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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24 views

Definition of teleparallel curve

I have heard that in a space with torsion teleparallel curves and geodesics are different and they coincide when the torsion vanishes. But I couldn't find any definition for the teleparallel curve. ...
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Just what really are Pp-waves, and how are they theoretically formed?

I have heard of this odd wave known as the Pp-waves, and are seemingly some form of mix of electromagnetic and gravitational waves. One infamous example is the Wave of Death, one that can destroy ...
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Higher-order variation of an action

In general relativity, the first-order variation of a point particle action gives the geodesic equation while a second-order variation gives the geodesic deviation equation. Similarly, is there any ...
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46 views

Is it possible to construct a geodesic for an anholonomic system?

For anholonomic system, i.e. Gravitation Eq. 9.22 $$[e_\mu, e_\nu] =c_{\mu\nu}^\alpha e_\alpha$$ where $$[e_\mu, e_\nu]\neq 0$$ for some $\mu,\nu$, the states of the system dependent on its path. ...
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58 views

Does modifying the geodesic Lagrangian $L$ with a smooth function $f(L)$ give the same geodesic curves as solutions?

Mathematical side of the problem Given the metric $$ds^2 = dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2$$ we can easily construct the action of a free particle $$S=\alpha \int d\tau \underbrace{\...
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1answer
52 views

Null-geodesics vs null-killing vectors

Consider a null-killing vector $\xi^{\mu}$. Now due to the killing equation we have $$\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu} = 0$$. Now I constract one of the index with $\xi^{\mu}$ to obtain $$\...
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45 views

Null Killing vectors constrain the space-time? [closed]

I have heard that spacetimes which admit null Killing vectors are sort of constrained. I wish to know how and why? What makes null Killing vectors so special?
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1answer
86 views

Equations of motion describing a great circle

I'd like to argue that equations of motions of the form $$\ddot \varphi = 0 \quad \text{and} \quad \ddot\theta = \sin\theta\cos\theta\dot\varphi^2$$ describe a great circle. I think the standard ...
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Are there spacetimes with zones that can reverse the relationship between proper time and coordinate time?

On a surface embedded in Euclidian space, where the metric signature is all positive, it is possible for a particle traveling along a geodesic to encounter a curved bit and then get turned around so ...
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1answer
82 views

Can I get turned around backwards in time? [duplicate]

On a surface embedded in Euclidian space, where the metric signature is all positive, it is possible for a particle traveling along a geodesic to encounter a curved bit and then get turned around so ...
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1answer
50 views

Proving worldline is geodesic

I am using coordinates $\{t,x,y,z\}$ and a metric $$ds^2=-dt^2+f(t,z)dx^2+f(t,z)dy^2+dz^2$$in which $$\Gamma^\mu_{tt}=0\quad\text{for all }\quad\mu=t,x,y,z.$$ I am then asked to show that a worldline ...
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Proper time experienced by an observer moving along a NON-geodesic

For an observer moving along a time-like geodesic $x^{\mu}(\lambda)$ (parametrized by $\lambda$) the geodesic equations are satisfied $$ \ddot{x}^{\mu}(\lambda) + \Gamma^{\mu}_{\ \; \nu\rho} \; \dot{x}...
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Null geodesics in the equatorial plane of Schwarzschild geometry

In Chandrasekhar's The Mathematical Theory of Black Holes, the various classes of null geodesics in the equatorial plane of Schwarzschild geometry are as in the following two figures: In the figures, ...
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Calculating the distance on a cone or a sphere and the implications for understanding the geometry of space

I am reading Eddington’s book on space and time https://www.gutenberg.org/files/29782/29782-pdf.pdf And on page 71 he gives the formula for the distance, $ds$, between two points on a spherical ...
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Trouble understanding Caroll's explanation on why geodesics maximize proper time

I'm reading Caroll's Lectures on GR 2 on pages 71-72, he states: Let’s now explain the earlier remark that timelike geodesics are maxima of the proper time. The reason we know this is true is ...
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1answer
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Solving the geodesic equation for a Schwarzschild metric [closed]

Using the Schwarzschild solution is there a simple differential equation describing the four position of a particle influenced by a Schwarzschild metric using the geodesic equation. How would the ...
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43 views

Coordinate-independent metric implies constant metric along geodesic

I am struggling to understand a problem I was given. The problem is as follows: $ {}$ Show that if the metric does not explicitly depend upon a coordinate (e.g. $x^1$) then the term $g(\dot{x}, \...
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Force on a particle from geodesic equation

Say I had an action for a scalar field where the matter action is $$S_m=\int d^4x \sqrt{-\tilde{g}}\mathcal{L}_m(\psi,\tilde{g}_{\mu\nu})\tag{1}$$ such that matter will follow geodesics according to $\...
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Is a reasonable spacetime geodesically connected?

