# Tagged Questions

For questions involving consideration of the shortest (or longest) path between two points in a curved space (e.g. a straight line between two points on the surface of a sphere such as the earth).

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### GR and my journey to the centre of the Earth

[General Relativity] basically says that the reason you are sticking to the floor right now is that the shortest distance between today and tomorrow is through the center of the Earth. I love ...
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### The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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### What does this depiction of a black hole in the movie Interstellar mean?

I was expecting a whirlpool in 3D and the matter glowing from friction as it nears the center, as I expected a event horizon to be negligible visually. How does this depiction work? How big is the ...
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### How can a point-like particle “feel” gravity, if locally the curvature of spacetime is always flat?

I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that Locally, as expressed in the ...
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### Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2$. ...
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### Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: $$ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2)$$ ...
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### In general relativity, are light-like curves light-like geodesics?

Just as the title. If a curve is light-like, i.e. a null-curve, is it definitely a null geodesic?
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### “WLOG” re Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem ...
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### Local inertial coordinates/Fermi normal coordinates

It is said that we can introduce local inertial coordinates/Fermi normal coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do ...
Considering the fact that electrons tend to take the maximum conductance path to flow from A to B. This is justified by saying that $\vec{E}$ is larger in conductors. But once similarly it was thought ...
### Sign of $dr$ in Schwarzschild geodesics
There is an equation that relates energy $E$, angular momentum $L$ and other constants and variables to find $\left(\frac{dr}{d\tau}\right)^2$ in a plane. \left(\frac{dr}{d\tau}\right)^2=\frac{E^2}{...