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:
It is well known that in wave coordinates, vacuum Einstein field equations are equivalent to the following so called reduced Einstein equations:
$$R_{\mu\nu}+g_{\sigma(\mu}\Gamma_{,\nu)}^{\sigma}=0.$$
But I don't know how to show this fact. The…
How can we verify the result that
$$[f\textbf{X},\textbf{Y}]= f[\textbf{X}, \textbf{Y}] - \textbf{Y} \left( f \right) \bf{X}$$
Where:
f- Some function
$\textbf{X}, \textbf{Y}$ - Some vector fields
I am reading a paper in which the author is deducing from the following differential equation
$$\dot{v} + \Big(\frac{\dot{a}}{a} + \frac{\dot{f_2}}{f_2}\Big)(1-v^2)v = 0, $$ that $$\frac{v}{\sqrt{1-v^2}} \propto (af_2)^{-1}.$$
For context, $a$…
I know there are already some questions about this very topic, but my question here is going to be more specific. Are these books the right ones to understand the fundamentals of general relativity? If not, what important topics are not covered by…
I recall in GR class learning that Schwarschild solution was a radially symmetric solution to the field equations, independent of the time parameter $t$ with a coordinate singularity at $r=2m$ and a genuine curvature singularity at $r=0$. Thus we…
Suppose I have some tensor equation which deals with rank-2 tensors of two variables $x,y$,
$$ \frac{\partial K^{ab}}{\partial x^k} g^{kc} = \frac{\partial g^{ab}}{\partial x^k} K^{kc} $$
This is obviously satisfied for $K^{ab} = g^{ab}$, but is…
In General Relativity, it is assumed that the metric can LOCALLY, be always transformed to a metric with Lorentzian signature: $(+,-,-,-)$.
Given a certain metric at a certain space point - What does the transformation, which transforms the metric…
If in Euclidean space one has an equation of the form $f(p) = 0$, then the solutions $\lbrace p\in \mathbb{R}^{n}: f(p) = 0\rbrace$ determine a hypersurface in Euclidean space. Given a Lorentzian manifold, how does one generalise this in terms of…
In General Relativity there is a helpful procedure which is called "Perturbative approach".
Given a chart, and forgetting the basis, the metric is $g_{uv}$ and we can express it as: $$g_{uv}=\eta_{uv}+h_{uv}$$
From a physical point of view the…
I have to prove that $A_{\mu} B^{\mu}$ is Lorentz invariant, and I'd like to check my understanding with you if you don't mind.
My first question is about the definition of Lorentz invariance. Does it mean the following:
$A'_{\mu} B'^{\mu} = A_{\mu}…
So i was given this question:
• If two people are on a train that is travelling at 200km/hr westward along the equator and they decide to play catch. The person throwing the ball is at the back of the train car and the person catching the ball is…
Can anyone point out the real difference between Einstein-Cartan Theory and Metric Affine Gravitation Theory?
Both of them rely on a pseudoriemannian metric $g$ and generalised affine connection $\Gamma$ (which is not the Christoffel symbol) and the…
I'm looking for some clarification on what each of the terms in the relativistic velocity transformation law are.
The formula is: $s = (v+u)/(1 + uv/c^2)$
It would be really great if you could give me some kind of example to explain what u, v and s…
In the Space-Time diagram (in the rest frame) we often take the Space axis as the horizontal axis and the time axis as the axis perpendicular to it as in the given figure.
While there are other models to graphically represent space-time, my…
