Mordred

Quite correct. Though I am going to add some detail for the posters clarity. Lets start with time is \[10^{-43}\] seconds. The volume of the Observable universe is so miniscule that curvature loses any meaning. At the extreme hot temperatures at this time roughly \[10^{19} Gev \] all particle species will be at thermal equilibrium. As no particle species are distinguishable from one another. One can describe the this state via its temperature. As photons mediate temperature (temperature being part of the EM field) the the effective degrees of freedom is 2.

There is a particular section in the Lineweaver and Davies paper that is related to this discussion "These velocities are with respect to the comoving observer who observes the receding object to have redshift, z. The GR description is written explicitly as a function of time because when we observe an object with redshift, z, we must specify the epoch at which we wish to calculate its recession velocity. For example, setting t = t0 yields the recession velocity today of the object that emitted the observed photons at tem. Setting t = tem yields the recession velocity at the time the photons were emitted (see Eqs. A.3 & A.10). The changing recession velocity of a comoving object is reflected in the changing slope of its worldline in the top panel of Fig. 1.1. There is no such time dependence in the SR relation. Despite the fact that special relativity incorrectly describes cosmological redshifts it has been used for decades to convert cosmological redshifts into velocity because the special relativistic Doppler shift formula (Eq. 2.2), shares the same low redshift approximation, v = cz, as Hubble’s Law (Fig. 2.1). It has only been in the last decade that routine observations have been deep enough that the distinction has become significant." The First equation I posted on this thread has the terms for the corrected and currently used equation from the originally used cosmological redshift equation given below. The equation below that you see in everyday links, textbooks etc for cosmological redshift quickly becomes increasingly inaccurate at greater distances. \[1+Z=\frac{\lambda}{\lambda_o} or 1+Z=\frac{\lambda-\lambda_o}{\lambda_o}\] further details can be found here https://arxiv.org/pdf/astro-ph/9905116.pdf

Well you have quite a bit to go through, but one side note. As an accredited Cosmologist your really going about trying to learn cosmology in a rather haphazard manner. The reason I state haphazard is that using random google searches and subsequent articles will really twist you around a great deal of convoluted paths. My biggest suggestion is to pick up a couple of cosmology textbooks. For example the zero energy universe, while it is the most popular choice for a universe from Nothing model. Has the inherent problem in that it will only would in Euclidean spacetime. Although your statement of the universe arising from quantum fluctuations is commonly accepted. The problem with negative energy and negative mass is that those terms and applications are negative to a (and this is a very important point) NON ZERO baseline. True Negative energy and negative mass however isn't viable under GR. Though antiparticles were once thought of as one possibility. An antiparticle has the same mass density as its positive partner. We will start there for now edit I should add I can easily counter a contracting universe by simply pointing out its effect on the Blackbody temperature history of the CMB. If the universe was contracting the Blackbody temperature would be increasing and not decreasing in accordance with the ideal gas laws.

