Mordred

I'm assuming your asking when a vector vs a scalar quantity best describes the system state. This is a seemingly easy question but one has to be careful on answering. The care being what the scalar or vector quantity being measured represents. Certain quantities such as temperature and spin zero particle fields suit being scalar fields as there is no inherent direction to those fields. A spin zero particle field being the Higg's field. The electromagnetic field however has charge, this charge has two inherent directions adding degrees of freedom to the system state. This type of interaction is best described by vector fields. Now particle color, and flavor adds additional degrees of freedom. You can think of color and flavor as a form of charge, but in the case of color you have 3 charges. Which describes your strong force. Gravity suits spin 2 statistics, which is rather tricky to describe. You would have to google quadrupole wave to see what I mean. However that's the spin aspects of the Bose-Einstein and Fermi-Dirac statistics...which I assume the question relates to. So essentially as the temperature rises, more and more particles reach thermal equilibrium. This reduces the number of degrees of freedom. As the guage bosons for each force reach thermal equilibrium, we lose the interactions of those forces. When the electomagnetic, weak force and strong force all unify we no longer have any charge dynamics. This system can now be modelled as a scalar field regardless of scale. In terms of thermodynamics this state occurs at the Vacuum expectation value or VeV. The equation of state for a scalar field is the last formula on this page. https://en.wikipedia.org/wiki/Equation_of_state_%28cosmology%29 under scalar modelling... For the four forces treating gravity as a force. electromagnetic spin 1 weak spin 1 strong spin 1. gravity spin 2. As you study the Weise paper chapter 3 and 4 pay attention to the degrees of freedom due to the boson to fermionic interactions in terms of spin. Ie spin 1/2 electron with photon spin 1 as the guage boson... Higgs field isn't a force but its field is spin 0.

A crust would indicate a bound universe or finite universe. We don't know if the universe is finite or infinite. The metrics we show for expansion is our Observable portion of the entire universe. Thankfully those metrics still work regardless if the universe is finite or infinite. The finite point of the BB being smaller than an atom is only our observable portion in the past. By the way +1 on your comprehension thus far.

Very much so in terms of geometric seperation distances and angles between measurement points. Just remember there is no outside in the raisin bread analogy. It also shows the pressure relations to seperation relations. take each raisin as a measurement point. Each raisin has the same surrounding pressure (the bread dough). By Newtons laws there is no net directional force. The seperation is due to density reduction of the dough. Yet no raisin can gain inertia by Newtons laws as they have a surrounding fluid that is uniform in density.

Yes people get distracted by pop media coverage of dark matter and dark energy when it comes to expansion. Those people tend to gloss past the related math, so seldom see the thermodynamic relations. Another pop media distraction is the entropy arrow of time. In statistic mechanics this is simply time reversal symmetry. Everyday statistic mechanics is an essential aspect to understand expansion of the universe. Baryonic matter, dark matter and radiation are all described. Dark energy aka the Cosmological constant is still giving scientists trouble. Not because its present but that it is so fine tuned and constant as the volume expands. Unlike any other particle field in statistic mechanics. However if the Higgs metastability proves accurate, the cosmological constant and inflation could very well be explained via the Higgs field in SO (10) MSM. MSM is minimal symmetric model (minimal standard model). Just remember the universe follows an adiabatic expansion ( no inflow or outlow) there is no need for an outside of the universe. The container walls are determined by other particles and the rate of particle movement in a medium. (those sound waves for example of the CMB.) Now here is a key detail.... In a homogeneous and isotropic distribution. There is no net directional flow, nor is there any density gradient. So although temperature or pressure can perform the work for expansion, expansion is not described accurately as a higher density flowing to a lower density. Though lower density is a result, there is no flow. The expansion is a relationship between the universe self gravity vs the inherent kinetic energy of the particle contributors. Much like the same relation in the matter dominant and Jeans instability relations I described previously. PS sounds to me that you have a decent understanding of the ideal gas laws and thermodynamics. This is excellent for understanding Cosmology. It places you ahead of many members that ignore those thermodynamics. When you get down to it, statistical mechanics is just as important as relativity and particle physics in cosmology.