There are some theorems concerning whether a spacetime is geodesically connected (whether any two points $p, q \in M$ admit a geodesic connecting them) or not, ie [1][2], but all of these are ...
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1answer
32 views

Superluminal speed in anti de sitter

This a bit of an elementary question, but I would like to understand how one correctly computes velocities in anti de sitter. It is well known that photons, traveling on null geodesics, will actually ...
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1answer
43 views

What does an outgoing null geodesic for FLRW metric look like?

I know it involves setting the line element to zero, as well as setting $d\phi$ and $d\theta$ to zero. But other than that I do not know what to do. I get an expression involving $a(t)$, the scale ...
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1answer
44 views

Is there work done on a mass from matter-antimatter pairs?

Imagine I have a vaccuum at very low temperature and I put a single neutron in, then gamma rays interact to form matter-antimatter pairs within this vaccuum and assume that this happens extremely ...
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86 views

Geodesic Incompleteness and the Kretschmann Scalar

If my understanding is correct, singularities (whether time-like or space-like) are defined by geodesic incompleteness. Since it is easier, we use the Kretschmann scalar $R^{\alpha\beta\gamma\delta}...
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55 views

Geodesics through the wormhole

To define a traversable wormhole, there should be some conditions on the metric components, such as: I) No event horizon, II) Minimum value for the shape function (considering a spherical ...
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Doubt regarding particle dynamics and hydrodynamics in Schwarzschild geometry

The effective potential for particle orbits in the equatorial plane of a Schwarzschild black hole in units $G=M=c=1$ is given by $$V_{\textrm{eff}}=\sqrt{\left(1-\frac{2}{r}\right)\left(1+\frac{l^2}{r^...
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Geodesic Equation Derivation

I am having some issues completing the derivation of the geodesic equation using the Lagrangian and also trying by differentiating the metric with respect to the path length parameter. When ...
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1answer
113 views

Christoffel symbols from Geodesic equation for a metric with non diagonal elements

In the case of a diagonal metric, \begin{align} \mathrm{d}s^2=g_{\mu\nu}\mathrm{d}{x}^\mu\mathrm{d}{x}^\nu, \end{align} it is relatively straightforward to find the Christoffel symbols by comparing ...
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General Relativity: Why is the energy $E = - p_{\mu} U^{\mu}$?

This follows Carroll's Gravity book (page 110). An observer with four-velocity $U^{\mu}$ (such that $g_{\mu\nu}U^{\mu}U^{\mu}=-1$) measures the energy of a particle along a geodesic $$p^{\mu} = \frac{...
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1answer
78 views

Geodesic equation in the language of frames [closed]

How to write the geodesic equation in the language of frame fields or tetrads in terms of spin connection?
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1answer
67 views

Massive particles in de Sitter space

Can massive particles in de Sitter space move faster than light? For the radial coordinate (in static coordinates) I have got the hyperbolic expression $$r(\tau)\propto \sinh\left(\sqrt{\frac{\...
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1answer
86 views

Confusion about derivatives along worldline in general relativity

I just began learning general relativity and I am quite confused with what I'm learning. I understand that in special relativity, the four-velocity is $$U^\mu=\frac{\text{d}X^\mu}{\text{d}\tau}$$ and ...
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1answer
79 views

Physical intuition for the Geodesic equation derivation via Action Principle

The most commom derivation I've seen of the geodesic equation of a massive particle is by the use of the Variational Principle. My problem is that I can't realize what the meaning of find a spacetime ...
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296 views

How to recover the Newtonian gravitational potential from general relativity? [duplicate]

To take the newton limit of the geodesic equation: $$\frac{\mathrm{d}^2x^\mu}{\mathrm{d}\tau^2}+\Gamma^\mu_{\nu\rho}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\rho}{\mathrm{d}\tau}=0 $$ ...
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Is the Schwarzschild horizon lightlike?

