Geodesics Definition and 182 Threads

Hi, I am currently doing a course in GR and have just gone over a proof of the focusing theorem.. now this relied on the fact that geodesics do not cross. But I could not see clearly the contradiction if geodesics did happen to cross? any help would be greatly appreciated. Maki :smile:

Hi all. This is one of the problems that I was asked to do for my General Relativity class. I know this may look a little long, but if anyone can help me out with ANYTHING about this problem, I will greatly appreciate it. Homework Statement Consider the family of hypersurfaces where each...

I was looking into the geodesic equations for the Schwarzschild metric and I noticed a discrepancy between two sources: According to http://www.mathpages.com/rr/s5-05/5-05.htm (near the bottom) the second derivative of the azimuth angle is \frac {d^{2} \phi}{d \lambda ^2}=\frac {-2}{r}...

Homework Statement Hi, I am reading Ray d'Inverno's book, 'Introducing Einstein's Relativity' and there is a particular derivation of the geodesic equation that I get stumped on (chapter 7). It is a variational method and the final equation is df/dx_alpha-d/du{df/dx_alpha_dot}=0 where...

In another thread the subject came up regarding whether the affine connection was more "general" in defining geodesics than the metric tensor. Hurkyl provided an illustrative example in the post https://www.physicsforums.com/showpost.php?p=1783469&postcount=116 Hurkyl - Let me take this...

Consider this real situation: Light from a far-background galaxy is observed to be lensed by a closer massive galaxy. Individual photons that graze the closer galaxy travel to the observer along various geodesic paths, as prescibed by GR. Now consider a very hypothetical situation. Suppose a...

Homework Statement In comoving coordinates, a one dimensional expanding flat universe has a metric ds^2 = -c^2dt^2 + at(t)^2dr^2. Derive an expression for a conserved quantity for geodesics in terms of a, \tau and r, where \tau is the time measured in the rest frame of the freely falling...

Hello, Suppose that R^2 is provided with the following metric ds^2 = dx^2 + (\cosh(x))^2 dy^2 Can we find a general exact formula \alpha(t) for the geodesics (starting at an arbitrary point) ? The geodesic equation gives x'' - \cosh(x)\sinh(x) (y')^2 = 0 y'' + 2...

Hi , everyone I have a problem with geodesic equation . I know the method of solving it , but I can't understand the solutions . When I tried to solve it for a torus I arrived at : \dot{u} = \frac{k} {{\left( {c + a\cos v} \right)^2 }} \] \[ \dot v = \pm \sqrt { - \frac{{k^2 }}...

Hi, So I am going over on how to find a geodesic from any metric, esp. on a 2-sphere. I have been looking at my lecture notes and am confused as to how my professor solves for the equation in terms of the variables, i.e. (\theta , \phi) . If i use the 2-sphere as an example here where the...

For a course on tensor analysis, we were asked to perform some calculations regarding a 2D space embedded in 'regular' Euclidean 3D space. I've gotten some results, but I've hit a snag... Homework Statement The following functions relate the coordinates in the embedding (3D, Euclidean) space...

Hi, everyone: I was just going over some work on Hyperbolic geometry, and noticed that the geodesics in the disk model are the same as the geodesics in the upper- half plane, i.e, half-circles or line segments, both perpendicular to the boundary. Now, I know the two...

Distant mass A, causes a gravitational field which travels along a geodesic to mass B. Mass B experiences a force due to gravitational attraction along a specific vector. Would light also travel along the same geodesic form mass A to B causing the vector of the visible image of distant mass A...

What are you trying to do when you find parametric equations for a geodesic lines on a surface? Take the metric ds^2 = dq^2 + (sinh(q)*dp)^2 Are you simply trying to get q as a function of s? and p as a function of s? If so, why? Thanks

How many geodesics (up to affine reparametrization) connect two arbitrary points in arbitrary spacetime ? On the 2D sphere for example, you have two geodesics which are the parts of a the great circle connecting the two points but circulating in opposite directions. I think one of them...

