Mordred

If you cannot get a specific value at a specific point on any graph does it matter ?

Already provided a dataset use that it incorporates the full FLRW metric with the relations I described previously and used the Lineweaver and Davies paper for its benchmark tests.

would you like me to prove you wrong under full blown GR treatment ? I can readily provide the Full GR treatment for each case if you like. of course that will barrage you with a large list of equations as I would also have to include the mathematical proof under 4 momentum along the null geodesics involved.

Ok I'm done with attitude in a post you have my answer it's your choice to understand the answer or ignore it. The lineweaver paper your comparing to uses factors you choose to ignore. I provided the relations of those factors to the redshift relation Z. You cannot inverse the Doppler shift relation including the relativistic Doppler and expect to get the same results as the Cosmological redshift because the Cosmological redshift relation involves factor not included in relativistic Doppler the very terms you choose to ignore. I'm tired of repeating myself as well Ever stop to wonder why your receiving negative rep points ? For the record the Moderator you just talked back to also has a PhD in physics

Still waiting for it but I already identified where your error is and you just admitted you did not account for the evolution of the scale factor. You didn't build the first top graph I already knew that however I know how that graph was derived and the originator used the details I mentioned but just on the GR line though as per the comment on the paper provided (Lineweaver and Davies) The equations specifically describing the difference between SR and what they termed the GR line is described at the beginning of the article To quote "conversion from cosmological redshift to recession velocity is different from the conversion from an SR Doppler shift to peculiar velocity"

Great mathematically include it in your graphs by simply knowing it but not knowing how to incorporate it into your the graphs. Not going to work you must know how to include those variations mathematically

How will that integral work if you do not include how the scale factor and the goemetry changes over time? It won't period in order for the integral to work you must include and account for all time dependent variations. That has been precisely what I have been trying to get through to you but you keep missing that very point.

Would you prefer I provided simply provide the end result equations without any other detail relating to their mathematical proof ? It would do you absolutely no good if I simply posted say the last equation using the Natural logarithmic functions without describing mathematically how it works vis the equations of state would it ? It would do no good to simply provide the equations of state and not include the relations to the critical density formula. Nor would it do any good to simply state that the scale factor, Hubble parameter varies over time and are not constant so any attempt to use the value today to calculate z=1100 will get the wrong answer. Hence I provided the equations showing why and how that occurs instead of just stating such and expecting you to automatically take my word for it. For the reasons I provided the Hogg paper as it's directly related for precisely the same reasons. Hence I am being thorough and providing the necessary equations and relations to understand the key distinctions between the Minkowskii metric and the FLRW metric. (That distinction directly relates to the influence upon the FLRW metric due to the acceleration equation over the time period from now to the CMB. ) The cross over for the matter dominant era to the Lambda dominant era occurs after the CMB so there is also a shift in expansions rates due to that crossover.

Why are you mixing plots between Minkowskii to the FLRW metric to begin with ? That is simply wrong Perhaps you should look directly at why the the two are distinctive from one another If you like I will post the Christoffel symbols for each perhaps that will help

I described the details needed to properly use the FLRW metric. If you don't take into account the differences in the equations of state and the different rates they evolve as expansion occurs you will get the wrong answers. You also will also get the wrong answers if you mix observer types and different metrics or ignore the distinctions between Minkoskii Euclidean and the FLRW metric

As mentioned numerous tines those details need to be included here for discussion. I was being polite by even looking at your link I don't expect others to do so nor should I have to jump links to keep track of a conversation.

WHT don't you describe what that inconsistency is. As the inconsistency is your claim and not mine. Secondly I never down vote that's another reader of the thread.

