OK lets do some math to show some details first lets describe how the Bose Einstein works in a heuristic manner ie a more familiar examiniation
[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. You can fit that into the previous post where I detailed those statistics. This shoulld better help you understand that portion.
Now lets supply some details on GR in particular the Newton limit
Central potential and Newton limit
In the presence of matter or when matter is not too distant physical distances between two points change. For example an approximately static distribution of matter in region D. Can be replaced by the equivalent mass
[latex]M=\int_Dd^3x\rho(\overrightarrow{x})[/latex] concentrated at a point [latex]\overrightarrow{x}_0=M^{-1}\int_Dd^3x\overrightarrow{x}\rho(\overrightarrow{x})[/latex]
Which we can choose to be at the origin
[latex]\overrightarrow{x}=\overrightarrow{0}[/latex]
Sources outside region D the following Newton potential at [latex]\overrightarrow{x}[/latex]
[latex]\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}[/latex]
Where [latex] G_n=6.673*10^{-11}m^3/KG s^2[/latex] and [latex]r\equiv||\overrightarrow{x}||[/latex]
According to Einsteins theory the physical distance of objects in the gravitational field of this mass distribution is described by the line element.
[latex]ds^2=c^2(1+\frac{2\phi_N}{c^2})-\frac{dr^2}{1+2\phi_N/c^2}-r^2d\Omega^2[/latex]
Where [latex]d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2[/latex] denotes the volume element of a 2d sphere
[latex]\theta\in(0,\pi)[/latex] and [latex]\varphi\in(0,\pi)[/latex] are the two angles fully covering the sphere.
The general relativistic form is.
[latex]ds^2=g_{\mu\nu}(x)dx^\mu x^\nu[/latex]
By comparing the last two equations we can find the static mass distribution in spherical coordinates.
[latex](r,\theta\varphi)[/latex]
[latex]G_{\mu\nu}=\begin{pmatrix}1+2\phi_N/c^2&0&0&0\\0&-(1+2\phi_N/c^2)^{-1}&0&0\\0&0&-r^2&0\\0&0&0&-r^2sin^2(\theta)\end{pmatrix}[/latex]
Now that we have defined our static multi particle field.
Our next step is to define the geodesic to include the principle of equivalence. Followed by General Covariance.
Ok so now the Principle of Equivalence.
You can google that term for more detail but in the same format as above [latex]m_i=m_g...m_i\frac{d^2\overrightarrow{x}}{dt^2}=m_g\overrightarrow{g}[/latex] [latex]\overrightarrow{g}-\bigtriangledown\phi_N[/latex]
Denotes the gravitational field above.
Now General Covariance. Which use the ds^2 line elements above and the Einstein tensor it follows that the line element above is invariant under general coordinate transformation(diffeomorphism)
[latex]x\mu\rightarrow\tilde{x}^\mu(x)[/latex] Provided ds^2 is invariant [latex]ds^2=d\tilde{s}^2[/latex] an infinitesimal coordinate transformation [latex]d\tilde{x}^\mu=\frac{\partial\tilde{x}^\mu}{\partial x^\alpha}dx^\alpha[/latex] With the line element invariance [latex]\tilde{g}_{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g_{\alpha\beta}x[/latex] The inverse of the metric tensor transforms as[latex]\tilde{g}^{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g^{\alpha\beta}x[/latex] In GR one introduces the notion of covariant vectors [latex]A_\mu[/latex] and contravariant [latex]A^\mu[/latex] which is related as [latex]A_\mu=G_{\mu\nu} A^\nu[/latex] conversely the inverse is [latex]A^\mu=G^{\mu\nu} A_\nu[/latex] the metric tensor can be defined as [latex]g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\mu[/latex] where [latex]\delta^\mu_nu[/latex]=diag(1,1,1,1) which denotes the Kronecker delta.
Finally we can start to look at geodesics.
Let us consider a free falling observer. O who erects a special coordinate system such that particles move along trajectories [latex]\xi^\mu=\xi^\mu (t)=(\xi^0,x^i)[/latex] Specified by a non accelerated motion. Described as [latex]\frac{d^2\xi^\mu}{ds^2}[/latex] Where the line element ds=cdt such that [latex]ds^2=c^2dt^2=\eta_{\mu\nu}d\xi^\mu d\xi^\nu[/latex] Now assume that the motion of O changes in such a way that it can be described by a coordinate transformation. [latex]d\xi^\mu=\frac{\partial\xi^\mu}{\partial x^\alpha}dx^\alpha, x^\mu=(ct,x^0)[/latex] This and the previous non accelerated equation imply that the observer O, will percieve an accelerated motion of particles governed by the Geodesic equation. [latex]\frac{d^2x^\mu}{ds^2}+\Gamma^\mu_{\alpha\beta}(x)\frac{dx^\alpha}{ds}\frac{dx^\beta}{ds}=0[/latex] Where the new line element is given by [latex]ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu[/latex] and [latex] g_{\mu\nu}=\frac{\partial\xi^\alpha}{\partial\xi x^\mu}\frac{\partial\xi^\beta}{\partial x^\nu}\eta_{\alpha\beta}[/latex] and [latex]\Gamma^\mu_{\alpha\beta}=\frac{\partial x^\mu}{\partial\eta^\nu}\frac{\partial^2\xi^\nu}{\partial x^\alpha\partial x^\beta}[/latex] Denote the metric tensor and the affine Levi-Civita connection respectively.
now as the topic of Higg's came up lets supply some details on this.
