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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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How does the Ricci tensor describe the changing separation of two airplanes flying from the equator? Conceptually understanding the Ricci tensor

I'm trying to understand the concept of the Ricci tensor and its physical implications using a concrete example involving two airplanes. Suppose two airplanes start at the equator, separated by a ...
bananenheld's user avatar
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47 views

Are there exactly solvable problems in curved space, except for cases of constant curvature of space?

I have two questions. I know the expressions for geodesic distance in Minkowski, de Sitter and anti de Sitter space-time and their Euclidean analogues $R^n$, $S^n$ and $H^n$ [1]. For what other curved ...
grodta's user avatar
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Does gravity accelerate you towards the geodesic of light between you and the mass?

If there's a planet far away, you will accelerate straight towards it due to gravity. If you place a Schwarzschild black hole right in the middle between you and the planet (the distance between the ...
Zach's user avatar
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1 answer
52 views

Geodesic variation and the (Riemann) curvature tensor -- what about uniform gravitational fields?

BACKGROUND: The equation of geodesic variation is in almost every GR book. Suppose $x(s)$ and $x(s)+\epsilon(s)$ are two nearby geodesics at $x$, with $\epsilon(s)$ small, and also $d\epsilon(s)/ds$ ...
Khun Chang's user avatar
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1 answer
87 views

Derivative of line element in general relativity is zero?

The Lagrangian for a point particle in general relativity is $$ L= -m \sqrt{-g_{\mu\nu}\dot{x}^\mu \dot{x}^\nu} $$ where $x^\mu(\lambda)$ is the world line of a particle with mass $m$. The world line ...
jojo123456's user avatar
3 votes
0 answers
77 views

Connection between the metric tensor and mass

The general expression of a line element in a space with metric tensor $g_{\mu \nu}$ is $$ds = \sqrt{ g_{\mu \nu} dX^{\mu} dX^{\nu} }$$ If we consider a curve $X^{\mu}(\tau)$ parametrised by $\tau$, ...
pll04's user avatar
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60 views

Dirac "GTR" Eq. 27.11 -- how to show that a boundary term vanishes?

In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation $\delta(I_g+I_m)=0$ of the actions ...
Khun Chang's user avatar
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54 views

R-W Metric and null geodesic path of photon

I was reading through Introduction to Cosmology, on Chapter 3, it gives me the R-W Metric: $ds^2 = -c^2dt^2+a(t)^2[dr^2+{S_κ}(r)^2dΩ^2]$ ${S_κ}(r)$ is a function related to the curvature of space, κ ...
Polaris5744's user avatar
2 votes
1 answer
118 views

Equation for the curve of a free falling particle in Kruskal diagram

I'm currently trying to help a friend who is taking a GR course (she is an experimentalist, and I'm rusty as hell, please be patient). We got stuck in an issue concerning Kruskal coordinates. We are ...
André Noronha's user avatar
5 votes
3 answers
1k views

Is the definition of geodesics different if the electromagnetism is added to GR?

In plain GR, geodesic are defined by: $$ \nabla_{u} u^{\,\mu} = 0 $$ where $u^{\,\mu}$ is the four-velocity of the particle. Now, I don't understand if this holds true also in the electromagnetic case,...
Aleph12345's user avatar
1 vote
2 answers
108 views

Necessity of equivalence principle

Is the equivalence principle necessary to formulate general relativity or is it possible to formulate general relativity without it?
IcBR_1's user avatar
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1 answer
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What mistake did Einstein make in 1911 when he miscalculated the light deviation?

When Einstein published the general relativity theory in 1911, why was the light deviation not predicted correctly? What was the incompleteness of the theory when he published it in 1911? When and how ...
iVenky's user avatar
  • 131
2 votes
1 answer
191 views

How does this canonical transformation on a Schwarzschild black hole work?

In this paper "Holography of the Photon Ring" the authors use a canonical transformation in section 2.4 in eqs. (2.52)-(2.55). It is basically a transformation from spherical coordinates for ...
Geigercounter's user avatar
3 votes
1 answer
210 views

Definition of free fall in GR

I apologize if this question is very elementary. Somewhere I've found the following: Freely falling observers (resp. photon) move on timelike (resp. null) geodesics. Please note that by definition, ...
Matha Mota's user avatar
1 vote
2 answers
86 views

Maximizing proper time with parabolic trajectory in uniform gravitational field

In Feynman Lectures, Vol II, Chapter 42, he states, "In a uniform gravitational field the trajectory with the maximum proper time for a fixed elapsed time is a parabola." How can I prove ...
sku's user avatar
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1 answer
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Geodesics on a gravitational wave

$$ \newcommand{\dot}[1]{\overset{.}{#1}} \newcommand{\ddot}[1]{\overset{..}{#1}} $$ Consider the following metric, which describes a linearized plane gravitational wave: $$ g_{\mu\nu} = \eta_{\mu\nu} +...
paulina's user avatar
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1 vote
3 answers
159 views