Your sort of on the right track. In cosmology we have to make certain adjustments due to expansion as well as relativity. So we have a couple of different distances. We have the commoving distance as well as the proper distance. The proper distance is the invariant distance (same for all observers). The wiki coverage isn't bad, not nearly as good as a textbook but it will do in this case. https://en.wikipedia.org/wiki/Comoving_and_proper_distances The Lineweaver and Davies article in the link below has a handy graph of both commoving distance and proper distance. page 8 https://arxiv.org/pdf/astro-ph/0402278v1.pdf below is a copy from the calculator in my signature. All calculations are in Proper distance. The row 0.000 is time now rows after that are in the future while rows previous to that is in our past up till the CMB. I can set the calculator to examine prior to that but for this purpose its unnecessary. Included is the distance to the particle horizon aka the Cosmological event horizon \[{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&Scale (a)&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&H(t) \\ \hline 1.09e+3&9.17e-4&1.09e+3&3.72e-4&6.27e-4&4.53e+1&4.16e-2&5.67e-2&8.52e-4&1.55e+6\\ \hline 7.39e+2&1.35e-3&7.40e+2&7.10e-4&1.17e-3&4.50e+1&6.09e-2&8.33e-2&1.66e-3&8.32e+5\\ \hline 5.01e+2&1.99e-3&5.02e+2&1.34e-3&2.16e-3&4.47e+1&8.91e-2&1.22e-1&3.20e-3&4.51e+5\\ \hline 3.39e+2&2.94e-3&3.40e+2&2.49e-3&3.95e-3&4.42e+1&1.30e-1&1.79e-1&6.11e-3&2.46e+5\\ \hline 2.30e+2&4.34e-3&2.31e+2&4.59e-3&7.18e-3&4.36e+1&1.89e-1&2.61e-1&1.15e-2&1.36e+5\\ \hline 1.55e+2&6.40e-3&1.56e+2&8.40e-3&1.30e-2&4.29e+1&2.74e-1&3.80e-1&2.16e-2&7.49e+4\\ \hline 1.05e+2&9.44e-3&1.06e+2&1.53e-2&2.34e-2&4.20e+1&3.97e-1&5.53e-1&4.01e-2&4.15e+4\\ \hline 7.09e+1&1.39e-2&7.19e+1&2.77e-2&4.22e-2&4.10e+1&5.70e-1&8.00e-1&7.40e-2&2.31e+4\\ \hline 4.77e+1&2.05e-2&4.87e+1&5.00e-2&7.58e-2&3.97e+1&8.14e-1&1.15e+0&1.36e-1&1.28e+4\\ \hline 3.20e+1&3.03e-2&3.30e+1&9.01e-2&1.36e-1&3.81e+1&1.15e+0&1.65e+0&2.48e-1&7.15e+3\\ \hline 2.14e+1&4.47e-2&2.24e+1&1.62e-1&2.44e-1&3.61e+1&1.61e+0&2.35e+0&4.53e-1&3.98e+3\\ \hline 1.42e+1&6.59e-2&1.52e+1&2.91e-1&4.38e-1&3.38e+1&2.23e+0&3.31e+0&8.22e-1&2.22e+3\\ \hline 9.29e+0&9.71e-2&1.03e+1&5.22e-1&7.84e-1&3.09e+1&3.00e+0&4.61e+0&1.49e+0&1.24e+3\\ \hline 5.98e+0&1.43e-1&6.98e+0&9.35e-1&1.40e+0&2.75e+1&3.94e+0&6.31e+0&2.69e+0&6.94e+2\\ \hline 3.73e+0&2.11e-1&4.73e+0&1.67e+0&2.50e+0&2.33e+1&4.92e+0&8.42e+0&4.86e+0&3.90e+2\\ \hline 2.21e+0&3.12e-1&3.21e+0&2.98e+0&4.37e+0&1.83e+1&5.69e+0&1.09e+1&8.73e+0&2.23e+2\\ \hline 1.18e+0&4.60e-1&2.18e+0&5.21e+0&7.34e+0&1.24e+1&5.71e+0&1.33e+1&1.56e+1&1.33e+2\\ \hline 4.75e-1&6.78e-1&1.47e+0&8.79e+0&1.11e+1&6.06e+0&4.11e+0&1.53e+1&2.73e+1&8.74e+1\\ \hline 0.00e+0&1.00e+0&1.00e+0&1.38e+1&1.44e+1&0.00e+0&0.00e+0&1.65e+1&4.63e+1&6.74e+1\\ \hline -3.19e-1&1.47e+0&6.81e-1&1.97e+1&1.63e+1&4.93e+0&7.23e+0&1.71e+1&7.51e+1&5.99e+1\\ \hline -5.36e-1&2.15e+0&4.64e-1&2.61e+1&1.70e+1&8.54e+0&1.84e+1&1.72e+1&1.18e+2&5.73e+1\\ \hline -6.84e-1&3.16e+0&3.16e-1&3.27e+1&1.72e+1&1.11e+1&3.50e+1&1.73e+1&1.81e+2&5.64e+1\\ \hline -7.85e-1&4.64e+0&2.15e-1&3.94e+1&1.73e+1&1.28e+1&5.95e+1&1.73e+1&2.74e+2&5.62e+1\\ \hline -8.53e-1&6.81e+0&1.47e-1&4.60e+1&1.74e+1&1.40e+1&9.55e+1&1.74e+1&4.11e+2&5.61e+1\\ \hline -9.00e-1&1.00e+1&1.00e-1&5.27e+1&1.74e+1&1.48e+1&1.48e+2&1.74e+1&6.11e+2&5.60e+1\\ \hline -9.32e-1&1.47e+1&6.81e-2&5.93e+1&1.74e+1&1.54e+1&2.26e+2&1.74e+1&9.05e+2&5.60e+1\\ \hline -9.54e-1&2.15e+1&4.64e-2&6.60e+1&1.74e+1&1.58e+1&3.39e+2&1.74e+1&1.34e+3&5.60e+1\\ \hline -9.68e-1&3.16e+1&3.16e-2&7.27e+1&1.74e+1&1.60e+1&5.06e+2&1.74e+1&1.97e+3&5.60e+1\\ \hline -9.78e-1&4.64e+1&2.15e-2&7.93e+1&1.74e+1&1.62e+1&7.51e+2&1.74e+1&2.90e+3&5.60e+1\\ \hline -9.85e-1&6.81e+1&1.47e-2&8.60e+1&1.74e+1&1.63e+1&1.11e+3&1.74e+1&4.26e+3&5.60e+1\\ \hline -9.90e-1&1.00e+2&1.00e-2&9.27e+1&1.74e+1&1.64e+1&1.64e+3&1.74e+1&6.27e+3&5.60e+1\\ \hline \end{array}}\] if your reading the correct line you will see at time now. The distance to the cosmological even horizon from our location is 46.3 Gly.

