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).
1
vote
1answer
35 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
81 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
112 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 ...
2
votes
1answer
43 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
1answer
64 views
Why four velocity under covariant differential is considered to be zero?
In Einstein's general theory of relativity the elements of four velocity $U^{\mu} (\gamma c, \gamma v)$ under covariant differential is considered to be zero, why?
$$\mathcal{D} U^{\mu}=0$$
in other ...
1
vote
2answers
155 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 ...
1
vote
2answers
65 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 ...
2
votes
1answer
71 views
The role of the affine connection the geodesic equation
I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is ...
1
vote
0answers
45 views
Naked singularity and null coordinates
I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually ...
2
votes
0answers
48 views
Naked singularity and extendable geodesics
I'm currently trying to understand the notion of a naked singularity. After consulting books by Wald and Choquet-Bruhat, it seems that for a naked singularity one must have that the causal curves can ...
4
votes
2answers
281 views
Action for a point particle in a curved spacetime
Is this action for a point particle in a curved spacetime correct?
$$\mathcal S =-Mc \int ds = -Mc \int_{\xi_0}^{\xi_1}\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$
0
votes
1answer
53 views
Can you enter a timelike hypersurface?
As I understand it, a timelike hypersurface is one that has only spacelike normal vectors. But does this not imply that a the geodesic of a particle crossing it must be spacelike at that point? But ...
4
votes
1answer
374 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 ...
6
votes
1answer
43 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:
...
3
votes
0answers
56 views
Gravitational effects and metric spaces
Could somebody please explain something regarding the Nordstrom metric?
In particular, I am referring to the last part of question 3 on this sheet -- about the freely falling massive bodies.
My ...
0
votes
1answer
116 views
Homogeneous gravitational field and the geodesic deviation
In General Relativity (GR), we have the geodesic deviation equation (GDE)
...
2
votes
3answers
146 views
Is the path of stationary action unique? What are the physical implications of $L_{\dot{x}}=L_x$
Below, for any function $Q$ the notation $Q_x$ means $\frac{\partial Q}{\partial x}$, and $Q_{xx}$ means $\frac{\partial^2 Q}{\partial x^2}$.
In physics, the trajectory of a particle is given by the ...
2
votes
1answer
80 views
How far does typical view of clouds/atmosphere extend?
The specific "sub questions" I'm asking are:
When you are looking at clouds just on the horizon, how far away would they be?
How wide (in km) is that total field of vision at roughly cloud height.
...
8
votes
4answers
332 views
To which extent is general relativity a gauge theory?
In quantum mechanics, we know that a change of frame -- a gauge transform -- leaves the probability of an outcome measurement invariant (well, the square modulus of the wave-function, i.e. the ...
1
vote
1answer
229 views
Potential Energy in General Relativity
I often hear about how general relativity is very complicated because of all forms of energy are considered, including gravitation's own gravitational binding energy. I have two questions:
In ...
3
votes
1answer
257 views
Problem with convergent geodesics at 2D sphere
There is a chapter on general relativity in the book Spacetime Physics Introduction To Special Relativity by Taylor and Wheeler, which qualitatively explains how attractive gravitational force can be ...
0
votes
0answers
47 views
Cauchy Problem in Convex Neighborhood
While reading the reference
Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available here,
there is something that I don't quite understand.
...
4
votes
1answer
213 views
Geodesic Equation from energy-momentum conservation
I've been reading the excelent review from Eric Poisson found here.
While studying it I stumbled in a proof that I can't make... I can't find a way to go from Eq.(19.3) to the one before Eq.(19.4) ...
0
votes
2answers
240 views
How to think of the harmonic oscillator equation in terms of “acceleration = gradient”
This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as:
$$\ddot{x}+kx=0$$
What little physics I grasp centers on ...
4
votes
1answer
628 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 ...