Higg's field details that will make understanding the Higg's itself simpler. Keep in mind I am using Lewis Ryder "Introductory to General Relativity" for this. You may find more recent articles with slightly different metrics. (PS this will take me some time to type in and latex)First we need to notice that there is actually 4 field quanta in electro-weak theory. [latex]\gamma, W^-, W^+, and, Z^o.[/latex] notice the second and third is an antiparticle pair. Now the problem is we need a mechanism to give the neutrinos mass without giving photons mass. This is where the Higg's mechanism steps in. To start with Peter Higg's looked at superconductivity. The defining characteristic of conductivity is that at a temperature below a critical temperature [latex]T_c[/latex] some metals lose all electrical resistance. Resistance literally becomes zero, not merely very small. [latex](E=Rj) =j=\sigma E[/latex] where [latex]\sigma[/latex] is the conductivity. A metal in conductivity state then exhibits a persistant current even in no field:[latex]j=\not=0[/latex] when E=0. The key to understanding superconductivity is to describe the current as supercurrent [latex]j_s[/latex]. But unlike the equation above to realize this is proportional not to E but to the vector potential A. [latex]j_s=-k^2A[/latex] with a negative proportionality. This is the London equation. The relevant property we however are seeking is the Meissner effect, which is a phenomena that the magnetic flux is expelled from superconductors. Higg's then showed that suitably transformed into a relativistic theory, this is the equivalent to showing the photon has mass. (just not rest mass lol) The reasoning goes as follows. First the London equation explains the Meissner effect, for taking the curl of Amperes equation[latex]\nabla*BB=j[/latex] gives [latex]\nabla(\nabla^2B=\nabla*j[/latex] noting that [latex]\nabla*B=0[/latex] (no magnetic monopoles) gives [latex]\nabla^2B=k^2B[/latex] which is equal to [latex]\nabla^2A=k^2A[/latex]
In one dimension the solution to this is [latex]B(x)=B(0)exp(-kx)[/latex] which describes the Meissner effect-the magnetic field is exponentially damped inside the superconductor, only penetrating to a depth of order 1/k. This however is still non relativistic. To make it relativistic [latex]\nabla^2[/latex] is replaced by the Klein_Gordon operator [latex]\Box[/latex] and A by the four vector [latex]A^\mu=(\phi,A)[/latex]
giving [latex](\frac{1}{c^2}\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial x^2})A^\mu=k^2A^\mu[/latex] the vector potential is a field but we are currently interested in the photon, the quantum of the field. so we make the transition to quantum theory by the usual description.
[latex]\frac{\partial}{\partial t}\mapsto-\frac{i}{\hbar}E, \frac{\partial}{\partial}{\partial x}\mapsto\frac{i}{\hbar p_x}....[/latex]etc giving the quantum of the field [latex]A^\mu[/latex], [latex]E^2-p^2c^2=k^2c^2\hbar^2[/latex] where E is the total, including rest energy of the field quantum an p isits momentum comparison to [latex]E^2-p^2c^2=m^2c^4[/latex] implies that the mass of the quantum in a superconductor is [latex]m_\gamma=\frac{k\hbar}{c}[/latex] the photon behaves as a massive particle in a superconductor. This is the import of the Meissner effect. Now we need to make a further connection to the Bardeen-Cooper_Schreiffer (BCS theory) of superconductivity which is a microscopic theory that accounts for superconductivity by positing a scalar field [latex]\phi[/latex] (spin zero for scalar fields). Which describes a Cooper pair of electrons, the pairing is in momentu space rather than coordinate space. You can correlate the many particle wave function of Cooper pairing with the above. I'm trying to save time here lol and this is already getting lengthy. The main difference between a superconductor and the Higg's field is that the Higg's field is all pervasive unlike (unlike BCS which is inside a superconductor) The Higg's field through treatment gives rise to the mass of the above neutrinos in the same manner but not to photons. In point of detail the Higg's field can be treated as 4 separate fields one for each of the above. latex]\gamma, W^-, W^+, and, Z^o.[/latex] Now the Higg's potential when [latex]t<t_c[/latex] has a maximum at [latex]\phi=0[/latex] and two minima at [latex]\phi=\pm A[/latex] when[latex] t>t_c][/latex] there is only a minimal at [latex]\phi=0[/latex] THIS is the Mexican hat potential. [latex]V \phi=\frac{m^2}{2}\phi^2+\frac{\lambda}{4}\phi^4[/latex] where [latex]\phi^4[/latex] is the quartic self interaction.. The extremal values of [latex]V\phi[/latex], given by [latex]\partial V/\partial \phi=0[/latex] becomes[latex]\phi=0,\pm\sqrt{\frac{-m^2}{\lambda}}=0,\pm a[/latex]when there is no field [latex]\phi=0[/latex], the energy is not a mimimal but at a maximal, further more the lowest energy is a state in which the field does not vanish and is also two fold degenerate. I hope that helps better understand the Higg's field and how it came about ie was derived in the first place. Section 10.10 Lewis Ryder "Introduction to General Relativity"..