No problem, now in Cosmology we deal more with the global influences rather than localized. The equations above provide an adequate demonstration of how large scale structure formation affect the global density of matter. Rather than use the numerous localized hydrodynamic equations, we can average the matter/radiation etc influence into a value called critical density. The critical density without the cosmological constant, is a calculated value that represents the turning point between a contracting or expanding universe. The actual density average compared to the critical density, gives us the universe curvature constant. k I wrote an article covering this in my universe geometry article on the site in my signature. http://cosmology101.wikidot.com/universe-geometry page 2 http://cosmology101.wikidot.com/geometry-flrw-metric/ I didn't go into detail on the FLRW fluid equation nor the deceleration equation as I focused on the distance metrics. However the Weise article has those key formulas the equation you want in this instance is the thermodynamic equation of an adiabatic fluid. Meaning no net inflow or outflow. I wrote this earlier to demonstrate how the radiation and matter equations of state are derived from the first law of thermodynamics. [latex]DU=pdV[/latex]. First take the first law of thermodynamics. [latex]dU=dW=dQ[/latex] U is internal energy W =work. As we dont need heat transfer Q we write this as [latex]DW=Fdr=pdV[/latex] Which leads to [latex]dU=-pdV.[/latex]. Which is the first law of thermodynamics for an ideal gas. [latex]U=\rho V[/latex] [latex]\dot{U}=\dot{\rho}V+{\rho}\dot{V}=-p\dot{V}[/latex] [latex]V\propto r^3[/latex] [latex]\frac{\dot{V}}{V}=3\frac{\dot{r}}{r}[/latex] Which leads to [latex]\dot{\rho}=-3(\rho+p)\frac{\dot{r}}{r}[/latex] We will use the last formula for both radiation and matter. Assuming density of matter [latex]\rho=\frac{M}{\frac{4}{3}\pi r^3}[/latex] [latex]\rho=\frac{dp}{dr}\dot{r}=-3\rho \frac{\dot{r}}{r}[/latex] Using the above equation the pressure due to matter gives an Eos of Pressure=0. Which makes sense as matter doesn't exert a lot of kinetic energy/momentum. For radiation we will need some further formulas. Visualize a wavelength as a vibration on a string. [latex]L=\frac{N\lambda}{2}[/latex] As we're dealing with relativistic particles [latex]c=f\lambda=f\frac{2L}{N}[/latex] substitute [latex]f=\frac{n}{2L}c[/latex] into Plancks formula [latex]U=\hbar w=hf[/latex] [latex]U=\frac{Nhc}{2}\frac{1}{L}\propto V^{-\frac{1}{3}}[/latex] Using [latex]dU=-pdV[/latex] using [latex]p=-\frac{dU}{dV}=\frac{1}{3}\frac{U}{V}[/latex] As well as [latex]\rho=\frac{U}{V}[/latex] leads to [latex]p=1/3\rho[/latex] for ultra relativistic radiation. Those are examples of how the first law of thermodynamics fit within the equations of state. There is more intensive formulas involved. In particular the Bose-Einstein statistics and Fermi-Dirac statistics

http://www.wiese.itp.unibe.ch/lectures/universe.pdf forgot to drop quote marks

Good question. As quarks are fermionic this will be when the quark epoch occuurs according to this SO (5) chronology. The lepton family drops out later. https://en.m.wikipedia.org/wiki/Chronology_of_the_universe The supersymmetric model however has earlier possible stages. The exact timing is model dependant. A good coverage between models is http://pdg.lbl.gov/2011/reviews/rpp2011-rev-guts.pdf GRAND UNIFIED THEORIES http://arxiv.org/pdf/0904.1556.pdf The Algebra of Grand Unified Theories John Baez and John Huerta A good textbook coverage is chapter 3 below http://www.wiese.itp.unibe.ch/lectures/universe.pdf:"Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis This chapter does an excellent job covering the Boltzmann aspects and additional degrees of freedom as particles drop out of thermal equilibrium. It details the Bose-Einstein and Fermi-Dirac statistics leading to the Maxwell Boltzmann statistics. It includes GR and the FLRW metric as well. http://www.wiese.itp.unibe.ch/lectures/universe.pdf "Particle Physics of the Early universe" by Uwe-Jens Wiese Thermodynamics, Big bang Nucleosynthesis

Not a bad correlation on the snowflake analogy, the localized hydrodynamic equations assist the filament developments. These are typically covered in decent astrophysics textbooks. A good resource being "Elements of Astrophysics". Jeans equation above is one example of localized hydrodynamics due to gravity and particle movement of pressure less dust. The FLRW metric used in cosmology essentially averages these localized influences onto a global scale. This is where the Cosmological principle takes hold. At some point of volume where a homogenous and isotropic average occurs decides the scale from local to global measurements and influence. This scale being roughly 100 Mpc in the universe today. Less than 100 Mpc that region is not considered uniform. What I described above is essentially showing how a matter dominated universe can expand, however we only considered matter. photons and other forms of radiation influence expansion via their own equations of state. An equation of state correlates density to pressure relations of each particle species. GR curvature is generally speaking on the localized scale, where universe curvature is a thermodynamic history of expansion rates. This distinction is important when it comes to gravitational redshift, as opposed to cosmological redshift... The universe undergoes three primary eras, radiation dominant BB to time of last scattering (CMB,), matter dominant followed by the current Lambda dominant. Each era has its own thermodynamic relations which is an average dominant influence.

