# Questions tagged [geodesics]

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).

811 questions
Filter by
Sorted by
Tagged with
24 views

### How do tidal forces on incomplete geodesics determine extendability?

Why can we be sure that the manifold with the metric $(M,g)$ does not have a geodesically complete extension if it has an incomplete timelike geodesic along which the tidal force blows up? Does this ...
• 21
38 views

### Hamiltonian for the time-like particle on the geodesic

I am trying to reproduce the results from this paper. On page 2 of the paper, they have an equation: $$2 H=-\frac{\dot{r}^2}{g(r)}-L \dot{\phi }+E \dot{t}=\epsilon\tag{9}$$ where they make a comment ...
• 105
132 views

### How much longer is the path through spacetime of a mass that falls freely compared to a resting mass?

A mass that falls to Earth follows a shortest path through spacetime. If a mass falls from a 1km high building, how much longer will its path be compared to a mass resting on a table?
61 views

### Geodesic: maximal aging versus extremal aging

From Exploring Black Holes, by Taylor and Wheeler, page 1-7: Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of ...
• 716
26 views

• 39
37 views

### Confusions about Schwarzchild Geodesic Deviation

In schwarzchild metric space gets bigger as you approach the horizon, if you build shells around a black hole infinitesimally small distance apart you can build an infinite number of such shells. ...
• 495
169 views

### What is the relative acceleration composition law in General relativity?

In Euclidean geometry we have the following relative acceleration composition law: $$\vec a_{DE} + \vec a_{EF} = \vec a_{DF}$$ Where the relative acceleration between $i$ and $j$ for any $i$ and $j$ ...
• 1,063
36 views

### Calculating coordinate increase of light ray escaping black hole

Consider a light ray near a black hole described by Eddington-Finkelstein coordinates $(v,r,\theta,\varphi)$. My aim is to calculate the increase of the coordinate $v$ along a radial path from the ...
• 295
39 views

### How to plot off-equatorial orbits in Kerr metric?

I am trying to plot the time-like trajectories in Kerr metric. I have taken the equations from here. I am trying to reproduce the off-equatorial trajectories from Chandrasekhar's textbook, e.g., the ...
• 1,754
1 vote
25 views

### Showing the equivalence principle mathematically [duplicate]

Given the geodesic equation $$\ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\lambda}\dot{x}^{\nu}\dot{x}^{\lambda} = 0$$ I wish to find a co-ordinate system around a point $x_0$ such that the geodesic equation ...
• 912
195 views

### Removing a Coordinate Singularity of a 2D metric

While trying to find the null geodesics of the metric $$ds^2 = (r^2 - 1)dt^2 - (drdt + dtdr)$$ gives $$\frac{dt}{dr} = \frac{2}{r^2-1}$$ which is singular at $r=1$. However, we know that this is a ...
64 views

### Infinitesimal geodesic motion directly from the metric?

How can I see---directly from the Schwarzschild metric---that initially stationary (w.r.t. Schwarzschild coordinates) inertial test clocks will begin to fall toward e.g. the Earth (i.e. far outside ...
• 398
205 views

### Can the geodesic equation be understood in plain english to articulate the radial attraction of gravity?

I'm looking to gain an intuitive understanding of the geodesic equation (which incorporates the Christoffel symbol) and how it is used to calculate the radial attraction of gravity. In its native form ...
• 181
65 views

### How does the book prove that "a free stone moves so that its wristwatch time along each segment of its worldline is a maximum"?

In the book Exploring Black Holes in the second chapter they say over and over again that "The Principle of Maximal Aging tells us that a falling stone moves so that its summed wristwatch time is ...
• 3,682
1 vote
100 views

### Understanding Proper Time Parametrization

I am having trouble understanding parametrizing a path in proper time. So my understanding is that using proper time to parametrize a path corresponds to the rest frame of the particle whereby it ...
• 891
40 views

### Does gravitational lensing make objects that would be obscured behind other objects completely visible?

Reading a small amount about gravitational lensing and viewing many of the visualizations, it appeared that bodies directly behind other massive objects (from some point of view; namely galaxies ...
• 277
84 views

