Mordred Posted May 21, 2023 Author Posted May 21, 2023 All good, it took me several months studying various literature directly relating to CKMS for me to finally fill in the blanks and be comfortable working with it. It has been one of my goals in this thread. (Still is but now that I figured out how the cross sections connect to to the CKMS for both left and right hand particles. I can now look at the supersymmetric partners.
Baron d'Holbach Posted May 22, 2023 Posted May 22, 2023 4 hours ago, Mordred said: left and right hand particles Have you looked into cosmic microwave background radiation and how it affect it? It may have influenced the generation and the amplification of chiral asymmetry and may have played a role in the emergence of chirality overall.
Mordred Posted May 22, 2023 Author Posted May 22, 2023 (edited) Leptogenesis and baryogenesis would occur at the initial electroweak symmetry breaking stages prior to the dark ages where the mean free path of photons due to overall density is less than 10^-30 metres. The CMB data would unlikely be able to preserve any evidence as the expansion and slow roll stages of inflation would cause supercooling followed btmy reheating. However I'm not trying to solve either leptogenesis and baryogenesis. I already know the cross scatterings show that the right neutrino mixing angles would be insufficient in quantity via the Higgs seesaw to account for either That possibility is already well researched. 10 minutes ago, Baron d'Holbach said: Have you looked into cosmic microwave background radiation and how it affect it? It may have influenced the generation and the amplification of chiral asymmetry and may have played a role in the emergence of chirality overall. However there is current research studying neutrino oscillations itself that may or may not provide insight to the above. Edited May 22, 2023 by Mordred
Baron d'Holbach Posted May 22, 2023 Posted May 22, 2023 5 hours ago, Mordred said: supersymmetric partners. Why are you looking into this? It technically do not exist? Or you just want a philosophical footing?
Mordred Posted May 22, 2023 Author Posted May 22, 2023 (edited) To better understand the Weinberg mixing angles with regards to the CKMS matrix and to further examine the aspects of the seesaw mechanism of the Higgs field. Assuming supersymmetry though you would have supersymmetric Higgs partners as well. Supersymmetry though hasn't been disproven yet and is still viable. However our colliders are still too low an energy level to produce a supersymmetric particle. Were on the minimal border line however. From what I see the supersymmetric partners do not work in the current CKMS matrix so you would need a different matrix to account for them. That is what I'm confirming. On 5/21/2023 at 8:44 PM, Baron d'Holbach said: Why are you looking into this? It technically do not exist? Or you just want a philosophical footing? I was correct you need a super-CKMS matrix for supersymmetry. Details here https://arxiv.org/pdf/0810.1613.pdfc Bose Einstein QFT