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).

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31 views

Parallel transport and Geodesic deviation

We know that when we derive the Geodesic equation, we want to actually understand the geometrical meaning of the Riemann tensor. We see from the geodesic equation that the second derivative of the ...
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2answers
418 views

Is work done by gravity equal to zero in general relativity?

I've heard it said that, according to Einstein's equations, gravity isn't actually a force but is instead an effect of the curvature of spacetime. In other words, a body in a gravitational field ...
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How to check if a parametrized curve is a geodesic

Given some parametrization in the plane, e.g., $x = \sin \lambda$ and $y = \cos \lambda$, how can I know if that curve is a geodesic? $x$ and $y$ constitute a circle in the plane, centered at $(0,0)$ ...
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How does the fluid approximation affect the geodesic equation?

So general relativity strictly speaking is a theory where only fields are allowed. To make calculations in it we often use the fluid approximation and think the matter surrounding us instead of stars, ...
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Entanglement entropy at finite temperature in different coordinates

I have read the seminal papers abous entanglement entropy by Takayanagi and still do not understand several statements. I would be grateful for any reasonable comments. First, consider Minkowskian AdS$...
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70 views

Cosmology - The Geodesic Equation Derivation

My question relates to the derivation of the Geodesic Equation given on pages 121-122 of Baumann's cosmology. $$\frac{d^2x^i}{d\tau^2} = -\partial^i \Phi (dt/d\tau)^2 + O(\Phi^2). \tag{A.2.43}$$ On ...
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1answer
44 views

ISO an elegant derivation of the Geodesic Equation from the Einstein Equation

Assuming a particle filled with perfect pressure-less fluid can the Geodesic Equation be derived from the Einstein Equations?
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41 views

Covariant derivative and vector fields along a curve

I have to prove a relation that involves the covariant derivative induced by the Levi-Civita connection. With the reference on N. Straumann General Relativity Given $T=\frac{\partial x^\alpha}{\...
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1answer
104 views

Can you help me resolve this paradox for a geodesic orbit? Does it maximize proper time to sit still rather than orbit for 1 year? [duplicate]

The usual simple statement of geodesics is that when moving from point $(x_o,y_o,z_o,t_o)$ to point $(x_1,y_1,z_1,t_1)$ the object will follow the path of greatest proper time experience, i.e., a ...
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1answer
45 views

Spacetimes where $R_{ij}\neq 0$ but $R_{ij}V^iV^j=0$ on a timelike and/or null geodesic?

Do there exist spacetimes with a timelike and/or null geodesic $\gamma$ with tangent vector $V$ for which $R_{ij}\neq 0$ on the geodesic, but $R_{ij}V^iV^j=0$ on it? If so, are there any general ...
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If a light beam is sent tangent across earth would it curve at 9.8 $\rm m/s^2$? [closed]

Just to see if my understanding of the curvature of light is correct.
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2answers
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Units in the geodesic equation / Schwarzschild metric

Most textbooks define the geodesic equation for a particle with unit mass, such that it looks like: $$ \ddot{x}^{\mu} + \Gamma^{\mu}_{\alpha \beta} \dot{x}^\alpha\dot{x}^\beta = 0$$ Where "dot&...
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Questions on Affine Parameter

According to p.75 of Hobson's General Relativity book, the author defines an affine parameter along a curve by a parameter that makes the length of the tangent vectors constant along the curve. By ...
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1answer
80 views

Proper time of a timelike geodesic

In the contest of the newtonian limit in general relativity, if I consider a timelike geodesic that can represent the motion of a free falling particle under the influence of the gravitational force ...
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1answer
132 views

Lie brackets of vector fields along a geodesic to obtain the Jacobi equation

I have done this question in mathstack but someone has suggested me it is more appropriate to ask this here. With reference in https://archive.org/details/GeneralRelativity/page/n82/mode/1up, where it ...
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Misunderstanding of $\mathrm{AdS}_3$ spacetime

In my research I deal with $\mathrm{AdS}_3$ spacetime. It is convenient for me to use Poincare coordinates, which means that interval is given by $$ds^2 =\frac{1}{z^2}(-dt^2+dx^2+dz^2).\quad \tag{1}$$ ...
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How to retrace the paths of light back in time, for light around a black hole?

