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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Snell shortest path : would the path depend on the observer classically?

Fermat principle for deriving Snell's law reads formally as minimizing : $$T=\frac{x}{v_1\sin i}+\frac{L-x}{v_2\sin r}$$ Where $\sin i=\frac{x}{\sqrt{h^2+x^2}}$ is the incidence angle sine and $\sin r=...
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Can the Lorentz force equation in curved spacetime be derived from the Einstein-Maxwell equations?

Given the Einstein field equations, $$R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = \kappa T_{\mu\nu}$$ that imply in particular that $\nabla_\mu T^{\mu\nu}=0$, one can show, using the explicit form of $T^{\...
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Time taken to travel by light a minimum or maximum?

I'm stuck on the following. We know that the path taken by light to travel between 2 points A and B corresponds to the path which minimises time elapsed. However, from relativity we also know light ...
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A question for expert in geometrical method and Riemannian metrics

I'm a physical oceanographer with great interest in Theoretical Geophysical Fluid Dynamics. I have some ideas on the possibility to derive the so-called: geostrophic equilibrium (i.e. on a rotating ...
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Vanishing expansion of a geodesic congruence

I am considering a timelike geodesic in the outer region of the Schwarzschild spacetime, whose tangent vector field is denoted by $X$. I know that we can construct geodesics moving on a plane $\{\...
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2 votes
1 answer
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How to simplify the process of calculating spacetime geodesics?

I want to study the movement of a particle along geodesics in an expanding universe with metric (FRW metric) $$ ds^2 = -dt^2 + a^2(t) \left( \dfrac{1}{1-kr}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\...
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Geodesics of a rotating sphere

I know when we live on a rotating sphere we feel virtual forces. By these forces how can we define our geodesics? l think these geodesics are different with the geodesics of a non-rotating sphere.
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Calculating divergence and flux of geodesic word lines

Given a family of neighbouring geodesic word lines, is there a way of calculating properties such as their divergence or flux? maybe by converting the tangent vectors of the world lines to a vector ...
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2 votes
1 answer
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Can point masses following geodesics and orbiting one another emit gravitational radiation?

I am a bit confused about this situation: according to general relativity, when two masses orbit one another, they emit graviational waves, which carry away certain energy. For example, check out ...
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Geodesic deviation and Lie dragging

Suppose that $𝑥_\mu(\lambda,𝑠)$ represents a family of curves. Let $𝑣_𝜇$ represents the the tangent vector to a curve $𝑥_𝜇(\lambda,𝑠_0)$ with $𝑠_0$fixed that is $𝑣_𝜇=∂𝑥_𝜇/∂\lambda$ and ...
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Electromagnetism as curvature of space-time? [duplicate]

If gravity can be expressed as a curvature of spacetime, can electromagnetism similarly be expressed as curvature on higher dimensions of spacetime? Also (I'm not sure if the 2 questions are linked), ...
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Rotation and Killing vectors in Minkowski spacetime

There are $3$ Killing vectors in the Minkowski spacetime related to the conservation of angular momentum. Sometimes it is mentioned that it is related to the rotational symmetry of the spacetime. But ...
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1 answer
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Geodesic equations with varying mass and the variational principle

Consider the action, $$ S = \int d\lambda\ \phi(x) \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda} \right)^{1/2}. $$ Using the variation principle we obtain, $$ \delta S = \int d\...
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Particle in "external potential" VS particle on "curved surface": equivalence?

Let's consider a non-relativistic particle - its position is $x(t)\in \mathbb{R}^n$ - in an external potential $\phi$, with Lagrangian $$L=\dot{x}^i \eta_{ij}\dot{x}^j/2 - \phi(x),\tag{1}$$ where $\...
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Airy's Water Filled Telescope: help required to fully understand the calibration procedure followed

I am trying to understand Airy's Water Filled Telescope and the calibration procedure used (Airy, G. B., "History and Description of the Water Telescope of the Royal Observatory, Greenwich", ...
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Generalised Geodesic equation for forces

im wondering if there is a general geodesic equation that describes the forces and how they act. For example I started off with the original nieve derivation of the geodesic equation: $$\frac{d}{d\tau}...
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1 answer
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What kind of geodesics are possible in spacetimes of static spherically symmteric perfect fluid spheres?