format. \[|\vec{k_1}\vec{k_2}\rangle\hat{a}^\dagger(\vec{k_1})\hat{a}^\dagger(\vec{k_2})|0\rangle\] \[\Rightarrow |\vec{k_1}\vec{k_2}\rangle= |\vec{k_2}\vec{k_1}\rangle\] number operator \[\hat{N}=\hat{a}^\dagger(\vec{k})\hat{a}\vec{k})\] Hamilton operator \[\hat{H}=\int d^3k\omega_k[\hat{N}(\vec{k})+\frac{1}{2}]\] momentum of field \[\hat{P}=\int d^3k\vec{k}[\hat{N}(\vec{k})+\frac{1}{2}]\] renormlized Hamilton \[\hat{H_r}=\int d^3 k\omega_k\hat{a}^\dagger(\vec{k})\hat{a}(\vec{k})\] Edited March 26 by Mordred
Mordred Posted May 23, 2023 Author Posted May 23, 2023 (edited) Higgs again. \[m\overline{\Psi}\Psi=(m\overline{\Psi_l}\Psi_r+\overline{\Psi_r}\Psi)\] \[\mathcal{L}=(D_\mu\Phi^\dagger)(D_\mu\Phi)-V(\Phi^\dagger\Phi)\] 4 effective degrees of freedom doublet complex scalar field. with \[D_\mu\Phi=(\partial_\mu+igW_\mu-\frac{i}{2}\acute{g}B_\mu)\Phi\]\ \[V(\Phi^\dagger\Phi)=-\mu^2\Phi^\dagger\Phi+\frac{1}{2}\lambda(\Phi^\dagger\Phi)^2,\mu^2>0\] in Unitary gauge \[\mathcal{L}=\frac{\lambda}{4}v^4\] \[+\frac{1}{2}\partial_\mu H \partial^\mu H-\lambda v^2H^2+\frac{\lambda}{\sqrt{2}}vH^3+\frac{\lambda}{8}H^4\] \[+\frac{1}{4}(v+(\frac{1}{2}H)^2(W_mu^1W_\mu^2W_\mu^3B_\mu)\begin{pmatrix}g^2&0&0&0\\0&g^2&0&0\\0&0&g^2&g\acute{g}\\0&0&\acute{g}g&\acute{g}^2 \end{pmatrix}\begin{pmatrix}W^{1\mu}\\W^{2\mu}\\W^{3\mu}\\B^\mu\end{pmatrix}\] Right hand neutrino singlet needs charge conjugate for Majorana mass term (singlet requirement) \[\Psi^c=C\overline{\Psi}^T\] charge conjugate spinor \[C=i\gamma^2\gamma^0\] Chirality \[P_L\Psi_R^C=\Psi_R\] mass term requires \[\overline\Psi^C\Psi\] grants gauge invariance for singlets only. \[\mathcal{L}_{v.mass}=hv_{ij}\overline{I}_{Li}V_{Rj}\Phi+\frac{1}{2}M_{ij}\overline{V_{ri}}V_{rj}+h.c\] Higgs expectation value turns the Higgs coupling matrix into the Dirac mass matrix. Majorana mass matrix eugenvalues can be much higher than the Dirac mass. diagonal of \[\Psi^L,\Psi_R\] leads to three light modes v_i with mass matrix \[m_v=-MD^{-1}M_D^T\] MajorN mass in typical GUT \[M\propto10^{15},,GeV\] further details on Majorana mass matrix https://arxiv.org/pdf/1307.0988.pdf https://arxiv.org/pdf/hep-ph/9702253.pdf Edited May 23, 2023 by Mordred
Mordred Posted May 24, 2023 Author Posted May 24, 2023 (edited) Sterile Neutrino related research papers Next decade of sterile neutrino studies by Alexey Boyarsky, Dmytro Iakubovskyi, Oleg Ruchayskiy https://arxiv.org/pdf/1306.4954.pdf Detection of An Unidentified Emission Line in the Stacked X-ray spectrum of Galaxy Clusters Esra Bulbul, Maxim Markevitch, Adam Foster, Randall K. Smith, Michael Loewenstein, Scott W. Randall https://arxiv.org/abs/1402.2301 Neutrino Masses, Mixing, and Oscillations Revised October 2021 by M.C. Gonzalez-Garcia (YITP, Stony Brook; ICREA, Barcelona; ICC, U. of Barcelona) and M. Yokoyama (UTokyo; Kavli IPMU (WPI), UTokyo). https://pdg.lbl.gov/2022/reviews/rpp2022-rev-neutrino-mixing.pdf Edited May 24, 2023 by Mordred
