Mordred

You want to be able to plot every vector at every coordinate when you add [math]g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/math] that is where group symmetry comes into play every coordinate must be plotted under transformation. The number of coordinates increases the complexity.

here is an example the image on the left is a global vector field, the right is a local vector field. You want a set of equations to get from image left to image right. This is just U(1) image from Quantum Field theory Demystified, page number on image

that's one way to describe it, it may generate pretty pictures but you won't be able to calculate the path integral of a photon.

Unfortunately physics is concerned with all possible paths not just the ones you want. It must account for all to be fully predictive. Fractal geometry follows the same rules as the above....

Then why all the mistakes?

there ya go now that is just 2*8 by 2*8 not (8*8)*(8*8) of which every entry has two possible values 0 and 1.

Super just stop, you are causing more problems than its worth. When you have the tensor product of [math]E_8\otimes E_8[/math] this is the matrix of 8 columns and 8 rows multiplied by 8 columns and 8 rows of which the tensor product of such only shows the irreducible to zero portions. Every entry is multiplied to every other entry.

nope far far too low

You have absolutely no clue what your talking about. OK simple programming question. How many possible answers do you get when you add two 8 bit registers? (ALL POSSIBLE ANSWERS). that is tensor products.... ie [math]E_8\otimes E_8[/math]

OK lets do a simpler demonstration how many product under truth table is involved in a qubit multiplied by a qubit? write or the truth table. Now do the same truth table to the groups you just named. Your working with Tensor products, its just like Binary operations (more than one derivitave.....)

What good would that do? its just like programming in order to program a complex problem you must break that problem into simpler stages. Example flow charting.

Do you even know the Lemmas involved in the extended groups above? then your not working under the groups themselves and will get incorrect answers.

if you don't use integrals how do you plan on understand the very metrics that use them?

Do you ?, or do you only think you do? Can you describe the orthogonal group above under unitary groups ? Can you incorperate the right hand rule ? to the above?

No he is attempting the math, but unfortunately he isn't look deep enough to properly apply that math and yet your trying to advise him without even understanding the basics of the math. I lost track of the number of misconceptions in every post by you two that I cannot even list them all. You both assume you understand what is involved in string theory etc but I don't even know if you understand thee difference between Hilbert and phase space let alone how to describe a vector field. SO(1.3) is a Guassian vector field, Guassian is a particular treatment unto itself, the higher stage being Sturm Luiville. Niether of you can descibe the difference between a conformal and canonical field treatment. Neither has demonstrated a working knowledge of vector symmetry to apply it to a multiparticle system nor demonstrated how path integrals work under the various groups. Yet your both trying to model something as complex as the Schwartzchild metric which is far fa more advanced than the ds^2 line element shows. You both keep trying to apply those equations without understanding how they were derived in the first place. As such you get numerous errors.

He may very well have but your missing all the applicable axioms and lemmas of the lie groups under symmetry. This has to be studied seperately. For example the SO(1.3) lie group, do either of you know how matrix and tensors work or understand what the difference is between the two? Every coordinate in field theory has a vector assignment to it. (Every coordinate). That is how field theories work. Every coordinate is a function. That is true true complexity involved. Every group is its own algebra with specific lemmas and axioms specific to that group.

You two really don't believe in mastering the basic before tackling the complex do you? How do you you two ever plan on accomplishing your goals, if you don't ? I hate to break it to you both but both the mathematics I posted as well as Dubbelsix is really entry to intermediate level. Yet neither one of you could follow either and misapplied both examples by assuming you were understanding them. Yet your trying to handle higher dimension string theory without even understanding the lemmas and axoims involved in the group theory.

Good vids the second is a good explanatory to the Millenium vid. Also called Illustrius.

Lol you quoted before I removed the failed first link to the news post. By the way the simulation looks cool when you stream it to a big screen TV.

Well this particular image doesn't show the DM distribution. The image came from the Millennium simulation. http://www.illustris-project.org/ The simulation also show the DM distribution and tests the metalicity ratios.

One can have a complex variable they are not independant. Lets look at the simplest example. Spacetime... you have three independant variables x,y,z. Spatial dimensions.(space) two points with a seperation is a vector (magnitude and direction) Cauchy Schwartz inequality with the Kronecker delta describes this as the triangle inequality ie a^2+b^2=c^2 Pythagorous theorem (Euclidean geometry) Spacetime is any metric where time is treated as a dimension of length via ct. Time is an independant variable as it can change without affecting the others example Galiliean relativity (for Newton approximation under GR) Now for the Kaluzu Klien which leads to String theory charge is an independant variable. As charge is symmetric ie + or - the only difference in the two charge vectors is direction we only require one additional variable that can change without affecting the other variables. For the De-Sitter/Anti-Desitter it is a charged field ie vector field that can change without affecting other fields that they do not interact with. String theory must track particle interactions but not every particle interacts with every other particle. Key example neutrnos don't interact with the electromagnetic but in order to have a GUT you must track all interactions as well as isolate the non interactions. This corresponds to a particles degrees of freedom and its cross section. Hence why one must use higher dimensions 3 spatial, 1 time, 1 charge, 3 flavor, 3 color etc.

In essence your trying to model the universe via the Schwartzchild metric but there is a distinct difference. The Schwartzchild is a static metric (the coordinates do not change) however under the FRW the coordinates do change (commoving coordinates ) I posted the above to demonstrate the necessity of understanding the proofs behind the equations your trying to adapt. For example you tried to combine the Equations of state proofs via the Bose-Einstein statistics directly to the FRW metric but the statistics is part of the proof behind the FRW equations of state. ie matter w=0 Lambda w=1. The statistics is where those values come from in relation to the commoving volume itself. So combining the the two is incorrect as one is used to derive the other. The Higgs section was to demonstrate an example of how mass is attributed via field interactions mass = "resistance to inertia change"

Dimension under math is an independant variable. A variable that can change without affecting any other variable. Common example spatial dimensions. A given length x can change without affecting y. This applies even under string theory... it is not a seperate universe etc thats science fiction and pop media

I'm actually not asking you to. There is no way to teach that on a forum if you don't follow the math. I am asking you to start at the basics and giving you the tools to develop to the advanced. The choice is yours to accept the assist or not.

define a dimension under math... yes there is an answer