Mordred

Generate a graph then use the lightcone on my signature. Set your number of steps to 100. You will notice that there is further options such as graphing. The lightcone is done in proper distance. I suspect you may or may not see a deviation after Hubble Horizon but thats just from a glance on your last equation. Unfortunately the latex the lightcone generates doesn't post here on this forum. There is something that generates an error specific to the latex format here. The reason I suspect deviations beyond Hubble horizon is that z requires corrections beyond this point. Oh side note the lightcone has a couple of data options. Planck WMAP and combined from the 2012 Planck dataset. It will be closer than the calcs you have been using as it involves those data set parameters. Here is a correction workup I posted in another thread. ets look at the corrections to the redshift formula. First we define a commoving field. This formula though it includes curvature (global) you can set for flat spacetime. A static universe is perfectly flat. [latex]ds^2=c^2dt^2 [\frac {dr^2}{1-kr^2}+r^2 (d\theta^2+sin^2\theta d\phi^2)][/latex] we write [latex](x^0,x^1,x^2,x^3)=(ct,r,\theta,\phi)[/latex] we set the above as [latex]g_{00}=1,g_{11}=-\frac{R^2(t)}{(1-kr^2)},g_{22}=-R^2 (t)r^2, g_{33}=-R^2 (t)r^2sin^2\theta [/latex] the geodesic equation of the above is [latex]\frac {du^\mu}{d\lambda}+\Gamma^\mu_{\alpha\beta}\mu^\alpha\mu^\beta=0 [/latex] if the particle is massive [latex]\lambda[/latex] can be taken as the proper time s. If it is a photon lambda becomes an affine parameter. So lets look at k=0. we set [latex]d\theta=d\phi=0 [/latex] this leads to [latex]ds^2=c^2t^2-R^2 (t)dr^2=c^2dt^2-dl^2=dt^2 (c^2-v^2)[/latex] where dl is the spatial distance and v=dl/dt is the particle velocity in this commoving frame. Assuming it to be a massive particle of mass "m" [latex]q=m (\frac {dl}{ds})c=(1-\frac {v^2}{c^2})^{\frac{1}{2}}[/latex] from the above a photon emitted at time [latex]t_1[/latex] with frequency [latex]v_1 [/latex] which is observed at point P at time [latex]t_0 [/latex] with frequency [latex]v_0[/latex] with the above equation we get [latex]1+z=\frac {R (t_0)}{R (t_1)}[/latex] Please note were still in commoving coordinates with a static background metric. [latex]z=\frac {v}{c}[/latex] is only true if v is small compared to c. from this we get the Linear portion of Hubbles law [latex]v=cz=c\frac{(t_0-t_1)\dot{R}t_1}{R(t_1)}[/latex] now the above correlation only holds true if v is small. When v is high we depart from the linear relation to Hubbles law. We start hitting the concave curved portion. The departures from the linear relation requires a taylor series expansion of R (t) with the present epoch for this we will also need H_0. note the above line element in the first equation does not use the cosmological constant aka dark energy. This above worked prior to the cosmological constant Now for the departure from the linear portion of Hubbles law. [latex] v=H_Od, v=cz [/latex] when v is small. To this end we expand R (t) about the present epoch t_0. [latex]R (t)=R[(t_0-t)]=R(t_0)-(t_0)-(t_0)\dot {R}(t_0)+\frac {1}{2}(t_0-t)^2\ddot{R}(t_0)...=R (t_0)[1-(t_0-t)H_o-\frac {1}{2}(t_0-t)q_0H^2_0...[/latex] with [latex]q_0=-\frac{\ddot{R}(t_0)R(t_0)}{\dot{R}^2(t_0)}[/latex] q_0 is the deceleration parameter. Sometimes called the acceleration parameter. now in the first circumstances when v is small. A light ray follows [latex]\int_{t_1}^{t^0} c (dt/R (t)=\int_0^{r_1}dr=r_1 [/latex] with the use of this equation and the previous equation we get [latex]r=\int^{t_0}_t=\int^{t_0}_t cdt/{(1-R (t_0)[1-(t_0-t)H_0-...]}[/latex] [latex]=cR^{-1}(t_0)[t_0-t+1/2 (t_0-t)^2H_0+...][/latex] here r is the coordinate radius of the galaxy under consideration. Solving the above gives.. [latex]t_0-t=\frac {1}{c}-\frac {1}{2}H_0l^2/c^2 [/latex] which leads to the new redshift equation [latex]z=\frac {H_0l^2}{c+\frac {1}{2}(1+q_0)H^2_0l^2/c^2+O (H^3_0l^3)}[/latex] The last equation is the corrected redshift formula when recessive velocity exceeds c.. My suspicion is that you will match the linear portion from Hubbles law fairly close, but you will start deviating past Hubble horizon. PS most online calculators don't apply the corrections past Hubble horizon. Though the lightcone calculator is still to good approximation beyond Hubble horizon. It was compared graphically to the lightcones from Lineweaver and Davies. However it isn't 100% accurate either. Good luck and good job in requesting a comparison rather than stating your formulas is correct. Your showing proper methodology.+1 Oh I forgot to mention, it took me several years to find the corrections (last equation). Its not something included in the textbooks. Though they all state the cosmological redshift formula is only accurate when recessive velocity is small. I just wish I remember where I found those corrections. It was too long ago and I wrote them down in my notes but forgot to write down the source. (I saved the original paper I got it from on an old phone that died on me and haven't been able to relocate the original paper.) The source is somewhere on arxiv though. I only use peer reviewed sources I trust. Here is the paper we used for developing the lightcone calculator in my signature. http://www.google.ca/url?sa=t&source=web&cd=5&ved=0ahUKEwjG-_D-zJTQAhUC5mMKHcpMCOMQFggpMAQ&url=http%3A%2F%2Fwww.dark-cosmology.dk%2F~tamarad%2Fpapers%2Fthesis_complete.pdf&usg=AFQjCNHLzxKUp15sqgaDF2B8NU6i4xnBdg TM Davies. If you can match up with these lightcones your formulas are reasonably accurate.

