variation of ricci scalar

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Since the Ricci scalar is an invariant, so is \(a_\mathcal R\). our mathematical formalism does not work there). ), Dirac Delta Function and Schwarzschild Singularity.

By its own admission, Susskind has never been able to complete the entire derivation of the Einstein equation from this action, because it's too 'tedious'. Ricci scalar as a master ruler and clock: \(\,\)Anisotropic scaling in time direction, How variable \(c\) alters dilation of wavelength, Modifying the Lemaître redshift formula: A missing exponent of 3/2, Modifying the Hubble law: A reduced value \(H_o ≈ 47\), Fitting the Pantheon SNe1a data using Einstein’s Idea, New interpretation of the acceleration in Type Ia SNe based on Einstein’s Idea, Quadratic gravity: \(\,\)A simple solution, Mannheim’s theory of galactic rotation curves, Geodesics in the presence of variable \(c\), Action of Gravity coupled with fermion, gauge, and Higgs fields, \(\star\) Revisiting Einstein ‘1911 neglected idea: How it sheds new light on key outstanding problems in late-time cosmology, \(\star\) Lemaitre redshift formula in the presence of variable velocity of light: The case for a missing exponent of 3/2, \(\star\) Late-time cosmology: Distance-redshift relationships in the presence of variable velocity of light, \(\star\) Detailed analysis of the Pantheon SNe data in the presence of variable light speed: A new interpretation of the acceleration in late-time cosmology, \(\star\) Applying form invariance to early-time cosmology, \(\star\) Linking form invariance to Mannheim’s phenomenological theory of flattening galactic rotation curves, \(\star\) Action of form-invariant Gravity in vacuo and coupled with fields in Glashow-Salam-Weinberg theory. Thanks for contributing an answer to Physics Stack Exchange! Lawson & Michelsohn (1989), Theorem IV.4.4. Einstein’s desire to have \(c\) vary as a function of the gravitational field can be fulfilled if we make the Ricci scalar (being of dimension length\(^{-2}\)) be the scale setter for the physical processes. In late-time cosmology, as the universe expands inducing to a falling \(\mathcal R\) and hence a falling \(c\), this effect alters the redshift values of the supernovae and shows up in the behavior of the Type Ia SNe. j

We shall adopt his insight by making the Riemann tensor control the rate of evolution of physical processes. Our Lagrangian then is, L=R and we assume that the Einstein-Hilbert action could be epxressed as: Remark: This integral is taken over the whole of space-time if it …



the original partition function of QED can be made form-invariant: $$\begin{aligned}\mathcal{Z} &= \int\mathcal D \bar\psi \mathcal D \psi\mathcal D A_\mu\,\exp {\ i\frac{\hat{\mathcal{S}}}{\hat{\hbar}}}\\\hat{\mathcal{S}} &= \int d\tilde{t}\, d^{3}\tilde{x}\,\hat{\mathcal{L}}\\\mathcal{\hat{L}} & =\bar{\hat{\psi}}\left(i\hat{\hbar}\frac{1}{\hat{\tau}}\gamma^0\frac{\partial}{\partial\tilde{t}} -i\hat{\hbar}\,\hat{c}\,\vec\gamma \tilde{\nabla}-m\hat{c}^{2}\right)\hat{\psi} +e\,\bar{\hat{\psi}}\gamma^\mu\hat{A}_\mu\hat{\psi} -\frac{1}{4}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\\\hat{F}_{\mu\nu} &= \tilde{\partial}_{\mu}\hat{A}_{\nu}-\tilde{\partial}_{\nu}\hat{A}_{\mu}\end{aligned}$$, $$\begin{aligned}\tilde{\partial}_{0} &:= \frac{1}{\hat{c}\,\hat{\tau}}\frac{\partial}{\partial\tilde{t}}\\\tilde{\nabla} &:= \frac{\partial}{\partial\tilde{x}}\end{aligned}$$. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Among those who use index notation for tensors, it is common to use the letter R to represent three different things: These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Expressing the physical length \(x\) in terms of the length multiple \(\tilde x:=x/a_\mathcal R\) and the physical time \(t\) in terms of the time multiple \(\tilde t:=t/\tau_\mathcal R\), we recast the Dirac equation as: $$\left[\left(i\hbar\frac{\partial}{\tau_\mathcal R\,\partial\tilde{t}}+\frac{e^{2}}{a_{\mathcal{R}}\tilde{r}}\right)+i\hbar\, c\,\vec{\alpha}\frac{1}{a_{\mathcal{R}}}\frac{\partial}{\partial\vec{\tilde{x}}}-\beta mc^{2}\right]\psi=0$$, $$\frac{1}{a_{\mathcal{R}}}\left[\left(i\frac{\hbar}{\sqrt{a_{\mathcal{R}}}}\frac{1}{\tau_\mathcal R\, a_{\mathcal{R}}^{-3/2}}\frac{\partial}{\partial\tilde{t}}+\frac{e^{2}}{\tilde{r}}\right)+i\frac{\hbar}{\sqrt{a_{\mathcal{R}}}}\left(c\,\sqrt{a_{\mathcal{R}}}\right)\vec{\alpha}\frac{\partial}{\partial\vec{\tilde{x}}}-\beta m\left(c\,\sqrt{a_{\mathcal{R}}}\right)^{2}\right]\psi=0$$, Finally, we define 3 new constants: $$\hat{\hbar}:=\frac{\hbar}{\sqrt{a_{\mathcal{R}}}},\ \ \hat{c}:=c\,\sqrt{a_{\mathcal{R}}}, \ \ \hat{\tau}:=\tau_\mathcal R\, a_{\mathcal{R}}^{-3/2}$$, The Dirac equation becomes: $$\left[\left(i\hat{\hbar}\frac{1}{\hat{\tau}}\frac{\partial}{\partial\tilde{t}}+\frac{e^{2}}{\tilde{r}}\right)+i\hat{\hbar}\,\hat{c}\,\vec{\alpha}\tilde{\nabla}-\beta m\hat{c}^{2}\right]\psi=0$$. If we want to be formal, at $r=0$ there is a manifold singularity. ) "the resolution of the singularity" issues arise very unambiguously in the components of $R_{abc}{}^{d}$ that don't contribute to $R_{ab}$, all of which contain factors of $\frac{1}{r}$, and whose singularities are clearly physical, because they are coordinate invariant. Let us define at each given location a length numeraire \(a_\mathcal R\) which is tied to the Ricci scalar in the following fashion: $$a_\mathcal R:=\left|\mathcal R\right|^{-1/2}$$. Two identical clocks would revolve at different rates when they experience different values of the Ricci scalar; hence, they would accumulate 2 different total time counts after they have traversed through regions with different values of the Ricci scalar. For a closed Riemannian 2-manifold M, the scalar curvature has a clear relation to the topology of M, expressed by the Gauss–Bonnet theorem: the total scalar curvature of M is equal to 4π times the Euler characteristic of M. For example, the only closed surfaces with metrics of positive scalar curvature are those with positive Euler characteristic: the sphere S2 and RP2.

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