It isn't any one individual but rather the scientific community. Lets do a bit of history. The FLRW metric allows for positive, negative and flat spacetimes. \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] in the above equation k is the curvature term. Historically even after Hubble discovered the universe was expanding we still could not confirm whether or universe had curved or flat spacetime on the global mass distribution. This was a point of contention all the way up to the early 90's. We did not know the universe geometry in this aspect. However we knew that curved spacetime causes visual distortions. To understand this lets examine how those distortions would occur. If you take two parallel laser beams as the beams travel through spacetime you can have 3 results. If the beams remain parallel then the spacetime the beams travelled through is flat and have a Euclidean geometry with no overall time dilation effect. In this case you would have no distortions. However If the beams converge then you have a positive curvature and you would get distortions. The same would occur if you have negative curvature. To easily understand how this curvature affects images all one has to do is look through a convex or concave lens. Clear examples are gravitational lensing and the Einstein Ring around massive objects. How does this knowledge help up. Well we have a very handy stellar region we can use. The CMB (Cosmic microwave background). This was one of the primary goals of the COBE satellite was to use the CMB background to search for those distortions and help determine the geometry of the universe. Unfortunately the COBE images were too blurred due to not being sensitive enough to make a conclusive determination. However the WMAP images that came later were much clearer and showed no distortions caused by curved spacetime. Later on Planck also confirmed these findings. We still make use of how spacetime curvature affects images, redshift, luminosity etc to this very day. A great deal of the furthest distance that Hubble viewed was through usage of intervening gravitational lenses. In point of detail Hubble couldn't get many of its images without using gravitational lenses. We also use spacetime curvature to look for under dense and over dense regions using the Sache Wolfe effect. (this is an application of redshift). Now I want to consider one further detail. You cannot have curvature if the mass distribution is uniform. You can only get curvature by having regions with higher or lower mass densities. The reason I mention this is this is another examination. There is a relation between the luminosity and mass https://en.wikipedia.org/wiki/Mass–luminosity_relation with this tool we can further look for the mass distribution as a further confirmation. further details of universe geometry can be found here http://cosmology101.wikidot.com/universe-geometry page two further details on what effect curvature has on angles Pythagorus theorem is only accurate in flat spacetime and requires corrections in curved spacetime. http://cosmology101.wikidot.com/geometry-flrw-metric/ that is one of the effects of length contraction you mentioned above. So once again we can look for this effect.