Mordred Posted May 25, 2023 Author Posted May 25, 2023 (edited) seesaw mechanism righthand neutrino states with Higgs coupling \[f^v \varepsilon_{ab}\overline{L}^aH^bV_r\] which gives rise to Dirac mass term \[M_D(\overline{V_L}V_R+\overline{V}_RV_L\] Majorona mass terms \[M_{m1}\overline{V_L}V^c_L+M_{2}M^{-c}_RV_R+c.c\] \[\begin{pmatrix}\overline{V_L}\\\overline{V^c_R}\end{pmatrix}\begin{pmatrix}M_{m1}&M_D\\M_D&M_{M12}\end{pmatrix}(V^c_LV_R)\] eugenvalues \[\lambda^2=(M_{m1}+M_{M2})\lambda(M_{M1}M_{M2}-M_D^2)=0\] solution \[\lambda=\frac{(M_{M1}+M_{M2}\pm\sqrt{M_{(M1}-M_{M2}^2+4M_D^2}}{2}\] as one eugenvalue increases the other decreases. set \[M_{M1}=0,,,,M_{M2}>>M_D\] gives \[\lambda=M_{M2}(\frac{1\pm\sqrt{1+4}(\frac{M_D}{M_{M2}^2})}{2})\] \[\lambda_1\approx M_{M2},\lambda_2\approx \frac{M^2_D}{M_M^2}\] Edited May 25, 2023 by Mordred
Mordred Posted May 28, 2023 Author Posted May 28, 2023 (edited) Reminder notes Curl of a vector field definition if vector F equals P,Q,R as a vector field in R^3 and \[P_x,Q_y, R_z\] all exists the the curl F is defined as curl \[\vec{F}=(R_y-Q_z)\hat{i}+(P_z-R_x)\hat{J}+(Q_x-P_y)\hat{k}=(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})\hat{i}+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})\hat{J}+(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\hat{k}\] the curl of a vector is a vector field in contrast to divergence given as \[div \vec{F}=\vec{\nabla}\cdot\vec{F}\] \[\vec{\nabla}x\vec{F}\] \[\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\P&Q&R\end{pmatrix}\] with determinant loosely defined as \[(R_y-Q_z)\hat{i}-(R_x-P_z)\hat{j}-(Q_z-P_y)\hat{j}=(R_y-Q_z)\hat{i}+(R_x-P_z)\hat{j}+(Q_z-P_y)\hat{j}=curl \vec{F}\] above definitions from https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.05%3A_Divergence_and_Curl pursuant next study gravity is divergent free on one loop integrals but divergent on 2 loop Edited May 28, 2023 by Mordred
Mordred Posted May 31, 2023 Author Posted May 31, 2023 (edited) future references with regards to Einstein-Hilbert action one loop integrals and two loop integrals. https://arxiv.org/pdf/1706.02622.pdf https://cds.cern.ch/record/261104/files/CM-P00049196.pdf https://arxiv.org/abs/1207.2302 https://arxiv.org/pdf/hep-th/9605057.pdf Quantum geometrodynamics https://arxiv.org/abs/0812.0295 loop quantum gravity https://arxiv.org/abs/1201.4598 https://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf Edited May 31, 2023 by Mordred
Mordred Posted June 7, 2023 Author Posted June 7, 2023 (edited) BRST quantization aka Faadev Popoff Ghosts https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/henneaux.pdf https://arxiv.org/pdf/1407.7256.pdf appears to be the most common method of renormalization with regards to quantum gravity. Edited June 7, 2023 by Mordred
Mordred Posted June 10, 2023 Author Posted June 10, 2023 (edited) start of nucleosynthesis \[\rho_r c^2=\frac{3}{32\pi}\frac{c^2}{G_N}t^{-2}\] \[\rho_r c^2=\frac{3}{32\pi}(\rho c^2)_{PL}(\frac{t_{PL}}{t})\] \[K_b T\simeq 0.46 E_{PL}(\frac{t}{t_{PL}})^{-1/2}\] \[t_s\simeq \frac{10^{20}}{[T(K)]^2}\]\[\simeq (T_{bbn}=10^9 K)\] roughly \(10^2) seconds after BB Edited June 10, 2023 by Mordred