I'm not the one down voting. I rarely if ever do so

The first three equations when you solve them They are power law that I showed further solved to provide the natural logarithmic functions. \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] spacetime doesn't store anything how it evolves is the thermodynamic processes given by the equations of state. I would never ever describe spacetime as storing anything as it is a Metric its simply the arena where the standard model of particles reside. I certainly hope you don't consider spacetime as some fabric like entity and recognize that spacetime fabric is just an expression used to explain curvature to laymen. Where we choose to keep track of numbers and values and relations is our problem not the universes. It is the distribution and their momentum terms at any given point in time that determines what occurs and when. though if you noticed its oft easier to use the natural log functions with regards to the scale factor which I also provided

Where did you show that ? you need to include here if you recall. secondly we cross posted the calculator in in signature puts it all together (see above and notice the Stretch term S=1+z compared to Z. Thirdly if you included the acceleration equations why did you ask where I had described it in yesterdays post when 90 percent of that post directly applies the acceleration equation relations ? where is that included in your first post link ? ! Moderator Note you do need to bring them here as its a forum requirement that you agreed upon when you agreed to our forum policies edit forgot to add I set the calculator above on our cross post using the Planck Dataset+BAO 2018 and further set to GyR /Gly

Thats provided above under the thermodynamic sections by myself using the equations of state I also included the acceleration equation above. \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] it can also be described as \[\frac{\ddot{a}}{a}=-4\pi G(\acute{\rho}+3 \acute{p})\] as per wiki on the equations of state. The double overdot is the second order time derivative describing the acceleration term of the scale factor ( the equations you used in your graph did not involve acceleration) they used the first order relations of velocity. In the previous equation the single overdot is the time derivative for the velocity term of the scale factor. If you look again at the equations of state in the Lineweaver paper the single overdot used in the equations on the right hand side correspond by the same time derivatives. Everything I posted above in yesterdays post directly applies these equations of state... I simply showed how they were involved https://en.wikipedia.org/wiki/Equation_of_state_(cosmology) using those relations which the calculator in my signature includes from z=1100 to present \[{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&Scale (a)&S&T (Gyr)&R (Gly)&D_{now} (Gly)&V_{gen}/c&V_{now}/c&H(t)&Temp(K)&rho(kg/m^3)&OmegaM&OmegaL&OmegaR&OmegaT \\ \hline 1.09e+3&9.17e-4&1.09e+3&3.71e-4&6.25e-4&4.53e+1&2.12e+1&3.13e+0&1.56e+6&4.59e-18&2.97e+3&7.56e-1&1.29e-9&2.44e-1&1.00e+0\\ \hline 5.41e+2&1.84e-3&5.42e+2&1.18e-3&1.91e-3&4.47e+1&1.40e+1&3.09e+0&5.13e+5&4.94e-19&1.48e+3&8.62e-1&1.20e-8&1.38e-1&1.00e+0\\ \hline 2.68e+2&3.71e-3&2.69e+2&3.59e-3&5.64e-3&4.38e+1&9.51e+0&3.03e+0&1.73e+5&5.64e-20&7.34e+2&9.26e-1&1.05e-7&7.36e-2&1.00e+0\\ \hline 1.33e+2&7.47e-3&1.34e+2&1.07e-2&1.64e-2&4.25e+1&6.58e+0&2.94e+0&5.95e+4&6.66e-21&3.65e+2&9.62e-1&8.89e-7&3.80e-2&1.00e+0\\ \hline 6.55e+1&1.50e-2&6.65e+1&3.11e-2&4.73e-2&4.07e+1&4.59e+0&2.81e+0&2.07e+4&8.01e-22&1.81e+2&9.81e-1&7.39e-6&1.92e-2&1.00e+0\\ \hline 3.20e+1&3.03e-2&3.30e+1&8.98e-2&1.36e-1&3.80e+1&3.22e+0&2.63e+0&7.20e+3&9.73e-23&9.00e+1&9.90e-1&6.09e-5&9.65e-3&1.00e+0\\ \hline 1.54e+1&6.09e-2&1.64e+1&2.58e-1&3.89e-1&3.43e+1&2.27e+0&2.37e+0&2.52e+3&1.19e-23&4.47e+1&9.95e-1&4.99e-4&4.82e-3&1.00e+0\\ \hline 7.15e+0&1.23e-1&8.15e+0&7.39e-1&1.11e+0&2.89e+1&1.60e+0&2.00e+0&8.81e+2&1.46e-24&2.22e+1&9.94e-1&4.06e-3&2.39e-3&1.00e+0\\ \hline 3.05e+0&2.47e-1&4.05e+0&2.10e+0&3.12e+0&2.14e+1&1.14e+0&1.48e+0&3.13e+2&1.84e-25&1.10e+1&9.67e-1&3.22e-2&1.16e-3&1.00e+0\\ \hline 1.01e+0&4.97e-1&2.01e+0&5.80e+0&8.05e+0&1.12e+1&8.92e-1&7.73e-1&1.22e+2&2.77e-26&5.49e+0&7.86e-1&2.14e-1&4.67e-4&1.00e+0\\ \hline 0.00e+0&1.00e+0&1.00e+0&1.38e+1&1.45e+1&0.00e+0&1.00e+0&0.00e+0&6.77e+1&8.60e-27&2.73e+0&3.11e-1&6.89e-1&9.18e-5&1.00e+0\\ \hline \end{array}}\]

