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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41 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 ...
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71 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 ...
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0answers
19 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 ...
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1answer
67 views

Proving that Killing form contractions with geodesics are constants of motion

I want to prove the fundamental theorem of Killing forms, namely that $$\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) = 0 $$ If $P(\lambda)$ is a Geodesic ...
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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 ...
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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 $B$ ...
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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 ...
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1answer
145 views

Is there a Maupertuis principle for General Relativity?

The motion of a point particle in classical mechanics is given by Newton's equation, $\mathbf{F}=m\mathbf{a}$. Suppose all forces considered are conservative and we have a constant total energy $h$. ...
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1answer
455 views

Do light waves precisely follow null geodesic paths in General Relativity?

In special relativity one may show that a plane wave solution of Maxwell's equations (in a vacuum), of the form $A^a=C^a\mathrm{e}^{\mathrm{i}\psi}$ has the following properties: The normal ...
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1answer
85 views

How to proceed (Tough Problem) [closed]

The problem that I am considering is to find the shortest path (or geodesic) on a surface with the equation $z=f(x,y)$. The path is parameterized by $s$ so that the path goes from ...
1
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1answer
138 views

Null geodesic equations

If one is constrained to the $xt$ plane, one can define the intersection with that plane of the null hypersurfaces originating at some point $P$ as $$ g_{tt} \frac{d P^t}{d \lambda}\frac{d P^t}{d ...
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1answer
80 views

Geodesic equation (free particle)

How to find a coordinate system whose geodesic equation does not have the "Christoffel symbol" term? (i.e. free particle - generalized Newton's second law.)
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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 ...
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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} ...
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2answers
77 views

If curved paths imply that the vehicle is accelerated, how come do we assume that light gets curved whilst its speed is constant?

I don't understand how we can accept these two sentences at the same time: Light speed is constant, therefore experiences no acceleration. On the presence of a gravitation field, light path is ...
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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 ...
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0answers
54 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 ...
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1answer
130 views

Infalling light signals seen by a free falling observer

In this question/answer Does someone falling into a black hole see the end of the universe?, it is stated that an observer free falling toward/into a black hole will not see the end of the Universe ...
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0answers
35 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 ...
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0answers
35 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 ...
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1answer
85 views

Null geodesics vs timelike geodesics

I'm interested in the paragraph under equation (38) of this reference: ...
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3answers
158 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 ...
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1answer
61 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 ...
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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 ...
2
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1answer
81 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 ...
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0answers
50 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.
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0answers
34 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
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1answer
63 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 ...
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0answers
68 views

Existence of affine parametrization [closed]

This is a question from General Relativity by Wald Chapter 3, problem 5. Given either pseudo-Riemannian or Riemannian metric $g_{ab}$ and manifold $M$. Assume the $\nabla$ is compatible with the ...
3
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1answer
67 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 ...
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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.
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3answers
169 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?
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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 ...
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0answers
24 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
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0answers
71 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
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4answers
876 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
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1answer
127 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 ...
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2answers
127 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. ...
2
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2answers
697 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.
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1answer
89 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 ...
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1answer
177 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
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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 ...
2
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1answer
72 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, ...
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2answers
109 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 ...
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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 ...
2
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1answer
51 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
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1answer
152 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
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1answer
61 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
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1answer
191 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation ...
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1answer
198 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 ...