Articles of semi riemannian geometry

Index notation.

I have a very basic question concerning index notation, normally used in physic papers. I have never used index notation and it is very difficult to me to translate from index free notation, even in the case of very simple formulas. I would like to check if the following expression in index notation are correctly […]

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

How can the notion of “two curves just touching” (vs. “two curves intersecting”) be expressed for a given metric space?

A popular introductory description of a “tangent (in geometry)” is presented as “the tangent line (or simply tangent) to a plane curve at a given point is the straight line that “just touches” the curve at that point.“ I like to find out whether and how this description might be expressed or translated in the […]

Searching for a thesis.

In several articles about curves, etc, in Minkowski spaces $\Bbb L^3$ and $\Bbb L^4$, there is Walrave, J., Curves and surfaces in Minkowski space. Doctoral Thesis, K.U. Leuven, Fac. of Science, Leuven 1995. as a reference, but I’m having trouble finding said thesis in the internet. Since it dates from a long time ago, I […]

What is the exact motivation for the Minkowski metric?

In introductory texts about Lorentz Geometry, one always learns about the Minkowski space, i.e. $R^4$ with the Minkowski metric $$ m(x, y) := -x_0 y_0 + x_1y_1 + x_2y_2+ x_3 y_3 $$ Using this metric, one can define lightcones $\{x \in R^4 \mid m(x, x) = 0 \}$ and timecones $\{x \in R^4 \mid m(x, […]

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 $g_{\mu\nu}$ […]

Gathering books on Lorentzian Geometry

I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I’m talking about the Lorentzian manifolds and Lorentz-Minkowski spaces (some notations of it are $\Bbb L^n$, $\Bbb E^n_1$, etc). I know that the […]

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = \int_{t_1}^{t_2} \sqrt{1-[f'(t)]^2} \ dt $$ If I minimize $S$, I obtain the condition $f”(t)=0$, which implies $x=ct+x_0$, being a […]