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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68 views

What are the initial conditions for solving Schwarzschild geodesic equations?

I am trying to solve the Schwarzschild geodesic equations and trying to plot them. I am new to the subject, so I am struggling with the initial conditions that I need to feed my computer. For ...
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Intensity along null geodesic

Let $\mathcal P$ be a bundle of light with a continous spectrum of frequencies $\nu$ emitting from $x^\mu$ in a static spacetime $g_{\mu\nu}$ (e.g. kerr-schild) in the direction $n^\mu = \frac{p^\mu}{\...
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1answer
113 views

Curvature in the Newtonian Gravity

Let me give a little bit of insight. I was trying to calculate the geodesic of different curves when I realised some relation (if I can call it like that), between General Relativity and Newton's Law ...
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4answers
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Why is it said that, because gravity can bend light, that this entails a curvature of space-time?

Why can't gravity affect light directly without any reference to space-time? How does light bending show that space-time is curved? I would have thought that it shows that massless particles are ...
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1answer
58 views

What is the behaviour of massive particles traveling between tethered galaxies in the FLWR metric?

I was reading the following papers, arXiv:astro-ph/0104349 and arXiv:astro-ph/0511709. Both papers investigate the situation where two galaxies are held together by some sort of tether, such that the ...
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3answers
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Physical meaning of geodesic equation $p^\lambda \nabla_\lambda p^\mu=0$

In Sean Carroll's GR book pg. 109, it was said that the geodesic equation for timelike paths can be written in terms of the four momentum $p^\mu=mU^\mu=m\frac{dx^\mu}{d\tau}$: $$p^\lambda \nabla_\...
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2answers
464 views

Derivatives of Lagrangian for relativistic massive point particle

For a relativistic point particle with mass $m$ whose worldline is parameterized by $x(\lambda)$ the standard Lagrangian is: $$L(\dot{x}) = -mc\sqrt{g_{ab}\dot{x}^a \dot{x}^b} \tag1$$ where $g$ is a ...
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2answers
57 views

Calculating Distance between Prime Meridian and WGS84 location? [closed]

I'm trying to calculate the distance between the WGS84 meridian at 51°28′40.1″N 0°0′5.3″W and the actual prime meridian at 51°28′40.1″N 0°0′0″W in feet. Wiki claims the two points are 5.3'' from each ...
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1answer
36 views

Transforming absolute derivative of separation four-vectors in GR

Moore (A General Relativity Workbook) claims that the following implication holds, where $\boldsymbol{n}$ is the separation four-vector between two objects at a given proper time: $$ \frac{d\...
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1answer
208 views

Why would geodesics depend on velocity in general relativity?

In general relativity, it's the curvature of spacetime that gives the effect of gravity, due to objects following geodesics. What I learnt about geodesics is that they are like straight lines in ...
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1answer
42 views

Universe flatness

On: https://en.wikipedia.org/wiki/Shape_of_the_universe It is written: "The exact shape is still a matter of debate in physical cosmology, but experimental data from various independent sources (...
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1answer
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What is the physical meaning of Riemann normal Coordinates?

It's a beginner's question and hopefully not to trivial for this forum: The frame of Riemann normal coordinates (RNC) with regard to a point $P$ in a given metric $g$ is often said to be the reference ...
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1answer
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E.O.M of a free- particle with "dynamical mass" from action principle in General Relativity

If one tries to obtain the E.O.M of a (massive) free-particle, one should extremize the action: $$ S_{m} = \int ds = - \int m \sqrt{-g_{\mu \nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}} d\...
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Resolving GR with two dropped objects

I'm trying to understand GR and geodesics in relation to a specific example. Two objects, suspended at different heights so that one is over the other while both of their velocities relative to the ...
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2answers
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The time component of a geodesic [closed]

A geodesic is the path taken by a falling object so if the person is standing still he is not following a path, he is stationary on a path. This could be called "standing still" on the ...
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1answer
135 views

Newtonian motion from a simplified Schwarzschild's metric in 1+1D

I've read that the simplified Schwarzschild's metric in the $t$-$z$ space $$ds^2=\left(1+2\phi\right)dt^2-dz^2$$ could (approximately) reproduce the classical motion of a particle in a constant ...
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Stationary on a geodesic

If the Earth were not spinning or orbiting the Sun would we still feel gravity? I ask because it seems to me that in that case we would not be accelerating and we would still be standing on a geodesic ...
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4answers
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How does the curvature of space-time explain a ball falling in reverse?

