Mordred

Pilot wave is considered more an interpretation much like the Copenhagen interpretation under QM. Its premise is more deterministic than probabilistic however the Copenhagen is what is considered more in alignment with QM. There were numerous issues with pilot wave in so far as entanglement and hidden variables as one of the reasons as to why the Copenhagen interpretation became more accepted. Personally I dont see any means where it would tighten constraints that are not already accomplished by statistical weighted average for most likely position of a particle. Perhaps looking at how each interpretation would work with Dirac Delta functions might provide some insight.

You dont require billions of years to toy model cyclic universe models. There are numerous models avaliable that have accomplished this. Cyclic universe models is nothing new to physics. Alternately there are numerous bounce cosmology models. However toy modelling requires mathematics and for this application applicable geometric treatments. I would suggest application of the Raychaudhuri equations would be incredibly useful. ( though that methodology has already been done)

Lets start with the correct definition of mass. Mass is the resistance of inertia change or acceleration for short. It is a property of a system or state/particle. It isn't something that exists on its own. Energy is also a property,, Energy being the property of a system, state, classical objects, fields, particles etc describing the ability to perform work. Once again Energy does not exist on its own. It may surprise you but e=mc^2 is just the invariant (rest mass) it is not the full equation. The full equation is the energy momentum relation \[E^2=(pc)^2+(m_o c^2)^2\] https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation

Agreed it's not perfectly flat with regards to the Rheimann tensor as to the earlier statement I made "where (tS ) is the scale factor and k is a constant which denotes the spatial curvature of the three-space and could be normalized to the values +1, 0, –1. When k = 0 the three-space is flat and the model is called Einstein de-Sitter static model, when k = +1 and k = –1 the three-space are of positive and negative constant curvature; these incorporate the closed and open Friedmann models respectively." note the statement 3 space which also agrees with your earlier statement. https://mpra.ub.uni-muenchen.de/52402/1/MPRA_paper_52402.pdf Obviously we're both aware a static solution is considered impossible. The other use for the critical density formula as shown above ties into the fate of the universe.

I found after some digging a useful article showing some of the corrections mentioned earlier for higher redshift distances https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf One thing I should mention often textbooks etc on a topic gives you the first order formulas for various things like redshift, luminosity distance, angular diameter distance etc. You rarely find the more advanced formulas in common literature. Those formulas tend to be something that the instructor will have you derive yourself. The above lecture lesson is an example We have to be careful here K is a specific relationship with the critical density formula. When K=0 precisely then the energy mass density equals the critical density. If K=1 then this describes a closed universe where the energy density is greater than the critical density. If K=-1 then the mass energy density is less than critical density.( open universe). Though open closed universes are an older application of the critical density formula. With regards to inflation one of the problems inflation addresses is the fatness problem related to the above. K value remains unchanging throughout the universes expansionary history. For example if in the first case k is precisely zero the universe is static neither expanding nor contracting.

Yes there is superluminal expansion during inflation however the period during such time would be more problematic as the mean free path of photons would be too short to receive signals from emitter to observer aka the dark ages. Mean free path time estimate 10^{-32} seconds. So you wouldn't be able to recieve signals between two inertial frames of reference Good point

yes changes to the scale factor evolution is non linear over the Universes history this leads to numerous adjustments that must be made to linear relations (first order formulas) where second order relations must be incorperated example of second order being acceleration example accelerating expansion. various measurements as a result of the above non linear expansion rates that require corrections is angular diameter size, angular diameter distance, look back time (ie age of universe at a given distance usually as a function of redshift), redshift corrections, luminosity distance corrections. the above corrections must apply the equations of state for matter, radiation, Lambda and any applicable curvature term. for example if one tried to take the time dilation formula under SR once you get to recessive velocity greater than c then that equation will give wrong answers. I know your not strong in the mathematics but If your interested in the corrections for higher recessive velocities they are here corrections to look back time I would have to dig up the corrections for angular diameter distance , luminosity distance etc but the previous two examples show how the equations of state are involved. There are also other factors such as the relation between angular size and angular distance. One counter intuitive example is that above redshift 1.5 Z approximately the angular size increases rather that decreases.

Lmao @StringJunky beat me to the punchline

No worries one detail when dealing with probability distributions or multi measurements over an ensemble of measurements. The area of the distribution ie highest distribution is what becomes relevant. For example if you take 100 samples and 20 of those samples are in close proximity to one another while the rest are scattered in without a discernible pattern. The area of those 20 samples is your higher probability region Here is a simple example of gaussian distribution. https://introcs.cs.princeton.edu/python/appendix_gaussian/

You run into ppl like that. Its one of the reasons I try to supply reference papers for statements I make. However some ppl fail to even look at those reference papers or fail to understand them. However I always consider adding them useful for other readers of the thread as well. Glad to hear you learned something from that thread. You asked earlier on this normal distribution. As it is a probability density function you won't have a negative curve. All probability functions regardless of type are positive norm. However I should note some terminology is a little loose. For example the Dirac Delta function used to describe point mass isn't a true function but a measurement distribution. As such it's handled a little differently via Lebesque integration. Example here https://arxiv.org/pdf/2508.11639 Edit forgot to note a simple function has a finite range this isn't the case with Dirac Delta functions

Not quite accurate any solution to the Dirac equation is a solution of the Klein Gordon equation. It is treated as a foundation equation of QFT. Though today it's main use is bosons such as Higgs.(scalar) Once you involve spin (under Dirac use of spinors) then the Dirac equations are used. For other readers