The last post makes the most sense.... Yes expansion involves the ideal gas laws at every stage of its evolution. Every particle whether its fermionic or bosonic contributes to expansion. Even gravity itself aids expansion though not in the way many think. Take a homgeneous and isotropic initial condition. As inhomogeniety occurs, the average density of matter drops, due to pooling into large sale structure formation. This means gravity has less influence on the voids away fro the LSS. The very fact that gravity tends to condense into the LSS regions alone aids expansion. However we would need some rules governing LSS formation. For this we will use strictly a non relativistic matter only fluid. Just matter.... Well one set of rules has to do with Jeans mass, Primordial density fluctuations expand linearly at a rate slower than the rate of expansion. This induces localized anistropy regions that sets up two possibilities locally. Inflow and outlfow of matter. the dividing line between the two possibilities can be found by the following argument. Let the time of freefall into the overdense region be [latex]t_g=1/\sqrt{G_\rho}[/latex] sound waves in a medium propogate with velocity [latex]c_s=\sqrt{\frac{\partial p}{\partial\rho}}[/latex] so they move one wavelength in the time [latex]t_s=\lambda/c_s[/latex] when t_g is shorter than t_s the fluctuation is unstable and will continue to grow until it collapses locally setting T_g as equal to T_s we find the Jeans instability, Which correlates the rate of expansion and collapse locally due to gravity itself (locally only) [latex]\lambda_j=\sqrt{\frac{\pi}{G\rho}}c_s[/latex] So we can see from this that expansion and gravity both aid in large scale structure formation. That large scale structure formation in turn helps the rate of expansion by the following equation. [latex]H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}[/latex] as expansion increases and the LSS develop the matter density evolves by the ratio in the last equation. This alters the rate of expansion compared to the rate today by the last equation. You will note radaition and matter both evolve but the cosmological constant does not. If you truly look and study the thermodynamics of particles, you will find nothing is more natural than an expanding or collapsing universe. Expansion is literally a thermodynamic process. (Though were still trying to fit the cosmological constant under thermodynamic process) possible solution is the Higgs metastability. Essentially from the above if the matter wavelength (Jeans mass wavelength) is significantly smaller than the Hubble expansion. Locally gravity will collapse, this local collapse will in turn aid expansion by lowering the global mass density average. So from above we see that a matter only universe can still expand when we involve nothing more than gravity and matter. Radiation has a different ratio of contribution so does the Cosmological constant. ill regardless the rate of expansion still boils down to potential vs kinetic energy relations of the ideal gas laws and thermodynamics. We don't need anything more than GR and thermodynamics to explain expansion. Regardless if the universe is finite or infinite.

No other types of galaxies weren't generated afiak. It is an immensely huge dataset. Funny I don't find expansion unimaginable. Of course I've spent years studying expansion. Quite frankly nothing is more natural than the development of those strands and expansion. It follows the rules of an adiabatic and isentropic fluid and particle physics in exquisite detail. If your intetsted in the math I can provide several key formulas to large scale structure formation later on.

Tar that simulation tests more than just expansion. In excellent agreement it also tests our knowledge of nucleosynthesis in the metalicity details. Also it generated all known galaxy types. How expansion works with regards to gravity is a rather tricky process to fully appreciate. With somewhat surprising results. First off the range from gravitationally bound objects where the cosmological constant takes over, though this isn't only cause of expansion. To calculate the distance has to do with relative field strength per cubic metre. Gravity locally is far stronger than the cosmological constant.

I have a huge collection of books in Cosmology/relativity and particle physics. None of the books I have or read cover Bundles. My collection is well over 150 books. Arxiv has provided the only details I know on bundle treatments

the size of this model is 130 Mpc if I recall correctly

Not really his section on tangent and fibre bundles was limitted.

Or as Sean Carroll wrote " Fibre bundles can be thought of as the internal vectors within a group".

