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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55 views

Is there a GR solution for the geocentric system?

It is possible to get the Schwartzschild metric assuming spherical symmetry, vacuum solution and Minkowski spacetime when $r \to \infty$. Is it possible an analytic solution for a geocentric system? I ...
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56 views

Throwing darts in a rocket ship

Far from major gravitational sources... A man is standing throwing darts at a dartboard in a rocket ship. The thrust is upwards so he feels almost like he would on Earth but the G-force is slightly ...
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4answers
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Hartle's definition of the Lagrangian when discussing geodesics

When Hartle discusses the geodesics in his Gravity: An Introduction to Einstein's General Relativity book he uses the following definition for the Lagrangian: $ L \Big(\frac{d x^\alpha}{d \sigma}, x^\...
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From geodesic motion of photons to Maxwell equations in curved spacetime

In curved spacetime, 1.) a photon is supposed to move along a null-geodesic, i.e., a trajectory $x^{\mu} = x^{\mu}(\lambda)$ satisfying $$\frac{d^{2}x^{\rho}}{d\lambda^2} + \Gamma^{\rho}{}_{\mu\nu}\...
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Rindler metric and equivalance principle [duplicate]

I'm trying to understand the connection between The Equivalence principle and the Rindler space. According to Einstein the inhabitants of the elevator should feel the acceleration. However, for an ...
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Does a free particle always follow the trajectory of shortest distance? [duplicate]

Some context for the above question is warranted. While reading Hartle's Gravity, the following statements made recurring appearances: "Gravity is not a force, it is the geometry of 4D spacetime. ...
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1answer
53 views

Why does homogeneity imply $d P^{\mu} / dX^{i} = 0$ along geodesic?

I'm reading the Cosmology lecture notes Daniel Baumann and there they describe the path of a freely-falling particle along a geodesic, which is denoted by the curve $X^{\mu}(\tau)$, $\tau$ being ...
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55 views

Deriving the effective potential from the Schwarzschild metric

I am attempting to derive the effective potential for a test particle around a massive object from the Schwarzschild metric; however, an extra term has appeared in my solution compared to Wikipedia's, ...
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1answer
38 views

How Would Projectile Motion Be Described in Non-Euclidean Space?

If I and a friend found ourselves in a world described by spherical geometry (as simulated in this linked video https://youtu.be/yY9GAyJtuJ0 ) how would the kinematics equations need to be augmented ...
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1answer
37 views

Synge's World function generates geodesic flow

Let $(Q,g)$ be a Riemannian manifold and let $q_0,q_1\in Q$ be two points that are joined by a unique geodesic $\gamma$ (this holds in particular if $q_1$ belongs to a normal neighborhood of $q_0$). ...
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Transverse metric being 2-dimensional in null case

In Wald section 9.2 page 221 he says that We turn our attention; now , to null geodesic congruences. Again, we parameterize the geodesics by an affine parameter $\lambda$, but , unlike the timelike ...
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Is there anything such as gravitational field-lines in GR similar to the electric/magnetic field lines in electromagnetism?

I sometimes mistake space-time curvature for gravitational field lines. Do geodesics in some ways represent $g$-field lines? Why is not it traditional to show $g$-field lines around a massive object ...
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Is this result valid only for a non-rotating system?

I am reading this paper "Quantum Raychaudhuri equation" by S.Das. In this paper, the author derives a quantum version of the Raychaudhuri equation which has nice implication on the focussing ...
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“And God said … and the universe was …” What does this equation mean?

There is a T-Shirt with this equation on it: It says: And God said ... and the universe was ... It looks like something related to General Relativity but I don't know what is it? Could you help me? ...
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1answer
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Inverting the parallel transport via path-ordered exponential

In Riemann geometry one can formally solve the parallel transport equation $$ \dot{v}^\mu + \Gamma^\mu_{\rho\sigma} \, u^\rho \, v^\sigma = 0 $$ of a vector $v$ along a curve with unit tangent vector $...
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1answer
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How to derive this relation?

According to A relativist's toolkit by Poisson, the expansion of null radial geodetic in the Schwarzschild spacetime is $$\theta=\dfrac{2}{r}$$ How to derive this expression? The expansion is defined ...
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Radial outgoing null geodetic in Kerr spacetime

According to Poisson's book A relativist's toolkit, pag 52, with the Schwarzschild metric I can define outgoing radial null geodesic as follows: $$u=t-\int f(r)^{-1} dr$$ where $f(r)=1-\dfrac{2m}{r}$. ...
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86 views

How to calculate spacetime interval for Schwarzschild metric? [closed]

Sorry, my question may be feel dumb but still I am finding it hard to understand & calculate, how can one calculate $s$ from Schwarzschild metric equation $ds^2=-c^2(1-M/r)dt^2+r^2(d\theta^2+sin^2\...
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Generalizing a flat-spacetime-approach for time dilation to curved spacetimes

I would like to discuss an idea to generalize a flat-spacetime-approach for time dilation to arbitrary curved spacetimes. Starting Point Suppose we have - in flat spacetime - one inertial observer ...
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1answer
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Representing bound geodesics in Penrose diagrams

I recently started reading about conformal (Penrose) diagrams and have since been faced with a couple of conceptual doubts. Based on the coordinate transformations, null curves in Penrose diagrams are ...
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3answers
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Does the Ricci tensor actually change volume of a body in reality?

