# Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### If two hypersurfaces are conformally related and one of them is a Cauchy surface, is the other a Cauchy surface too?

I'm reading Wald's "General Relativity". In appendix D he states that conformal transformations preserve causal structure. Does this mean that they also preserve when a hypersurface is a ...
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### Christoffel symbols of Poincaré metric using orthonormal tetrad formalism

I want to calculate the Christoffel symbols for the Poincaré metric using the orthonormal tetrad formalism. $$ds^2 = y^{-2}dx^2 + y^{-2}dy^2.$$ I introduce a non coordinate orthonormal basis with one-...
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### Riemann curvature tensor is the only tensor that you can write down that has two derivatives of the metric tensor

Is the statement in the main question correct? Can someone send me a proof (link, pdf file etc.)
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### What are the analogues of $F_{\mu\nu}$ in General Relativity?

In electromagnetism, the measurable gauge-invariant quantities are the electric and magnetic fields or the six independent components of the field strength tensor $F_{\mu\nu}$. What are the analogues ...
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### Why is Divergence of a vector field which is decreasing in magnitude as we move away from origin positive at points other than origin?

divergence of $\frac{\hat{r}}{r}$ is positive, $\frac{\hat{r}}{r^2}$ is zero and $\frac{\hat{r}}{r^3}$ is negative at points other than origin. As I have studied, divergence in 3D space tells whether ...
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### Intrinsic definition of canonical vector field on tangent bundle

I have been trying to solve the following question's last part (iv): on giving an intrinsic definition to the vector field whose local definition is that in part (iii). I believe this is called the ...
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### About the existenece and uniqueness of time-like vector fields in a time orientable manifold

I'm reading Wald's book "General Relativity" and I'm having a bit of trouble understanding his proof on the following lemma: Let $(\mathcal{M}, g_{ab})$ be time orientable. Then, there ...
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### Ricci scalar of AdS in $D$ spacetime dimensions from structure equations

Starting from the AdS metric in $D$ spacetime dimensions in Poincare coordinates $ds^2 = \frac{R^2}{(x^3)^2}\eta_{\mu\nu}dx^\mu dx^\nu$ (R here is the AdS radius), I would like to compute the ...
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### Transformation law for the Levi-Civita symbol under a change of basis

I'm trying to prove that the Levi-Civita symbol $\epsilon_{i_1 ... i_n}$ is a tensor density of weight $w=-1$. For this purpose, it has to be shown that the transformation law for the components of ...
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### Why would geodesics depend on velocity in general relativity?

In general relativity, it's the curvature of spacetime that gives the effect of gravity, due to objects following geodesics. What I learnt about geodesics is that they are like straight lines in ...
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### Dependence of entropy on topology?

In black hole thermodynamics, there is a nontrivial dependence of thermodynamical entropy on the area of the event horizon which is a geometric object. I want to know how this may be intuitively ...