To add to this. The deviations to Pythagoras theory for Shwartzchild metric (the curvature) is due to strictly stress-energy-momentum/tensor gradient. Which causes time dilation. In the FLRW metric the curvature gradient is a history of expansion and contraction change. The volume is changing by the scale factor a(t). not time dilation. Your deviations from Pythagoras theory is due to this history curvature. In a homogeneous and isotropic universe the global geometry at any time slice is uniform in mass/energy distribution. There is no inherent vector quantity due to gravity. Gravity in this particular case a scalar value. GR deals with the conservation of the four momentum/four velocity. This is expressed by the stress/momentum tensor. In this static case the stress tensor value is zero. There is no deviation in the four momentum/four velocity. Parallel transport of two light rays will always remain parallel with no deviation. So a static uniform mass distribution there is no length contraction nor time dilation. As it is static we also have no expansion/contraction volume change. When you have a localized difference in mass distribution, from the uniform global distribution this changes. Now we have a definable direction to the four momentum etc. The stress tensor is no longer zero, this causes changes to the Reimann curvature. This is where you see time dilation. In the FLRW metric case, The global density lowers, however this change is global as time progresses. At any particular time slice gravity is static. There is no vector component to the average particle momentum. NO change to the stress/momentum tensor. Which means no Reimann curvature. No time dilation. The volume change does not influence the conservation of four momentum except via temperature change (kinetic energy). The changes to separation distance are due to expansion and contraction. Not the energy/momentum tensor. Particles of the present don't interact with particles in the past. The change in density from one moment in time cannot influence a past density.

No the global density is still uniform. I recognize this is tricky. Its common to think higher density means slower time. In the Schwartzchild example this is true. However this isn't true when its the Global geometry changing density through volume change. (we currently have one poster in Speculations trying to prove the above is wrong. Though he realized his math is wrong so is now rethinking) For example Ask yourself Why we do not measure time dilation of standard candles the further away they are from Earth. They are emitting from a higher global density past. So each standard candle or any other interactive process should be more and more time dilated the further you look. Yet we don't see that Cosmological redshift is due to volume change not time dilation.

No you can never observe yourself in the past. Nor can you ever observe the same object in both the present and past. Its always one or the other never both. Not counting reflections

Lets look back at this OP. We need to clear up a few misconceptions on time. Inhomogeneous mass distribution is needed for time dilation. However time still functions. So if you take a homogeneous and isotropic fluid two static observers in that fluid regardless of location will measure no time dilation regardless of location. Time still functions. Many ppl tend to believe that a higher density past compared to density now but this isn't true. The hyperslice of an event is homogeneous and isotropic. Now in an expanding volume how causal connection is defined gets a little more complex than simply considering just the speed of light and time it takes for information exchange. We must also account for the expansion. Now the distinction above may be better explained by the following. example A. Use the Schwartzchild metric ie a Bh or planet. The Global geometry is a homogeneous and isotropic fluid. It will have an "as close to universal density as possible". |||The Schwartzchild metric assumes zero value for the vacuum|||. However this isn't true in other metrics. So in this case your global hyperslice is uniform. Homogenous and isotropic. Any two or more static observers, observing each other will have the same time rate. A global observer can set this as a universal "now". the global geometry will be [latex]ct,x,y,z [/latex] which is Euclidean flat. The observer at the EH will also use the same geometry for his reference frame. When you draw a line between the Global to local observer the lightpath is curved by the localized anistrophy. However in this case the volume isn't changing... In the case of the FLRW metric the observers global event is still a homogenous and isotropic fluid but the change in density is due to a change in volume. Not the gravitational potential. in other words what is curved from one observer to the other is density change due to change in volume. Where as in the first case the volume is constant. This is an important distinction between the Einstein field equations and the FLRW metric.

I assume your talking about your own formulation by Hubble shift doesn't work. This is where you might want to study Hubble expansion in some detail. Its not constant... I believe on this thread Ive already discussed that.

only the enclosed triangles are observable by the present day observer.

What event type is defined by the red line? a) spacelike b) timelike c) lightlike What event types of the above choices represent the inner cone region? How does the Worldline terms and line element relate? study hint if your unsure hint How does the above relate to Pythagoras theory.?

Migl an important study hint to my last post. Worldline and line element in relation to lightcones. https://en.m.wikipedia.org/wiki/Line_element https://en.m.wikipedia.org/wiki/World_line

for example the lightcone on this link is an Euclidean lightcone. https://en.m.wikipedia.org/wiki/Light_cone however the lightcone examples on this Cosmology page are different than the above. http://www.astro.ucla.edu/~wright/cosmo_03.htm cross posted with Migl

Its better to think of it as casually connected to an observer. Rather than directly observable. This becomes an important distiction when you involve expansion and contraction. Which can play havoc on lightcone worldlines. Yes each observer will have his own set of possibly causally connected events. Or lightcone. look at how the 45 degree line is determined as an event type boundary. This 45 degree line will change when you start adding curvature influences. I'm hinting at an important distinction between SR lightcones to other types ie GR lightcones. Too often ppl set in stone the 45 degree cone and ignore influences upon that 45 degree line. Ie GR itself.