Mordred

I'm well aware of how the transverse traceless gauge for the quadrupole plane waves propogate. The expression is highly misleading.

How do you go sideways in time? lol forgot to let phone update to the adfitional posts

Oh forgot the most important detail. The dot product of two vectors is a scalar. The cross product of two vectors is a vector.

Yes very much like curl more accurately rotatons under symmetry where the symmetry applies directly the applicable conservation laws etc. Ie for GR freefall has conservation of energy/momentum under a constant velocity and a change in velocity via acceleration is a rotation (rapidity). For conservation of angular momemtum under the closed angular momemtum system where the conservation law applies torque is zero. When torque is applied you undergo a rotation of the symmetry groups. The generator matrixes are rotations under symmetry. Recall under electromagnetism the magnetic field is 90 degrees out of phase to the electric field. So you first define the symmetry between the two fields then apply the rotation translation. For time you need to apply a time translation which will again be a rotation under symmetry. Hence cross products

Close but not quite under Banack spaces which if I understand correctly you want. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www2.physics.umanitoba.ca/u/khodr/Publications/inner-product-ANS-2015.pdf&ved=0ahUKEwi3_Zroy7_XAhWC0hoKHRBLBYUQFgggMAA&usg=AOvVaw2uUcHQ_2e_rfYt0L6ZYYbQ start with ijk coordinate basis ie i follows x, to value 1 on x coordinate, j follows the y axis to value 1 while k follows the z coordinate axis l. See last link on Cartesian Calculus 1 chapter uses Kronecker Here see 1.57 https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www2.physics.umanitoba.ca/u/khodr/Publications/inner-product-ANS-2015.pdf&ved=0ahUKEwi3_Zroy7_XAhWC0hoKHRBLBYUQFgggMAA&usg=AOvVaw2uUcHQ_2e_rfYt0L6ZYYbQ Here Vector calculus see section on cross product and its association to Levi-Civita 1.58 Chapter Calculus II applying Levi Civita for spherical and polar coordinates https://www.google.ca/url?sa=t&source=web&rct=j&url=http://physics.csusb.edu/~prenteln/notes/vc_notes.pdf&ved=0ahUKEwiJ1qns1b_XAhVBiRoKHYFfABYQFggwMAU&usg=AOvVaw2yHnhKwx8JCDebWO7Wxz5i This last should give you the details you need to get the triple inner product for your [math]E_8[/math] ie the dyad and quaturnions which should lead to your octonionic projective plane

Good luck but what Dubblesix has is missing a key the k coordinate basis for the Levi Cevita. He has the Kronecker affine connections [math]\delta_{ij}[/math] which works under [math]\eta_{\mu\nu}[/math] you need the Levi-Cevita [math]\delta_{ijk}[/math] to include tidal force under GR in order to get the full Schwartzchild

Nothing is made up of energy. It is a property not a thing. "energy is the ability to perform work" nothing more

Group theory itself can be tricky two completely different objects that have absolutely nothing to do with one another can be described by the same symmetry group depending on how the vectors, scalars and spinors. That is all a group is "a representation" it is a means to organize all the degrees of freedom upon its vectors being applied. x posted lol nice picture

Well a large part of the problem with changing metrics is that you also change the axioms associated with each metric. For example in QM and QFT you have completely different operators. The problem isn't as pronounced under GR to SR primarily in GR all frames are inertial which is a bit different than SR with a rest frame. One must examine the axioms behind each metric before trying to mix them. It would be like trying to mix canonical and conformal treatments, particularly since several of the transposed vectors themselves may be different. In particular when dealing with complex conjugates etc... By the way naturally if you apply the Strong force to photons as per c you will get the wrong answer. I included the above to demonstrate the two primary types of generators used in particle physics ie the standard model itself. Photons don't interact with the strong force so are not part of the SU(3) group. It is described under the SU(2) and but not SU(3). Different fields can readily overlap the same volume without influencing one or the other.

Here this will help with the above Exceptional lie algebra SU(3) and Jordon pairs it details the group reductions https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1403.5120&ved=0ahUKEwiniNOlnL_XAhUSFuwKHURyAckQFgghMAI&usg=AOvVaw3bvor0uH-6fTf3Hp9JxCyR Note it is arxiv on phone atm This will help as it provides the Jordon Schrodinger equation "On a Jordan-algebraic formulation of quantum mechanics: Hilbert space construction" https://arxiv.org/abs/hep-th/9304124

Ok I suggest you look specifically at the generators for the SU(2) then the 3^2=8 generators for SU(3) the previous uses the Pauli matrices ie Dirac. While the latter uses the Gell-Mann matrices. The strong force for example uses the latter via the eightfold Wayen.