It is often said that the event horizon of a Schwarzschild black hole is lightlike. Is this correct and, if so, what exactly does this mean? Intuitively, this may mean that any two points on the ...
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When are conics geodesics?

Does there exists a Riemannian metric on $\mathbb{R}^2$ (or an open subset of it) such that the geodesics are exactly the conics whose one focus is the point $(0,0)$ (which are the Newtonian orbits of ...
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4answers
313 views

Entering a black hole, does the incident angle matter at all?

Inside the black hole (as you enter the EH), all objects (massive or massless) must move towards the singularity. The singularity becomes a moment in future. In the context of general relativity, ...
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1answer
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Curved spacetime and geodesics

Is it right to take in account that the Universe might be similar to the Earth regarding curvature so when we look at two galaxies equidistant to Earth but at a right angle observed from Earth their ...
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95 views

Connection between affine parameter and time in general relativity

Setup Consider the metric for a spherically symmetric and isotropic spacetime: $$ds^2 = -B(r)c^2dt^2 + A(r)dr^2 + r^2d\Omega^2.\tag{$\Delta$}$$ With this metric we can find (with the geodesic ...
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1answer
87 views

Null Geodesics in Anti-de Sitter space time [closed]

Would anyone be able to explain how the step was taken in getting the final equation with $R \tan(t/R)$ I understand the steps before where we are finding the null geodesic equation for the AdS space ...
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60 views

Geodesic with decreasing value of time coordinate

Does there exists an example of geodesic for an exotic space-time manifold in which even though the proper time on the geodesic is increasing but still the time coordinate of the geodesic in global ...
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1answer
48 views

Why are the Schwarzchild or Eddington-Finkelstein not maximal?

In the book by d'Inverno, he writes the definition of a manifold being maximal as "A manifold endowed with an affine or metric geometry is said to be maximal if every geodesic emanating from an ...
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1answer
111 views

Is the Borde-Guth-Vilenkin Theorem applicable in cosmologies based on Einstein-Cartan Theory?

At https://arxiv.org/pdf/1403.1599.pdf, Vilenkin, in 2014, used the Borde-Guth-Vilenkin theorem's premise that the "bouncing" local universes of an expanding or contracting multiverse would ...
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125 views

Eddington-Finkelstein coordinates and radial null geodesics

Looking at Carroll's chapter 5.6 he derives the Eddingtion-Finkelstein coordinates and writes the Schwarzschild metric out, resulting in ($v-r$ coordinates) $$\mathrm{d} s^{2}=-\left(1-\frac{2 G M}{r}\...
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110 views

Geodesic equations in Schwarzschild geometry

I am a little bit confused about the following equations. In the lecture we derived the following four equations for a geodesic motion of a particle in Schwarzschild geometry using the Lagrangian ...
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Why are massive bodies following a different trajectory in a gravity field than light?

I often read, that light is being bent in a gravity field and so are the paths of massive bodies (e.g. planets or stars). I also read that the curvature of space (caused by some other mass) is the ...
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58 views

What is actually travelling in a straight (geodesic) line?

I know light travels in a straight line and experiment (such as the double slit experiment) had shown photon can take any path so my question is for a single photon, what is it that is actually ...
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1answer
128 views

Not sure why the geodesic derivation equation involved second ordinary derivative

My question is related to this link (18:10) https://www.youtube.com/watch?v=oQZTYt_Pxcc&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx&index=28 Recently I have watched a video about volume ...
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41 views

Simplest GR geometrical model for bending of light [duplicate]

How to find orbit of bent light in the simplest gravity model by solving the null geodesic equation either by Newton's physical or by Einstein's GR? Bent Light To compare we have the simplest model ...
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1answer
89 views

Is a brachistochrone a straight line in curved space?

Please bear with me, and don't get upset if i have lack in knowledge about spacetime. Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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1answer
74 views

Particle falling into a Kerr black hole

Let's say that a particle starts a radial free fall towards a Kerr black hole with zero initial energy at $r\rightarrow\infty$. The initial angular momentum of the particle is zero ($p_\phi = 0)$. ...

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