For reference an extremely handy Feymann rules listing https://porthos.tecnico.ulisboa.pt/CTQFT/files/SM-FeynmanRules.pdf

Your idea of considering the effects of relativity in regards to expansion related measurements is something that has already been examined so take heart in that. edit I should add that one of the biggest pieces of evidence of our expanding history is its metallicity history. Factors such as the density of hydrogen, lithium etc in our evolution history. The Saha equations in the nucleosynthesis link apply there. As well as the Bose_Einstein and Fermi-Dirac statistics.

All observations are calibrated to include any relativistic effects. The evidence of expansion goes well beyond simply relativistic effects. They also go beyond those involved in cosmological redshift. These methods include those such as interstellar parallax, the various methods are collectively called the cosmological distance ladder. https://en.wikipedia.org/wiki/Cosmic_distance_ladder Physicist can never rely on any single methodology in any given observation or experiment. In order to become Robust any theory must match any number of experimental and observational evidence. As far as the Hubble contention, there is ongoing evidence supporting that our local region is under-dense which has ramifications with regards to the near and far Hubble rates. This is something not mentioned in pop medial coverage of the JW telescope findings here is the related paper https://arxiv.org/abs/1907.12402 here is a later counter paper https://arxiv.org/abs/2110.04226 I post these to show other ongoing research beyond what you see in pop media. As shown there are other possibilities for the contention that go beyond claims of LCDM being incorrect or the BB itself. One thing most people also are not aware of is that the Hubble constant evolves over time. We call it a constant strictly in the historical sense. It isn't constant over time but merely constant everywhere at a given time. It should really be treated as simply a parameter. The Hubble parameter itself is in actuality decreasing in time in our Universe evolution history. the formula as a function of redshift is given by \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] this formula accounts for how matter density, radiation density and the Cosmological constant evolves overtime and its subsequent effects on our universe expansion rates. To understand how this formula is derived you would also require the equations of state for each which are a direct result of thermodynamics under the ideal gas laws. https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) these equations of state are further applied to the FLRW deceleration oft call acceleration equation. https://en.wikipedia.org/wiki/Friedmann_equations included in last link. That link also ties into the first link. You will note that relativity is inclusive the GR EFE equation used is in the Newtonian limit. Here is the route to the equation. FLRW Metric equations \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as \[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\] \[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\] which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0. the related GR solution the the above will be the Newton approximation. \[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\] Thermodynamics Tds=DU+pDV Adiabatic and isentropic fluid (closed system) equation of state \[w=\frac{\rho}{p}\sim p=\omega\rho\] \[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\] as radiation equation of state is \[p_R=\rho_R/3\equiv \omega=1/3 \] radiation density in thermal equilibrium is therefore \[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \] \[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\] temperature scales inversely to the scale factor giving \[T=T_O(1+z)\] with the density evolution of radiation, matter and Lambda given as a function of z \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] This I already had posted on this site under a thread I currently have underway regarding nucleosynthesis in our expanding universe and an examination of the various processes as a result. https://www.scienceforums.net/topic/128332-early-universe-nucleosynthesis/ The only reason I posted this thread in Speculations is that I wish to maintain the Privilege of toy modelling. However every single formula in that thread are commonly used and are main concordance formulas.