I have been thinking about black-hole raytracing, and have found this writeup which talks about deriving the shape of a photon's orbit around a Schwarzchild BH, enabling raytracing around a BH from a ...
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2answers
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Curvature Singularities in Geodesically Complete Manifolds

Do there exist manifolds which are geodesically complete, and yet have a curvature singularity? While I don't believe this is the case, I have yet to find a proper proof of the same.
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Is this a null geodesic?

I'm studying general relativity and when talking about null geodesics my teacher put this example of energy-momentum tensor $$T^{\mu \nu} = C n^\mu n^\nu,$$ where $C$ is a constant and $n$ is a ...
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1answer
43 views

How can I get an intuitive explanation of the visualization of geodesics in a cone?

I calculated that the geodesics in a cone satisfy the following formula: $$r=\frac{1}{A\cos(\omega \phi + \alpha)}.$$ The cone is parametrised by taking spherical coordinates and fixing $\theta=\...
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Action for massless particles in GR [duplicate]

Relativistic action for a massive point particle is defined to be $$S=-mc\int d\sigma \sqrt{-g_{\mu \nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}.$$ In David Tong's lecture notes on GR If all we want to do is ...
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62 views

Null Second Fundamental Form Expansion & Null Geodesic Congruence [closed]

I'm a mathematician delving into the world of physics and I've come across what seem to be two closely related concepts but I can't piece them together. I'm particularly interested in defining a ...
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Deriving Geodesic Deviation in Newtonian Gravity

I am trying to derive the Newtonian Geodesic Equation: $$\frac{d^2\xi^i}{dt^2}=-\phi_{,ij}~\xi^j$$ from the relativistic version: $$\nabla_{\vec{V}}\nabla_{\vec{V}}~\xi^\alpha=R^\alpha_{~\mu\nu\beta}V^...
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5answers
120 views

Is it a postulate that light travels on a geodesics?

In arbitrary spacetime, light travels on geodesics. Is this a postulate or can it be derived from a more fundamental principle?
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27 views

Do all null geodesics on a marginally trapped surface remain on or inside the surface?

The definition I've seen and used is essentially "if every congruence orthogonal to a spacelike two-surface has negative expansion, then that surface is said to be trapped." Does this ...
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1answer
36 views

Geodesics equation in a 2-space with a certain $ds^2$

This is exercise 3.20 of Hobson's general relativity. It's presented as follows: In the 2-space with line element $$ds^2=\frac{dr^2+r^2d\theta^2}{r^2-a^2}-\frac{r^2dr^2}{(r^2-a^2)^2}$$ Where r>a, ...
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2answers
64 views

Alternative form of the geodesic equations

In Hobson's General Relativity for physics, a geodesic is described as a curve satisfying $\frac{d\boldsymbol{t}}{du}=f(u)t$, where $u$ is a general parameter, $f(u)$ is some function of $u$ and $\...
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3answers
115 views

Affine and metric geodesics

In D'Inverno's "Introducing Einstein's Relativity", an affine geodesic is defined as a privileged curve along which the tangent vector is propagated parallel to itself. Choosing an affine ...
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1answer
55 views

Trouble solving the geodesic equation? [closed]

I am working through Moore's "General Relativity Workbook" and am stuck on a problem (8.4.2). I believe I am missing something quite trivial, but I can't figure it out. Any help is ...
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1answer
56 views

Jacobi Matrix between Cartesian and Schwarzschild coordinates

Let $\mathcal P$ be a photon at position $\vec x =(x,y,z)$ with 3-velocity $\vec v=(v_x,v_y,v_z)$, where both are given in local Cartesian coordinates. I want to follow the photons geodesic by ...
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1answer
55 views

Show that a geodesic remains of the same type at all points by first proving that $v^a w_a$ remains constant when parallel-transported

The title basically says it all. Given $\boldsymbol{v}$ and $\boldsymbol{w}$ with contravariant components $v^a$ and $w^a$, I'm asked to show that $v^a w_a$ after being parallel-transported along a ...
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On the solving of the geodesic equations

I'm reading Hobson's Introduction to General Relativity for physicists. On chapter 3, the geodesic equation is presented in several forms, one of them being: $\ddot{x}^a+\Gamma^a_{bc}\dot{x}^b\dot{x}^...
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2answers
98 views

Timelike geodesics of 2-dimensional de Sitter spacetime

The de Sitter spacetime in two dimensions ($\text{dS}_2$) can be spanned by two coordinates : $\tau$ and $\phi$ (can be viewed as a cylinder). The metric is then defined as follows : $$ds^2=-d\tau^2+\...
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27 views