Yes but Dubblesix isn't working under SU(3) Hermitean he is working under SU(2) hermitean. If I picked up the exceptional E_8 group its irreducible is SU(3) ie the Jordon identities are 3×3 hermitean matrix. Granted that is a quick study of the E_8 group ie it uses the Gell Mann matrixes . The other problem I can surmise on this quick study is E_8 has a triple inner product which certainly isn't what Dubblesix is applying.

took me a bit to go through this post but looks to me you arrived at the solution on your last pot on that thread. String theory happens to be one of my weaker topics however. (never really had much interest in it) so how SO(32) is handled to the unitary group reductions I can't be of much help other than stating any orthogonal group regardless of number of dimensions can be expressed with unitary groups. An orthogonal group is a double cover group. However simply do not have familiarity with SO(32)

Little side note hint, the corresponding blackbody temperature of the universe will be roughly the inverse of the scale factor. (the above should explain why inverse)

It takes time to fully understand but your not doing too bad. Hopefully you picked up the [math]\dot{a}[/math] is a time derivative while [math]\ddot{a}[/math] is the second time derivative. now the scale factor Hubble parameter is defined as [ [latex]H=\frac{\dot{a}(t)}{a(t)}[/latex] recessive velocity [latex]v_{rec}=\frac{\dot{a}(t)}{a}[/latex] so here is some examples, note a(t) today is 1.00 see stretch 1.00 column. a(t) at z=1090 s 0.001 this is the CMB surface of last scattering. the a with values greater than 1 is future expansion. (in this case roughly 88 Gyrs into the future lol. here perhaps this will help understand scale factor. taken from the user guide for the cosmological calculator example given below though the calc has numerous more columns and rows when fully set up. "The LightCone tabular cosmological calculator is a versatile tool for understanding the expansion history of the universe: past, present and future. Stages in expansion history are designated by the corresponding scale factor a, which is the size of a generic distance compared with its size at present. For instance, a=1 denotes the present and a=0.5 the stage when cosmic distances were half their present size. In the same way, a=2 refers to a future stage when distances will be twice what they are now." http://cosmocalc.wikidot.com/lightcone-userguide Marcus. Jorrie and I wewnt through a lot of effort to simplify how we wrote the guides as we encountered numerous posters where the textbook descriptives were difficult for the average poster to understand (hope this helps) here is an example table. [latex]{\small\begin{array}{|c|c|c|c|c|c|}\hline T_{Ho} (Gy) & T_{H\infty} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}[/latex] [latex]{\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&z&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&H/Ho \\ \hline 0.001&1090.000&1089.000&0.000373&0.000628&45.331596&0.041589&0.056714&22915.263\\ \hline 0.001&683.804&682.804&0.000810&0.001326&44.962398&0.065753&0.089864&10859.192\\ \hline 0.002&428.979&427.979&0.001722&0.002756&44.477683&0.103683&0.142116&5224.758\\ \hline 0.004&269.117&268.117&0.003606&0.005666&43.849475&0.162938&0.224202&2541.361\\ \hline 0.006&168.829&167.829&0.007463&0.011563&43.042568&0.254948&0.352603&1245.393\\ \hline 0.009&105.913&104.913&0.015309&0.023478&42.012463&0.396668&0.552333&613.344\\ \hline 0.015&66.444&65.444&0.031211&0.047518&40.702622&0.612585&0.860719&303.042\\ \hline 0.024&41.683&40.683&0.063355&0.095974&39.041469&0.936624&1.332155&150.041\\ \hline 0.038&26.150&25.150&0.128224&0.193578&36.938267&1.412573&2.043059&74.389\\ \hline 0.061&16.405&15.405&0.258995&0.390062&34.278330&2.089532&3.094542&36.917\\ \hline 0.097&10.291&9.291&0.522342&0.785104&30.917756&3.004225&4.606237&18.342\\ \hline 0.155&6.456&5.456&1.051751&1.575989&26.679131&4.132295&6.685941&9.137\\ \hline 0.247&4.050&3.050&2.109877&3.133394&21.362526&5.274330&9.344906&4.596\\ \hline 0.394&2.541&1.541&4.180384&6.013592&14.827243&5.835394&12.323993&2.395\\ \hline 0.627&1.594&0.594&7.955449&10.346218&7.320583&4.592515&14.935503&1.392\\ \hline 1.000&1.000&-0.000&13.787206&14.399932&0.000000&0.000000&16.472274&1.000\\ \hline 1.585&0.631&-0.369&20.956083&16.410335&5.731185&9.083316&17.046787&0.877\\ \hline 2.512&0.398&-0.602&28.694196&17.063037&9.638020&24.209612&17.203810&0.844\\ \hline 3.981&0.251&-0.749&36.601471&17.239540&12.159687&48.408586&17.239540&0.835\\ \hline 6.310&0.158&-0.842&44.553231&17.284732&13.760162&86.820752&17.284732&0.833\\ \hline 10.000&0.100&-0.900&52.516301&17.296130&14.771503&147.715032&17.296130&0.833\\ \hline 15.849&0.063&-0.937&60.482221&17.298988&15.409856&244.229762&17.298988&0.832\\ \hline 25.119&0.040&-0.960&68.448857&17.299697&15.812667&397.196249&17.299697&0.832\\ \hline 39.811&0.025&-0.975&76.415673&17.299867&16.066830&639.632027&17.299867&0.832\\ \hline 63.096&0.016&-0.984&84.382534&17.299901&16.227197&1023.866895&17.299901&0.832\\ \hline 100.000&0.010&-0.990&92.349407&17.299900&16.328381&1632.838131&17.299900&0.832\\ \hline \end{array}}[/latex] This will give you some useful relations to compare wiki has not too bad a coverage for equations of state remember the cosmological constant is constant over time... https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) Hopefully it is enough to show that the scale factor conpares the radius of the sphere today with the radius of the sphere at the time being examined. It is simply a ratio of change between two measurement events.

