Mordred

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

Fine dont wish to believe me I provided 2 links clearly showing where your wrong both professionally peer reviewed. All you have to do is read them the first link clearly shows conformal time and proper time on 2 seperate graphs. Is Lineweaver and Davies wrong ? The second article is also peer reviewed and clearly states commoving distance as being used to calculate age of the Universe. So what about those objects prior to the CMB are they ageless as the CMB doesn't exist then ? But hey I guess every professional physicist is wrong simply because you dont agree with them. Lmao all you had to do was look at how the scale factor is derived which is required for your formula to realize the CMB itself is irrelevant when the scale factor uses the particle aka cosmological event horizon.

Evidently you have no clue how the metric actually works. Observer is us on Earth, the CMB surrounds Earth and exists everywhere in the Universe today. Its current blackbody temperature is 2.73 Kelvin. It did not exist at every moment in the past ie previous to 380 million years after BB. Resorting to try and insult me doesn't cut it. Particularly since I do have credentials in the field of Cosmology. However as you cannot counter my points I made with anything related to actual physics then its pointless for this thread to continue. For the third time conformal time is not proper time. I provided you with the proper time corrections as well as the reason why they are required.

Why ? What about a geometry without a CMB the metric is perfectly capable of accurate conformal time simply using observer ie the commoving observer now and emitter to any past object including those objects prior to the CMB. You dont require some special period in our universe history for the equations as is to work. Why would you want to restrict the flexibility it has and apply unnecessary limits ? You can literally take any object at a given redshift and use that as well as the expansion history with the equations above and get an approximate age of when the signal was sent. See the calculator in my signature( it has that very feature). After all the only two requirements is a geometry and a change in volume over a given time period. The rest of the formulas allows you to determine the volume at a given redshift to determine the scale factor. You dont require a CMB for that Thanks merry Xmas and happy new year to you as well

Its because if your following what Im describing instead of delivering straight answers you might think about it. It seems your trying to find preferred frame. You dont want that. Keep it emitter/ observer. As observer you already have a frame of reference which is already assigned by the usage of the scale factor to some other past moment provided by the redshift function. The CMB is only 1 possible past moment and even the surface of last scatterring spans a number of years. Which arbitrary point will you choose ? Its isn't some discrete point in time. I doubt you've worked with the Saha equations with regards to hydrogen dropping out of thermal equilibrium which traps the free electrons. For example the aforementioned 3000 kelvin mentioned this thread represents the temp where 75% of the hydrogen formation. At 4000 kelvin it's 50% etc. Its not some one point in time. After all the static on your radio is the noise from the CMB today and not the CMB at say Z=1050 or 1100 Which time during the CMBs presence will you choose? As it's still present today as well as 13+ billion years ago ? Keep in mind I could have stated on my first post that how the age of the Universe was determined is already using conformal time from my first post. However it was clear you were not aware of that nor aware of the distinction between conformal time vs proper time as it pertains to GR four momentum.

were discussing proper age vs cosmological age. Conformal time does not describe proper time and its age is in cosmological time as per its usage in the FLRW metric equations which uses the commoving observer on a commoving coordinate system. Proper time is coordinate independent. Conformal time is not, it relies on those previously mentioned criteria. That should answer your last question as I did mention commoving observer and conformal time uses commoving coordinates. Proper time however does not and that distinction is extremely important when it comes to how GR or SR applies to the FLRW metric.

if you were to apply the Lorentz transformations under SR once you exceed the Hubble Horizon then the recessive velocities given by Hubble's law will exceed c. At Z=1100 for example the recessive velocity is 3.2 c. To get the corrections you have to apply the evolution densities of matter, radiation and the cosmological constant to account for beyond the Hubble horizon the the cosmological event horizon or particle horizon. The Hubble horizon is z=1.46 here is the methodology for the corrections a couple of posts up in that thread. I also have the more accurate or modern used look back time corrections its compatible with Peeple's equation 14

No it's not do you want the proper time corrections beyond Hubble Horizon for when the recessive velocity exceeds c ? SR without those corrections will give you the wrong answer. Same as GR without accounting for those higher recessive velocities. Hint proper time uses proper distance not commoving distance. Cosmological time uses commoving distance to a commoving observer. Age of the Universe is determined by the latter not the former

cosmology like to allow for all 3 possibilities hence its versatility and you still missed that we already use conformal distance to calculate the age of the universe. equation 14 and 30.... hence cosmic time is not the same as proper time

only in a spatially flat universe not a curved. see above you missed my edit

when you realize that the radius of the universe given by Google is the proper distance and not the commoving distance which is required by conformal time which would not be 47 Gyrs edit sorry other way around the point you should be seeing is commoving time is distinct from proper time when it comes to SR and GR treatments. Conformal distance is a rescaling in that regard. Your opening post argued that proper distance is preferred over commoving distances but obviously you didn't look at those 2 formulas and determined which is being applied did you ?

Tell you what take equation 14 and apply it to equation 30 https://arxiv.org/pdf/astro-ph/9905116 this is your graph here https://ned.ipac.caltech.edu/level5/Hogg/Hogg10.html then look at the difference using proper distance vs commoving distance each has its use but look back time is how you determine the age of the universe see first link for equation and relations with an multicomponent universe like our own. That should answer why you need all components contributing to expansion >particularly when it comes to angular diameter distance etc.

edit forgot to add why not just use radiation also pertains to angular diameter distance. You need to account for all influences when it comes to distance measurements.

Radiation is not the driving force to expansion today that is the cosmological constant. Our universe underwent 3 distinct era... radiation dominant, matter dominant and today matter dominant. I hope you realize that conformal time specifies the use of commoving distance and not proper distance if not see the first 2 graphs here https://arxiv.org/abs/astro-ph/0310808

sigh obviously were going to use the evolutionary history of our universe over time which I already know your using for your conformal distance to time relations. ie the radius of the observable universe in this case. Which is preciseley my point to my previous statements. The temperature will vary with the scale factor. Your formula you provided specified that above. Now back to my question which you evidently don't want to answer. If you removed matter your evolutionary history of the rate the scale factor will change..... so why would you use that as a benchmark for age of the universe with a non linear curvature to the scale factors evolution through time given varying rates of expansion.

oh gee all physical processes such as temperature change can be related to age. Age is not strictly a factor of matter only. it is a duration of events. the evolutionary history of our universe can be segmented using age. Nuclear decay is simply one type of aging

do yourself a favor calculate the look back time for a radiation only universe. You will not get 47 Gyrs. Nor will you get 13.8 Gyrs I've given you a source material to see what I mean

you do understand that a radiation only universe would not expand at the same rate as a multi component universe do you not ? That would effect your scale factor \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{r}(1+z)^4+\Omega_{\Lambda}}\]

either way my question remains even if you apply a radiation only solution to the FLRW metric radiation still causes expansion. Barbera Ryden's introductory to cosmology will give you the look back times for single component universe. If you apply a single component universe your expansion rates will differ significatly and you would not get 47 Gy for your conformal time as matter contributes to expansion. The expansion contributors are matter, radiation lambda and curvature terms though for our close to flat universe curvature can be safely ignored. So the conformal time you have would be incorrect as you did not calculate the expansion rate for a universe with no matter.