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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4answers
940 views

Lagrangian for relativistic massless point particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of ...
0
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1answer
47 views

Questions about null geodesic

Show for the null geodesic in 3D flat spacetime using polar coordinates so the line element is $ds^2=-dt^2+dr^2+r^2d\phi^2$. Do light rays move on straight lines? My question is that I only learned ...
2
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1answer
31 views

Energy conservation around a black hole

In the Schwarzschild black hole, the Killing vector "time translation" $k^a$, so that the following quantity is conserved along a geodesic: $$E = -g_{ab}k^au^b = (1 - ...
2
votes
1answer
42 views

Difference between Fermi and Riemann normal coordinates

What is the difference between Fermi normal coordinates and Riemann normal coordinates? Which one of them is related to the vanishing of the Christoffel symbols?
0
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0answers
31 views

Propagator in AdS

Can someone explain me why I can write, in some limit, the two point function in a field theory, as a path integral representation over the geodesics?
2
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3answers
164 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 ...
1
vote
0answers
29 views

Geodesic tangent vector in a Riemannian 4-space

I am doing a question in Lewis Ryder's introduction to General relativity. I am very close to the answer but not quite there. The question is: A Riemannian 4-space has metric $$ds^2 = ...
1
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1answer
59 views

On the proof of the existence of geodesics coordinates

From "Introducing Einstein’s Relativity" by Ray D’Inverno page 77-78 In my calculation, the process is $$\frac{\partial{x^{'a}}}{\partial{x^d}}=\frac{\partial{x^{a}}}{\partial{x^d}}+\frac{1}{2} ...
53
votes
4answers
3k views

GR and my journey to the centre of the Earth

[General Relativity] basically says that the reason you are sticking to the floor right now is that the shortest distance between today and tomorrow is through the center of the Earth. ...
1
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2answers
30 views

Why do relativistic wormholes have to be brought together to make a “time machine”?

In "From wormhole to time machine: Comments on Hawking’s Chronology Protection Conjecture" by Matt Visser (http://arxiv.org/abs/hep-th/9202090), he summarizes how "time machines" may be created from ...
1
vote
0answers
28 views

Null geodesics in uniform gravitational field metric

I'm trying to understand the null geodesics in the metric: $$\mathrm{ds}^2 = -(1+gz)^2 \mathrm{dt}^2 + \mathrm{dz}^2 + \mathrm{dx}^2$$ In particular I'm wondering if the following intuition is ...
0
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0answers
34 views

Geodesic equation in normal coordinates [closed]

I've come across the following statement, about the components of the Levi-Civita connection in normal coordinates: Lemma: $\Gamma^{\mu}_{(\nu\rho)}(p)=0$ in normal coordinates at $p$. For a ...
6
votes
2answers
383 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 $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). ...
0
votes
1answer
58 views

Straight lines in general relativity

This question stems from a possibly misguided attempt to understand General Relativity. I am about to leave High school for college, I do however have a rudimentary understanding of tensors, and I ...
1
vote
1answer
56 views

Spacetime manifold surgery: is this result still a valid etc. spacetime?

Given a valid classical GR spacetime manifold $M$ (i.e. 4D, Lorentzian, Hausdorff, paracompact, ?etc.), and $B\subset M$, a closed spatial subset (e.g. a closed ball at fixed $t$) to be excised, ...
0
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0answers
35 views

GR - curve (in)completeness & (in)extendibility

Seeking clarification of the distinction between completeness of geodesics/extendibility of curves in GR spacetimes? (Confirm: not the geodesic completeness of a spacetime but the completeness of an ...
0
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2answers
53 views

Bending of Light in General Relativity using Perturbation

It is standard textbook calculation (e.g. Schutz's First Course in General Relativity page 294) that we can find a total angular change in light deflection due to gravity to be ...
5
votes
2answers
57 views

Wald's General Relativity, section 6.3 Page 144

I cannot understand how he reaches the conclusion in equation 6.3.36 and 6.3.37; even the terminology is somewhat confusing. This is a problem of bending of light under gravitational field. This is ...
1
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0answers
31 views

Einstein-Infeld-Hoffman-Lagrangian for a Test-Particle as Limit of Schwarzschild-Geodesic

Consider a test particle of mass $m$ which is in orbit around a spherical-symmetric body with mass $M$. It therefore has a position as described by the coordinates $r,\phi$, and its motion can be ...
1
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1answer
251 views

Null geodesics vs timelike geodesics

I'm interested in the paragraph under equation (38) of this reference: ...
1
vote
0answers
47 views

Using geodesic deviation for freely falling particles when gravitational waves comes through

Suppose we have a gravitational wave which gives us the following metric $$ds^2=-dt^2+(1+h_+\cos(\omega(t-z)))dx^2+(1-h_+\cos(\omega(t-z)))dy^2+dz^2$$ I want to calculate the time it takes for a ...
3
votes
2answers
113 views

Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr). I know that the direct way is via the geodesic equation ...
0
votes
1answer
294 views

How do objects even move due to gravity?

I am an newbie general relativistic learner and I learnt that gravity is bending of space-time and since objects move in straight-lines but since its curved they follow curved movement through space ...
0
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1answer
38 views

Simplify calculation of geodesics from action principle

I don't understand a step with the calculation of geodesics equations from action principle on this link : demo geodesics equations My issue is the following step : $$\int ...
0
votes
2answers
53 views

The shortest path among two points inside Earth [closed]

I have this idea and I don't know how to process, explain or question it. I hope you can understand these images and help me formulating a good question. This is like a gravitational train but it ...
1
vote
1answer
44 views

Representing 1+1 Minkowski space as a surface in 3D Euclidean space

In 1+1 Minkowski space the distance between two points is given by$$ (x_1 -x_2)^2 -(t_1 - t_2)^2.$$ This is different from the Euclidean distance. But is it possible to come up with a 2D surface ...
0
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0answers
43 views

Anti de-Sitter Geodesics

Timelike geodesics in anti de-Sitter space cannot reach infinity. I believe this has something to do with Clairaut's relation. I'm pretty sure it's true though as the analogy with conservation of ...
0
votes
1answer
56 views

How do I derive geodesic equation using variational principle? [duplicate]

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got ...
18
votes
1answer
517 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 ...
1
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0answers
47 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 ...
0
votes
0answers
90 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 ...
7
votes
2answers
276 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 ...
1
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3answers
182 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
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0answers
22 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 ...
2
votes
1answer
74 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 ...
7
votes
2answers
728 views

Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: \begin{equation} ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2) \end{equation} ...
1
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0answers
44 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
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1answer
34 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 $B$ ...
5
votes
1answer
153 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 $h$. ...
1
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0answers
41 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 ...
0
votes
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 ...
4
votes
0answers
64 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 ...
1
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1answer
163 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 ...
3
votes
0answers
40 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 ...
6
votes
3answers
336 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 ...
1
vote
1answer
95 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
48 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} ...
9
votes
3answers
7k views

What is the physical meaning of the affine parameter for null geodesic?

For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is ...
1
vote
2answers
79 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 ...
1
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0answers
50 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 ...