Mordred

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.

47 gly is the conformal time of the universe today correct ? As that's what you have on you opening post. That scale factor you used is the universe today. Did you apply the conformal time to age integral ? At BB n=0 Dt=a(n)dn T(n)=int_0^ n a(\prime{n})d\prime{n}

Are you sure about that ? Better supply a reference of where I am in error. Lookback time The lookback time tL to an object is the difference between the age to of the Universe now (at observation) and the age te of the Universe at the time the photons were emitted (according to the object). It is used to predict properties of high-redshift objects with evolutionary models, such as passive stellar evolution for galaxies. Recall that E(z) is the time derivative of the logarithm of the scale factor a(t); the scale factor is proportional to (1 + z), so the product (1 +z) E(z) is proportional to the derivative of z with respect to the lookback time https://ned.ipac.caltech.edu/level5/Hogg/paper.pdf

Glad to see your studying, a couple of points on conformal distance vs proper distance the latter being the actual physical distance while the conformal distance is rescaled, with the addition of the scale factor. Through this rescaling this simplifies redshift relations, angular diameter distance etc to a fundamental observer. Just a side note the calc in my signature for example initially uses conformal distance then converts to proper distance. There is a link for the tutorial on how to use it including which formulas are used This article may help understand how it simplifies some key distance relations. Though careful one detail this uses a rescaled proper distance in essence the equivalent of a commoving distance as opposed to the actual physical (proper distance). It specifies that detail and discusses it https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf

Yw

Not really, once you delve deep enough you start to learn how useful conformal time is with regards to measurements via luminosity distance, angular diameter distance or redshift.

Commoving time and conformal time including the formula for age of the Universe is all forms of coordinate time. When you apply a scale factor to a metric in the case of the FLRW you are specifying you are a commoving observer and the radius for that scale factor is also a form of coordinate. So the change in radius is a commoving event.. The proper time being along the worldline or null geodesic between emitter and receiver. The age of the Universe ie for example 10^-43 sec after the BB occurs prior to the formation of the CMB. For that matter it would extend beyond the Cosmic neutrino background assuming we can ever eventually measure it. The time periods prior to nucleosynthesis is needed as it provides a timeline for inflation and electroweak symmetry breaking both which occur prior to nucleosynthesis which forms the CMB. Now consider the following argument as to why conformal time is preferred ? Take a redshift value you can establish a distance as well as time the signal is emitted but that depends on coordinates between observer and emitter and not some clock following the null geodesic. As to an observer at CMB ? Well see above I already explained that the calculation for the age of the Universe is not the proper time age. The wiki link I posted yesterday specifically stated it's conformal time. The Peebles article further highlighted that detail. Look back time is the formula used for age of universe and it accounts for expansion which entails commoving coordinates

https://en.wikipedia.org/wiki/Proper_time Search Proper timeArticle Talk Language Download PDF Watch Edit In relativity, proper time (from the Latin proprius, meaning own) along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time Now given the above definition is commoving coordinate independent of its geometry ? Or is that now a coordinate time ? Any location you choose for a reference point for the emitter or observer is a coordinate dependant event location. The proper time follows the ds^2 line element aka the world line or null geodesic Here is the FLRW metric Chrisoffel not that it's needed now given the definition above of proper time.

Great where is proper time in line element ?

Good glad you recognize time derivatives now take GR line element and distinguish between proper time and coordinate time under GR...where does proper time under GR reside? As opposed to coordinate time. One is invariant to all observers the other is not ....

Are you intentionally being obtuse? What is the distinction between conformal time and proper time when it comes to distance measures. Do I need to repeat the distinction between the two a 5th time ? No I do not have a problem with superluminal recessive velocity I know the professionally accept corrections for that and I posted the professionally accepted corrections. You however dont want to grasp why proper distance and commoving distance are distinct when it comes to geometry and refuse to acknowledge that conformal time is specific to commoving coordinates not proper distance for proper time. Even though I've supplied literature specifically showing that distinction. You are aware I hope the Christoffels symbols I mentioned are used to derive the Ricci tensor. Including the one in that wiki link.... Why not write out the ds^2 line element for Cartesian coordinates then compare that to ds^2 line element of the FLRW metric after all we may as well look at those Christoffel symbols in greater detail so you can understand the Ricci tensor solution thst the wiki link posted. Are you familiar with the overdot notation of that Ricci tensor for example which overdose dot describes the acceleration of the scale factor and which overdot describes the velocity ? I ask to make sure you understand that notation

Yes exactly you have to derive how to fit the scale factor into GR field equations finally your getting it. Now every thing I stated is covered by the Lineweaver Davies dissertation. I suggest you read it including where it discusses superluminal recessive velocity. Literally every single statement I mentioned this thread is covered under that dissertation papers. Did you not understand why I stated the FLRW metric is a special class of solution of GR which is what your link is highlighting

Fine see chapter 4.2 https://people.smp.uq.edu.au/TamaraDavis/papers/thesis_complete.pdf Go ahead show me one GR textbook or article that shows the metric tensor with the inclusion of the scale factor as per Einstein field equation. Feel free to post that reference. If You like I can provide you thr FLRW metric Christoffel as well as the Minkowskii Christoffel and from that you can readily see the difference What are your 4 dimension that make up the metric tensor does it include a scale factor go ahead post me a reference showing that the scale factor is included in the metric tensor Alternately show where the scale factor is included in GR's four momentum

Sigh perhaps you should read what I wrote the FLRW metric us a special class of solution that uses GR however the field equations themself does not include a scale factor. Go ahead look it up

No your obviously ignoring what I've shown you. Tell me does the detail that there is no scale factor in SR or GR elude you ? Both metrics uses proper time