I ponder about geodesics in static spherically symmetric perfect fluid spheres. My first thought was that only radial geodesics, i.e. geodesics with zero angular momentum ($l=0$) are possible because ...
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If $F^2 = g_{pq} \dot{x^p}\dot{x^q}$ , where $g_{pq}$ is a metric tensor, then find $\frac{\partial F}{\partial{\dot{x^k}}}$

I am trying to find the geodesics in a Riemannian space, using Tensor analysis. I am also using the Principle of Variation. I want to minimize the geodesics integral whose integrand is $F$. Then, ...
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Distance between two points on Earth (arc length) doesn't increase with altitude (radius) according to GPS

Consider the following example: Point A has coordinates 45 lat, 0 long. Point B has coordinates 45 lat, 2 long. Both points are 5000 ft above sea level. The distance between them is X. Point C has ...
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2 answers
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What are geodesic curves in static perfect fluid sphere?

I have read that in perfect fluid only dust particles follow geodesics. If there is pressure in fluid, the particle trajectories are not geodesics [1]. My intention was to describe a static spacetime ...
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On the capacity for equations of motion to be contained in field equations

I've heard that the equation of geodesic motion can be derived from the vacuum Einstein field equations, although there appears to be some debate about how rigorously this can be proved, due to a ...
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Metric compatibility from existence of normal coordinate

It is well known that for a Riemannian manifold $(M, g)$, one can define a torsionless and metric-compatible connection $\nabla$, and then use $\nabla$ to construct geodesics and normal coordinates. ...
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Find resultant vector of light ray under the influence of a Schwarzschild black hole?

I'm working on implementing a GR (general relativity) raytracer for the purposes of displaying realistic visuals of a black hole against some fixed imagery (pictures of stars, for example.) I don't ...
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2 votes
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Schwarschild radius and paramaterizing path

Consider the metric $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.$$ Suppose a particle very large starts at the initial radius $...
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1 vote
1 answer
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Does AdS/CFT help solve the singularity of a black hole?

How does thinking of a black hole as encoding its information in its surface help solve what happens inside it, more specifically geodesic incompleteness. Doesn't it tell us that if we can see how ...
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1 vote
1 answer
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Regular vs. General Geodesic Equation

I'm reading Wald, and I've just got up to the Geodesic equation: $$T^a \nabla_a T^b = 0.\tag{1}$$ Right after, Wald says that "one might require only that the tangent vector to the curve point in ...
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1 answer
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Can we derive the geodesic equation from the Lie derivative?

I have seen everywhere about derivation of geodesic equation from covariant derivative, variational principle but my question is: can we derive geodesic equation from the Lie derivative also?
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1 answer
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Geodesic deviation equation with lowered indices

I am studying General Relativity from Schutz's book. In chapter 7 he starts with the geodesics equation with lowered indices: $$ p^{\alpha}p_{\beta;\alpha} = 0 \qquad(7.25) $$ and goes on to derive ...
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1 answer
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A Lagrangian of sufficient generality

Consider the Lagrangian of the form \begin{equation} \mathcal{L} = \frac{1}{2}g_{\beta\gamma}\dot{q}^\beta\dot{q}^\gamma+A_\beta\dot{q}^\beta-V \end{equation} where $g_{\beta\gamma} = g_{\gamma\beta}$ ...
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Taut strings in curved spacetime

In general relativity, a straight line is defined (or, in a sense, replaced?) by the path a light beam takes. In Newtonian physics, another way to approximate a straight line between two points is to ...
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Quaternion formulation for parallel transport along a curve on a 2-sphere

One image that is often used to illustrate curvature in general relativity is the triangle on a 2-sphere, made out of great circle arcs. At the end of a geodesic transport along this triangle, the ...
2 votes
1 answer
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Photon momentum, null geodesics and FRW metric

I have some homework that I am really struggling with (this class is above my level), and I would hope I could get some clarification. I am given the geodesic equation $$\frac{\mathrm{d}^{2}x^{\mu}}{\...
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Geodesics from a specific metric

I am currently working through the exercises from "problem book in relativity and gravitation". I am stuck at a particular question. The question: Calculate the time like geodesics of the ...
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2 votes
2 answers
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Assumption of affine parametrization in the geodesic equation derivation

The derivation of geodesic equation is straight from Padmanabhan's book on General relativity. Consider the action $$A = \int d\tau=\int\sqrt{-g_{ab}dx^adx^b}.\tag{4.39}$$ We impose the condition $\...
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Lagrangian mechanics and geodesics in configuration space? [duplicate]

In lagrangian mechanics Is the path that take a System in the configuration space between initial and final state is identical to the geodesic which connect the two points?
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4 votes
3 answers
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On an infinite plane, with gravity the same of that of Earth, how far could light at an arbitrary angle travel until bending to hit the plane?

