# 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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### Calculating Christoffel symbols from Lagrangian

I was given the following metric for a sphere $$g_{\mu\nu} = diag(1, r^2, r^2\sin^2\theta)$$ and tasked to calculate the Christoffel symbols. There are 2 ways that I know of to calculate them. One ...
131 views

### Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
130 views

### Free-falling from rest into a Kerr black hole

Is it impossible for a particle (with zero angular momentum) to free-fall from rest at infinity into the ergosphere of a Kerr black hole? It seems like it is very easy to show this is the case, but ...
237 views

### How warped spacetime bends trajectories of light and moving objects?

I fail to see why the light follows something like the blue line and not the green line on the attached image. Figure 1 - light bends around warped spacetime Afaik. something similar happens ...
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### Principle of Sufficient Reason on light travelling in straight line

I was reading a book Laws and Symmetry by Bas C. Van Fraassen I found that there is an argument for arguing that light travel in straight line: Leibniz's reconstruction of these arguments goes ...
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### Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
233 views

### Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
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### Lagrangian, geodesics and relativity [closed]

My background is in maths, but I have been studying some basic physics with occasional input from a friend who is studying for a physics PhD. Due to my background, I am keen to visualize things ...
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### Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
141 views

### Geodetic effect and Frame dragging

Two gyroscopes pointing perpendicular to each other were housed inside Gravity Probe B which performed polar orbit around Earth to test Einstein's theory of relativity. As the probe is orbiting ...
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### Deriving the geodesic equation [closed]

I having been reading a general relativity book, but when in comes to the geodesic equation, it is not derived. How does one go about doing this?
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### Null geodesics in FRW metric: why angular coordinates are constant?

Consider a ray passing through $r=0$ in the FRW metric $ds^2 = -dt^2 +a(t)^2(\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin{\theta}^2d\phi^2))$ The geodesic curve is parametrized by the affine parameter ...
137 views

### Schwarzschild metric circular orbits and kepler's 3rd law

I have been looking at the Schwarzschild metric presented to me as the following within lectures: ...
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### Is the Weyl Postulate correct?

The Weyl postulate in cosmology states that worldlines do not intersect but it can be shown in GR that using Raychaudhuri equation that geodesics can intersect if there is curvature so I'm really ...
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### Why is $p_\phi$ conserved in a Schwarzschild orbit?

This arises from the question What is the relationship between $a$ and $m$, which I'm afraid I answered just by looking it up in Schutz's book. However Schutz (as he frequently does) glosses over ...
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### Are time and gravity affected when at rest compared to free fall?

A falling object moves along a geodesic path ('straight path') in spacetime. When it comes to rest it now follows a 'curved path' through spacetime. Is the passage of time and force of gravity ...
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### Curved paths through spacetime when standing still?

I have heard that falling objects fall at the same rate irrespective of their mass. They are 'following straight line paths through curved spacetime'. Does this mean that objects that accelerate in ...
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### If gravitation is due to space-time curvature, how can a body free-fall in a straight line?

According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ...
162 views

### How close can an observer approach the black hole in an unpowered flyby without falling into it?

In classical mechanics by choosing the right trajectory you can approach a planet arbitrarily closely, if there is no atmosphere or anything to slow you down, you can approach the surface then fly ...
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### Free fall coordinates/Fermi (normal) coordinates

It makes sense intuitively given the equivalent principle, and I've seen many times it stated, that for a free fall (geodesic) path in an arbitrary spacetime, we can choose our coordinate system to ...
476 views

### Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
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### Is it possible to express various nonlinear motions as straight lines in transformed spacetime?

I am trying to understand simple examples of space-time curvature. Assume for the moment that $c$ is infinite (classical curvature due to Newton's laws). Also, I will only consider 1+1-dimensional ...
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### All geodesics are inextendable?

I think the title is true, because geodesics has a tangent vector with a constant length parametrized by an affine parameter. Probably, it is easier to think about timelike or spacelike geodesics. ...
3k views

### 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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### Is the orbit in Schwarzshild metric a path with maximal proper time?

In curved spacetime particles follow timelike geodesics, which should have maximal proper time (at least locally). I thought this path usually corresponds to a global maximum, and there are only ...
127 views

### Moving From Schwarzchild Geodesic Equations to Equations of Motion

So I am a student and decided (for some bizarre reason) to attempt to tackle general relativity for my final astrophysics and computational physics project this term. I have been doing a lot of ...
80 views

### What happens as the stable orbital velocity approaches the speed of light?

Based on my understanding of the relationship between planetary mass, orbital radius and the velocity for stable orbit, a satellite orbiting a mass equivalent to Earth with an altitude of ~5mm would ...
86 views

### Path of light in Kerr metric? [closed]

How can one find the trajectory of light in various direction in the Kerr metric? Just wondering if there are some classes of solutions, I don't need exact formula. Are there different classes than ...
313 views

### Ray tracing in General Relativity

I would like to find out what one would see at the Schwarzschild radius of a massive non-rotating black hole, if the black hole is surrounded by a bright ring. For that, I would place the observer at ...
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### Why don't objects following a geodesic maintain their rotational state?

If I throw a ball into the air, it comes back down because that is the shape of spacetime and the ball is just following it. But if I paint a spot on the ball and throw it upwards with no rotational ...
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### Circular orbit in Schwarzschild coordinates [closed]

This was an example in a general relativity textbook which I've been trying to work through myself. A spaceship uses its rocket engine to maintain a circular orbit around a Schwarzschild black hole ...
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### Geodesic deviation equation - why does the ordinary second derivative give the correct answer?

I've calculated the correct answer to my problem, but don't understand one of the assumptions I made when doing so. I used the geodesic deviation equation ...
64 views

### Does velocity determine a geodesic?

If the gravity effect we witness is due to objects travelling along geodesics, why is the geodesic different for objects with different velocities if there is no gravitational force as such? For ...
I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: $$ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2)$$ ...