Its a slightly different form in the first equation. However you will probably find this link helpful. http://web.mit.edu/~emin/www.old/writings/quantum/quantum.html If I'm not mistaken this relates to the first equation. I believe the first equation relates to the Heisenburg uncertanty. However my higher QM is rusty lol

that makes no sense, I think you better understand what a waveform means in physics. Might try starting with an electromagnetic wave.

incorrect this can happen in both the finite or infinite case.

Too close to flat, that it could be either negative or positive. Its that close that an exact determination is tricky. Though the datasets at present indicate a leanings towards the positive side. The best indicator is CMB distortions which is tricky to exactly determine

One day you will show how you derived the equations your posting. One doesn't just willy nilly new equations by simple replacement. Every equation you have posted for several months now is useless without showing how you derived them. In other words its been useless to merely post new equations. They have no meaning until you perform the taylor series expansion using kinematics and freefall motion. You have not shown a single corresponding geodesic equation of motion to even define your geodesic path. Which is of fundamental importance to relativity of simultaneaty. So how can you possibly conclude the above as being accurate? This has already been pointed out by Bignose. Try listening to what is being told to you. Do you even understand what |v| means ? it has specific meanings which you obviously ignored. As I cannot determine which usage your using. Choices are 1) absolute value: the magnitude of a real number without regard to its sign. Speed is the absolute value of velocity. If you have |v| as an absolute value then it has no direction. no longer a vector.. 2)determinant: determinant of matrix V 3)parallel: self explanatory 4)cardinality: the number of elements of set V none of these makes sense in your equation above Which is it? as your definetely not using the inner/outer Minkowskii dot product. Which you should be using.