Here is an example of how to examine Bohm Non locality with regards to Bells inequality. This is the sort of math one can apply inclusive to experimental results using said math. https://arxiv.org/pdf/1807.01568.pdf

then explain how your holonomic toroid applies and show that it is truly holonomic. While your at it you can further apply Bells inequality via the following. set a correlation function between Alice and Bobs measurement apparatus and how a measurement on Bobs apparatus does not depend on a measurement on Alice apparatus. You can use spin of electron so you should have a statement such as used in Bells inequality as \[P(A,B|a,b\lambda)=P(A,|a,\lambda|)P(B|b,\lambda)\] where\[\lambda\] is any possible hidden variable if hidden variables are involved the outcome would give \[E^(HV)(ab)=\int d\lambda f(\lambda)\bar{A}(a,\lambda)\bar{B}(b,\lambda)\]

I would suggest you examine the Bose Einstein and Fermi Dirac statistics and how it applies to blackbody temperatures in particular interest is that all effective degrees of freedom inherent in any particle species do contribute to that temperature and further allow those statistic to predict the number density of particle states at any given blackbody temperature. They are direct applications of the ideal gas laws and can work at any temperature extreme (at least until you hit a singularity state)

Look simple geometry and how you mathematically describe a triangle has little to do with your claims. You claim a Holonomic guiding wave which you call your travelling wave has an interaction that violates the maximum rate of information exchange. Referring to its hidden variable characteristic yet have not even told us what that variable is or how it would even apply to the state of a quantum system. You have event described any quantum system state. You cant even validate your travelling wave is holonomic as you claim. There is a very good reason Why Bohm himself couldn't do what you claim to do. That very reason is that the mathematics do not support your theory.

considering I hold degrees in Physics and actually understand and regularly use the mathematics of the theories you mentioned. I would suggest I understand those theories far better that you do. Hence why I have provided you the opportunity to prove me wrong. For example you cannot have a holonomic state that had any dependency on coordinates. Any state in curved spacetime would inherently have this dependency unless you can ensure translational and rotational invariance. Another related key detail is that one entangled particle does not cause a change of state in another entangled particle. Entanglement requires a correlation function, this is part of statistical mechanics. Positive, negative and no correlation has no need for any causation. a simple example is place an orange on one bag and an apple in another. Give a bag to two other people. You have a 50 % chance the bag held by Alice has an apple as well as a 50% chance its held by an orange. So the apple/orange probability state is in superposition. Once you open Alice bag that superposition state collapses as you have now determined the state and also automatically know the state in Bobs bag. If Alice had an orange then Bob can only have the apple. Particle entanglement is much the same way. In order for two particles to become entangled they must have first interacted in some fashion. An example of interaction is parametric down conversion used in entanglement experiments. Parametric down conversion further results in numerous conservation laws being applied. Conservation of energy/momentum, isospin, color, flavor, charge, lepton number etc. From these details and the experimental setup ie number and position of detectors. One then sets up a correlation function. So from this you can easily see that there is literally zero logic in trying to interpret Bohmian non locality, EPR, Copenhagen interpretations with regards to any experimental evidence and ignore the mathematics of those theories.

Don't pay to much attention to the pop media coverage of findings from the James Webb telescope for starters. No result from the James Webb telescope tells us that the universe isn't expanding. lets look at your scenario for a second, Expansion is roughly 70 km/Mpc/sec. That is extremely slow relative to the speed of light, especially considering that a single Mpc has roughly \[ 3.262*10^6\] light years. It is only at extreme distance that expansion becomes measurable, also it only occurs in regions that is not gravitationally bound. The evidence of an expanding universe occurs in a great deal of observational evidence. For example the CMB wouldn't even exist if expansion didn't occur. The temperature history which shows the universe cooling down as a direct result of expansion and how it applies to the ideal gas laws is another key piece of evidence. Cosmological redshift is another but certainly not the only piece of evidence

So mathematic were supplied by him yet even knowing that the mathematics were required in that scenario. You still can't see that you may require the mathematics for your theory ? In order to be a theory mathematics are required its a simple fact.

