Questions tagged [gauss-bonnet]
The gauss-bonnet tag has no usage guidance.
21
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Calculating Gaussian Curvature for metric
I am trying to calculate Gaussian curvature of an optical metric
$$
d \sigma^2=\frac{r\left(\omega_{\infty}^2-\omega_e^2\right)+2 m \omega_e^2}{(r-2 m) \omega_{\infty}^2}\left(\frac{d r^2}{1-\frac{2 m}...
- 21
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86
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Gauss-Bonnet term in tetrad formalism
In eq. 4 of https://arxiv.org/abs/hep-th/9508128 it is stated that the Gauss-Bonnet term for gravity in 4d can be written as
$$
S=\frac{1}{4}\int d^{4}x\,\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}R_{\...
- 21
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63
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Invariance of the Euler characteristic of a manifold with boundaries
It is known that in a compact manifold of dimension $d=4$, the following integral is invariant under small deformations of the metric tensor
$$
\int_\mathcal M \sqrt{g}(R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\...
- 123
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73
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Simplest way to see that the Euler density is topological
I’m looking for a simple way to see that the integrated Euler density $\sqrt{g} \, E_{2n}$ is topological (i.e. metric-independent in general even dimension $2n$). I can see it it must be true in two ...
- 169
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72
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Duality and corrections to second-order gravity without and with torsion terms
Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like:
$$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{...
- 6,611
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1
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461
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Calculating Gaussian Curvature in 4D
I am kind of confused by the vast number of formulas for computing the Gaussian Curvature. Having a metric tensor / an expression for the line element in 4D (e.g. $t,x,y,z$ or in spherical coordinates ...
- 69
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Gaussian curvature in a weird metric
Consider a disformal metric $\tilde{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$, with $g_{\mu\nu}$ being the Schwarzschild metric tensor perturbed by $h_{\mu\nu}$, a generic second-rank symmetric tensor. ...
- 69
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180
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Gauss-Bonnet topological invariant at linear order
I have a question about the Gauss-Bonnet invariant $T$,
\begin{equation}
T=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-4R_{\mu\nu}R^{\mu\nu}+R^2
\end{equation}
It is known that $T$ is a topological ...
- 91
2
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163
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Variation of the Gauss-Bonnet term [closed]
We have the Gauss-Bonnet term
$$L_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$
where $R$, $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ are the Ricci scalar, the Ricci ...
- 43
3
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192
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Graviton propagator, and Gauss-Bonnet gravity
Let's say we consider Einstein's Lagrangian from GR. In linearized gravity, we would expand the Ricci scalar to quadratic order in the perturbation parameter to find the propagator.
My question is as ...
- 1,182
3
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1
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183
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Is the Palatini-Lovelock action of order $k$ topological in $2k$ dimensions?
I am interested in Lovelock actions in the metric-affine (or Palatini) formalism. It is well-known that the metric version (starting from the Levi-Civita curvature) of the Lovelock lagrangian of order ...
- 567
3
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1
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388
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What is the physical significance of Gaussian curvature in condensed matter physics?
In basic models concerning two-level systems, we deal with manifolds such as the Bloch sphere and torus. I believe that the Chern number is what dominates the theory in terms of ties to differential ...
- 591
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260
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Euler charateristic of the universe
According to Gauss-Bonnet theorem, the volume integral
$$\int_M d^4x\sqrt{-g}(R^{abcd}R_{abcd}-4R^{ab}R_{ab}+R^2)=\chi(M)$$
$\chi(M)$ is the Euler characteristic of the manifold $M$.
Let us now ...
- 1,487
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How to show the Gauss-Bonnet term is a total derivative?
It is well-known that the Gauss-Bonnet term
$$\mathcal L_G =R^2 -4 R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\tag 1$$
does not contribute to the equations of motion when adding it ...
user146681
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Gauss-Bonnet correction to Carnot's efficiency
Could anyone tell explicitly, how did the author get the $T_c$ vs. $\alpha$ plot given in Figure 3 in Gauss-Bonnet Black Holes and Holographic Heat Engines Beyond Large N.
- 31