5
votes
1answer
53 views

What's the definition of incompleteness of a coordinate system and a spacetime?

I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This ...
0
votes
1answer
68 views

What are great circles of 2-sphere?

What exactly are great circles, and how does one derive them? Given that the Lagrangian is: $$ L =\frac {1}{2}(\dot\theta^2 + \sin^2\theta\dot\phi^2)$$ it was written that the great circles were ...
1
vote
0answers
52 views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
0
votes
1answer
86 views

Geodesic deviation on a unit sphere

Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation ...
3
votes
2answers
351 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
118 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 ...
0
votes
0answers
36 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
71 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?
2
votes
1answer
104 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
82 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 + ...
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
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 $\vec{U}$, ...
8
votes
2answers
353 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
142 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
1answer
107 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 ...
14
votes
4answers
420 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
0
votes
1answer
92 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 ...
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
235 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. ...
3
votes
1answer
134 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: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
6
votes
1answer
347 views

Two formulas for a particle's acceleration

While on a class my teacher was taking about particle's motion in space. At some point she said the following: Consider that the particle's path is described by a curve in space defined by the ...
2
votes
1answer
160 views

Can two parallel lines meet? [closed]

My physics teacher talked about the meeting of 2 parallel lines, and he said that it may occur in the infinity or something. I know that 2 parallel lines can meet in spherical geometry, (thanks to ...
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
76 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 ...
10
votes
3answers
449 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
103 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 ...
1
vote
1answer
820 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
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
405 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 ...
1
vote
2answers
156 views

What is path of light in the accelerating elevator?

Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator? What is the difference between an ordinary derivative and covariant derivative (which is ...
3
votes
2answers
955 views

Null geodesic given metric

I (desperately) need help with the following: What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$ I don't know how to transform a metric into a geodesic...! There is no need to ...
6
votes
1answer
2k views

Physical significance of Killing vector field along geodesic

Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter. What physical significance do the scalar quantity ...
13
votes
3answers
1k views

Why do objects follow geodesics in spacetime?

Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation ...
5
votes
1answer
2k views

Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment. I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must ...
6
votes
1answer
92 views

How does a geodesic equation on an n-manifold deal with singularities?

My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds and then find applications to theoretical physics by: ...
4
votes
1answer
479 views

How do we know the geodesic is a minimum?

The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum. The introductory GR ...