Well there is alternate models where BB is a cyclic process of expand/collapse of our universe. In this scenario one can nearly remove the BB in a sense. It depends on how you define the BB in this case. LQC has a mathematical method to handle the singularity problem via "bounce" which is essentially the above. However as we cannot measure far enough back in time due to the mean free path of light prior to CMB. The mean free path was too short due to too much interferance from other particles (surface of last scattering). Hopefully if we can develop a reliable method of detecting neutrinos. We may be able to see much closer to the BB but not quite all the way as neutrinos dropped out of thermal equilibrium slightly later. Between this and LHC studies we can hopefully garnish a solution to BB

Absolutely, by the way I don't see anything outside of a common mainstream question so far in this thread. If you like I can move it to Astronomy forum but I'll let you decide

We can only conclusively confirm our observable portion. Due to lack of net flow either towards or inward flow, indicates via thermodynamic flow that the portion outside our Observable portion should be in roughly the same thermodynamic state. However we can't conclusively determine that. So yes the universe can be finite or infinite. The only viable means that I know of to determine one or the other is to solve the BB itself. We simply cannot measure the entire universe. Universe geometry isn't conclusive enough

light beams only return to origin on positive curvature. Fairly cut and dry, your back of the head scenario is only viable on that scenario.

It would never return in a perfectly flat scenario. Just continue a straight line path. A return path is only viable under positive curvature.

I believe there are others though less active lol. I'm still not positive on my interpretation of that paper. The spin statistics aspect is covered in numerous "Introductory to GR" textbooks. It is one of the lessons to learning GR. The chapter that covers this is usually under GR waves. What would really help on that paper is someone who better understands some of the gauge group symbology in that paper. Although I understand gauge groups to a certain extent. There is numerous relations symbols used on that paper that I don't recognize. More precisely two specific symbols. I'm not even sure I can latex them lol. Ah found them. [latex]\Upsilon, \Xi [/latex] they appear to be unique to this paper, collectively he has a group [latex]\begin{pmatrix}-\Upsilon&0&0&0\\0&-\Xi&0&0\\0&0&-\Xi&0\\0&0&0&-\Xi\end{pmatrix}[/latex] The majority of the rest is standard Euler-Langrange and Hamilton equations so those parts I'm familiar with. For your benefict "Introductory to Langrange mechanics" http://www.google.ca/url?sa=t&source=web&cd=1&ved=0ahUKEwifs6zvno7QAhVL6WMKHSogDIoQFggaMAA&url=http%3A%2F%2Fwww.macs.hw.ac.uk%2F~simonm%2Fmechanics.pdf&usg=AFQjCNHZnAntVnyYJnhX0bQrDFbA6n46QA as Dirac was mentioned some of the Dirac notation is used. http://www.google.ca/url?sa=t&source=web&cd=13&ved=0ahUKEwjH1PGnoI7QAhUK0WMKHX1mCeoQFghAMAw&url=http%3A%2F%2Fwww.users.csbsju.edu%2F~frioux%2Fdirac%2Fdirac.pdf&usg=AFQjCNEv9MysNDmO-XWbIhz6QftvngBTWA Another required study to understand paper is Hamilton. http://www.google.ca/url?sa=t&source=web&cd=1&ved=0ahUKEwihgu--oY7QAhUDxGMKHXwLCLoQFggaMAA&url=http%3A%2F%2Fwww.damtp.cam.ac.uk%2Fuser%2Ftong%2Fdynamics%2Ffour.pdf&usg=AFQjCNE1AIMv-gse0hNgko8_XvYxW2RXHA Naturally you need a good understanding of tensors. The majority of that paper is fairly decent. The problem I'm having on full comprehensive understanding of it is the group above. Though those details are likely within the paper itself but I would have to study it in greater detail to know for sure.

Well I will be the first to admit that philosophy isn't my strong suit. I'm not sure what you mean by cover up. Anyways I spent some time studying numerous block style arguments. There is numerous key aspects shrouded by those articles. For example determinism and reversible processes. Yet when I mention those aspects, they were either ignored or split off... Much like presentism isn't compatible with relativity. Block itself doesn't work well with "probalistic observers". This is a specific observer used in evolving block. Which isn't identical to growing block. All of this put aside, as an off and on assistant instructor. I found my students "light bulb go on" when you detail the thermodynamic aspects of GR. Proper understanding of the ideal gas laws in GR removes the majority of the mystery behind spacetime curvature. Its too bad many forum members ignore this truth. Not just this forum, for some mysterious reason thermodynamics is too mundane a topic. They rather have the mystery. Little hint, if you truly want a comprehensive knowledge of block arguments, study the terminology including those relating to key thermodynamics. Quite literally when I read terms such as deterministic, reversible and irreversible processes etc. I literally see the related formulas. Lol what I find truly amusing, is that I posted some mathematics showing how Lorentz Ether could viably work under. Yet it 100% ignored. Imagine that.....so much for properly examining the two models... Lorentz ether vs SR ah well. You once mentioned that the mathematician in me interferes with understanding block. Quite the opposite, it allows me to better comrehend block and discern the quality of various papers on the subject.

If the universe has a slight positive curvature. Two parallel light beams will gradually converge. The light path will not be straight but slightly curved. A perfect flat universe, the light beams will stay parallel. In a negative curvature the light beams will diverge. Now assuming expansion stopped, with the current miniscule deviation from a flat universe in the Planck dataset. If you fire an ideal laser beam. It would take roughly 880 Billion light years for the laser beam to return to its original point. Of course we know its highly unlikely expansion will stop lol. Key note, at one time it was once thought that if you have a positive universe, the universe would be bounded. This however isn't true due to the cosmological constant. The universe can be bounded or unbounded.

I like the form you use of that equation, there is another form from Ryden that I find more useful. For other readers I will detail the equations with some explanation. First thing to understand is that the critical density formula and the one posted by Imatsfaal is the GR aspects of the FLRW metric, Other key aspects is the acceleration equation and the ds^2 line element of the metric. However for this post I'm just going to focus on the two equations posted above. Essentially those two equations are derived by inserting the Einstein field equations into the FLRW metric. This is for all contributors (photons, matter, radiation etc). So first we replace [latex]\rho(t)[/latex] mass density with energy density in the form [latex]\epsilon(t)/c^2[/latex] the GR form of the Freidmann equations is in the Newton limit in GR, this is low gravity such as stars, galaxies, LSS etc. It is a specific class solution in GR. This gives the form of [latex](\frac{\dot{a}}{a})^2[/latex][latex]=\frac{8\pi G}{3}\frac{\epsilon(t)}{c^2}[/latex][latex]-\frac{kc^2}{R_0^2}\frac{1}{a^2(t)}[/latex] If [latex]k\le0[/latex] and the energy density is positive, then the R.H.S of the last equation is always positive. This is an expanding universe that will expand forever. If matter is the dominant form of energy, as opposed to radiation this implies [latex]\epsilon\propto \frac{1}{a^2(t)}[/latex]. If k=+1 then the R.H.S must eventually reach 0, after which the universe will contract. To get to the density parameter we can substitute [latex]H(t)=(\frac{\dot{a}}{a})^2[/latex] and we can rewrite the above equation into the Hubble parameter. (note I hate calling it constant, as its only constant at a particular moment in time) [latex]H(t)=\frac{8\pi G}{3}\frac{\epsilon(t)}{c^2}[/latex][latex]-\frac{kc^2}{R_0^2}\frac{1}{a^2(t)}[/latex] if k=0 then [latex]\rho_c(t)=\frac{e_c(t)}{c^2}=\frac{3H^2(t)}{8\pi G}[/latex] with the following density parameter relations [latex]\Omega=\frac{\epsilon}{\epsilon_c}=\frac{\epsilon}{c^2}*\frac{8\pi G}{3H^3}[/latex] note how we correlate the constant c in the the above. The cosmological constant isn't included in the above, essentially the Cosmological constant leads to an increase rate of expansion from the above relations. Also as it is constant as far as we can tell, this universe will continue to expand. This is what I should have taken the proper time to post. Again thanks Imatsfaal for catching the above. Busy work week lol The thermodynamic details take a bit to explain but from the above and using the equations of state for each contributor one can determine the deceleration equation. Wiki has a decent enough coverage. https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology) https://en.m.wikipedia.org/wiki/Friedmann_equations

