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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20 views

With north and south poles fixed, do all geodesics have constant $\theta$ and $\phi$?

I was going thorough reading Kolb and Turner's The Early Universe where in Section 2.2 it starts by asking the following question. For a comoving observer with coordinates $(r_0,\theta_0,\phi_0)$, ...
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34 views

Radial null geodesics in Schwarzschild de Sitter space

I am currently studying the geodesics of different type of spacetimes and I'm not sure if I'm doing it in the correct way for Schwarzschild de Sitter space (SdS). The metric in SdS is given by: $$ ds^...
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91 views

A Potential Euler-Lagrange Equation Alternate Derivation?

Can the Euler Lagrange Equation be derived with this overall strategy? Step 1 – Define a geodesic in flat space to be $\frac{d}{dy} \frac{ds}{dx} = \frac{d}{dx} \frac{ds}{dy}$, where $ds$ represents ...
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49 views

Geodesic Equation from Coordinate Transformation

Let $\xi^a$ be the usual coordinates and $x^\mu$ the new coordinates, both flat. Now we know that since the metric is flat, $$ \frac{d^2\xi^a}{d\tau^2} = 0 $$ $$ \Rightarrow \ \frac{\partial}{\...
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23 views

Explicit form of the Christofffel Symbol used in Geodesic Equation

One way to motivate the Christoffel symbol is to consider the partial derivative of a tensor, $T_\alpha$ $\frac{\partial T_\alpha}{\partial x^\gamma}=\frac{\partial^2 x^\beta}{\partial x^\alpha \...
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74 views

Geodesics in general relativity changing from timelike to spacelike?

Is it possible for geodesic in general relativity to be time-like, and at some point cross null-plane and become space-like?
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1answer
45 views

Understanding the geodesic equation in a Wikipedia article

I was reading this Wikipedia article which attempts to motivates some concepts key to General Relativity in the Newtonian setting first. However I was not able to understand one of the equations ...
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78 views

Confusion re: geodesics, connections, and straightness

In my readings in GR I often come across geodesics characterized as "straightest possible curves." This characterization confuses me. I'd like some clarification as to whether I'm understanding the ...
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What do we mean by straight line in Inertia Law, autoparallel curve or extremal line?

As usually stated: Newton’s first law states that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed ...
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69 views

Geodesic equation in Einstein-Cartan manifold

In GR we use the Riemannian manifold without any torsion to describe the theory. Hence, the geodesic equation can be interpreted as "a trajectory of a free falling particle" or "equation of motion". ...
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165 views

Light does not always travel on null geodesic?

I am currently reading Wald's General Relativity and a result of section 4.3 stomped me. Part of Maxwell's equation in GR may be written as $$\nabla^a F_{ab} = \nabla^a \nabla_a A_b - R^a_b A_a= 0.$$ ...
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Geodesics and first fundamental form

If we consider Hamiltonian $H(p,q)=(1/2)p^TAp$. where $p=(p_1,p_2)$ and $A$ is a symmetric matrix $A_{ii}=e^{q_i^2}+1$ and $A_{12}=A_{21}=1$. Then how I proceed to find the first Fundamental form of a ...
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207 views

Minimizing proper time

I've started studying general relativity course and now I have a question about proper time. Consider functional $$S[x]=-\int_A^B ds,$$ where $A$, $B$ are fixed points of the space-time and $ds^2=dt^2-...
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41 views

Inertial frame and its transformation in anti-de Sitter spacetime

From the wikipedia, I learned and was able to follow mathematically the definition of anti-de Sitter space. As the maximally symmetric solution to field equations with negative cosmological constant, ...
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49 views

geodesic equation for electromagnetic field

I am trying to derive geodesic motion for photons from the Lagrangian of electromagnetism coupled to General Relativity. I tried to use the covariant conservation of the Stress energy tensor: $$\...
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70 views

Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$ I interpret the metric to be a spacetime,...
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56 views

Why does things travel in a straight line in inertial frames?