I have learned about the Ricci tensor and I was wondering if it actually changes volume in real-life physics. The question I know the Ricci tensor is a mathematical object but does it only change the ...
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Interpretation of $t$ coordinate for charged black hole in Kerr-Schild form

In GP coordinates, the $t$ coordinate matches the proper time of a free-falling observer starting at infinity with zero velocity. Now consider the Reissner–Nordström metric in Kerr–Schild form: $$ g_\...
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Proper time, geodesic and external force

Deriving the geodesic equation $$ (\nabla_u u)^\mu = 0 $$ one uses the variation of proper time $\tau[C]$ along $C$ $$ \delta \int_C d\tau = \int_C d\tau \, g_{\mu\nu} \, \delta x^\mu \, (\nabla_u u)^\...
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1answer
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Describing the deformation of a medium as a diffeomorphism

In this paper online, the author models the deformation of a medium as a diffeomorphism $ \mathbb{R}^3 \rightarrow \mathbb{R}^3$ as given by: $$ y^i \mapsto x^i(y)=y^i + u^i(x) $$ as given by equation ...
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Singularity in the Expansion of Geodesic congruences

My question is regarding the following graph, The $X$-axis is the $r$ coordinate of the $(t,r,\theta,\phi)$ system, and the $Y$-axis is the expansion $\Theta$ of a congruence of weakly bound, ...
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1answer
73 views

How can I interpret the equation of the orbit in Schwarzschild metric?

Given the standard geodesic equation: $$\frac{d^2 x^\mu}{d\lambda^2}+\Gamma ^\mu _{\sigma \rho}\frac{d x^\sigma}{d \lambda}\frac{d x^\rho}{d \lambda}=0$$ we want to apply it to the Schwarzschild ...
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5answers
319 views

Why do photons follow the geodesic curvature of the gravitational field instead of the spacetime curvature? [closed]

If mass merely 'curves' spacetime, why do photons follow the geodesic path of the gravitational field (path A) instead of the spacetime curvature itself (path B)? It seems, as if, the gravitational ...
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2answers
75 views

Runaway solutions of geodesic equations [closed]

Given a metric tensor $g_{\mu\nu}$ it is possible to calculate the geodesic equations from: $$\dfrac{d^2x^{\mu}}{ds^2}=\Gamma^\mu_{\nu \eta}\dfrac{dx^\nu}{ds}\dfrac{dx^\eta}{ds}$$ where the $\Gamma^\...
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Parametric Geodesics of a 2-sphere

The geodesic equations of a 2-sphere are the following: $$\ddot\theta - \sin\theta \cos\theta\dot\phi = 0$$ $$\ddot\phi + 2\cot(\theta)\dot\theta \dot\phi = 0$$ For $\theta$=$\pi$/2, $\phi(s)$=a$s$+b. ...
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297 views

Why does a gravitational field permanently alter the direction of photons?

Given we observe that light do follows path A (gravitational lensing) instead of path B, is there any direct empirical evidence about how photons and gravity interact, other than stating that photons ...
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45 views

How to check if some metric has a particular type of geodesics?

For instance, we want to know if cylindrically symmetric de Sitter-type spacetime has an axial geodesic. This is the metric I am interested in $$ds^2= \cos^{\frac{4}{3}}\left(\frac{\sqrt{3 \Lambda}}{2}...
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1answer
72 views

Geodesics and constraints on the parameterization

I am following an introductory course on General Relativity based on the work of Sean Carroll in: Spacetime and Geometry. After a lot of trouble we get to the following differential equation: $$\frac{...
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3answers
143 views

Why do we fall towards earth and not hover during free fall as per General relativity? [duplicate]

So this is what I understand from General Theory of Relativity: A body freely falling towards earth's surface would be in an inertial frame of reference (air removed) with zero net force acting on it. ...
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How does posing $\sqrt{\dot{x}\eta \dot{x}}=c$ not contradict the fact that we are looking for variations?

For example, if I have a function which gives the radius of a circle dependant on $x$ and $y$: $$ R[x,y]=\sqrt{x^2+y^2}.\tag{1} $$ Then its partial derivative with respect to $x$ will be $$ \frac{\...
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Can we reverse the geodesic equation to find a metric for the theory?

The geodesic equation describes the motion of a particle moving in a straight line embedded in a curved geometry. $$\frac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\...
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3answers
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How does Fermat's principle of least time for light apply to curved spacetime?

In a region of space which has no massive object light rays travel parallel to each other or ,simply, in a straight line. However, in a positively curved region of space (like near a planet or a star),...
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Can geodesic deviation in free fall sometimes be indistinguishable from mutual gravitation?

Suppose you are in radial free fall at some point outside the event horizon of a Schwarzchild metric. The strong equivalence principle implies that locally you would be unable to discern whether you ...
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Can the image of a spacetime geodesic be characterized through Synge's world function?

Since a geodesic is understood to be a map $\gamma : \text{real number interval} \rightarrow \text{spacetime } \mathcal S$, with certain additional properties, the image of a particular geodesic is ...
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27 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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31 views

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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1answer
52 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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1answer
64 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{\sqrt{\...
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1answer
61 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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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
93 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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0answers
19 views

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
88 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
60 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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1answer
31 views

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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