Geodesic equation: Bridging the gap between inertia and gravity

The geodesic equation can be expressed as: $$\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\tau ^2} + \Gamma _{\alpha \beta} ^\mu \frac{\mathrm{d}x^\alpha}{\mathrm{d}\tau} \frac{\mathrm{d}x^\beta}{\mathrm{d}\...
RJurjevic's user avatar
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Nature of the spacetime trajectory (worldline) described by $\frac{d^2x^\mu}{d\tau^2}=0$

The covariant equation of motion of a free particle, in flat Minkowski spacetime and Cartesian coordinates, reads $$ \frac{d^2x^\mu}{d\tau^2}=0, \tag{1} $$ with $\mu=0,1,2,3$, and has the solution $$ ...
Solidification's user avatar
2 votes
3 answers
155 views

Geodesic in flat space in spherical coordinates

let's consider the expression, where $u^\mu$ is the tangent vector to the geodesic $\theta = \nabla_\mu u^\mu$....scalar $\Rightarrow$ valid in every coordinate system So in flat space in Cartesian ...
Coderboy's user avatar
1 vote
0 answers
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Gravitational Time Dilation: How to find the time difference between orbits at different radii? [closed]

I want to calculate the difference in time measured by a clock at on earths surface (r=6000km), and a geostationary satellite (r=26000km). My approach is as follows: For simplicity, we consider curves ...
John Grace's user avatar
4 votes
2 answers
628 views

GR contribution to time dilation when both clocks are falling freely

When reading simplified explanations of time dilation experienced by satellites, such as those used for the GPS and other satellite navigation systems, the time dilation is often presented as having ...
Jeppe Stig Nielsen's user avatar
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Is it possible to refind entry trajectory after leaving a rotating black hole?

I'm asking about the case where an infalling object travels a path right through a rotating black hole, intact. I want to provide a simple parallel for the purposes of question clarity: A horse can ...
Wookie's user avatar
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2 votes
2 answers
124 views

Derivation of the geodesic equation. Why do we start with the special relativistic action?

I'm working on a derivation of the geodesic equation from the action functional. In special relativity, specifically for flat spacetime, we assume that the metric tensor is constant (not necessarily ...
DingleGlop's user avatar
2 votes
1 answer
67 views

Borde-Guth-Vilenkin (BGV) Theorem equation for the motion free falling observer

In the derivation of the BGV theorem paper: https://arxiv.org/abs/2307.10958 There is the following relation (numbered (8)): $\dfrac{d}{d s}\left[a^2(t) \dfrac{d \mathbf{x}}{d s}\right]=0$ It's very ...
Vincent ISOZ's user avatar
1 vote
1 answer
72 views

Why future infinity have no future end points?

I am studying Hawking's area theorem from the book the large scale structure of spacetime by Hawking and Ellis. At the end of page#318, it said: null geodesic generators of future infinity have no ...
Talha Ahmed's user avatar
6 votes
1 answer
129 views

Does a localised particle created with a QFT on curved spacetime follow geodesics?

I was wondering if there is a result, analogous to the Ehrenfest theorem in quantum field theory (QFT), and in particular if the QFT is on a curved spacetime. In the last case, I would expect to ...
Sebastiano Tomasi's user avatar
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0 answers
24 views

Weinberg gravitation variational principle in free falling bodies [duplicate]

In weinberg's gravitation and cosmology in page 77 appears this I can't see why the equation and the symmetry of Christoffel symbols and equation 3.3.5 makes that equation 3.3.10 appears I ask my ...
Alberto Alejandro Blanco Rojas's user avatar
1 vote
1 answer
95 views

Reaching a turning point in photon trajectory

Given the geodesic equations for a photon in a Schwarzchild or Kerr metric (provided by a near BH for example), the radial equation has usually two possible signs: \begin{equation} \dfrac{dr}{d\tau}= ...
gravitone123's user avatar
1 vote
1 answer
96 views

Orthogonal self-intersection of geodesics

I learnt that geodesics parallel transport their velocity vectors. Does that mean a geodesic cannot intersect itself orthogonally?
SX849's user avatar
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1 answer
63 views

Don't Geodesics change due to other geodesics?

So the geodesics that point towards the Earth brings space-time towards the Earth and then back out again, but then the moon has its own geodesics so wouldn't it be kind of like geodesics affecting ...
Roghan Arun's user avatar
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2 votes
2 answers
199 views

In general relativity vizualizations what are the equations of grids?

I am wondering what curved 3D grids in GR outreach correspond to in equations. How would I compute such a curved grid if I know the metric tensor $g_{\mu\nu}$ in every point of space? What would be ...
Vincent's user avatar
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5 votes
4 answers
2k views

Aren't places where geodesics end singularities?

So of course when stuff falls into black holes, the geodesic for anything ends in that singularity. However, isn't it technically true that a light ray that originates from the sun and then hits the ...
Roghan Arun's user avatar
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1 answer
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Inertial and a gravitational component of two observers

Started reading the book "How Einstein found his field equations" by Janssen and Renn and I am already blocked on chapter 1. What do the authors means after the "split" below?: &...
gianpaolo's user avatar
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-2 votes
1 answer
47 views

If every particle always follows a geodesic, why do we make any sort of differentiation between geodesics and anything else?