I'll grant the second upvote as its well deserved

Not nearly detailed enough to convey any real conclusion. It certainly wouldn't turn any heads sufficient to sway too many opinions on BHs

anti symmetric for the ricci tensor see Einstein Cartan "The Ricci tensor is no longer symmetric because the connection contains a nonzero torsion tensor;" https://en.m.wikipedia.org/wiki/Einstein–Cartan_theory A little hint look at conservation of angular momentum and how torsion is involved in a specifically closed system. Also look specifically at the mathematical definition of a conserved quantity in terms of symmetry ie invariance.

yeah you have to be careful particularly if your jumping from canonical to conformal states etc that and not all papers get various aspects of group theory entirely correct depending on which lemmas they are following. I myself have oft gotten confused when looking at different treatnents. Its often very frustrating but challenges are fun lol

yep basically correct, we had a small cross post I added to my last reply for skew symmetric lemma I like the reply on the Levi-Cevita given in the last link. (It is why I wanted to see you looking into the Levi-Cevita applications glad to see you are.

I really like that last link extremely informative gonna study it a bit myself. This lemma will help the difference of a square matrix and its conjugate transpose is skew Hermitian [math]A=-A^\dagger[/math] being anti Hermitean

Well he has done something odd ball see equation 3 of this article showing the EFE as per natural units and his equation 2 https://www.google.ca/url?sa=t&source=web&rct=j&url=https://www.seas.upenn.edu/~amyers/NaturalUnits.pdf&ved=0ahUKEwihusegv7nXAhUB-2MKHd5bBR0QFghjMBE&usg=AOvVaw23tw7F_yapqigznAYED9oW the latter paper gives it as [math]R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi}{m^2_p}T_{\mu\nu}[/math] where he gives [math]\frac{8\pi l_p}{m_pc^2}T_{\mu\nu}[/math]

lol Anytime someone refers strictly to his own papers as support I immediately question. To state the paper is too short is an understatement. The problem is in Planck units G=1 it is already in natural units before even using the equation above. Or rather it appears he is mixing natural and cgs units for G. c=g=h=1 his notes on the first references support that. In order to derive the natural units in the first place.

Well I for one cannot find any consistency in the math posted. The formulas provided are literally useless. The explanations and replies by the OP clearly shows that the OP doesn't even understand the basic definitions of the terms he is using. However On that I am not surprised, its very common to see posters believe they can rewrite physics without even having a basic understanding of physics with no rigor to applying even high school definitions taught in high school physics. lets start with provide a mathematical definition for each term Swansont has mentioned as the OP is not using them in any recognizable standard (not that a pile of zeroth order formulas could ever have any hope of rewriting physics) there isn't a single vector in the above, all values above are apparently scalars.) I wouldn't even call them first order approximations until I see more rigor.

Sorry forgot to answer your first question. The FRW metric uses normalized units [math]c=g=\hbar=1 [/math] unity under math means 1 in normalized units and De Moivre numbers as per http://mathworld.wolfram.com/Unity.html