Articles of general relativity

Tensor equations. Can I change an equation from covariant to contravariant?

Say I have a tensor equation like $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$. Does this also imply that $G^{ab}=R^{ab}-\frac{1}{2}g^{ab}R$?

What is the difference between intrinsic and extrinsic curvature?

In general relativity, energy bends spacetime. However, this doesn’t mean that a fifth dimension for spacetime to “bend into” exists.” That is, spacetime isn’t embedded in a higher dimensional space, Instead, the curvature is said to be intrinsic. But what does that mean? One could imagine the sphere on a ball as an example of […]

Calculating Christoffel symbols using variational geodesic equation

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 $$L = g_{ab} {\dot{x}}^a {\dot{x}}^b = e^v \dot{t}^2 – e^{\lambda} \dot{r}^2 – r^2 \dot{\theta}^2 – r^2 \sin^2 \theta \dot{\phi}^2.$$ […]

Differentiating a function of a variable with respect to the variable's derivative

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 […]

Most succinct proof of the uniqueness and existence of the Levi-Civita connection.

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.

Does the Riemann tensor encode all information about the second derivatives of the metric?

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 $g_{\mu\nu}$ […]

Open problems in General Relativity

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?

Tensor Calculus

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 […]

Riemannian Geometry book to complement General Relativity course?

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 […]

Solving ODE with negative expansion power series

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 […]