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).

learn more… | top users | synonyms

0
votes
0answers
40 views

Acceleration in AdS

I've been reading some notes ("Anti-de Sitter space" by Bengtsson) on anti-de Sitter space. It is shown in equation 152 that timelike observers at fixed radial distance from the origin experience a ...
0
votes
0answers
43 views

Geodesics on dS and AdS

I have been reading through the following paper "The dS and AdS Sightseeing Tour" (http://www.bourbaphy.fr/moschella.pdf) but I am not clear about why geodesics appear as hyperbola in the dS case (see ...
1
vote
1answer
307 views

Null geodesics vs timelike geodesics

I'm interested in the paragraph under equation (38) of this reference: ...
0
votes
3answers
219 views

What is the meaning of “a straight line”? [duplicate]

The book "The Dancing Wu Li Masters" page 189 talking about General Relativity says "A geodesic is not always a straight line". Is that true? What is a definition of "straight line" that makes sense ...
0
votes
1answer
74 views

The Ricci tensor and its relation to volume

From Wikipedia's entry on Ricci tensor, In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in ...
0
votes
0answers
103 views

How to calculate photons/light trajectory under gravity

I'm aware many questions are out there asking similar questions about photons and gravity and I got the basic concept by searching through them. I will just call it light although it may not be the ...
2
votes
1answer
98 views

Derivation of geodesic deviation equation from two neighbouring geodesics

I'm stuck trying to follow Foster and Nightingale's derivation of the geodesic equation from two neighbouring geodesics $x^{a}\left(u\right)$ and $\tilde{x}^{a}\left(u\right)$ joined by a ...
0
votes
0answers
51 views

How would you describe what the affine parameter is in layman's terms? [duplicate]

I've been trying to learn it from other sites, but I'm not well-versed enough in mathematics to understand.
1
vote
0answers
35 views

Null geodesic equation with affine parameters [duplicate]

A photon's geodesic equation is defined by re-parameterizing the geodesic equation to some parameter other than proper time. This is done because $ds=0$ for the photon. Again if we use affine ...
2
votes
1answer
68 views

Movement of bodies in space, affected only by gravity

I have been extensively studying General Relativity for some time now. Recently I asked myself a question which I can't answer. If the gravitational metric is determined by the Energy content of the ...
3
votes
1answer
68 views

Can you recover a spacetime from its null geodesics?

So, I know that you can learn a lot about a spacetime from its causal structure, but can one completely recover the metric of a spacetime, just knowing the equations for the null geodesics in it? If ...
0
votes
0answers
54 views

Is it true that particles propagate on geodesics in Yang-Mills theory?

I mean free particles. Sorry for the inaccurate wording, I'm new in this field of physics.
1
vote
3answers
186 views

Path of light as it travels between two black holes

What would happen to light passing through a narrow space between the event horizons of two equal-mass black holes? Would it deviate or follow a straight path?
1
vote
1answer
50 views

Why doesn't light vibrate in-situ?

Light always moves in a straight geodesic path (shortest distance between 2 points in flat space where gravity is homogeneous) across 3 dimensions of space and 1 dimension of time. It is consists of a ...
1
vote
0answers
26 views

The null geodesic for given geodesic [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
1
vote
0answers
74 views

What is the null geodesic equation? [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
2
votes
4answers
904 views

When space bends, what are the lines that are being bent?

In an electric field diagram, the lines represent the electrostatic force vector at the position. These lines are bent when you place a charge into the system. What is the equivalent description ...
5
votes
1answer
130 views

How is the Lagrangian defined in GR?

Reading about the Schwarzschild metric in general relativity I see that sometimes $$L=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$$ and sometimes $$L=\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ Which is ...
0
votes
2answers
135 views

How can light enter a black hole if it cannot get out?