Scalar field Dynamics here we need to couple the scalar field to gravitation.
[latex]\frac{1}{2}\dot{\phi}^2+\frac{1}{2}(\triangledown\phi^2)+V(\phi)[/latex] and the dynamics can be described by two equations. ::Friedmann equations [latex]H^2+\frac{k}{a^2}=\frac{8\pi}{3M^2_P}(\frac{1}{2}(\dot{\phi})^2+V(\phi)[/latex] and the Klein Gordon equation obeys the scalar fields [latex]\ddot{\phi}+3H\dot{\phi}+\acute{V}(\phi)=0[/latex] if the [latex]\phi_a[/latex] is large we have [latex](\triangledown \phi_a^2)<<V(\phi_2)[/latex] the speed of expansion [latex]H=\frac{\dot{a}}{a}[/latex] is dominated by the potential [latex]V(\phi_a)[/latex] in equation[latex]H^2+\frac{k}{a^2}=\frac{8\pi}{3M^2_P}(\frac{1}{2}(\dot{\phi})^2+V(\phi)[/latex] the advantage of Higg's inflation is that inflation is readily modelled using just the standard model of particles. We do not need k-Fields, inflatons, curvatons, Quintessence or any other quasi particle or field. Secondly we can model inflation as a symmetry phase transistion which is extremely important as we tie inflation with the electro-weak symmetry breaking itself.
Higg's inflation.
Higg's field. Is a complex scalar field [latex]SU(2)_w[/latex] doublet. [latex]\phi=(\begin{matrix}\phi_1 & \phi_2 \\ \phi_3& \phi_4 \end{matrix})[/latex] the vector bosons (guage bosons) interact with the four real components [latex]\phi_i[/latex] of the [latex]SU(2)_{w^-}[/latex] symmetric field [latex]\phi[/latex] false vacuum corresponds to [latex]\phi=0 or \phi_1=\phi_2=\phi_3=\phi_4=0[/latex] the true vacuum corresponds to [latex]\phi_1=\phi_2,,,\phi_3^2=\phi_4^2=constant>0[/latex]
assign V on the Y axis, [latex]\phi_3[/latex] on the x axis, [latex]\phi_4[/latex] on a 45 degree between the x and Z axis. when you have conditions [latex]\phi_4=0,\phi_3>0[/latex] then the rotational symmetry is spontaneously broken. The Higg's boson becomes massive as well as the vector bosons W+,W-Z and photons the two neutral fields [latex]B^0 and W^0[/latex] form the linear combinations [latex]\gamma=B^0 cos\theta_w+W^0sin\theta_w[/latex] [latex]Z^0=-B^0sin\theta_w+W^0cos\theta_w[/latex] where Z becomes massive. whee as our ordinary photon [latex]\gamma[/latex] remains massless as the photon does not interact with the electro-weak Higg's field. It is electro-weak neutral. The electroweak symmetry is given by [latex]SU(2)_w\otimes U(1)_{b-L}[/latex] as time decreases the vacuum expectation value [latex]\theta_0[/latex] decreases. (expansion in reverse) the true minimal of the potential is [latex] \phi=0[/latex] this occurs above the critical temperature [latex]T_c=\frac{2\mu}{\sqrt{\lambda}}[/latex] at this point the field interactions take on in essence superconductivity properties.
Now isn't that far more precise than images???
Every formula in physics has a mathematical proof, every definition has a mathematical precision.
This is only a very miniscule portion of developing an effective GUT>>>
We haven't even touched on [math]SO(10)\otimes SO(5)\otimes SO(3)\otimes SO(2)\otimes U(1)[/math]
Do you actually believe your images compares to the mathematical precision involved in symmetry breaking via the SO(10)??? How does one make a single prediction with nothing more than images? How can you possibly determine all possible paths involved in a two particle interaction let alone a multi particle system?