The first graph is correct The details you are missing in the correct graph which your attempted graph did not reproduce lies in the relations I posted above. Which describes how the commoving coordinates EVOLVE over proper time. This is what is missing from your graph not the graph on top. Now lets understand the correct graph ( the top graph showing the distinctions between Doppler, SR and GR) Notice the cutoff in SR where velocities approach c ? This detail highlights that The Lorentz transformation laws have an effective cutoff when v=c described by its light cone its Global metric is Minkowskii. The FLRW metric using GR graph applies with regards a commoving (conformal Observer). The commoving observer is also inertial as well as the The object being observed. It is the class of observer as well as the coordinate choice and evolution over time that gives the distinction between SR treatment under Minkowskii which The FLRW metric is only locally equivalent and not globally equivalent. The weak field Newton limit directly relates to this detail as it employs the Minkowskii tensor but the FRLW metric has key distinctions on how it evolves over time. Hence the use of a commoving observer. The Doppler relations given in the top graph do not include the Gamma factor and its influence on the coordinate changes. The SR version does Now I cannot stress the Hubble parameter ( value used evolves over time) that is this key relation \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] So if your applying strictly Hubble's law and the SR transformations your ignoring the coordinate transformations as well as the observer in accordance to the coordinates. Hubble's Law \[V_{recess}=H_o D\] That's why your graph doesn't work it doesn't take in the observer type nor the coordinate transformations itself described via the acceleration equations that your graph doesn't include. The proper time relations provided in the Lineweaver paper. This equation is the line element but does not include how the FLRW metric it describes evolves over time. \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] to include how the above changes over time you need the FLRW metric acceleration equations. So you cannot ignore the expansion history and expect to get the correct results as that is an essential component as to why the metric works the way it does. It is also why the Cosmological redshift equation is strictly Doppler shift. The geometry itself changes over time due to thermodynamic evolution

Just an aside many will miss the significance of this equation \[H=\frac{d}{dt}ln(\frac{a(t)}{a_0}=\frac{d}{dt}ln(\frac{1}{1+z})=\frac{-1}{1+z}\frac{dz}{dt}\] The natural log function has a term being the "natural log function" \[log_e(x)\] where the irrational number is 2.71828 this is identical to the e-folds term e used by inflationary theories. This relation carries on into the expansion rate today as well. As you can see both the scale factor and cosmological redshift are Natural logarithmic functions defined by number of e-folds. Aka your accelerating separation distance Recall I highlighted proper time being \{c^2(dt)\} that's a second order term. Velocity and equations involving strictly velocity relation such as v/c is a first order relation. While we see from the above relation Hubble constant, Z and scale factor are second order. (Acceleration in this case) Now consider Hubble law \[v_{recessive}=H_O d\] The recessive velocity is shown to be accelerating it's not an inertial velocity but is oft described as a peculiar velocity based on separating distance. The accelerating expansion. While the value of the Hubble constant is decreasing in time shown by the relation above of Hubble parameter to redshift relation in the previous post.

