Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).
Mathematical topics within the theory or relativity include:
Say we have a Riemannian manifold $(M, g)$ with vector field $Y$, obeying:
$g(Y, Y) = 1$; and
the $1$-form $\varphi(X) = g(X, Y)$ is $d$-closed, $d\varphi = 0$.
I know that the integral curves of $Y$ are geodesics, i.e. $D_Y Y = 0$. Does it follow…
Say we have a Riemannian manifold $(M, g)$ with vector field $X$ obeying the following:
$g(X, X) = 1$; and
the $1$-form $\varphi(Y) = g(Y, X)$ is $d$-closed, $d\varphi = 0$.
Does it necessarily following that the integral curves of $X$ are…
This questions arises from working through the 2015 Heraeus lectures on gravity, specifically tutorial 2 exercise 2, so this is why I post it as a physics question although it is purely mathematical.
When we represent the Moebius strip as a…
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…
I've read that when a hypersurface within a manifold contains a curve, if the curve is a geodesic in the manifold, it is also a geodesic in the hypersurface.
This is quite abstract for me, I've only recently started learning GR, could someone…
In Minkowski spacetime, is it true to say that a null vector is orthogonal to itself? Why or why not? Can a
timelike vector be orthogonal to a null vector? Can a timelike vector be orthogonal to another timelike vector?
The plus and cross polarizations of a gravitational wave are at 45 degree to each other. However, I find no explanation of this angle. Can somebody help?
I want to ask some very naive questions in general relativity. I have the background of PDE and few Riemannian geometry.
After Schwazchild gave a solution, people study its singularity and predict the black hole, but for some PDEs, people can get a…
I know that the Schwarzschild line-element is:
$$ds^2=c^2\left(1-\frac{2\mu}{r}\right)dt^2-\left(1-\frac{2\mu}{r}\right)^{-1}dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2$$
Dividing by $d\lambda^2$, where $\lambda$ is an affine parameter, I…
I am stuck varying an action.
This is the action $$S=\int\mathrm d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2)$$
And this is the solution, $\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h $.
This is what I get…
Consider a closed causal curve $\gamma(t)$, $\gamma(0) = \gamma(1)$, with $\gamma'$ causal everywhere. If we take this curve to be contractible (that is, homotopic to a single point), can it be contained entirely within a single coordinate patch to…
I am trying to understand FRW model and how one interpreted it. I missed a lecture and I am now trying to go through a friends notes. The teacher presented this metric as an example
$$ds^2=-dt^2+\frac{t}{c}(dx^2+dy^2+dz^2) $$
where $c$ is a…
In Robertson-Walker universe, light is emitted from a star with spatial coordinates $(r_s,\theta_s,\phi_s)$. It travels radially inwards and is received by an observer situated at the origin $(r=0)$. Show that the ratio of the observed wavelength…
How to derivate (covariant derivative) the expressions which is function of Ricci scalar?
Also, if R is Ricci scalar,
what would be ∇i∇j F(R) ?, where ∇i is covariant derivative.
The schwarzschild metric is given by:
$ds^2=-(1-\frac {2GM}{r})dt^2+(1-\frac{2GM}r)^{-1} dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$
Here is the well known geodesic equation:
$0=\frac d{d\tau}(g_{\mu \nu}\frac{dx^{\nu}}{d\tau})-\frac…