### Schwarzschild's null-geodesic new form or an error?

My question is whether or not this form (radial acceleration of a photon) $$\ddot{r}=\frac{L^2}{r^4}(r-3M)$$ is correct ? Recall the standard set of second-order ODE for the Schwarzschild metric (for ...
• 23
1 vote
88 views

### Changing Coordinates, and the Geodesic Equation in GR

I have a question about doing Lorentz-like coordinate transformations in general relativity. I will try not to get into too much detail about what exactly I am trying to do to not muddy the waters. ...
• 891
92 views

### Spacelike geodesics of FLRW radiation universe

I'm having an interpretation problem with the radial spacelike geodesics in the flat radiation dominated universe. I'm using standard conformal coordinates $\eta \ge 0$ and $\chi \ge 0$ for the ...
• 6,558
88 views

### What are the meaning of geodesics?

I was trying to learn General Relativity. While I was learning about curved geometry, I found out this post. As said by the author, those lines are geodesics on a 2-manifold. I think I don't quite ...
162 views

### How is proper time extremized?

I just completed an exercise that asked me to prove that, in special relativity, free particles move with uniform velocity on geodesics that are straight lines. After doing this problem, I was ...
47 views

### Hyperbolic disks in AdS/CFT

The embedding of AdS space into Minkowski spacetime describes a hyperboloid as e.g. shown in the corresponding Wikipedia article on AdS space. Now my questions are: How does this relate to the ...
22 views

### Would gravity pulling person toward Earth change if velocity of Earth changes? [duplicate]

So I was reading the Albert Einstein's theory of how gravity works. From my understanding, the more mass an object has, the more space-time around it it bends. All objects travels in completely ...
22 views

• 705
1 vote
71 views

### Time taken to travel by light a minimum or maximum?

I'm stuck on the following. We know that the path taken by light to travel between 2 points A and B corresponds to the path which minimises time elapsed. However, from relativity we also know light ...
• 1,357
98 views

### A question for expert in geometrical method and Riemannian metrics

I'm a physical oceanographer with great interest in Theoretical Geophysical Fluid Dynamics. I have some ideas on the possibility to derive the so-called: geostrophic equilibrium (i.e. on a rotating ...
36 views

• 2,968
21 views

### Airy's Water Filled Telescope: help required to fully understand the calibration procedure followed

I am trying to understand Airy's Water Filled Telescope and the calibration procedure used (Airy, G. B., "History and Description of the Water Telescope of the Royal Observatory, Greenwich", ...
• 11
45 views

### Generalised Geodesic equation for forces

im wondering if there is a general geodesic equation that describes the forces and how they act. For example I started off with the original nieve derivation of the geodesic equation: \frac{d}{d\tau}...
• 593
1 vote
66 views

### What kind of geodesics are possible in spacetimes of static spherically symmteric perfect fluid spheres?

I ponder about geodesics in static spherically symmetric perfect fluid spheres. My first thought was that only radial geodesics, i.e. geodesics with zero angular momentum ($l=0$) are possible because ...
• 1,197
29 views

### If $F^2 = g_{pq} \dot{x^p}\dot{x^q}$ , where $g_{pq}$ is a metric tensor, then find $\frac{\partial F}{\partial{\dot{x^k}}}$

I am trying to find the geodesics in a Riemannian space, using Tensor analysis. I am also using the Principle of Variation. I want to minimize the geodesics integral whose integrand is $F$. Then, ...
49 views

### Distance between two points on Earth (arc length) doesn't increase with altitude (radius) according to GPS

Consider the following example: Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X. Point C has ...
115 views

### What are geodesic curves in static perfect fluid sphere?

I have read that in perfect fluid only dust particles follow geodesics. If there is pressure in fluid, the particle trajectories are not geodesics . My intention was to describe a static spacetime ...
• 1,197
1 vote
34 views

### On the capacity for equations of motion to be contained in field equations

I've heard that the equation of geodesic motion can be derived from the vacuum Einstein field equations, although there appears to be some debate about how rigorously this can be proved, due to a ...
• 2,002
It is well known that for a Riemannian manifold $(M, g)$, one can define a torsionless and metric-compatible connection $\nabla$, and then use $\nabla$ to construct geodesics and normal coordinates. ...