Lastly observational evidence shows that our universe is very close to being critically dense with a very slight curvature term. In essence the global spacetime metric is close to flat. K=0 this is the correct term for our universe according to observational evidence. This was determined via CMB lack of distortions that curvature would cause. By the way there is a professional paper by Bunn and Hoggs that tried to equate cosmological redshift as Doppler shift however it required using a continous series of infinitisimals to get it to work. Here is the paper https://arxiv.org/abs/0808.1081 As you hit your 5 post first day limit so will check if you reply tomorrow.

Not necessarily, a critically dense universe is one where the actual density perfectly matches the critical density. If the actual density is greater than the critical density then you have positive curvature for the FLRW metric. If the actual density is less than the critical density then the curvature term is negative curvature. However with combination of the Cosmological Principle the mass density of the overall universe (global metric is uniformly distributed) the 2 terms describing this is homogeneous and isotropic. Meaning no preferred location and no preferred direction. If you apply Newtons shell theorem to this mass distribution gravity equals zero.... You have no center of mass term nor any directional vector field. So you do not have any divergence term such as the gradient term of gravity. These details is why your gravitational redshift formula does not work regardless of your using the relativistic Doppler shift. There is no gravitational nor inertial mass term to begin with The recessive velocity is due to the separation distance described by Hubbles law : the greater the separation distance the greater the recessive velocity. That is certainly not an inertial velocity that is described by kinematics as per GR. see below for the key relations of the FLRW metric the proper time component for a commoving observer being \{c^2d{t^2}\} of the lime element (worldline) next equation below 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 last equation shows how the equations of state evolve due to expansion and how that affects the Hubble parameter as a function of redshift. This form is the professionally used form that you rarely see in introductory textbooks as its at a more advanced level of understanding. This will further equate to any luminosity distance formulas as well as affect how we determine conformal time A more detailed analysis here including the curvature term k Cosmological Principle implies \[d\tau^2=g_{\mu\nu}dx^\mu dx^\nu=dt^2-a^2t{\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\varphi^2}\] the Freidmann equations read \[(\frac{\dot{a}}{a})^2+\frac{k}{a^2}=\frac{8\pi G}{3}\rho\] for \[\rho=\sum^i\rho_i=\rho_m+\rho_{rad}+\rho_\Lambda\] \[2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2}=-8\pi Gp\] for \[p=\sum^ip_i=P_{rad}+p_\Lambda\] with conservation of the energy momentum stress tensor \[T^{\mu\nu}_\nu=0\] \[\dot{p}a^3=\frac{d}{dt}[a^3(\rho+p)]\Rightarrow \frac{d}{dt}(\rho a^3)=-p\frac{d}{dt}a^3\] \[p=\omega\rho\] given w=0 \(\rho\propto a^{-3}\) for matter, radiation P=1/3 \(\rho\propto{-3}\), Lambda w=-1.\(p=-\rho\) for k=0 \[H_o^2=\frac{k}a^2_O=\frac{8\pi G}{3}(\rho^0+\rho_{rad}^0\\rho_\Lambda)\] dividing by \(H^2_0\) and \(P^0_{crit}=\frac{3H^2_0}{8\pi G}\) gives \[1=-\frac{k}{h_0^2a^2_0}+\Omega^o_m+\Omega^0_{rad}=\Omega^0_\Lambda\] \[\Omega_k^0=-\frac{k}{h^2_0a^2_0}\Rightarrow 1=\Omega_k^0+\Omega^0_{rad}+\Omega^0_\Lambda\] densities can be written as \[\rho_{rad}=\rho^0_{rad}(\frac{a_o}{a})^4=\frac{3}{8\pi G}H_0^2\Omega^0_{rad}(\frac{a_o}{a})^4\] \[\Omega_m=\rho^0_m(\frac{a_o}{a})^3=\frac{3}{8\pi G}H_0^2\Omega^0_{rad}(\frac{a_o}{a})^3\] \[\rho_\Lambda=\rho_\Lambda^0=\frac{3}{8\pi G}H_o^2\Omega^0_\Lambda\] \[-\frac{k}{a^2}=\overbrace{-\frac{k}{a^2_0H_o^2}}^{\Omega^0_k}H^2_0(\frac{a_o}{a})^2\] with \(1+z=\frac{a_0}{a}\) densities according to scale factor as functions of redshift. \[\rho_{rad}=\frac{3}{8\pi g}H^2_o\Omega^0_{rad}(\frac{a_o}{a}^4=\frac{3}{8\pi G}H^2_0\Omega^0_{rad}(1+z)^4\] \[\rho_m=\frac{3}{8\pi g}H^2_o\Omega^0_m(\frac{a_o}{a}^3=\frac{3}{8\pi G}H^2_0\Omega^0_m(1+z)^3\] \[\rho_\Lambda=\frac{3}{8\pi G}H_0^2\Omega^0_\Lambda\] \[H^2=H_o^2[\Omega^2_{rad}(1+z)^4+\Omega_m^0(1+z)^3+\Omega_k^0(1+z)^2+\Omega_\Lambda^0]\] the Hubble parameter can be written as \[H=\frac{d}{dt}ln(\frac{a(t)}{a_0}=\frac{d}{dt}ln(\frac{1}{1+z})=\frac{-1}{1+z}\frac{dz}{dt}\] look back time given as \[t=\int^{t(a)}_0\frac{d\acute{a}}{\acute{\dot{a}}}\] \[\frac{dt}{dz}=H_0^{-1}\frac{-1}{1+z}\frac{1}{[\Omega_{rad}(1+z^4)+\Omega^0_m(1=z0^3+\Omega^0_k(1+z)^2+\Omega_\Lambda^0]^{1/2}}\] \[t_0-t=h_1\int^z_0\frac{\acute{dz}}{(1+\acute{z})[\Omega^0_{rad}(1+\acute{z})^4+\Omega^0_m(1+\acute{z})^3=\Omega^0_k(1+\acute{z})^2+\Omega^0_\Lambda]^{1/2}}\] here is the curvature term from the critical density relation \[\Omega_k^0=-\frac{k}{h^2_0a^2_0}\] note the logarithmic functions here \[H=\frac{d}{dt}ln(\frac{a(t)}{a_0}=\frac{d}{dt}ln(\frac{1}{1+z})=\frac{-1}{1+z}\frac{dz}{dt}\] in point of detail the cosmological redshift is a logarithmic function

