@ACuriousMind I made it through the first three chapters (review of GR and geometry) unscathed. However, I will have to carefully study their discussion of the perfect fluid in the Lagrangian prescription with pencil and paper tomorrow. The equations got unmanageable for mental math.
The notation is great, the only thing that bothers me is their use of $\mathrm{D}/\partial t$ for the cov. derivative along a curve.
@Danu Well, fermion spin sure isn't what you would call "normal" by any conventional 3D definition. H multiplications just have a nice twist on rotating an object in which first you rotate it in 4D space, then with second multiplication you twist the object back into 3D space -- but also add a rotation when you do so.
It's more just an interesting analogy about how something that is at first very "non-3D" turns into a quite normal 3D rotation operation when you do it twice (granted, with a pre-post switch. H is just a fun space to rotate things in, bottom line.
@ACuriousMind The standard line with two origins. Every other physics book I've seen that defines manifolds precisely doesn't bother explain why Hausdorff is important.
@0celo7 It's an interesting mode of delayed conversation, makes me feel a bit like one of those folks wandering down the street talking on a Bluetooth to invisible people... :)
@Danu I'm not sure what I was going for there! If I had to give an example of geometry, one would be that the fundamental fermions map in an almost distressingly simple way onto the vertices of a corner-balanced cube, with vertical as electric charge and horizontal (horizontal in a 4D slice actually, it's not really an ordinary cube) as strong charge. Dunno if those same cube-like symmetries show up explicitly in the SM or not. Probably not? It's a nice visual mnemonic in any case.
@vzn Yes, CG is the most common challenge posted because it's generally pretty easy. I disagree entirely with both quotes you present. From my experience, the guys there dislike the 'lulz' and 'cute' stuff for more numerical and algorithmic challenges. I also don't think that any of it is truly serious in the sense that it's at all 'cutting edge.' It's all fun and games.
@0celo7 "While we expect that most of our readers will have some acquain- tance with General Relativity, we have endeavoured to write this book so that it is self-contained apart from requiring a knowledge of simple calculus, algebra and point set topology." - HE
@0celo7 True! Well I liked rotations from the beginning though. Like of all those trig identities, most stem from the angle addition formulas. Which can also be written as R(q)R(f)=R(q+f) for a 2D rotation matrices R(theta). Which (since each e^(iq) can be associated uniquely to a rotation matrix R(q) by associating to i, {{0,-1},{1,0}} and to 1 {{1,0},{0,1}}) is equivalent to expressing the fact that e^(iq)*e^(i*f)=e^(i(q+f)). I like that.
plus the fact that all of those 4D rotating cubes don't look neat because it's a 4D projection
they look neat because they let you see the 4D rotation group!
it's nice to have this whole narrative of rotations join together
although my dad was not satisfied when he asked me to explain e^(i pi)=-1 and I launched into a rant about lie groups
@Icosahedron Sure thing, bud. Go ahead and read it. The preface never lies.
There's a reason Wald refers to it for the difficult proofs.
@NeuroFuzzy My dad keeps asking me when I'm going to learn useful things like Fortran and numerical PDEs. He couldn't give a flying flip about Euler identity. (Despite him taking complex analysis in grad school.)
@Danu I think the absolute exterior differential is a special case of that on the tensor/tangent bundle.
@ChrisWhite @JimtheEnchanter Since the energy-momentum tensor of dark energy does not satisfy the energy conditions of the singularity theorems, does it invalidate them for our universe? Or is the cosmological constant small enough so that the total energy-momentum tensor (baryons + DM + radiation + other) is positive definite?
I'm not sure if Hawking & Ellis will discuss the possibility of a cosmological constant in the singularities chapters.
@0celo7 I don't know what you mean by "absolute exterior differential"
In any case, the map I was talking about was not some special case of a map defined on the tangent bundle. Rather, the map on the tangent bundle is the "trivial case" of the map I was talking about.
user54412
@0celo7 The relative fraction of energy density due to $\Lambda$ is large and growing. But while it violates the strongest conditions (I always forget which is named what), it only saturates the... weak?... one.
@Danu The absolute exterior differential acts on tensor-valued forms. It's square on a vector-valued 0-form is the Riemann form.
@JimtheEnchanter Only matter? How is a vacuum energy distinguished from matter? This is not made clear in Wald.
@ChrisWhite Do you know of any recent articles on the subject (preferably no paywall)?
@Danu I didn't say you're talking about a special case. I said that I know of an operator that acts twice on forms and gives the curvature. Perhapsthis is a special case of what you're talking about.
@0celo7 I'm talking baryonic/dark matter. You don't include dark energy on scales the size of a black hole because gravity is overwhelmingly dominated by matter on those scales. Now, black holes aren't my speciality, but I believe the energy conditions of the singularity theorems hold as long as gravity bends light inwards.
Let me be more scientific. All ordinary matter follows the strong energy condition. Vacuum energy does not, but with the exception of the inflationary regime, it's not an issue
So long as the dominant energy condition is not violated, there's no problem
Black holes only require the weak condition to be satisfied, which is just the the local gravity bends parallel light rays together.
That said, the fact that scalar field vacuum energy violates the dominant energy condition allows us to interpret that as there being no singularity corresponding to inflation
@0celo7 Oh, the matter energy of the universe was all contained in scalar fields and so it violates the dominant energy condition, which allows us to smooth over the singularity. Most of us still say there was a curvature singularity, but some theories allow for this smooth singularity. It's not extremely important in the grand scheme of things
@0celo7 Nope, I know of no non-paywall papers. And only some inflation theories have no singularity. Others have a singularity, and others still have nothing even coming close to that situation entirely
@Danu @0celo7 The square of the covariant exterior derivative w.r.t. a connection is always the wedge with the curvature of that connection. Nothing special for the Levi-Civita connection/Riemann tensor here ;)
user54412
7:06 PM
@DanielSank what do the angled brackets mean here?
@ACuriousMind No wedge with the curvature tensor involved in the thing I'm talking about
We didn't really have very clear terminology, so I don't know what you mean by the covariant exterior derivative. We just called it the "generalization of ..."
@Danu I have a question from HE that might have a simple solution. Suppose we have an ODE of the form $A''=-RA$. We have the initial conditions $A(0)=1$ and $A'(0)$. What does the solution look like if $A'(0)$ is large vs. small? (Note: in the real problem, A is a matrix and $R$ the Riemann tensor.)
I know that in the simple case I can solve it for a SHO, but I'm not sure how to look at it in the matrix case.
They say that if the initial data for the derivative is very large, the RHS of the ODE becomes "irrelevant".