Mordred Posted June 11, 2023 Author Posted June 11, 2023 (edited) photon propogator \[\frac{i}{k^2}[-g^{\mu\nu}+(1-\zeta)\frac{k^\mu k\nu}{k^2}]\] in Feymann gauge \(\zeta=1\) gives \[-\frac{i}{k^2}g^{\mu\nu}\] polarization states of photon \[\epsilon_1=\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\] \[\epsilon_2=\begin{pmatrix}0\\0\\1\\0\end{pmatrix}\] normalization given by \(\epsilon_1 \cdot \epsilon_2=g^{\mu\nu}\) Electron/positron propogator \[\frac{i(\gamma^\mu q_\mu+m)}{q^2-m^2}\] delta function \((2\pi)^4\varphi ( p_1-p_2-q)\) Edited June 11, 2023 by Mordred
Mordred Posted June 14, 2023 Author Posted June 14, 2023 (edited) Slow Roll single scalar field perturbation \[[\delta\frac{\tilde{p}}{\rho}]^2=\frac{k^3}{2\pi^3}\int d^3 xe^{i\vec{k}\cdot \vec{x}}\langle \frac{\partial \rho}{\rho}\vec{x},t \frac{\partial \rho}{\rho}\vec{O},t\rangle\] \[[\delta\tilde{t}(\vec{k})]^2=\frac{k^3}{(2\pi)^3}\int d^3xe^{i\vec{k}\cdot\vec{x}}\langle \partial t\vec{x}\partial{t}\vec{O}\rangle\] Edited June 14, 2023 by Mordred
Mordred Posted June 22, 2023 Author Posted June 22, 2023 (edited) Building the full Pontecorvo-Maki-Nakagawa-Sakata matrix from six independent Majorana-type phases https://cds.cern.ch/record/1127373/files/GetPDFServlet.pdf further examining the following from the article \(\frac{N_b}{N_\gamma}=(6.1^{+0.3}_{-0.2})*10^{-10}\) Quote In this framework a CP asymmetry is generated through out-of-equilibrium L-violating decays of heavy Majorana neutrinos [38] leading to a lepton asymmetry which, in the presence of ðB þ LÞ-violating but ðB LÞ-conserving sphaleron processes [53], produces a baryon asymmetry. In the single flavor approach, with three singlet heavy neutrinos Ni, thermal leptogenesis is insensitive to the CP-violating phases appearing in the PMNS matrix. In this case there is complete decoupling among the phases responsible for CP violation at low energies and those responsible for leptogenesis [54,55]. hrrm this seems to imply this Cern paper considers right hand neutrinos accounting for leptogenesis. Edited June 22, 2023 by Mordred
Mordred Posted March 23 Author Posted March 23 For reference an extremely handy Feymann rules listing https://porthos.tecnico.ulisboa.pt/CTQFT/files/SM-FeynmanRules.pdf
Mordred Posted April 3 Author Posted April 3 Just pulling a link to an older thread with some articles and references to use for this thread.
Mordred Posted April 4 Author Posted April 4 \[R^{\mu'}_{\phantom{\mu'}\nu'\alpha'\beta'}=\dfrac{\partial x^{\mu'}}{\partial x^\mu}\dfrac{\partial x^\nu}{\partial x^{\nu'}}\dfrac{\partial x^\alpha}{\partial x^{\alpha'}}\dfrac{\partial x^\beta}{\partial x^{\beta'}}R^\mu_{\phantom{\mu}\nu\alpha\beta}\]