Now, I'm a complete idiot, so bear with me. I've recently come across the idea that standing an infinite flat Earth would in theory appear the same as standing inside a hollow earth, since light would,...
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3 votes
0 answers
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Which geodesics light travels on: metric or affine in Einstein-Cartan theory?

In a generic Lorentzian spacetime solving the Einstein equations since both the metric and affine geodesic coincide this question doesn't arise. But in the Einstein–Cartan theory it is not very ...
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Geodesic equation using 4-momenta

I know the geodesic equation in the form $$\frac{d^2X^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\alpha}\frac{dX^\nu}{d\lambda} \frac{dX^\alpha}{d\lambda} = 0,$$ where $\lambda$ is a parametrization of the ...
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Eddington-Finkelstein and extension of the Schwarzschild spacetime

I'm trying to understand how the Eddington–Finkelstein coordinates lead to the extension of the "original Schwarzschild spacetime" (which I understand to be the region with $r > 0$ in the ...
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2 votes
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Geodesic equations and carter constant: how can I derive the right equations?

Introduction Suppose the Hamilton-Jacobi equation: $$\frac{\partial S}{\partial \lambda} + \frac{1}{2}g^{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}}= 0. \tag{1}$$ Now,...
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Inverse haversine formula - Change in geocentric coordinates from great circle distance and bearing?

I think this question best fits here - it's reasonably mathematical in nature, but is formulated as a practical kinematics problem; I'm happy to move it if I'm wrong. I'm trying to build a plane ...
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3 votes
1 answer
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Christoffel symbols in normal coordinates

I'm confused by the proof of $\Gamma^\lambda{_{\mu \nu}}(p) = 0$ in the normal coordinates. Let $p \in M$ be a point with an orthonormal basis $e_a$, and one considers the exponential map $\exp_p: T_p ...
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Knowing the position of an object at different times, can we figure out the possible shapes of spacetime and object's trajectory?

I'm trying to figure out if it's possible to apply general relativity to determine the probable future locations of an object. After seeing this excellent illustration of geodesics, it made me think ...
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5 votes
2 answers
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Do test particles initially comoving with a black hole accelerate away from it?

This question feels ridiculous, but I really am confused. If you Google Image search "schwarzchild light cones" it shows how, relative to the frame comoving with the singularity, the speed ...
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Confusion with timelike geodesics in de Sitter space

Without deriving the whole geodesic, I'm thinking I should be able to qualitatively see how the geodesics will curve just by looking at the connection coefficients. Given the static coordinates $$ds^...
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Ryden Eqn 5.49, Proper distance with changing scale factor

In Chapter 5 on Barbara Ryden Intro to a Cosmology 2nd ed, she presents this equation for calculating the proper distance at current time ($t_0$) to a distant galaxy whose light emitted at time $t_e$ ...
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Is refraction just an application of minimising the distance $s$ in 4D spacetime?

I apologise that I don’t know how to fully put into words the thought I’m having so it may be a bit rambly. In school we’re taught that refraction of light happens because as light enters a denser ...
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1 answer
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How to explain the trajectory of object going up then come down with curvature of spacetime?

I can understand that the Moon is orbiting the Earth because it is going in a straight path within the distortion of spacetime caused by the Earth's mass, so not outside force is required. However, if ...
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4 votes
4 answers
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Why do object accelerate towards the Earth in general relativity?

In general relativity something in free fall, that appear to accelerate towards the earth, is actually not accelerating at all but moving along a geodesic so why does it appear that it is accelerating ...
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Acceleration equation for the geodesic deviation equation

Defining a family of geodesics by $\gamma_s$, parametrized by the affine parameter $t$, the coordinates are defined by $x^{\alpha}(s,t)$. The vector tangent to the geodesics is defined as $u^{\alpha} =...
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