Mordred

One can assume the unobservable portion near our region of shared causality (observable universe) will have the same laws of physics. Ie GR still applies (including the speed limit)

That is a common error. We can predict far more than we can accomplish through technology. We know the limitations of space travel. We also know the consequences of near c velocities. The near c or greater than c velocities are incredibly problematic when you study the physics and time aspects. Even the Alcubierre drive has consequences such as gamma ray production. ( It could literally radiate your destination)

Yes constant velocity for flat spacetimes. Curved spacetime is a curved geodesic and includes acceleration.

As to geodesic paths three good guidelines to follow. 1) The speed limit of all possible exchange of c. (This also applies to possible worldlines. 2) A field in GR amounts to a set of potential values at each event. 3) greater potential difference in the field potentials result in greater energy/momentum. ( A curved worldine is a worldine with acceleration) (has rapidity) which can be screw symmetric) a freefall worldline has constant velocity)(symmetric under Lorentz transforms) Most important mass is simply the resistance to inertia change or acceleration while energy is the ability to perform work. The classical definitions do not change. Of fundamental importance time is described by the interval between two events. Ie time for a signal to transverse between those two events.

The Principle of maximal aging as it is also called is reducible to Newtonian mechanics including the principle of least action. So both answers are valid. Edit keep in mind under GR freefalling objects ie constant velocity have no force acting upon it. Force of gravity being described by spacetime curvature. Ie an accelerating worldwide.

Markus beat me to it with his excellent explanation. Here is a pdf on the principle of extrenum aging. https://www.google.com/url?sa=t&source=web&rct=j&url=https://ocw.mit.edu/courses/physics/8-224-exploring-black-holes-general-relativity-astrophysics-spring-2003/assignments/gravitationalforcesnotes3a.pdf&ved=2ahUKEwicqrOjyuXpAhVTrZ4KHTXlCT0QFjAEegQIARAB&usg=AOvVaw0bgkXdM3RXpKmchukb5Doi

Well I'm not familiar enough with the Wilson loop methodology itself. Although I have studied it a bit I prefer the perturbation methodologies of QFT So other than seeking obvious mistakes I wouldn't be a great help.

Muffler for a tuba ? @🤣

Good points above. Black holes cannot drive expansion of the universe through Hawking radiation any more than stars can drive expansion through their much higher radiation emissions. The radiation emitted by stars and BH's are miniscule compared to the mass density of the universe. Secondly radiation falls off in density as you move further from the source. Lastly the cosmological constant whatever the cause has been around long before the first black holes even existed. Although miniscule in effect as the two prior eras (radiation dominant, matter dominant eras) overpowered DE. By this I will use the matter dominant era as an example. Radiation obviously existed however the main contributor to expansion during the era is matter. The cosmological constant was around as well. The same goes for the previous era (radiation dominant) the other two contributors still existed. Just that their contributions can be ignored. Now here is where I really muddy the waters. The Hubble parameter is decreasing but the rate of expansion via the recessive velocity formula is increasing Yet the cosmological constant stays constant in energy/Mass density.... At time Z=1080 The Hubble parameter is roughly 20,000 times greater than its value today

You are to me, Myself I concentrate more on the canonical treatments under QFT.

Well an obvious asymmetry is any treatment involving inner products as opposed to cross products. Differential geometry applies to any field treatment. Hence Kronecker delta and Levi Cevitta applications are essential along with the holomorphic (holonomy etc) connections. An obvious necessity in higher dimensional applications along with the limits of any applicable equation. (Highly common application phase polarities and other applicable wavefunction). This leads back to my first post on the observer orientation aspects. Which I need to clarify with the topology application as being defined as a fully coordinate independent treatment. (QFT for example has a coordinate dependence (strongly allied in the weak field appromation as per SR) though second order to QM first order treatments). Cross product fields require an velocity operator to comply to the right hand rule. Under group applications this is often the [math]\ mathbb{Z][/ math] this is also applicable to parity operators.

Einstein summation has its topological applications. The summation specifically involves the covariant and contravarient terms of each group. The superscript is being the covariant terms. The subscript contravarient while a mixed group will have both. The full Kronecker delta is an Ideal example to study. Granted the Levi Cevita adds additional degrees of freedom. PS I tend to think more gauge group than topological, while Studiot for example thinks the latter. ( I haven't seen enough of your posts but I am thinking your more the latter as well) @the OP I have zero problems with applying Wilson loops to the SM model In an entirely. It is a viable alternative. So I support the OP on thus methodology though myself I am more up to date on canonical treatments as per GFT. Doesn't invalidate other treatments. I fully support you in showing the Langrangians as per observable vs propriogator action particularly in terms of show to apply the Ops model to the QED and QCD applications. (The Higgs can be addressed later ).