Mordred Posted April 5 Author Posted April 5 (edited) Christoffels for the FLRW metric in spherical coordinates. \[ds^2=-c(dt^2)+\frac{a(t)}{1-kr^2}dr^2+a^2(t)r^2 d\theta^2+a^2(t)r^2sin^2d\phi\] \[g_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\0&\frac{a^2}{1-kr^2}&0&0\\0&0&a^2 r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\] \[\Gamma^0_{\mu\nu}=\begin{pmatrix}0&0&0&0\\0&\frac{a}{1-(kr^2)}&0&0\\0&0&a^2r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\] \[\Gamma^1_{\mu\nu}=\begin{pmatrix}0&\frac{\dot{a}}{ca}&0&0\\\frac{\dot{a}}{ca}&\frac{a\dot{a}}{c(1-kr^2)}&0&0\\0&0&\frac{1}{c}a\dot{a}r^2&0\\0&0&0&\frac{1}{c}a\dot{a}sin^2\theta \end{pmatrix}\] \[\Gamma^2_{\mu\nu}=\begin{pmatrix}0&0&\frac{\dot{a}}{ca}&0\\0&0&\frac{1}{r}&0\\\frac{\dot{a}}{ca}&\frac{1}{r}&0&0\\0&0&0&-sin\theta cos\theta \end{pmatrix}\] \[\Gamma^3_{\mu\nu}=\begin{pmatrix}0&0&0&\frac{\dot{a}}{ca}\\0&0&0&\frac{1}{r}\\0&0&0&cot\theta\\\frac{\dot{a}}{c}&\frac{1}{r}&cot\theta&0\end{pmatrix}\] \(\dot{a}\) is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term Newton limit geodesic \[\frac{d^r}{dt^2}=-c^2\Gamma^1_{00}\] Christoffel Newton limit \[\Gamma^1_{00}=\frac{GM}{c^2r^2}\] Covariant derivative of a vector \(A^\lambda\) \[\nabla_\mu A^\lambda=\partial_\mu A^\lambda+\Gamma_{\mu\nu}^\lambda A^\nu\] Edited April 23 by Mordred
Mordred Posted April 21 Author Posted April 21 (edited) Palatini Higgs Induced gravity scenario \[S=\int d^4x\sqrt{-g}[\frac{\xi h^2}{2}R-\frac{1}{2}(\partial h)^2-\frac{1}{4}h^4-\frac{1}{4}f_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}h^2B_\mu B^\mu-i\bar{\psi}{\not}\tiny\,\normalsize\partial\psi-\frac{\gamma}{\sqrt{2}}h\bar{\psi}{\psi}]\] {\not}\tiny\,\normalsize\partial scalar field h, vector field \(B_{\mu\nu}\) fermion field \(\psi\) above Abelion with standard \(B_{\mu}B^{\nu}\) kinetic terms \(F_{\mu\nu}F^{\mu\nu}\) scalar field expectation value \[G_{n,eff}\equiv\frac{1}{8\pi\xi h^2}\] to keep \(G_{n,eff}\) well behave non-minimal coupling \(\xi\) is constrained to positive values for semi-positive definiteness of the scalar field kinetic term. shown by a field redefinition \(h^2\rightarrow h^2\xi\) Einstein-Hilbert frame redefinition of the metric terms. \(g_{\mu\nu}\rightarrow \Theta g_{\mu\nu},,\Theta \equiv \frac{F^2_\infty}{h^2},,,F_\infty\equiv\frac{m_P}{\sqrt{\xi}}\) with rescaling of the vector and fermion fields \(A_\mu \rightarrow \Theta^{-1/2},,,,\psi \rightarrow \Theta^{-3/4}\psi\) \[S=\int d^4 x[\frac{M_p^2}{2}R-\frac{1}{2}m^2_P K(\Theta)(\partial\Theta)^2-\frac{\lambda}{4}F_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}F^2_\infty B_\mu B^{\mu}-i\bar{\psi} {\not}\tiny\,\normalsize\partial \psi-\frac{\gamma}{\sqrt{2}}T_\infty \bar{\psi}\psi]\] contains non-canonical term for the \(\Theta\) with kinetic coefficient \[K(\Theta)\equiv\frac{1}{4|a|\Theta^2}\] quadrupole at \(\Theta=0)\) and a constant \[a\equiv\frac{\xi}{1+6\xi}<0\] canonical field correction via field redefinition \[\Theta^{-1}=exp(\frac{2\sqrt{|a|\phi}}{M_P})\] \[ S=\int dx^4 x\sqrt{-g}[\frac{M_P^2}{2}R-\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}F^4_\infty-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}F^2_\infty B_\mu B^\mu-i\bar{\psi}{\not}\tiny\,\normalsize\partial \psi+\frac{y}{\sqrt{2}}F_\infty\bar{\phi}\phi ]\] https://arxiv.org/abs/1807.02376 Edited April 21 by Mordred