Geodesic and Acceleration

The following is from Elementary Differential Geometry by Pressley. He says "A bug living in a surface and traveling along a curve $\gamma$ would perceive only the component of acceleration ...
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1answer
82 views

Solving Geodesics

I'm wondering how to solve the geodesic equation to get a null geodesic. I know the two equations $$\frac{d^2x^\mu}{ds^2}+{\Gamma^\mu}_{\nu\lambda}\frac{dx^\nu}{ds}\frac{dx^\lambda}{ds}=0$$ and the ...
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1answer
59 views

Worldline action of point particle in gravitational field

In my GR lectures on the derivation of geodesic equations via extremal length, my lecturer wrote that the worldline action $S$ of a point particle with mass $m$ is given by $$S=-m\int\sqrt{-g_{\alpha\...
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1answer
64 views

Geodesic equation in terms of four velocity

I am trying to show that for timelike paths, we can write the geodesic equation in terms of the four-velocity $U^\mu=\frac{dx^\mu}{d\tau}$ as $$U^\lambda\nabla_\lambda U^\mu=0.$$ In other words, ...
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1answer
61 views

How does Eddington-Finkelstein Coordinates solve coordinate Singularity

Hi i am reading about Eddington-Finkelstein coordinates and i read that they do remove the coordinate singularity at $r=r_s$ but still there is some problem with these coordinates which can be removed ...
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4answers
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Least Action in General Relativity

For an affinely parameterised geodesic we can form the Lagrangian: $$ \mathcal L = g_{ab}\dot x^a\dot x^b = \text{constant} $$ The Lagrangian is constant by the fact that the geodesic parallel ...
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Friedmann Metric and congruence

I am told to show that the congruence in the Friedmann Metric is geodesic. I am really confused how to solve it, because i am not sure how to "find the congruence". I thought that i could ...
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1answer
48 views

Trajectory of light ray in constant gravitational field

A light ray is shot from a laser at angle $\theta$ from the ground (upwards). Assuming a constant gravitational field, what shape is the trajectory? Looking at Wikipedia-Rindler Coordinates, I find an ...
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2answers
78 views

Why do we use null geodesics?

Why do we use the null geodesic equation not only for the photon but for all GR? I thought the null geodesic was only it to the photon, How do these things intertwine? Null geodesic equation And not ...
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43 views

Affine Parameter along Null Geodesic

I want to integrate the Boltzmann-Equation along a photon (null) geodesic, $ \begin{align} \frac{df}{d\lambda} &= p^\alpha\frac{\partial f}{\partial x^\alpha} +\frac{\partial p^k}{\partial\...
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1answer
75 views

Combining the Einstein field equations with the geodesic equation

I've seen this question How does Einstein field equations interact with geodesic equation?, but it doesn't make any sense to me. If spacetime is a Lorentzian manifold, then surely one thing general ...
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Geodesics in an Alcubierrie Warp Drive metric

The geodesics in an Alcubierrie Warp Drive metric are given by $$\dot{t} = 1 \quad \dot{x} = X\dot{t} \quad y, z = \text{constant}$$ where $X = \dot{x}_s(t) f(r_s)$. Here $f(r_s)$ is a shape function ...
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3answers
274 views

Is an orbiting object traveling along a geodesic in general relativity?

I am getting a layman's understanding of General Relativity. I understand that gravity is understood to be an objects propensity to travel along a straight line in curved spacetime. Would an orbiting ...
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1answer
177 views

Geodesic equations without solutions

Is there a nontrivial Riemannian metric for which a geodesic equation doesn't have any solutions? What would such a metric mean physically?
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64 views

Can someone explain why does these terms get cancel out while deriving Riemann curvature tensor from geodesic deviation?

I was trying to derive Riemann curvature tensor from geodesic deviation and I came across these terms with the derivatives of seperation vector. I made some research and seems like they should cancel ...
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30 views

Geodesic equation with respect to a Riemannian diagonal metric

I am asked to prove that the differential equations of geodesics on an open set of a pseudoRiemannian manifold $(M, g)$ of dimension $n$ where $g_{ij}=0$ if $i \neq j$ are given by $$ \frac{d}{ds}(g_{...
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117 views

Area element from tangent vectors

I am working on a problem in the framework of General Relativity, where light is emitted from a point source and the light bundle is described as a congruence of geodesics. Consider taking two spatial ...

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