his neighborhood and at this event the horizon is a radially sphere which is expanding past him. He has no idea that the shell is coming yet because its moving at the speed of light and his first encounter with it is still some time in his future! The dotted red line, plus part of the EH, is the world sheet of a sphere which contracts onto this event at the speed of light, i.e. the "past light cone" of this event.

Again, the point is to sharply contrast an important local concept (curvature) with an important global concept (the event horizon).

 

[Thrice]

It really is beautiful stuff, although certainly not the only beautiful theory out there.

I actually can't think of any off the top of my head. You know of some I could use to pass time?

 

[Alabran]

Bummer

I wasn't able to talk near as much as I wanted to. A firedrill cut into 15 of my 50 minutes. I think I was able to get the concept across pretty well, I made a cool little simulator of space-time fabric (a stretched sheet with a globe and several ball-bearings.) I was able to get across the idea of what gravitation "really" was. We never arrived upon Tidal Forces, which irked me. So, now the mathmatic portion of the final will be simply the use of Lorentz Transformation.

Oh well, it was still enjoyable, and better still, I think I kept my audiance interested (except for during the mandatory "history" section, which is utterly uninteresting.)

 

[Chris Hillman]

Seriously?

I actually can't think of any off the top of my head. You know of some I could use to pass time?

I hope you are joking, but if not, start another thread in the "General Math" forum (because there is no "General discussion" subforum of the "General Physics" forum, and because when I say "theory" in generic non-PF contexts, I often mean a mathematical theory), and I'll name some theories which are fully as beautiful, subtle, and fascinating as gtr.

 

[Chris Hillman]

Congragulations on giving your first lecture!

I wasn't able to talk near as much as I wanted to. A firedrill cut into 15 of my 50 minutes.

Welcome to the real world of classroom teaching!

With experience you learn to expect the unexpected.

I think I was able to get the concept across pretty well, I made a cool little simulator of space-time fabric (a stretched sheet with a globe and several ball-bearings.)

As you may know, electrostatics is formally equivalent to Newtonian gravitostatics (both are governed by a partial differential equation called the Poisson equation, which is also important in gtr and many other places in math). In the early days of automotive engineering, when they needed to design a headlight they simulated the electric field using a rubber sheet pulled or pushed into the desired shape, and then they rolled ball bearings to see where the electrons would go. In old textbooks you can sometimes find pictures of such engineering design devices--- we might call them "analog MAD", as compared to "digital CAD".

We never arrived upon Tidal Forces, which irked me.

With experience, you learn to cut your estimate of what you can hope to say by half. Then cut it again. And again.