4
votes
1answer
99 views

Is the zero acceleration path also the shortest path between two points?

In flat, free, Euclidean space, the shortest path and the zero acceleration path are the same path, which is a straight line. However, in general relativity, is the zero acceleration path also the ...
0
votes
1answer
132 views

How do objects even move due to gravity?

I am an newbie general relativistic learner and I learnt that gravity is bending of space-time and since objects move in straight-lines but since its curved they follow curved movement through space ...
3
votes
2answers
348 views

Which of these two textbook equations of geodesic deviation is correct?

My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner et al's Gravitation (p34) gives the geodesic ...
1
vote
0answers
114 views

Textbook disagreement on geodesic deviation on a 2-sphere

Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation ...
1
vote
0answers
48 views

Radial Null Geodesics in Static Maximally Symmetric DeSitter Space

Given a DeSitter-space metric from the line element: $$ ds^2=\left(1-\frac{r^2}{R^2}\right)dt^2-\left(1-\frac{r^2}{R^2}\right)^{-1}dr^2-r^2d\Omega^2 $$ Where $R=\sqrt{\frac{3}{\Lambda}}$, and ...
0
votes
0answers
31 views

Caustic and Singularities in General Relativity

What is the relation between the formation of caustics of a family of null geodesics and the existence of an incomplete null geodesic?
1
vote
2answers
70 views

Static geodesics in GR

Can we find static geodesics of the type $$x^{\nu}=x_0^{\nu}+\delta_0^{\nu}\tau$$ in some space-time other than Minkowski's?
-1
votes
1answer
81 views

Does the formula $ \theta = \frac{v}{c} $ to find out deflection of light make sense?

I read in reliable sites that GR and classical physics calculate the angle of deflection in the same manner. The formula is almost identical: $$\theta = \frac{4GM}{c^2*r} \rightarrow \frac{4GM}{c*r} = ...
1
vote
0answers
54 views

Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess ...
4
votes
1answer
72 views

Geodesics in Kerr

I'm interested in plotting the trajectories of null geodesics near an uncharged rotating black hole (described by the Kerr solution) which involves a system of first order differential equations. Kerr ...
1
vote
3answers
108 views

Do massless particles follow the curved spacetime or not?

I am assuming that zero (rest) mass particles don't interact gravitationally with each other and other particles. Does that mean they experience a "flat" spacetime instead of a curved one? I find it a ...
2
votes
1answer
102 views

Computing the Christoffel symbols with the geodesic equation

I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ...
3
votes
1answer
81 views

Geodesics in AdS3

I'm having some trouble doing an easy computation with the AdS space. I'm considering $\text{AdS}_3$ space with the Poincaré coordinates, so the metric reads $$ds^2 = \frac{R^2}{z^2}(dz^2 - dt^2 + ...
3
votes
2answers
160 views

Can a curvature in time (and not space) cause acceleration?

I realize that the curvature of space-time causes acceleration (gravity). Is it possible to have a curvature only of space, or a curvature only of time? If so, would a curvature only of space, or a ...
6
votes
2answers
150 views

Killing vectors in flat FLRW metric

I have the flat FLRW metric, $$ ds^2=-dt^2+a(t)^2(dx^2+dy^2+dz^2) $$ and a geodesic $\gamma(s)=(t(s),x(s),y(s),z(s))$ with parameter $s$. Two of the Killing vectors of the metric are $ \partial_x$ ...
2
votes
2answers
57 views

Does geodesics from solving full field equations are same as path from energy-momentum tensor?

As we know, if we had an energy-momentum tensor in all space-time we could obtain the metric tensor by solving field equations. Also i think if we had an energy-momentum tensor then we have ...
2
votes
0answers
48 views

Geodesic Deviation between Test Particles from Gravitational Wave

I'm having trouble understanding how Carroll (Spacetime and Geometry p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, ...
2
votes
1answer
110 views

Wave Packet in Curved Spacetime

It is known that the classical equation of motion for a scalar field wave packet on a curved spacetime background gives the geodesic trajectory (the e.o.m. is $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). ...
0
votes
1answer
52 views

How exactly can we describe the normal force on a static person standing on earth's surface using general theory of relativity?

For planetary motion I can understand that the planets move along the geodesics e.g. the warped space-time geometry. Imagine that the moon gets suddenly stopped by some external force and comes to ...
8
votes
2answers
351 views

AdS Space Boundary and Geodesics

I'm new to working with AdS space and am primarily concerned with black holes. I'm just playing round with the metric for AdS$_4$ $$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\zeta^2$$ for $f(r)=r^2+m $, ...
2
votes
1answer
133 views

Stuck following derivation of geodesic equation

In the book "Reflections on Relativity" by Kevin Brown, there is a chapter called "Relatively Straight", in which he derives the geodesic equations using the Euler equation. Online version Just ...
0
votes
0answers
19 views

Angular and luminosity distance in general?

Consider a non-Friedmannian Universe in which we know the trajectories of photons, ie in which we know null geodesics $\left(\eta, x^{1}, x^{2}, x^{3}, a, z\right)$ where : $\eta$ is the conformal ...
2
votes
1answer
125 views

Trajectory of a photon around a Schwarzschild black hole?

Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see this for illustration). Its impact parameter is $b$ and its distance ...
2
votes
1answer
117 views

Relation between impact parameter and distance of closest approach of a light ray in Schwarzschild Geodesics

The following wikipedia articles are incompatible : Two body problem / bending of light by gravity Schwarzschild geodesics / bending of light by gravity According to both articles, the equation ...
0
votes
1answer
105 views

Geodesic devation on a two sphere

So the geodesic deviation equation gives the relative acceleration between two geodesics in motion. But given a pair of geodesic (let's say on the two sphere) that start at the equator, separated by ...
0
votes
1answer
91 views

Null Geodesics in flat 2+1 dimensional Minkowski space

For a given line element in flat 2+1 dimensional Minkowski space $$ g = ds^{2} = − dz \otimes dz + dx \otimes dx + dy \otimes dy .$$ The null geodesics are supposedly given by: $$ x = lu + l' $$ ...
3
votes
1answer
112 views

Can geodesics in a Lorentzian manifold change their character?

From a physics perspective, it's pretty easy to see why a a massive particle will be restricted to timelike paths, etc. but does the math guarantee that on its own or do we have to impose it? More ...
2
votes
1answer
47 views

Ensuring globally hyperbolic geodesically-complete spacetimes

Let's say we have an incomplete spacetime A that is globally hyperbolic, does there necessary exist a globally hyperbolic completion? My guess is no, in which case what further restrictions can be ...
1
vote
1answer
59 views

Unable to resolve 2 equivalent geodesic equations

A free particle moves along geodesics, one form being \begin{split} \ddot x^\mu &= -\Gamma^{\mu}_{\sigma \rho} \dot x^\sigma \dot x^\rho \\ &= -\frac{1}{2}g^{\mu \nu}(\partial_\sigma g_{\rho ...
4
votes
2answers
211 views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
4
votes
1answer
106 views

Sign of $dr$ in Schwarzschild geodesics

There is an equation that relates energy $E$, angular momentum $L$ and other constants and variables to find $\left(\frac{dr}{d\tau}\right)^2$ in a plane. ...
5
votes
1answer
92 views

Geodesics in a point mass universe

This question may reflect my (lack of) knowledge about general relativity, please ask for any clarifications or note any corrections in the comments and I'll try to address them. The Schwarzschild ...
3
votes
1answer
132 views

Schwarzschild geodesics

I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
1
vote
3answers
205 views

What makes matter travel along geodesics?

The relativistic explanation of gravity is geometric, the motion of a body in a field of space-time distortion can be described as being at rest and travelling along a geodesic of that field, but why ...
12
votes
1answer
355 views

How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?

In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure ...
0
votes
0answers
83 views

How to use The Schwarzchild Metric formula to get distribution representing “free-fall”

Given formula: How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should ...
2
votes
1answer
173 views

About the geodesics in general relativity [duplicate]

I'm learning general relativity from the book " Einstein's General Theory of Relativity - Øyvind Grøn and Sigbjorn Hervik". The field equations are derived by the Hilbert - Einstein action and are ...
6
votes
1answer
382 views

In general relativity, are light-like curves light-like geodesics?

Just as the title. If a curve is light-like, i.e. a null-curve, is it definitely a null geodesic?
5
votes
3answers
3k views

Why does light always travel in a straight line?

No matter the frame light is in, it always moves in a straight line in that frame. Why is that? It doesn't seem like something to me that should necessarily be true. If some one runs forward and sends ...
1
vote
0answers
74 views

Preservation of a scalar along geodesic trajectory

Let $u^\mu$ be the velocity of a particle , and $\xi^\mu$ be a killing vector. would taking a contravariant derivative of to scalar product $\xi_\mu u^\mu$ , and showing that it equals to 0 shows that ...
1
vote
1answer
70 views

Maximum aging and path of rock

When a rock falls from a ledge, why does it head to the surface and not up to where time runs faster? If a rock, free from forces, follows a worldline of maximum aging, why would that rock approach ...
10
votes
3answers
443 views

Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $. ...
2
votes
1answer
98 views

From Euler-Lagrange equation to non affine geodesic equation

I have some problems to demonstrate the non affine geodesic equation from Euler-Lagrange's equations. I start defining the Lagrangian $L=\sqrt f$, but then I'm not able to find the Christoffel ...
2
votes
1answer
623 views

Equation for null geodesic around schwarzschild metric?

I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself. I just ...
1
vote
1answer
810 views

Can anyone please explain Hawking-Penrose Singularity Theorems and geodesic incompleteness?

Can anyone please explain Hawking-Penrose Singularity Theorems and geodesic incompleteness? In easy to understand plain English please.
1
vote
2answers
292 views

Solving a light ray worldline with the geodesic equation

I'm having trouble solving the geodesic equation for a light ray. $$ {d^2 x^\mu \over d\tau^2} + \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} = 0 $$ I apologise, but I'm a ...
1
vote
1answer
142 views

How to find distance of closest approach for a Schwarzschild geodesic?

What is the distance of closest approach in this Wikipedia article? I can't seem to find its definition, and this other question doesn't have an answer I can understand.
1
vote
1answer
130 views

Killing vector argument gone awry?

What has gone wrong with this argument?! The original question A space-time such that $$ds^2=-dt^2+t^2dx^2$$ has Killing vectors $(0,1),(-\exp(x),\frac{\exp(x)}{t}), ...
3
votes
2answers
403 views

Geodesic equations

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2 dimensional space such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ the timelike geodesics ...
4
votes
1answer
116 views

“WLOG” re Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem ...