# Tagged Questions

Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a [tag:calabi-yau] manifold.

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### $f^{\prime}(R)=0$ in $f(R)$ gravity

Suppose in a certain $f(R)$ gravity theory, $f^{\prime}(R)=0$ for some finite value of $R$. (e.g. let $f(R)=R+\alpha R^2$ with $\alpha<0$. $f^{\prime}(R)=0$ at $R=-\frac{1}{2\alpha}$.) Also ...
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### Does mass compress space-time?

My understanding of relativity explains that the presence of mass warps space-time so that light travelling through the warp follows at straight line but the warp itself is curved and therefore the ...
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### How would it be to live in a very small universe, let´s say 20x20 square meters?

Let´s consider a curved universe that is very small, say 20 square meters and not expanding. If you stood at the middle of this tiny universe and looked forward, you wouldn´t see any walls, since it ...
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### Gauss-Bonnet term in Physics

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
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### Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
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### Examples of manifolds (not) being: flat, homogeneous and isotropic

I am looking for (at least) one example of the following manifolds: Flat, homogeneous and isotropic Curved, homogeneous and isotropic Flat, non-homogeneous and isotropic Flat, homogeneous and non-...
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### de Sitter and anti de Sitter metric

Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
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### If gravity is due to curvature, how does gravity work in situations with no curvature?

The strength of the gravitational field falls off as the inverse square of the distance from a spherical source. It only falls off as the inverse of the distance from an extended cylindrical or line ...
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### Is there inflationary solution in $R^2$ theory in Jordan frame?

In the Starobinsky $R^2$ inflation model, one usually uses a conformal transformation from Jordan frame to Einstein frame in which the action can be written just like Einstein action + scalar field ...
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### Proper time and asymptotic flatness

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time $\tau$ is infinite for a timelike geodesic, does it mean ...
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### Zero mean curvature and maximal volume

I have a 1+1 asympotically flat spacetime system (spherical symmetry) for which I'm trying to find the hypersurface which maximizes the volume enclosed in a sphere of a given radius $r=R$. **It seems ...
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### Scattering theory of Dirac equation in curved space-time in presence of a strong magnetic field

What is the exact solution of the Dirac equation in curved space-time in the presence of a strong magnetic field? The solution should be in momentum space for simplicity to calculate scattering cross ...
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