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).

learn more… | top users | synonyms

1
vote
0answers
30 views

Null geodesic equation with affine parameters [duplicate]

A photon's geodesic equation is defined by re-parameterizing the geodesic equation to some parameter other than proper time. This is done because $ds=0$ for the photon. Again if we use affine ...
4
votes
2answers
156 views

Examples in which the light maximizes the optical path length

I posted a similar question about geodesics on Math.SE. Many sources (Wikibooks for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually ...
0
votes
0answers
46 views

How would you describe what the affine parameter is in layman's terms? [duplicate]

I've been trying to learn it from other sites, but I'm not well-versed enough in mathematics to understand.
2
votes
1answer
51 views

Derivation of geodesic deviation equation from two neighbouring geodesics

I'm stuck trying to follow Foster and Nightingale's derivation of the geodesic equation from two neighbouring geodesics $x^{a}\left(u\right)$ and $\tilde{x}^{a}\left(u\right)$ joined by a ...
2
votes
3answers
125 views

Is it possible to express various nonlinear motions as straight lines in transformed spacetime?

I am trying to understand simple examples of space-time curvature. Assume for the moment that $c$ is infinite (classical curvature due to Newton's laws). Also, I will only consider 1+1-dimensional ...
2
votes
1answer
48 views

Movement of bodies in space, affected only by gravity

I have been extensively studying General Relativity for some time now. Recently I asked myself a question which I can't answer. If the gravitational metric is determined by the Energy content of the ...
2
votes
4answers
480 views

If gravitation is due to space-time curvature, how can a body free-fall in a straight line?

According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ...
2
votes
1answer
177 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
3
votes
1answer
253 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 $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). ...
1
vote
0answers
48 views

Existence of affine parametrization [closed]

This is a question from General Relativity by Wald Chapter 3, problem 5. Given either pseudo-Riemannian or Riemannian metric $g_{ab}$ and manifold $M$. Assume the $\nabla$ is compatible with the ...
4
votes
1answer
137 views

How warped spacetime bends trajectories of light and moving objects?

I fail to see why the light follows something like the blue line and not the green line on the attached image. Figure 1 - light bends around warped spacetime Afaik. something similar happens ...
3
votes
1answer
63 views

Can you recover a spacetime from its null geodesics?

So, I know that you can learn a lot about a spacetime from its causal structure, but can one completely recover the metric of a spacetime, just knowing the equations for the null geodesics in it? If ...
0
votes
0answers
52 views

Is it true that particles propagate on geodesics in Yang-Mills theory?

I mean free particles. Sorry for the inaccurate wording, I'm new in this field of physics.
0
votes
3answers
86 views

Path of light as it travels between two black holes

What would happen to light passing through a narrow space between the event horizons of two equal-mass black holes? Would it deviate or follow a straight path?
1
vote
1answer
49 views

Why doesn't light vibrate in-situ?

Light always moves in a straight geodesic path (shortest distance between 2 points in flat space where gravity is homogeneous) across 3 dimensions of space and 1 dimension of time. It is consists of a ...
0
votes
1answer
122 views

Carroll's derivation of the geodesic equations [duplicate]

In Carroll's derivation of the geodesic equations (page 69, http://preposterousuniverse.com/grnotes/grnotes-three.pdf), he starts with ...
0
votes
0answers
61 views

What is the null geodesic equation? [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
0
votes
0answers
22 views

The null geodesic for given geodesic [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
1
vote
0answers
96 views

Geodetic effect and Frame dragging

Two gyroscopes pointing perpendicular to each other were housed inside Gravity Probe B which performed polar orbit around Earth to test Einstein's theory of relativity. As the probe is orbiting ...
1
vote
1answer
71 views

Invariance in general relativity, university in problems question

From Problem #5 here, Free falling particles' worldlines in General Relativity are geodesics of the spacetime, i.e the curves $x^\mu(\lambda)$ with tangent vector $u^\mu=dx^\mu/d\lambda$, such ...
4
votes
4answers
327 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
2
votes
4answers
832 views

When space bends, what are the lines that are being bent?

In an electric field diagram, the lines represent the electrostatic force vector at the position. These lines are bent when you place a charge into the system. What is the equivalent description ...
5
votes
1answer
115 views

How is the Lagrangian defined in GR?

Reading about the Schwarzschild metric in general relativity I see that sometimes $$L=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$$ and sometimes $$L=\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ Which is ...
0
votes
2answers
104 views

How can light enter a black hole if it cannot get out?