Because all particles must follow their geodesics, is there any reason we make a differentiation between a geodesic and a "non-geodesic"? And is there any possible way to get a particle to ...
user avatar
1 vote
0 answers
32 views

Example of lightlike curve that's not a geodesic in Lorentz spacetime [duplicate]

Let $(M,g)$ be a 4 dimensional Lorentz spacetime. A smooth curve $\alpha:\ I\to M$ is called lightlike if $\alpha'(s)\in TM_{\alpha(s)}$ is lightlike for all $s\in I$, which means $$g_{\alpha(s)}\big(\...
PermQi's user avatar
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1 answer
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Brief process of solving Einstein field equations

Firstly, we solve the Einstein field equations to obtain the metric tensor. After that, we solve the geodesic equations to obtain the geodesics. Is it like this? What is the brief description of ...
Bik Kuang Min's user avatar
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0 answers
26 views

How do I find numerical solutions to geodesics when given initial four-position, initial four-velocity and a metric? (for a ray tracer)

I've had the idea to try and code a ray tracer that obeys laws of special/general relativity. In order to predict the motion of objects in the scene I'd need to compute geodesics with a user specified ...
DesertedGecko15's user avatar
1 vote
2 answers
95 views

Equations of motion in general relativity: Einstein field equations vs geodesic equation

It is said that the equations of motion of a theory are those whose solutions give the coordinates/trajectory of the system. I was wondering: which is the correct equation of motion in the theory of ...
Tomás's user avatar
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1 vote
0 answers
49 views

About the geometric optics approximation in Hawking radiation

I have read Hawking's famous paper Particle creation by Black Holes (Ref. 1) and I have some doubts about the geometric optics approximation and its implications in the argument being made. The ...
Mr. Feynman's user avatar
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1 vote
1 answer
63 views

Definition of surface gravity via the non-affine geodesic equation

I have found a discrepancy in the way different sources define surface gravity (or derive) via the non-affine geodesic equation satisfied by the a Killing vector $\xi$ on a Killing Horizon (KH), up to ...
Mr. Feynman's user avatar
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1 vote
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How increase in area of the horizon implies that the horizon in spacelike using Raychaudhuri equation?

In a talk The enigma of black hole horizons, (at 24:37), it is said that "Raychaudhuri equation implies, if the flux into H is positive, area increases and horizon is spacelike". How ...
apk's user avatar
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29 votes
2 answers
7k views

How does a laser from Earth manage to hit the Moon with precision?

A question I've been asked is how a laser, fired from earth, would hit the moon without "leading it" (or hit it with precision). When firing a laser at the moon, it takes about 3 seconds to ...
Marco Chacon's user avatar
0 votes
1 answer
53 views

Relationship between Hamilton's principle and covariant derivative

The first time I was introduced to the covariant derivative I didn't even realise that was another "kind" of derivative. Following Hamilton's principle taking an action such that: $$ S=\int ...
Álvaro's user avatar
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1 vote
0 answers
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How to get the quantity $\epsilon =-g_{\mu \nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}$ is constant along the geodesic? [duplicate]

S. Carroll in his book tells that the geodesic equation (together with metric compatibility) implies that the quantity $$\epsilon =-g_{\mu \nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}\tag{5....
Mahtab's user avatar
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1 answer
62 views

How does a proper antichronous transformation change the geodesic equation?

I want to apply a proper antichronous transformation to the geodesic equation in General Relativity and check if it is even or odd.
Manuel's user avatar
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3 votes
0 answers
221 views

How Do Gravitons Follow Geodesics?

It is known that all particles follow a geodesic in spacetime. Presumably gravitons follow geodesics as well. However, how does one describe that mathematically? For the case of other particles it is ...
physics_2015's user avatar
0 votes
3 answers
69 views

Geodesics on a 2D flat round metric/2 contradicting results

I was solving the geodesics of $$ds^2=dr^2+r^2d\phi^2$$ for an exercise. Using as an affine parameter the coordinate $\phi$ I arrived at the equations: $\frac{d^2r}{d\phi^2}-r=0$ $\frac{1}{r}\frac{dr}...
Knickers5637's user avatar
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1 answer
58 views

Euler-Lagrange confusion

Consider the action $S = \int dt \sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}.$ Now for computing the Euler-Lagrange equations, we need the time derivative of $\frac{\partial L}{\partial \dot{q}^c} = \frac{1}{\...
Geigercounter's user avatar
1 vote
0 answers
42 views

Doubts about plane waves in bitensor formulation

I am currently working with bitensors and plane waves but I'm getting some results which don't seem to make sense and can't figure out why. So first of all we know that Synge's world function $\Omega$ ...
Mario's user avatar
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33 views

Circular orbit in a stationary, axially symmetric spacetime

From Poisson's, A Relativists Toolkit, problem 1.12 on page 27: "A particle moving on a circular oribit in a stationary, axially symmetric spacetime is subjected to a dissipative force which ...
Gleeson's user avatar
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