I Agree excellent video one of the better ones Ive seen. I love how he went from classical wave theory, included SR to QM and then QFT in a very well laid out format. Lol literally covered several chapters of most textbooks in a short video. One added detail however the Schrodinger equation isn't lorentz invariant it doesn't work well with SR however the Klein Gordon equation used by QFT is. It does so by factoring in the mentioned energy momentum relation into its equations (in essence employs the 4 momentum.) Its an important distinction between QM and QFT. Some of you may have heard me mention the term canonical ( this is a quantized field theory) a conformal theory however isn't quantized. ( string theory as one example). Just some side tid bits

Assuming all particles reach thermal equilibrium the entropy can be safely described strictly via temp. As the mediator for temp is the photon entropy will end up being S=2 same as the entropy at 10^-43 seconds. However the problem at the low temperature end is that only massless particles travel at c and all massive particles will likely remain massive so wouldn't be in thermal equilibrium such as our universe beginning. Too many variables with regards to how particles would remain coupled for the mass terms to give any good guess. Will the coupling constants operate the same is anyone's guess. According to QM zero point energy you will always have quantum fluctuations hence absolute zero is impossible via current understanding of QM. Then there is still BH evaporation times to consider lol which is far greater than the time frame I mentioned above for a one solar mass BH. One could consider we understand electroweak symmetry breaking processes at the hot end better than we understand thermal equilibrium states on the cold end.

Keep in mind heat death is only one possibility. One that relies on the cosmological constant remaining constant. Its still viable at some point that this may no longer hold true and the universe could start to collapse. The key equation being the critical density relation which was originally used to determine the inflection point from and expanding universe to a collapsing universe. Aka cyclic bounce models Using Planck 2018+BAO dataset values roughly 45 B years into the future the Hubble constant will hit 55.7 km/Mpc/sec. It will remain roughly this value up to universe age 93 B years old. Thats as far as the cosmological calc in my signature goes. At that time the CMB balckbody temp will be roughly 0.0273 Kelvin. It will never hit absolute zero but that temp is still too warm for Bose-Einstein and Fermi-Dirac condensates so you will still have particles not in thermal equilibrium as per the standard model today. That of course is under the assumption the cosmological constant remains constant.

Wilsonian renormalization group with regards to Higgs https://www.physics.mcgill.ca/~keshav/675/wilsonianaturalness.pdf https://arxiv.org/abs/2310.10004 https://scoap3-prod-backend.s3.cern.ch/media/files/84579/10.1103/PhysRevD.109.076008.pdf https://www.db-thueringen.de/servlets/MCRFileNodeServlet/dbt_derivate_00035352/Sondenheimer_PhD-thesis.pdf

I recall that video always enjoyed Guths lectures as well as articles. Static vs inertial in terms of different observers can often give surprising results. Guth does an excellent job demonstrating some of the effects in that video

Several of the answers above have provided excellent clues into gravity vs density but let's refine that with mass density. Lets do a couple thought experiments and for simplicity we will keep the total mass constant in each case. Lets set at 1 solar mass ( mass of our sun). Case 1) spread that mass out evenly everywhere where no coordinate has greater mass than any other coordinate. No matter which location you choose you can state it's the effective center of mass. Gravity in the above case is zero everywhere. It does not matter what density of mass each coordinate has it could be as dense as one can fathom. As long as the mass density is uniform everywhere Newtons Shell theorem applies. Case 2) you have one region with higher mass density than other regions (anistropic distribution) Now you have a clear cut center of mass as the center of that region is clearly a higher density than the surrounding regions. Now you have gravity where the difference follows Newtons laws of gravity. Now Case 3 is rather special take that one solar mass above and let's assume it has the same volume as our sun. The strength of gravity one measures depends on the radius from the center of the sun. If however you collapse the radius of the sun below its Schwartzchild radius it becomes a blackhole. However the mass does not change. The radius where you can measure gravity has decreased so at the event horizon the strength is such that nothing can escape. Yet the force of gravity is still the same if you were to measure gravity from Earth. Hope that helps remember at no point did of the 3 scenarios change the total mass. It is the distribution of mass that leads to gravity and the radius from the center of mass.

Then what was the problem when I stated I had no problem with using conformal age that you felt it necessary to flame me in the manner you did since ?

Thank you so when you asked if I had a problem with conformal time as the age of the Universe. Did you specify proper age ? Instead of conformal age ? Both are valid conformal age has the side benefict of specifying what treatment your applying. Ie conformal coordinates

So age is on the rhs of the equal sign on that equation correct ? So why do you think 47 Gyrs is the age on the left hand side ?

Sounds like you already applied an age ie time zero to 13.8 haven't you

One last point if you used a(t) over the entire expansion history how did you end up with a singular value of 47 Gly the scale factor varies over time.

Thank you for admitting you are a sickpuppet account

No that formula does not today scale factor a=1 there is a difference between conformal time now and conformal time then. This will be my last post this thread be well.

Do you not understand the difference of conformal time today as opposed to conformal time during expansion history ? The equation you had during opening post only applied scale factor today not the scale factor of earlier times from BB forward hence the integral. I lost count the number of times I posted lookback time with E(z) including the evolutionary history of matter and radiation as terms. The formula you used in the opening post ignored the expansion history and you had already that the expansion history must be taken into account. Just calculating today's moment of the expansion radius is not including the expansion history. In essence the formula you used in the opening post was the equivalent of start of signal from an emitter today sending a signal to observer today how long would the signal take to arrive with no further expansion. In other words you did apply any form of look back time. Which I posted references to numerous times.