There is a few sections that give me difficulties in that paper. Its interesting but I don't agree with one section in that paper. In particular the spin 2 statistics section under sound waves. The quadrapole nature of GW waves does not require dark matter for the additional degrees of freedom. (I would have to spend more time studying it) If your interested, though lengthy and relatively complex I can post how spin 2 comes about on GW waves. At least according to examples found in textbooks.

Have you ever considered the detail that their is only two possible ways to look at time? absolute or variable? Is that not the only two possibilities? This is the main reason I feel philosophies are counter productive in many ways. They tend to blur the distinctions, with no to little math support. When you get right down to it. The best method to understand how time works under SR and GR is study the math, without trying to make it fit under a philosophy. How often do you see ppl trying to reinvent a model, based on their personal philosophical beliefs? Block is particularly problematic, their is 6 common variations. Not a mere two

I think one of the problems ppl have is understanding the difference between field potential and a charged field when it comes to trying to treat fields as a medium. A field doesn't involve any medium characteristics until you have an interaction.

Sorry your correct, was having a distracted moment. Concentrating on too many things at once.

nope doesn't work that way as there are examples of motion without any dilation

Your still involving time in merely taking snapshots. Regardless all that means is your measuring the same object at different measurement times nothing profound about that. Its done all the time in Astronomy. Even more important, the velocity value must be in units of metres/sec or equivalent to. Its inherent in the definition.

How? You need to determine the amount of time dilation to determine the relative portion. Even without relativity the units metres/second indicate velocity requires time by the very definition of velocity. Velocity=the rate at which an object changes its position. Rate itself is time.

Not really its fundamental in relativity. You must have a standard ruler (ct)(with t being proper time, not coordinate time) which equates your length. As c is invariant and t being time measured on the same frame of teference (at rest) is the only ruler all observers can agree on. Now I want you to think about the units of velocity.... Does those units suggest you don't need time or does those units indicate time is involved... Then think about different observers measuring the velocity of some massive object. 1) Do all observers agree on the velocity measurement? 2) If not why? 3) What was it about time that caused different observers to measure different velocities? (hint relativistic velocity addition)

Correct in gravity case the curvature is a result of time dilation/length contraction. This is localized anistropy regions. In Cosmology its the density change due to expansion/contraction over time. More accurately the curvature constant is determined by the following formula [latex]\rho_{crit} = \frac{3c^2H^2}{8\pi G}[/latex] k = curvature constant as a dimensionless value, k = critical density. As far as we can determine the curvature constant stays constant throughout our universes history. Prior to inflation the limited volume makes the curvature constant negligable. After inflation is when k can be determined. The following article is one I wrote using Barbera Rydens "Introductory to Cosmology" as a reference. Her methodology to explain the FLRW metric in terms of curvature was one of the easiest to relate to for the math challenged lol. http://cosmology101.wikidot.com/universe-geometry Page 2 with the FLRW metric coverage is here. http://cosmology101.wikidot.com/geometry-flrw-metric/

In this case the curve is in the expansion history. Following the cosmological principle. The mass density per time slice is uniform. The rate of expansion varies over time. This is what's curved is the history of expansion/contraction. Due to the expansion/contraction your light paths are also affected. Rough analogy light deflection due to moving medium.