Why does physical entities travel in straight paths in a flat space-time and in geodesic in curved spacetime? Is it due to Inertia? If it is, then why does waves also follow the same pattern?
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107 views

Acceleration in general relativity

Let's say that, from my point of view, another observer is accelerating. Now, from his point of view, he is standing still: all he feels is an overall fictitious force of gravity, which is just a ...
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82 views

How to formally integrate the geodesic equation given the Christoffel symbols?

Given the geodesic equation $$\ddot{x^a} + \Gamma^a_{\;bc}\,\dot{x}^b\dot{x}^c = 0$$ with initial conditions, say $x^a(\lambda=0)=0,\dot{x}^a(\lambda=0)=v^a$, and all the data of Christoffel symbols ...
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49 views

Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-...
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35 views

A Second Geodesic Equation?

I have a question about the geodesic equations. I understand the following formulation of it: $$\frac{d p^\mu}{d\tau} = - \Gamma^j_{v\mu}p^vp^\mu.$$ However, I was reading https://arxiv.org/abs/1305....
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Do virtual photons follow spacetime curvature?

I have read this question: https://link.springer.com/chapter/10.1007%2F978-3-319-13443-7_26 The electric field lines from a point charge — and the rays of light when the charge is replaced by a ...
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26 views

Gradient of the null affine parameter

For a timelike geodesic, we have $\frac{d}{d\tau}\tau=V^a \partial_a \tau=1$, where $\tau,V^a$ is the proper time and four velocity. It is thus natural to identify $\partial_a \tau$ as $-V_a$. My ...
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Interpretation of conserved covariant tangent vector norm in the presence of an EM field

The motion of a point particle in curved spacetime can be obtained by extremising $$S = \int L d\lambda= \int \left( \frac{m}{2}g_{\mu\nu}(x) \dot{x}^\mu \dot{x}^\nu \right) d\lambda,\tag{1}$$ where $...
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209 views

Does static gravity follow spacelike geodesics?

Thank @KyleKanos for a suggestion, I am rephrasing this question to hopefully make it more clear. According to this publication in Physics Letters, 2000: Aberration and the Speed of Gravity: It is ...
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Newtonian geodesic equation in barycentric frame

In a paper I was reading, I came across the Newtonian geodesic deviation equation $${\ddot{\eta}}^a + K{^a}{_b}{\eta^b}=0$$ Where $\eta{^a}$ are the components of the 3 -vector separating the two ...
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Minimum required initial conditions to uniquely solve geodesic equation

The geodesic equation is a 2nd order differential equation given as $$\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d} \lambda^2 }+\Gamma^\alpha_{\beta\gamma}\frac{\mathrm{d} x^\beta}{\mathrm{d} \lambda }\...
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54 views

Why is the Christoffel symbol in the geodesic equation for a test particle negative?

The geodesic equation is $$ {d^2 x^\mu \over {ds}^2}+\Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}=0\ $$ for some scalar parameter of motion s and connection coefficients of ...
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59 views

How does spatial curvature apply to the planets' orbits?

We all know that in the presence of large, massive objects, spacetime is positively curved, more so the more massive it is. This means that the path of an object without any forces on it is a straight ...
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42 views

Metric tensor and dependence on parameter

In "Tensors" by A. Das on p166 the author derives the first integral (5.202) of the geodesic equation. To achieve this, he uses the chain rule $$\frac{dg_{jk}}{d\tau} = \frac{\partial g_{jk}}{\...
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123 views

Spacelike, timelike and null geodesics

I have a question about an exercise in Misner, Thorne, and Wheeler's Gravitation. On page 321, exercise 13.5 says Show that a geodesic of spacetime which is timelike at one event is everywhere ...
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“Chain Rule” for functional derivatives in the context of a derivation of the geodesic equation by the stationary proper-time principle

I have been working on deriving the geodesic action via finding the stationary points of the proper-time integral for a massive point particle. Consider a space-time manifold $M$ ($\dim M=4)$ equipped ...
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73 views

GR visualization

I'm watching some GR lectures by Schuller (more or less rushing through them so bear with my ignorance here please) in Lecture 10: Metric Manifolds. He's talking about geodesics in a manifold with a ...
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26 views

Regd. derivation of some equations in “Bertrand Spacetimes” by Pelick

We are going through "Bertrand Spacetimes" by Dr Perlick, in which he first gave the idea of a new class of spacetimes named as Bertrand spacetimes after the well-known Bertrand's Theorem in Classical ...
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Line element of a field?

