# 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).

951 questions
Filter by
Sorted by
Tagged with
20 views

### 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 ...
• 2,035
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 ...
• 163
1 vote
50 views

### 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 ...
• 171
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 $d\epsilon(s)/ds$ ...
• 366
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 ...
• 115
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 $\tau$, ...
• 337
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 ...
• 366
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 ...
• 411
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 ...
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,...
• 770
1 vote
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?
• 31
1k views

### 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 ...
• 131
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 ...
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, ...
• 133
1 vote
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 ...
• 756
1 vote
84 views

87 views

• 143
77 views

### 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 ...
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 ...
1 vote
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 ...
• 309
1 vote
49 views

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 ...
• 1,989
1 vote
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 ...
• 1,989
1 vote
32 views

### 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 ...
• 293
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 ...
• 475
53 views

• 374
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.
• 476
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 ...
• 357
69 views

1 vote
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 $\Omega$ ...
• 49