I have known for a very long time that light cannot exit a black hole. I can even understand some of the simpler reasonning about it, such as escape velocity, or space geometry inside the black hole. ...
2
votes
2answers
123 views

What is a Null Geodesic? [duplicate]

What is a Null Geodesic? My textbook only explains it as the Minkowski metric which equals to zero, but I'd appreciate a more detailed explanation.
-1
votes
1answer
64 views

Is a planets orbit really a straight line through curved spacetime? [duplicate]

My understanding is that general relativity concludes that gravity isn't real because it does not exist in all frames of reference. Also that mass and energy warp spacetime into a curved geometry. ...
1
vote
0answers
48 views

Conformal time-like Killing vector near null geodesics in all spacetimes?

Is it true that in all spacetimes there is some conformal time-like Killing vector $\tau^a$ in the vicinity of null geodesics? If the above statement is true then can one argue that, for all ...
2
votes
1answer
65 views

If time-like paths are geodesics, what physical principle applies to space-like intervals?

If I have a number of particles interacting with one another locally, then the center of mass of the system moves along a geodesic. Taking this further with the particles interacting via an EM field, ...
-3
votes
2answers
92 views

Deriving the geodesic equation [closed]

I having been reading a general relativity book, but when in comes to the geodesic equation, it is not derived. How does one go about doing this?
0
votes
2answers
85 views

“Shortest” path in general relativity

My professor in mechanics course sneakily teach us some basic idea of general relativity. Which one of the basic assumption is particle walks in shortest world line. I understand shortest path in ...
0
votes
0answers
35 views

Vector fields corresponding to null geodesic congruences in general relativity

I'm working in Minkowski space, and I'm considering some 2D surface, $S$. On each point of the surface, I've computed a null vector, $k^a$, which is orthogonal to it. There will be a unique null ...
2
votes
1answer
38 views

When does light reach a shell observer in Schwarzschild metric?

I am trying to simulate the trajectory of light in the Schwarzschild metric (as seen by a far away observer) with fixed $\theta = \pi/2$. According to my source (Chapter 18, section 18.5) the ...
3
votes
1answer
44 views

Light trajectory

We have observed stars where "we should not" Some people say that gravity can alter light trajectory. Some people say that gravity actually alter the space on which light travels. Which one is ...
3
votes
1answer
132 views
0
votes
1answer
87 views

Free-falling from rest into a Kerr black hole

Is it impossible for a particle (with zero angular momentum) to free-fall from rest at infinity into the ergosphere of a Kerr black hole? It seems like it is very easy to show this is the case, but ...
0
votes
1answer
87 views

Calculating Christoffel symbols from Lagrangian

I was given the following metric for a sphere $$g_{\mu\nu} = diag(1, r^2, r^2\sin^2\theta)$$ and tasked to calculate the Christoffel symbols. There are 2 ways that I know of to calculate them. One ...
1
vote
1answer
122 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
3
votes
3answers
85 views

Principle of Sufficient Reason on light travelling in straight line

I was reading a book Laws and Symmetry by Bas C. Van Fraassen I found that there is an argument for arguing that light travel in straight line: Leibniz's reconstruction of these arguments goes ...
1
vote
1answer
167 views

Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
4
votes
2answers
2k 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
0answers
76 views

Lagrangian, geodesics and relativity [closed]

My background is in maths, but I have been studying some basic physics with occasional input from a friend who is studying for a physics PhD. Due to my background, I am keen to visualize things ...
2
votes
1answer
54 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
1
vote
1answer
52 views

Null geodesics in FRW metric: why angular coordinates are constant?

Consider a ray passing through $r=0$ in the FRW metric $ds^2 = -dt^2 +a(t)^2(\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin{\theta}^2d\phi^2))$ The geodesic curve is parametrized by the affine parameter ...
3
votes
3answers
182 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
0answers
86 views

Schwarzschild metric circular orbits and kepler's 3rd law

I have been looking at the Schwarzschild metric presented to me as the following within lectures: ...
1
vote
0answers
63 views

Is the Weyl Postulate correct?

The Weyl postulate in cosmology states that worldlines do not intersect but it can be shown in GR that using Raychaudhuri equation that geodesics can intersect if there is curvature so I'm really ...
3
votes
2answers
98 views

Why is $p_\phi$ conserved in a Schwarzschild orbit?

This arises from the question What is the relationship between $a$ and $m$, which I'm afraid I answered just by looking it up in Schutz's book. However Schutz (as he frequently does) glosses over ...
3
votes
2answers
780 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 ...
3
votes
1answer
73 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
0
votes
1answer
111 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 ...