I recently had a question ... Is there an intuitive way to go about the line-element of field? (without going into method below). The only way I can conceive going about this is to use Einstein field ...
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106 views

Derivation of the geodesic equations

Pg 79 of "Tensors, Relativity and Cosmology" In order to construct the geodesic equations which define the curve with a stationary arc length, we may choose the arc length itself as the action ...
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Geodesic equation — alternative form

From Thomas Moore A General Relativity Workbook I have the geodesic equation as, $$ 0=\frac{d}{d \tau} (g_{\alpha \beta} \frac{dx^\beta}{d \tau}) - \frac{1}{2} \partial_\alpha g_{\mu\nu} \frac{dx^\mu}...
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Some aspect of covariant derivative of point particle energy-momentum tensor

My question is related to Derivation of the geodesic equation from the continuity equation for the energy momentum tensor I need to understand one step in derivation. Let's consider the Energy-...
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107 views

Is there a known mechanism for mass-energy distorting spacetime?

I’ve been really interested in learning about the mechanisms behind physical phenomena that go beyond just learning to manipulate the equations and give a physical intuition about HOW something ...
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How to compute Kerr geodesics?

How would I start to numerically compute trajectories of Kerr geodesics with constants of motion like in this wikipedia page. I want to recreate trajectories like in this picture in Matlab.
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33 views

Derivative with respect to a coordiante differential (geodesic equation)

If the arc length is chosen to be the action integral, that is $$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$ Then Lagrangian is given by $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
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Sean Carroll GR - Ex.3.6 (b) & (c) [closed]

I'm working in the newtonian limit of GR with the metric $$ ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dr^2 +r^2d\theta^2+r^2sin^2\theta\;d\phi^2 $$ where $$\Phi = -\frac{GM}{r}.$$ We are first asked to ...
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118 views

Geodesics from variational principle with respect to coordinate?

I know you can find geodesic equations with respect to proper time $\tau $ using the variational principle, i.e. using Euler-Lagrange equations $$ \frac{\partial}{\partial x^{\mu}}L-\frac{d}{d\tau}\...
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Rotationally invariant metrics and conservation of angular momentum

This was prompted by an exam question, though the questions are more general: A 2D Riemannian space has the metric: $ds^2=dr^2 + \gamma^2 r^4 d\phi^2$ State what conserved quantity ...
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Proving that test particles in GR, follow spacetime geodesics

My question is pretty much in the title. According to this paper, this is not exactly proven rigorously yet. What I dont understand is what exactly is not proven. If I'm not too wrong, a test particle ...
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68 views

Radial infall in Schwarzschild

In Straumann's book on general relativity, one finds the following solution to the question of geodesic radial infall into the black hole: $$d\tau=(\frac{2m}{r}-\frac{2m}{R})^{-\frac{1}{2}}dr$$ To ...
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Why can we parameterize a null geodesic such that its velocity is four-momentum? [duplicate]

One principle in general relativity is that the wordlines of massless particles are null geodesics. It also seem to be a commonly stated fact (for instance see eq. (3.62) in Section 3.4 of Carroll's ...
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22 views

Fundamental Principle of Dynamics and equations of geodesics with proper time

I just wanted to have a little precision. In the expression below translating the PFD (Fundamental Principle of Dynamics) in tensor calculus (or more precisely the inertial principle) : $$a^{\nu}=\...
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36 views

Equality between derivatives of the metric

In one of my lecture, it is said: Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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129 views

How does General Relativity explain escape velocities?

In general relativity, objects follow the shortest possible path through curved space-time called a geodesic and that there exists no such force of gravity which pulls objects, it is just because ...