If we throw a perfect uniform ball up in the air without any rotation in a vacuum room in a perfect perpendicular direction of earths gravitational pull and mark the part of the ball that is pointing ...
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1answer
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Why do Fermi Coordinates describe a System, where an accelerated Point moving along a time-like geodetic is at Rest?

It is not yet clear to me, what Fermi coordinates really mean. They are constructed in a way, where three space-like coordinates $x'^\mu_a, a=1,2,3$ orthogonal to the time-like trajectory of an ...
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559 views

Actual astronomic latitude (direction of gravity) does not match calculations taking into account centrifugal force of earth

ABBREVIATED QUESTION/PROBLEM: The direction of earth's gravity vector does not point towards the center of the earth due to centrifugal force. Theoretically, it should point away by .1 degree. But it ...
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4answers
912 views

How do geodesics explain two identical balls thrown up at the different speeds? [duplicate]

As stated in the title, two identical balls, both thrown directly upward, but at different speeds. The slower ball will reverse direction at a lower height than the faster ball. But the curvature of ...
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3answers
51 views

Do massive objects follow geodesics in the same manner as electromagnetic waves?

In my understanding of GR, massive objects and light both follow the geodesic of whatever frame they are in. But I'm having a hard time understanding this in some specific situations. On earth, if I ...
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Basic confusion about interpretation of the 5th dimension in Kaluza-Klein theory

As has been mentioned in other posts, Kaluza originally didn't require the 5th dimension to be curled up/compactified. So how exactly would our 4D world emerge from a non-compactified 5D manifold? I ...
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2answers
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Can a Hamiltonian flow be rephrased into a geodesic flow?

Let us consider a classical Hamiltonian system composed by $n$ particles, namely, a Hamiltonian function $H:\Lambda\to\mathbb{R}$ where $\Lambda\subset\mathbb{R}^{2n}$ is the phase space. Then, let us ...
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Deriving the large $E$ expansion for geodesic boundary time from paper arXiv:2004.01192

In equation (14) of the paper "Holographic flows from CFT to the Kasner universe" https://arxiv.org/abs/2004.01192, they express the boundary time as $$\label{1} t(0) = -P \int^{r_{\star}}_{...
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Geodesic deviation equation with covariant derivative [closed]

Consider two wordlines $x^\mu(\sigma)$ and $x^\mu(\sigma)+\xi^\mu(\sigma)$, where $|\xi|<|x|$. The two wordlines fulfill the geodesic equation $$\frac{\text{d}x^\mu}{\text{d}\sigma^2}+\Gamma^\mu_{~\...
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Busemann function & entanglement entropy

In seminal paper by Ryu & Takayanagi it was proposed and shown that entanglement entropy for CFT$_2$ can be computed with help of AdS$_3$ consideration or, more conretely, $$S_{EE}\propto L(\...
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Deriving Newtonian Gravity from General Relativity? [duplicate]

Let me provide some insight, I was studiying about General Relativity, and I read that it "encapsulates" Newtonian Gravity. I understand this perfectly, because, Einstein used lots of ideas ...
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2answers
94 views

Falling body geodesics seem counterintuitive

We all know, a geodesic is the path an object follows to maximize its proper time. Geodesic equations of an object near a gravitational field show that it should fall towards the ground, which is the ...
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2answers
116 views

Are a tachyonic particle a mathematical impossibility (not just physical)?

I recently learned from a helpful SE user that, in general relativity, the "law of geodesic motion" is actually a mathematical law, not a physical one. That is, a "test particle" (...
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Geodesic: Why should an apple fall? [duplicate]

A geodesic is the path an object follows to extremise (maximise) its proper time. I am still confused why the final position of the apple should be near the earth. My argument: an apple falls so that ...
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1answer
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Doubt regarding the existence of conjugate points on timelike geodesics

The expansion of a timelike geodesic congruence with (normalized) tangent vector field $\xi^a$ is defined as $\theta=\nabla_a\xi^a$. Assuming the strong energy condition, $R_{ab}\xi^a\xi^b\geq 0$, and ...
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2answers
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Geodesic Ending Point

A geodesic is the path that an object follows in its world line. It maximizes its proper time given a starting point and an ending point. That's why things move in a straight line in a flat spacetime. ...
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0answers
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Warped space shouldn't cause movement in Gravity so what does? [duplicate]

I'm trying to get an intuitive understanding for how gravity works and have been reading various explanations from answers here to Sean Carroll and others' youtube videos. I have a set of university ...
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Expression for chain rule of a geodesic equation solution

I'm doing some work on General Relativity and I found that there's an identity that -if true- would really simplify my calculations. I feel it has to be true but I haven't been able to prove it. It ...
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1answer
61 views

Conserved quantities from Killing vectors in the presence of electric charge

I know that for particles carrying no electric charge, given a Killing vector $K_{\mu}$, we have a conserved quantity $K_{\mu}p^{\mu}$ along geodesics, where $p^{\mu}$ is a tangeant vector. However, ...
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2answers
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What is the proper procedure to compare observations of light bending around the sun to GR's predictions?

