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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How do tidal forces on incomplete geodesics determine extendability?

Why can we be sure that the manifold with the metric $(M,g)$ does not have a geodesically complete extension if it has an incomplete timelike geodesic along which the tidal force blows up? Does this ...
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1 answer
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Hamiltonian for the time-like particle on the geodesic

I am trying to reproduce the results from this paper. On page 2 of the paper, they have an equation: $$2 H=-\frac{\dot{r}^2}{g(r)}-L \dot{\phi }+E \dot{t}=\epsilon\tag{9}$$ where they make a comment ...
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4 votes
2 answers
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How much longer is the path through spacetime of a mass that falls freely compared to a resting mass?

A mass that falls to Earth follows a shortest path through spacetime. If a mass falls from a 1km high building, how much longer will its path be compared to a mass resting on a table?
3 votes
1 answer
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Geodesic: maximal aging versus extremal aging

From Exploring Black Holes, by Taylor and Wheeler, page 1-7: Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of ...
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Radial Null Geodesics in AdS space

Consider AdS spacetime in coordinates such that the metric is $$ ds^2 \enspace = \enspace - f(r) dt^2 + f(r)^{-1} dr^2 + r^2 d\Omega^2 \quad ,$$ where $$f(r) \enspace = \enspace 1 + \frac{r^2}{R^2} \...
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Imaginary velocity components in geodesic

When we try to find the geodesic of a partical at rest, in the second term of the geodesic equation we use dt/dtau = 1. Shouldn’t it be i (for imaginary number), since the time component of the 4-...
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1 vote
3 answers
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Can I really see what is on the opposite side of a black hole?

This question is only about objects outside the event horizon. Both the observer and the object are just outside the event horizon. I have read this question: An observer can see the back side of the ...
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131 views

Geodesic deviation using vector calculus for 2d surfaces in 3d space

Is it even possible to derive geodesic deviation equation using vector calculus, I haven’t been able to find it in any textbook of classical differential geometry. I found a website with derivation of ...
2 votes
1 answer
113 views

Relativistic Euler-Lagrange equation

I am confused from the equation 6, why we get Euler-Lagrange equation from equation 8 but not from equation 6? Why we need to use $\zeta$ as invariant parameter in equation 8 even we already have ...
1 vote
0 answers
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Geodesics Equation from Lagrangian [duplicate]

In the book Introduction to General Relativity Blackholes and Cosmology by Yvonne Choquet-Bruhat, she defines the length of a causal curve as $$\ell\equiv \int_a^b \left( -g_{\alpha \beta} \frac{d \...
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1 answer
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Confusions about Schwarzchild Geodesic Deviation

In schwarzchild metric space gets bigger as you approach the horizon, if you build shells around a black hole infinitesimally small distance apart you can build an infinite number of such shells. ...
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5 votes
2 answers
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What is the relative acceleration composition law in General relativity?

In Euclidean geometry we have the following relative acceleration composition law: $$ \vec a_{DE} + \vec a_{EF} = \vec a_{DF} $$ Where the relative acceleration between $i$ and $j$ for any $i$ and $j$ ...
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1 answer
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Calculating coordinate increase of light ray escaping black hole

Consider a light ray near a black hole described by Eddington-Finkelstein coordinates $(v,r,\theta,\varphi)$. My aim is to calculate the increase of the coordinate $v$ along a radial path from the ...
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2 votes
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How to plot off-equatorial orbits in Kerr metric?

I am trying to plot the time-like trajectories in Kerr metric. I have taken the equations from here. I am trying to reproduce the off-equatorial trajectories from Chandrasekhar's textbook, e.g., the ...
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1 vote
0 answers
25 views

Showing the equivalence principle mathematically [duplicate]

Given the geodesic equation $$\ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\lambda}\dot{x}^{\nu}\dot{x}^{\lambda} = 0$$ I wish to find a co-ordinate system around a point $x_0$ such that the geodesic equation ...
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2 votes
2 answers
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Removing a Coordinate Singularity of a 2D metric

While trying to find the null geodesics of the metric $$ ds^2 = (r^2 - 1)dt^2 - (drdt + dtdr) $$ gives $$\frac{dt}{dr} = \frac{2}{r^2-1}$$ which is singular at $r=1$. However, we know that this is a ...
2 votes
0 answers
64 views

Infinitesimal geodesic motion directly from the metric?