Mordred Posted April 21 Author Posted April 21 (edited) \[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\] Edited April 21 by Mordred
Orion1 Posted April 23 Posted April 23 (edited) On 4/4/2024 at 11:32 PM, Mordred said: Christoffels for the FLRW metric in spherical coordinates. [math]\color{blue}{\text{Your LaTex Karate has improved, what LaTex software are you using?, just some clarification on formal denotation.}}[/math] [math]\;[/math] [math]\color{blue}{g_{\mu \nu} \text{ and } g_{\alpha \beta} \text{ are formally denoted for the metric spacetime tensor in General Relativity.}}[/math] [math]\color{blue}{G_{\mu \nu} \text{ and } G_{\alpha \beta} \text{ are formally denoted for the Einstein tensor in General Relativity.}}[/math] [math]\;[/math] [math]\color{blue}{\text{The Friedmann–Lemaître–Robertson–Walker FLRW metric:} \; (\text{ref. 1})}[/math] [math]ds^2 = -c \; dt^2 + \frac{a\left(t \right)^2}{1 - k r^2} dr^2 + a\left(t \right)^2 r^2 d \theta^2 + a\left(t \right)^2 r^2 \sin^2 \theta \; d \phi^2[/math] [math]\;[/math] [math]\color{blue}{\text{The metric spacetime tensor in General Relativity for the FLRW metric:} \; (\text{ref. 3, sec. 3.2})}[/math] [math]g_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & \frac{a\left(t \right)^2}{1 - k r^2} & 0 & 0 \\ 0 & 0 & a\left(t \right)^2 r^2 & 0 \\ 0 & 0 & 0 & a\left(t \right)^2 r^2 \sin^2 \theta \end{pmatrix}[/math] [math]\;[/math] [math]\color{blue}{\text{The Einstein tensor in General Relativity:} \; (\text{ref. 2, ref. 3, eq. 3.17})}[/math] [math]G_{\mu \nu } = R_{\mu \nu } - \frac{1}{2} g_{\mu \nu } R[/math] [math]\;[/math] [math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math] [math]\;[/math] Reference: Friedmann-Lemaître-Robertson-Walker metric: (ref. 1) https://en.wikipedia.org/wiki/Friedmann-Lemaître-Robertson-Walker_metric Wikipedia - Einstein tensor (ref. 2) https://en.wikipedia.org/wiki/Einstein_tensor Relativistic Cosmology - M. Pettini: (ref. 3) https://people.ast.cam.ac.uk/~pettini/Intro Cosmology/Lecture03.pdf Edited April 23 by Orion1 source code correction...
Mordred Posted April 23 Author Posted April 23 (edited) Hey @Orion1 welcome back mate. No software I manually type in the latex. Lol thanks for the reminder to keep the metric tensor separate from the Einstein tensor lol Edited April 23 by Mordred
Mordred Posted April 23 Author Posted April 23 On 4/4/2024 at 11:32 PM, Mordred said: Christoffels for the FLRW metric in spherical coordinates. ds2=−c(dt2)+a(t)1−kr2dr2+a2(t)r2dθ2+a2(t)r2sin2dϕ gμν=⎛⎝⎜⎜⎜⎜−10000a21−kr20000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟ Γ0μν=⎛⎝⎜⎜⎜⎜00000a1−(kr2)0000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟ Γ1μν=⎛⎝⎜⎜⎜⎜⎜⎜0a˙ca00a˙caaa˙c(1−kr2)00001caa˙r200001caa˙sin2θ⎞⎠⎟⎟⎟⎟⎟⎟ Γ2μν=⎛⎝⎜⎜⎜⎜⎜00a˙ca0001r0a˙ca1r00000−sinθcosθ⎞⎠⎟⎟⎟⎟⎟ Γ3μν=⎛⎝⎜⎜⎜⎜⎜000a˙c0001r000cotθa˙ca1rcotθ0⎞⎠⎟⎟⎟⎟⎟ a˙ is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term Newton limit geodesic drdt2=−c2Γ100 Christoffel Newton limit Γ100=GMc2r2 Covariant derivative of a vector Aλ ∇μAλ=∂μAλ+ΓλμνAν Correction applied lol
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