I have known for a very long time that light cannot exit a black hole. I can even understand some of the simpler reasonning about it, such as escape velocity, or space geometry inside the black hole. ...
3
votes
2answers
2k views

What is a Null Geodesic? [duplicate]

What is a Null Geodesic? My textbook only explains it as the Minkowski metric which equals to zero, but I'd appreciate a more detailed explanation.
1
vote
1answer
95 views

Invariance in general relativity, university in problems question

From Problem #5 here, Free falling particles' worldlines in General Relativity are geodesics of the spacetime, i.e the curves $x^\mu(\lambda)$ with tangent vector $u^\mu=dx^\mu/d\lambda$, such ...
-1
votes
1answer
323 views

Is a planets orbit really a straight line through curved spacetime? [duplicate]

My understanding is that general relativity concludes that gravity isn't real because it does not exist in all frames of reference. Also that mass and energy warp spacetime into a curved geometry. ...
1
vote
0answers
74 views

Conformal time-like Killing vector near null geodesics in all spacetimes?

Is it true that in all spacetimes there is some conformal time-like Killing vector $\tau^a$ in the vicinity of null geodesics? If the above statement is true then can one argue that, for all ...
2
votes
1answer
76 views

If time-like paths are geodesics, what physical principle applies to space-like intervals?

If I have a number of particles interacting with one another locally, then the center of mass of the system moves along a geodesic. Taking this further with the particles interacting via an EM field, ...
0
votes
2answers
122 views

“Shortest” path in general relativity

My professor in mechanics course sneakily teach us some basic idea of general relativity. Which one of the basic assumption is particle walks in shortest world line. I understand shortest path in ...
2
votes
1answer
52 views

When does light reach a shell observer in Schwarzschild metric?

I am trying to simulate the trajectory of light in the Schwarzschild metric (as seen by a far away observer) with fixed $\theta = \pi/2$. According to my source (Chapter 18, section 18.5) the ...
0
votes
1answer
180 views

Carroll's derivation of the geodesic equations [duplicate]

In Carroll's derivation of the geodesic equations (page 69, http://preposterousuniverse.com/grnotes/grnotes-three.pdf), he starts with ...
3
votes
1answer
71 views

Light trajectory

We have observed stars where "we should not" Some people say that gravity can alter light trajectory. Some people say that gravity actually alter the space on which light travels. Which one is ...
3
votes
1answer
221 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation ...
0
votes
1answer
307 views

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 ...
1
vote
1answer
135 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 ...
0
votes
1answer
150 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 ...
7
votes
2answers
293 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 ...
3
votes
3answers
147 views

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 ...
2
votes
1answer
580 views

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 ...
1
vote
1answer
284 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 ...
2
votes
0answers
88 views

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 ...
2
votes
1answer
104 views

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 ...
1
vote
0answers
170 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 ...
-1
votes
2answers
148 views

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?
1
vote
1answer
92 views

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 ...
2
votes
0answers
162 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: ...
1
vote
0answers
69 views

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 ...
3
votes
2answers
114 views

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 ...
3
votes
1answer
87 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
1
vote
2answers
115 views

Can someone explain how Weinberg's definition of the affine connection for the geodesic equation matches the definition of an affine connection?

Consider the geodesic equation \begin{equation} 0=\frac{d^2 x^\lambda}{d\tau^2}+ \Gamma^\lambda_{\mu\nu} \frac{d x^\nu}{d\tau}\frac{d x^\mu}{d\tau} \end{equation} In Gravitation and Cosmology, on page ...
3
votes
1answer
180 views

Curvature of Light around a Black Hole [duplicate]

I am in a computer graphics class at my university and for my final project, I have chosen to create a program which renders a simple non-rotating black hole and models the curvature of light around ...
0
votes
1answer
92 views

Definition of the Lagrangian for a relativistic point particle in curved space

I have read that the Lagrangian in GR is defined as $L=\frac{\mathrm{d}s}{\mathrm{d}u}$, where $\mathrm{d}s = g_{ab}\mathrm{d}x^a\mathrm{d}x^b$ is the line element with the metric tensor $g_ab$ and ...
6
votes
3answers
348 views

Examples in which the light maximizes the optical path length

I posted a similar question about geodesics on Math.SE. Many sources (Wikibooks for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually ...
0
votes
1answer
34 views

Orbital variations [closed]

My question relates to relativity and the warping of spacetime. If the geodesics approaching the star are coplanar with the ecliptic how can polar planetary orbits form?