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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2
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3answers
142 views

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 ...
5
votes
1answer
312 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 ...
4
votes
1answer
330 views

Wave packet in curved spacetime

It is known that the classical equation of motion for a scalar field wave packet on a curved spacetime background gives the geodesic trajectory (the e.o.m. is $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). ...
3
votes
1answer
82 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} ...
2
votes
1answer
75 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 ...
0
votes
1answer
29 views

Comparing durations for two simply described motions in Schwarzschild geometry

I have some basic qualitative questions about the setup with: one satellite $A$ orbiting freely, on a stable circular path, a spherical non-rotating object of mass $M$; and another participant $B$ ...
0
votes
1answer
80 views

Null geodesics vs timelike geodesics

I'm interested in the paragraph under equation (38) of this reference: ...
0
votes
1answer
60 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 ...
3
votes
0answers
35 views

Sources for black hole geodesic orbits

I am looking for good sources that discuss both Kerr and Schwarzschild particle orbits (geodesics). Most sources write down the geodesic equations, constants of motion and the Hamiltonian, but do not ...
3
votes
0answers
53 views

Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
3
votes
0answers
248 views

How does one refute a Machian mechanism for inertial emergence?

Introduction: Consider the diagrams representing the duality between the weak and strong principles of equivalence. Now based on how these diagrams were taught to us (at least how they were taught ...
3
votes
0answers
85 views

Gravitational effects and metric spaces

Could somebody please explain something regarding the Nordstrom metric? In particular, I am referring to the last part of question 3 on this sheet -- about the freely falling massive bodies. My ...
2
votes
0answers
135 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: ...
2
votes
0answers
178 views

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 ...
2
votes
0answers
184 views

Radial Null Geodesics in Static Maximally Symmetric DeSitter Space

Given a DeSitter-space metric from the line element: $$ ds^2=\left(1-\frac{r^2}{R^2}\right)dt^2-\left(1-\frac{r^2}{R^2}\right)^{-1}dr^2-r^2d\Omega^2 $$ Where $R=\sqrt{\frac{3}{\Lambda}}$, and ...
2
votes
0answers
61 views

minimal proper time curves bounded on acceleration

Assuming Minkowski spacetime, we know that the longest proper time curve joining two points is the rect joinining both events, While the shortest time-like curve is not a compact set (because there ...
1
vote
0answers
40 views

Worldlines in Schwarzschild geometry

I have an observer and a photon on a hypersurface $ \theta=\pi/2$ . My observer has $e, l$ constants of motion (energy and angular momentum divided by mass) and photon has $e',l'$. What conditions ...
1
vote
0answers
18 views

Light Ray in AdS

On p77 of these lecture notes (http://arxiv.org/pdf/0712.0689v2.pdf), we are asked to check that a light ray takes infinitely long to reach the centre of AdS. 1, Why doesn't the Penrose diagram for ...
1
vote
0answers
37 views

Proper time and asymptotic flatness

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time $\tau$ is infinite for a timelike geodesic, does it mean ...
1
vote
0answers
37 views

Variation of Bazanski Lagrangian

The Bazanski Lagrangian is defined as $$ L=g_{\alpha \beta }U^{\alpha }\frac{D\psi ^{\beta }}{Ds} $$ and $$ U^{\alpha }=\frac{\mathrm{d} x^{\alpha }}{\mathrm{d} s} $$ $x^{\alpha }$ is the ...
1
vote
0answers
45 views

Existence of a solution for geodesic differential equations for a singular metric

In order to determine the geodesics, one must solve the following set of differential equations \begin{align} \frac{d^2 x^j}{ds^2} + {j\brace h\,\,k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0, \end{align} ...
1
vote
0answers
36 views

How does the expanding of null hypersurface orthogonal geodesic congruence imply a particular result?

Sorry that I do not know how to summarize my problem in the title. First, please go to the website here (free access, even though it looks otherwise) to download the paper done by R. Sashs on ...
1
vote
0answers
57 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 ...
1
vote
0answers
140 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
vote
0answers
68 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 ...
1
vote
0answers
162 views

Textbook disagreement on geodesic deviation on a 2-sphere

Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation ...
1
vote
0answers
110 views

Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess ...
1
vote
0answers
165 views

Derivation of equations of motion in Nordstrom's theory of scalar gravity?

Nordstrom's theory of a particle moving in the presence of a scalar field $\varphi (x)$ is given by $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
1
vote
0answers
100 views

Preservation of a scalar along geodesic trajectory

Let $u^\mu$ be the velocity of a particle , and $\xi^\mu$ be a killing vector. would taking a contravariant derivative of to scalar product $\xi_\mu u^\mu$ , and showing that it equals to 0 shows that ...
1
vote
0answers
146 views

Naked singularity and null coordinates

I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually ...
0
votes
0answers
70 views

Equation of motion of a free particle

We know that the equation of motion of particle can be derived from the respective action. But in the book I am reading, the author is saying: ... timelike worldline of a massive particle is ...
0
votes
0answers
34 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
34 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 ...
0
votes
0answers
88 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 ...
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.
0
votes
0answers
40 views

Vector fields corresponding to null geodesic congruences in general relativity

I'm working in Minkowski space, and I'm considering some 2D surface, $S$. On each point of the surface, I've computed a null vector, $k^a$, which is orthogonal to it. There will be a unique null ...
0
votes
0answers
136 views

How to use The Schwarzchild Metric formula to get distribution representing “free-fall”

Given formula: How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should ...