How can I see---directly from the Schwarzschild metric---that initially stationary (w.r.t. Schwarzschild coordinates) inertial test clocks will begin to fall toward e.g. the Earth (i.e. far outside ...
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4 answers
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Can the geodesic equation be understood in plain english to articulate the radial attraction of gravity?

I'm looking to gain an intuitive understanding of the geodesic equation (which incorporates the Christoffel symbol) and how it is used to calculate the radial attraction of gravity. In its native form ...
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1 answer
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How does the book prove that "a free stone moves so that its wristwatch time along each segment of its worldline is a maximum"?

In the book Exploring Black Holes in the second chapter they say over and over again that "The Principle of Maximal Aging tells us that a falling stone moves so that its summed wristwatch time is ...
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1 vote
1 answer
100 views

Understanding Proper Time Parametrization

I am having trouble understanding parametrizing a path in proper time. So my understanding is that using proper time to parametrize a path corresponds to the rest frame of the particle whereby it ...
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1 answer
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Does gravitational lensing make objects that would be obscured behind other objects completely visible?

Reading a small amount about gravitational lensing and viewing many of the visualizations, it appeared that bodies directly behind other massive objects (from some point of view; namely galaxies ...
2 votes
1 answer
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Schwarzschild's null-geodesic new form or an error?

My question is whether or not this form (radial acceleration of a photon) $$\ddot{r}=\frac{L^2}{r^4}(r-3M)$$ is correct ? Recall the standard set of second-order ODE for the Schwarzschild metric (for ...
1 vote
1 answer
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Changing Coordinates, and the Geodesic Equation in GR

I have a question about doing Lorentz-like coordinate transformations in general relativity. I will try not to get into too much detail about what exactly I am trying to do to not muddy the waters. ...
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3 votes
1 answer
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Spacelike geodesics of FLRW radiation universe

I'm having an interpretation problem with the radial spacelike geodesics in the flat radiation dominated universe. I'm using standard conformal coordinates $\eta \ge 0$ and $\chi \ge 0$ for the ...
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1 answer
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What are the meaning of geodesics?

I was trying to learn General Relativity. While I was learning about curved geometry, I found out this post. As said by the author, those lines are geodesics on a 2-manifold. I think I don't quite ...
3 votes
4 answers
162 views

How is proper time extremized?

I just completed an exercise that asked me to prove that, in special relativity, free particles move with uniform velocity on geodesics that are straight lines. After doing this problem, I was ...
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1 answer
47 views

Hyperbolic disks in AdS/CFT

The embedding of AdS space into Minkowski spacetime describes a hyperboloid as e.g. shown in the corresponding Wikipedia article on AdS space. Now my questions are: How does this relate to the ...
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0 answers
22 views

Would gravity pulling person toward Earth change if velocity of Earth changes? [duplicate]

So I was reading the Albert Einstein's theory of how gravity works. From my understanding, the more mass an object has, the more space-time around it it bends. All objects travels in completely ...
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0 answers
22 views

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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7 votes
0 answers
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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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4 votes
1 answer
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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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0 answers
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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 $\{\...
2 votes
1 answer
59 views

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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1 vote
0 answers
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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.
1 vote
1 answer
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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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3 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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0 answers
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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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0 answers
41 views

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 ...
1 vote
1 answer
104 views

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\...
3 votes
0 answers
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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}...
1 vote
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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0 answers
49 views

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 ...
0 votes
2 answers
115 views

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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1 vote
0 answers
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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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2 votes
0 answers
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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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