No that's is totally incorrect and not what is described by the scale factor of the FLRW metric to begin with. The curvature terms of the FLRW metric differs from GR in the sense that the curvature term is determined by the critical density formula relations. The GR solution for the FLRW metric is strictly in the Newtonian weak field limit. Where the stress energy momentum term that tells spacetime how to curve is essentially zero with the only term being the energy density in the T_00 component. The rest of your post can readily be shown erroneous by simply knowing that due to expansion the recessive velocity can be greater than c. That occurs beyond the Hubble horizon. In point of detail at z=1100 the recessive velocity is 2.3 c. So your graphs are in error as they do not show this detail. Due to the equations of state the rate of expansion is also not consistent the Hubble parameter decreases in time even though expansion described by recessive velocity is accelerating. This can readily be seen by the calculator in my signature with all the above. (That calculator only required 5 formulas )

That being said the first mistake of that link is that recessive velocity is not an inertial velocity and does not equate to the gravitational redshift formula. It is a relation involving separation distance and not spacetime curvature involving time dilation relations. The treatment is simply incorrect

Was just about to note the cross post lol were both essentially stating the same thing It's not that hard I've seeded data into it but in my case it was correcting a mistake

AI is easily fooled when you get right down to it. It's a fancy search engine that has some linguistic programming. You can readily seed false information into it and it likely wouldn't know any better. Pulse engines exist and a control loop is a type of circuit used in PWM (pulse width modulation) so it would be trivial to combine the two

Let's put it this way object A does not allow sufficient degree of independence to allow the time dimension to work with the tesseract. For that you want just the spatial components for the object itself then under translational and rotational invariance have that 3d object follow the worldline between events and that is where your time component gets factored. You must be able to separate the spatial components from the time components this is why the tesseract idea becomes impractical. The tesseract is useful to understand independent degrees of freedom but not useful to describe spacetime curvature ie gravity. Once you look into the mathematics this will readily become obvious