Running latex from a phone is something I have grown used to. I'm assuming your following standard notation on the transformation matrix [math]A_i^k[/math] which is a mixed group of covariant/contravariance (for other readers, OP knows Einstein summation) Are you applying gamma matrix as per [math]\gamma^5[/math] By the way welcome aboard I'm happy to get a good thread for discussion of this caliber. (Happens rarely). +1 Notation clarification (not one I recognize of the superscript.) Is this a parallel vs perpendicular ) commonly indicated on the subscript. If not please clarify. [math]A^|[/math]

Latex notation that you find handy the symbol commonly used for tensor multiplication. \otimes [math]\otimes[/math] You likely already know that but handy for other readers. However the group relations you have above is precisely what you looking at for the parity and helicity operations. Example (Hand fasteness) between left and right handed particles via the right hand rule. This is an essential part of your complex topology treatments. From the above for hermitean groups which is part of all special unitary groups be sure to understand the Kronecker delta relations. It will also apply to hermitean and orthogonality. (That includes Hamilton's)

See here for Lorentz invariance for the EM field https://www.google.com/url?sa=t&source=web&rct=j&url=http://physics.usask.ca/~hirose/p812/notes/Ch10.pdf&ved=2ahUKEwjS_OOfgdzpAhUDPH0KHSa5BNMQFjAAegQIAxAB&usg=AOvVaw3UyJ5bohB22t0xuLsS48fV Note the Stress energy momentum terms for the E and B fields.

I suggest you study the Maxwell equations for the phase angles between the E and B fields then under Lorentz invariance.

The use of R^2 isn't particularly practical for your first equation. I would recommend using the full ds^2 line element and applying the full four momentum and four velocity. Also the EM field has symmetric and antisymmetric that involve the Lorenz transforms from the E and B fields of the Maxwell equations. So to state the EM field is symmetric with spacetime isn't accurate. For example To to define the right hand rule for EM fields in your equations. Spacetime doesn't require the Right hand rule so you obviously have vector components of the EM field that is not symmetric with the spacetime metric. This will also become important for different observers/ detectors at different orientations. A key point being many of the Maxwell equations employ the cross product for the angular momentum terms. However the Minkowskii metric employs the inner products of the vectors. This is an obvious asymmetry between the two fields. Not to mention the curl operators of the EM field. Ie Spinors. You will find some of this important for the Pauli exclusion principle as well.

On the [math]P(n]:Q(n],[/math][math]\mathcal{C}[/math] and [math]\mathbb{Z}_2[/math] groups. Here is an article covering the Super lie algebra covering the above groups. https://www.google.com/url?sa=t&source=web&rct=j&url=https://core.ac.uk/download/pdf/81957395.pdf&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjAAegQIBhAB&usg=AOvVaw3fK1HirnZXUKfwWAxrV938 Here is an arxiv coverage via "Dictionary of Lie Superalgra" https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f You will find the details on the second article incredibly useful in the P(n) ,Q(n) in particular the even odd operators. As well as a wide collection of the superalgrebra groups. The CP(n-1) of the paper you linked I would think is a tangent from [math]C^\infty M[/math] but not positive on that. I would recommend studying Clifford and Lie algebra as well. If you haven't already. Edit found how CP(n-1) as per direct limit ie the use For the infinity suffix. https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f Which is a complex projective space. https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9607161&ved=2ahUKEwjS297p1NrpAhUHv54KHUJUASQQFjABegQIAhAB&usg=AOvVaw0r1Jr_tcDdLlP9PqtUwe2f Hope that helps.

As it's been awhile since I last looked into MSSM based models The notes you put above serves as useful reminders. It's been awhile since I looked at Grassmann variables. So I am currently looking for some useful direction in literature for the OP. At the OP have you studied Pati Salam via SO(10) MSSM ? Many of the group's used will be applicable particularly SU(N) and Z_2. I should have time tonight to do some digging as well as self reminders.

Here is an Caltech lecture note on the Langrangian formalism of GR. https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec33.pdf&ved=2ahUKEwjQn_i52tnpAhW1MX0KHcR9BAwQFjAAegQIBBAB&usg=AOvVaw0FPL8yDcjn0EG4tH5W8eZ5 Now a better way to learn this is through the Einstein Hilbert action which is mentioned in the above link however here is the MIT lecture note. https://www.google.com/url?sa=t&source=web&rct=j&url=http://web.mit.edu/edbert/GR/gr5.pdf&ved=2ahUKEwiH7vDZ29npAhUWIDQIHda1D54QFjAAegQICBAB&usg=AOvVaw0WRK38p_2yCObL796dRDk5

Clarify this is the correct paper. https://arxiv.org/abs/hep-th/0006025 You forgot to link the paper. I am also assuming you are looking at [math]CP^{n-1}[/math]

Accurate post well described.

I'm not sure there is any other way we can explain the above to him. This is a specific class of solution. For a large non rotating spherical symmetric object.

Nevermind figured it out. I will stand corrected on that. Missed a - sign on one of the SI unit powers. (14 hour work day so stupid mistake)

Lol honestly ? It's not ok to ignore units in an equation. They are a fundamental part of any equation. For example time is not dimensionless which your LHS of your equation. Neither mass nor radius. So go ahead try to prove a cancellation of your equation to end up with units on the RHS to equal the unit of seconds on the LHS. Impress me with your math. It literally makes no sense to state time is a dimensionless unit. Time has SI units of seconds end of discussion. However all that aside the Schwartzchild metric is only one class of solutions. As others has told you. The most common class of solutions is the weak field approximations. Which applies around the average star or planet. The Schwartzchild only applies to a static non rotating black hole. If you have rotation you need the Kerr metric. However in this case you have four event horizons and nearly BH observed in nature is rotating. Quite frankly this is getting pointless.