89 votes
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What is really curved, spacetime, or simply the coordinate lines?

Congratulations! You stumbled upon an important question of differential geometry: How can I know whether the curvature is caused by my choice of coordinates or the space I live in? As has been ...
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40 votes
Accepted

GR and my journey to the centre of the Earth

That is awesome! And it makes complete sense too! (other than a possible misusage of the word "distance"). Let's have a look at the equations of motion of you in Earth's curved spacetime, assuming ...
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30 votes

Is Fermat's principle only an approximation?

In general relativity, it's not entirely clear what "least time" means, since you have to ask "whose time are you talking about"? Are you talking about the time as measured by the emitter? The ...
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27 votes

What is the physical meaning of the affine parameter for null geodesic?

Following David Z's answer, the proof for the last paragraph is: since $t$ is an affine parameter it satisfies: \begin{equation} \frac{\mathrm d^2x^a}{\mathrm dt^2}-\Gamma^a_{bc}\frac{\mathrm dx^b}{\...
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  • 371
26 votes

Intuitively, why do attempts to delay hitting a black hole singularity cause you to reach it faster?

Actually, it turns out to be incorrect that the optimal strategy is to free fall. There is an optimal strategy for firing your rocket engine which maximizes your proper time from the event horizon to ...
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  • 68.6k
24 votes
Accepted

How close does a photon have to get to a black hole to do a full loop?

The motion of a photon in a Schwarzschild spacetime is described by $$ \frac{1}{L^2} \dot{r}^2 + V _{\text{eff}} (r) = \frac{1}{b^2}\,, $$ where $$V _{\text{eff}}(r) = \frac{1}{r} \left(1 - \frac{2GM}{...
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21 votes
Accepted

Is there something wrong with this numerical simulation of Schwarzschild photon orbits?

There seem to be several confusions here. Massive and massless particles behave qualitatively differently, even if the massive particle is traveling very fast. The minimum radius for a stable orbit ...
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  • 95.4k
19 votes

Trouble understanding Caroll's explanation on why geodesics maximize proper time

The fact that the curve doesn't have zero path length is identical to the following 'proof' that $\pi=4$. A detailed explanation can be found in this link, but the main idea is that the black line ...
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19 votes
Accepted

Can we bend a light ray into any closed loop?

Let's try to design an axisymmetric medium in which a concentric circle of radius $R$ is a possible light ray. The index of refraction is $n(r)$. In polar coordinates, light rays close to the desired ...
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  • 3,530
18 votes

What does this depiction of a black hole in the movie Interstellar mean?

The bright parts around the black hole are the accretion disk, which is in reality just a flat disk in the equatorial plane similar to the rings of Saturn, but is distorted visually by gravitational ...
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  • 6,050
18 votes

Chasing someone who has fallen into a black hole

Assuming that the black hole is large enough that one can cross the event horizon without being spaghettified by tidal forces, and the that when the accident happened, the both of you were hovering in ...
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17 votes
Accepted

Geodesic Equation from variation: Is the squared lagrangian equivalent?

The upshot of this answer is as follows: if a path satisfies the Euler-Lagrange equations for $L^2/2$, then it will satisfy the Euler-Lagrange equations for $L$, but the converse does not hold unless ...
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  • 54.8k
17 votes

Why are massive bodies following a different trajectory in a gravity field than light?

Light and matter both follow the curvature of spacetime when passing a massive object. The difference is that matter is ALWAYS slower than light, it will be in the more curved spacetime longer, so it'...
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17 votes
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"And God said ... and the universe was ..." What does this equation mean?

$\gamma:\mathbb R\rightarrow M$ is a curve whose image lies in the spacetime $M$, so $\gamma(t)$ is the event at parameter value $t$ along the curve. $\gamma'(t) \in T_{\gamma(t)}M$ is the tangent ...
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  • 53.6k
16 votes

Does gravity bend gravity?

I've had this doubt a while ago and I've asked an expert in my university, so this answer may not be up to date, although honestly I don't think anything has changed. This is actually an open question ...
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15 votes
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AdS Space Boundary and Geodesics

As far as I could understand, it seems that you want to know whether timelike geodesics can reach the conformal boundary of AdS. If that's the case (please do confirm), the answer is no - no timelike ...
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15 votes
Accepted

If the energy of the photon is conserved along a geodesic why is it redshifted

This is a good question and it tests ones understanding of red shift owing to gravity. The summary of what I am about to say can be stated as follows: the gravitational red-shift of light can best be ...
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14 votes

Geodesic equation from Euler - Lagrange

Let us do the RHS first. This just gives us a derivative on the metric: $$\frac{\partial L}{\partial x^\lambda}=\frac{1}{2}\partial_\lambda g_{\mu\nu}\dot x^\mu\dot x^\nu$$ The first derivative on the ...
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  • 8,433
14 votes

Least Action in General Relativity

OP's Lagrangian is indeed constant on-shell, but not necessarily off-shell. In contrast, the principle of stationary action compares various possibly off-shell paths.
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  • 172k
14 votes

How do geodesics explain two identical balls thrown up at the different speeds?

Two worldlines starting at the same event in spacetime but having different velocities are going in different “directions” in spacetime, even if they are going in the same direction in space. So their ...
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  • 68.6k
13 votes

Least Action in General Relativity

You could ask the same exact thing for the Lagrangian of a nonrelativistic particle. It's an exactly analogous situation. $$ L = \frac{1}{2} m \dot x^2 $$ Now, if we differentiate the Lagrangian with ...
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12 votes
Accepted

Difference between Fermi and Riemann normal coordinates

Take a future-directed timelike curve $\gamma= \gamma(\tau)$, $\tau$ being the proper time along $\gamma$ in the spacetime $M$. Assume that $p = \gamma(0)$ is the initial point of $\gamma$. Fermi ...
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12 votes

Is Fermat's principle only an approximation?

It is the least-time path. Fermat's principle was itself built on two earlier observations: on the one hand the Greeks had noticed that in reflections, light traveled along the least-distance path; ...
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  • 35.1k
12 votes
Accepted

Does an electromagnetic field affect neutral particles via the metric because of the EM stress-energy tensor?

The answer to your question is yes, the metric is influenced by the electromagnetic field, and a neutral particle will follow a geodesic of that metric. Thus, this implies that the neutral particle ...
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  • 6,499
11 votes

What is a Null Geodesic?

A null geodesic is a geodesic (that is: with respect to length extremal line in a manifold), whose tangent vector is a light-like vector everywhere on the geodesic (that is $x(s)$ is a geodesic and $...
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11 votes

What is really curved, spacetime, or simply the coordinate lines?

Both, actually. (of course these are completely different, but both are called "curvature") Coordinates are most definitely curved (that is why they are called curvilinear after all). But there is a ...
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11 votes

Trouble understanding Caroll's explanation on why geodesics maximize proper time

I think that all Caroll meant by this remark was that the stationary point has to be a maximum not a minimum of proper time. It can't be a minimum since there are nearby paths with lower (namely, zero)...
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