I'm going through the history of GR and trying to understand the specifics of the experiments which validated the theory. There were a number of duplicate experiments which got consistent results. I ...
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1answer
56 views

Deriving the Geodesic Equation using Euler-Lagrange

I have recently been reading up on GR and I'm currently deriving the Geodesic Equation using the principle of least action. When solving the Euler-Lagrange equation for $L=\frac{m}{2}g_{ij}(x)\dot{x}^...
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1answer
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Doubts about the Einstein "way" [closed]

I followed the whole Einstein/Schwarzschild derivation, and the very first thing I don't like in it, is that after emphasizing the equivalence-principle requirement, Einstein skips this as a boundary ...
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2answers
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Confusion regarding the equivalence principle

In section V.2 of Prof. A. Zee's book Einstein Gravity in a Nutshell, it is given that to get the action of a point particle in a gravitational field from that of the action in SR, one just replaces $\...
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1answer
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Geodesic deviation equation in flat spacetime $\sim$ divergence of geodesics

Consider the above two neighbouring geodesics $\mathcal{Y}$ given by $x^{\alpha}(\sigma)$ and $\mathcal{\tilde{Y}}$ by $\tilde{x}^{\alpha}(\sigma)$ for top and bottom curves respectively. Vector ...
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1answer
104 views

How to calculate relative velocity in curved spacetime?

Is there a nice geometric way to calculate relative velocity in curved space time? I'm looking from something similar to how the separation vector $n$ between $2$ neighboring geodesics obey: $$ \...
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After conformal compactification, do null geodesics intersect future null infinity nonasymptotically?

I was going through this paper and was worried about an assumption in the main proof (Theorem 3.1), where they assume null geodesics intersect the boundary extension after conformal compactification ...
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1answer
68 views

Geodesic deviation in flat space

Suppose that $x^\mu(t,s)$ represents a family of curves. Let $v^\mu$ represents the the tangent vector to a curve $x^\mu(t,s_0)$ with $s_0$ fixed that is $v^{\mu}=\partial x^{\mu} / \partial t$ and ...
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Does gravity bend gravity?

Let's say that there is a large mass $M$ a light-year or so away from a black hole merger, which causes a very large gravitational wave to be produced. When the gravitational wave reaches $M$, does it ...
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3answers
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The Meaning of the 'Scale' of Proper Time

In Thomas A Moore's General Relativity Workbook in Chapter 8 titled Geodesics, it is argued that the geodesic equation does not fix the scale of any worldline's proper-time $\tau$. The argument for ...
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1answer
86 views

Why Lagrangian has this form in general relativity?

One can derive the geodesic equation by Euler-Lagrangian equation, \begin{equation} \dfrac{\partial \mathcal{L}}{\partial x^\gamma} - \dfrac{d}{ds}\bigg(\dfrac{\partial \mathcal{L}}{\partial (dx^\...
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1answer
74 views

Geodesic incompleteness of static spherically symmetric solution

Static spherically symmetric solution of Einstein equations is given by the metric $$ ds^2=f(r)dt^2-\frac{dr^2}{f(r)}-r^2d\Omega^2, $$ where $f(r)=1-(kr)^2$, $d\Omega^2$ is the metric of unit sphere. ...
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Geodesics from a Lagrangian in a restricted space

Given a certain action (for instance), \begin{equation} S = \int_\alpha^\beta d\lambda ~ L(\dot x, x, \lambda) \end{equation} where $\lambda$ is some affine parameter. In order to minimise it, we ...
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1answer
77 views

Showing that null geodesics are incomplete

Given a metric: $$ds^2 = -dt^2 + t dx^2$$ for a manifold $M = \mathbb{R}^+ \times \mathbb{R}$. The geodesic equation for null geodesics is $$x = x_0 \pm \log t$$ for some constant $x_0$. Now I want to ...

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