Given the line element $$ds^2 = e^v dt^2 – e^{\lambda} dr^2 – r^2 d \theta^2 – r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. Our Lagrangian is $$L = g_{ab} {\dot{x}}^a {\dot{x}}^b = e^v \dot{t}^2 – e^{\lambda} \dot{r}^2 – r^2 \dot{\theta}^2 – r^2 \sin^2 \theta \dot{\phi}^2.$$ […]

Suppose $x:\mathbb{R}\to \mathbb{R}$ is parameterised by $\lambda$. What does it mean to take a derivative of a function $f(x)$ with respect to $\dot{x} = \frac{dx}{d\lambda}$. i.e. what does $\frac{df(x)}{d\dot{x}}$ mean? How do we compute it? Is $\frac{d}{d\dot{x}}=\frac{d}{d\frac{dx}{d\lambda}}=^{??} \frac{d\lambda}{dx}=^{??} 0$ ??? For example, how would one compute $\frac{d}{d\dot{x}} e^x$? (This question has arisen from an undergraduate […]

Seeing as proving the existence and/or uniqueness of the Levi-Civita connection seems to crop up in every single exam in Geometry and General Relativity, what is the most succinct proof of this, to memorize.

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with points $x\in\mathcal M$ and $y\in\mathcal N$. Then you can pick local coordinates on $\mathcal M$ and $\mathcal N$ such that the expressions for $g_{\mu\nu}$ […]

I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. Is there something that still needs to be justified mathematically in order to have solid foundations?

I am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III. I’ve started self studying tensor calculus, my sources are the video lecture series on the YouTube channel; “MathTheBeautiful” and the freeware textbook/notes; “Introduction to Tensor Calculus” by Kees Dullemond & Kasper […]

What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I’m in one right now, but I’d like a pure math perspective on the math that’s introduced as I can imagine, inevitably some things would be swept under the rug […]

Since on Mathematics stackexchange I didn’t get an answer, I’ll try it here, since people here are more familiar with this topic (general relativity related). I am reading a dissertation of Porfyriadis “Boundary Conditions, Effective Action, and Virasoro Algebra for $AdS_3$”, and I am trying to solve a system of DE to get the appropriate […]

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the n sphere $x_0^2 + x_1^2 + ….+x_n^2=R^2$, whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 d\phi_1^2+\sin{\phi_2}^2\sin{\phi_1}^2 d\phi_2^2+……)$$. I have done the same exercise for the 2 sphere, and found that the riemann curvature wih a lowered index is proportional to the product […]

Intereting Posts

How to find line parallel to direction vector and passing through a specific point?
Expected value of integrals of a gaussian process
Finite subgroups of the multiplicative group of a field are cyclic
Proof of Non-Ordering of Complex Field
Strictly associative coproducts
Quotient Space of Hausdorff space
Volume integral of the curl of a vector field
Are there any interesting semigroups that aren't monoids?
Delta Dirac Function
Functions of bounded variation on all $\mathbb{R}$
Diophantine equations solved using algebraic numbers?
Is it coherent to extend $\mathbb{R}$ with a reciprocal of $0$?
Rational map on smooth projective curve
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
How to prove that $\mathbb{Q}$ ( the rationals) is a countable set