Under GR every observer and event is inertial. There is no at rest frame. All frames of reference are also equally correct, there is no preferred frame of reference. Yes everything we see or measure is relative to the observer. However that was even true prior to GR/SR under Galilean relativity. That in and of itself is nothing new and has been understood for centuries. They key difference is time is absolute Under Galilean relativity.

pray tell how do you program a traveling wave without applying some form of mathematics ? I've done enough programming to know unlikelyhood of programming a travelling wave without some mathematics. Considering I've written programs in well over 30 different languages over the years. Yet another claim, here is the think Unless you can make testable predictions which requires mathematics. Any claim is worthless. That's the simple reality in Physics the very fundamental purpose is to be able to make testable predictions of cause and effect. That's precisely why The Pilot wave, Including its guiding wave has relevant formulas where the interpretations can potentially be testable. Aka EPR, vs Copenhagen vs Pilot wave etc tests constantly being conducted with each of the aforementioned having its own variations of mathematical treatments. The mathematics are readily available for each of the mentioned theories. Any interpretation of any given theory should always involve the mathematics of that theory otherwise what's the point.

This is oft something poorly understood. Geodesic paths are never truly straight nor smoothly curved. You learn to understand this via the Principle of least action which further applies calculus of variations. Take for example a thrown ball. At each infinitesimal of its flight, there will be variations in its direction. Those variations will result from the principle of least action. The geodesic path is the extremum of those variations. A common misunderstanding also includes trying to think in terms of 3d space which under graph vs a 4d spacetime graph can have rather surprising results. For example that thrown ball lets say you throw it a height of 20 meters and a distance of 10 meters on a 3d graph it would be a curved arced path. However lets say it took 1 second to travel there and 1 second back. Under a Minkowskii spacetime diagram using the interval (ct) you will find that the path is incredibly straight. Now further apply Interval (ct) to your different observers under the Minkowskii spacetime diagrams

A geometry requires a metric. I have yet to see any relevant metric. Please describe your theory under the mathematical holonomic restraints. this is a holonomic test equation see link https://en.wikipedia.org/wiki/Holonomic_constraints as the link shows the term holonomic has precise mathematical implications, including the time dimension. perhaps you can demonstrate your understanding of Bohmian nonlocality including holonomy under your metric. keep in mind Bohm was an excellent mathematician, he would also describe nonlocality under mathematical definition. Good example of how he goes about this is his pilot wave theory. So at least you will have a source of the relevant equations.

Probably easiest to simply examine what each means under the metric. One of the better ways has been how Barbera Ryden teaches each in her Introductory to cosmology. A key feature I haven't mentioned is how Pythagorus theorem applies in each case. Without trying to go too far astray one of my webpages uses her method and the paths descriptive. http://cosmology101.wikidot.com/universe-geometry this page just describes the critical density relations the next page has the needed images http://cosmology101.wikidot.com/geometry-flrw-metric/ note the following section Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices α β,γ obey the relation α+β+γ=π−AR2. a key detail to recall or understand is a negative curvature is a hyperboloid example the saddle image in the second link

That example doesn't particularly apply. The reason being that two falling objects as they approach either CoM of either BH would converge. They would also converge as they approached the barycenter. The most commonly known example is Anti-Desitter spacetime. This is used in the FLRW metric as one of the viable solutions historically it applied to an open as opposed to a closed universe. Which is another topic lol as explaining the two can get tricky. https://en.wikipedia.org/wiki/Anti-de_Sitter_space

While analogies can be useful, I've always found they tend to mislead. In this instance I have always found the method of describing spacetime curvature using geodesics paths of two parallel light beams far more useful. If the two beams remain parallel then you have flat spacetime. If the beams converge the positive curvature. If they diverge then you have negative curvature. It's a simple descriptive provided you include the details that mass is resistance to inertia change as well as ensuring that the reader further understands that the (ct) interval is what provides time with dimensionality of length.

Bump going to continue with this project

Been a bit busy with work will look through this when I can give it the proper attention. At first glance it's not bad but I will have to look at it closer

Both Ajb and I regularly discussed lie algebra. It's a very useful tool to understand physics in particular the standard model. However it's used in every major physics theory in general. I'm a little tied up atm but I will add more detail later on with regards to isospin and hypercharge.