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90 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 ...
Lukas Juhrich's user avatar
44 votes
Accepted

How does a laser from Earth manage to hit the Moon with precision?

Commenters on the first version of this answer (preserved in the edit history) did me the favor of finding the literature that I hadn't read for years, which contains a succinct answer to your ...
rob's user avatar
  • 91.1k
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 ...
Giorgio Comitini's user avatar
34 votes

In general relativity, why is Earth able to accelerate?

This is quite a common confusion for students new to general relativity, and it's because in GR the term acceleration means something slightly different from its everyday usage. In everyday usage we ...
John Rennie's user avatar
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 ...
Michael Seifert's user avatar
28 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}{...
Jacopo Tissino's user avatar
27 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 ...
Dale's user avatar
  • 102k
27 votes

Why does a piece of thread form a straight line when we pull it?

As @Fardin pointed out, you only get a straight line in the absence of gravity. While gravity is active, the string will form a catenary. In general, the string will try to follow the shortest ...
hdhondt's user avatar
  • 11k
23 votes

How does a laser from Earth manage to hit the Moon with precision?

As Rob mentions, when reflecting a laser off the Moon you do need to compensate by ~1.4 arc-seconds for the light-time delay. But the whole point of the Lunar Laser Ranging experiments (LLR) is to ...
PM 2Ring's user avatar
  • 12.2k
22 votes

Why does a piece of thread form a straight line when we pull it?

@hdhondt gave a good conceptual answer to this question, namely "if it wasn't the shortest path, the tension would try and shorten it." Here's the same answer, but less concise and in math. ...
linkhyrule5's user avatar
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 ...
knzhou's user avatar
  • 103k
20 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 ...
AccidentalTaylorExpansion's user avatar
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 ...
nanoman's user avatar
  • 3,870
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 ...
hmakholm left over Monica's user avatar
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'...
Adrian Howard's user avatar
17 votes
Accepted

"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 ...
J. Murray's user avatar
  • 70k
17 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 ...
Emanuele Giordano's user avatar
17 votes

Aren't places where geodesics end singularities?

Do not confuse geodesics (the abstract geometrical things) with the paths of particles. Particles follow geodesics, but the fact that a particle is created at some point and destroyed at another doesn'...
Eric Smith's user avatar
  • 9,256
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 ...
Andrew Steane's user avatar
15 votes

Aren't places where geodesics end singularities?

This doesn't seem to be so because geodesics are an intrinsic property of space time and not simply the path of a light ray or quantum. Indeed we might say that the light is no longer traveling along ...
Albertus Magnus's user avatar
14 votes

To which extent is general relativity a gauge theory?

The precise sense in which general relativity is a gauge theory has been known (but apparently largely overlooked) for decades. The original sources for what I'm summarizing in the following are a ...
ACuriousMind's user avatar
  • 126k
14 votes

Equivalence of the geodesic equation and the continuity equation for the energy-momentum tensor

Just be careful with what quantity depends on what argument, cf. above comment by user NowIGetToLearnWhatAHeadIs. Then it works like a charm: $$\begin{align} \nabla^{(y)}_{\mu} T^{\mu\nu}(y) ~=~&...
Qmechanic's user avatar
  • 205k
14 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 ...
user1379857's user avatar
  • 11.5k
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 ...
Dale's user avatar
  • 102k
13 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 ...
Valter Moretti's user avatar
13 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 ...
Cham's user avatar
  • 7,572
13 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)...
Andrew Steane's user avatar
13 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.
Qmechanic's user avatar
  • 205k
13 votes

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

The Schwarzschild radius, $$r_s = \frac{2GM}{c^2}$$ is the natural distance unit to use when discussing black holes. It's convenient to work in units where $r_s=1$. A Schwarzschild black hole is ...
PM 2Ring's user avatar
  • 12.2k

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