I am trying to figure out if a geodesic can be inflectional (in euclidean space...). I am not sure it even makes sens, from the definition of a geodesic, but it seems to me that a geodesic will not in general be extremal, but only stationnary. Is there a general theorem preventing a monster...

1) a. Show that if a curve C is a line of curvature and a geodesic then C is a plane curve. - Pf. Let a(s) be a parameterization of C by arc length (which I'm assuming always exists). then C is a geodesic if the covariant derivative of a(s) with respect to s is 0. We also know that if C is...

Geodesics of hyperbolic paraboloid (urgent!) Help me find the geodesics of the hyperbolic paraboloid z=xy passing through (0,0,0). I know that lines and normal sections are geodesics. Based on a picture, I think y=x and y=-x are 2 line geodesics. Then, maybe the planes in the z-y and z-x...

I have the following problem: Let L(q,\dot{q})=\sum g_{ij}(q)\dot{q}_i\dot{q}_j. And l(q,\dot{q})=\sqrt{L(q,\dot{q})}. Define the spaces \mathbb{X},\, \mathbb{Y} of parametrized curves \mathbb{X}=\{\gamma\,:\,[0,1]\rightarrow \mathbb{R}^n,\,\gamma \in...

Homework Statement Consider a 2-sphere of radius R parametrized by the 2 spherical polar coordinates θ and φ. Write down the standard metric in these coordinates. 1. Show that lines of constant longitude are geodesics, and that the only line of constant latitude is the equator. 2. How...

What are Jacobi Fields and how can I better understand geodesics in Minkowski spacetime by knowing what they are?

Hello everybody, I was studying the lecture notes about the schwarzschild solution for general relativity. In a particular example they calculate the equations of motion of a particle falling straight into a black hole. But there are some things about the calculation I really don't get...

Well since the denizens of the relativity forum don't like me, I thought I might ask here see if I get better replies.1) Why is it that in general geodesics are paths of "stationary character" rather than minimum? 2) http://img366.imageshack.us/img366/3280/math30016nx.jpg I can't quite...

How is the Earth on a geodesic? Forgive me if this is basic. I had associated the time dilation effect with acceleration & now it turns out gravity isn't a "force" even though GR in the equivalence principle models it as an acceleration. To rephrase, the Earth is constantly accelerating & yet...

Closed spacetime with its dispersed masses and open spacetime under dark energy interfere dynamically at the Euclidean boundary. This zero curvature surface distinguishes where the cosmic geometry contracts from where it expands, and represents a significant unexplored structure within familiar...

Hi, Hopefully someone can explain this to me in laymans terms... If I am understanding what I am reading correctly, gravity the result of the shape of space-time? I don't understand why this is considered a force at all if it is the result of the shape of space-time. Where does the...

The linear velocity of rotation of a spinning ball of the gravitational field matter (gfm) is faster at its edge than in its central part. According to Einstein's theory of the spinning disk, the spacetime curvature at the edge of the gfm ball is larger than that in the central part; and...

A commonly accepted modern SR/Gr statement is "timelike geodesics are maxima of the proper time". I really shall not dispute with these experts, who are all professors. But this statement is clearly questionable. It bothered me for a long time. The H & T experiment and GPS all show the...

Light cannot escape from black holes, hence their name. But Since light has the same speed everywhere does that mean that the space/time geodesics in and around black holes are closed loops?

Can you show that a closed spacetime may embody one (or even an infinity of) geodesics with infinite length? How does this influence local curvature?

Can you explain a little more about null and timelike geodesics (I think that's how you spell it)? I was reading Hawking and Penrose's The Nature of Space and Time, but it got a little technical. I would really like to know more about these though... thanks!

Hi all, I am trying to reformulate the axioms of Special Relativity. It seems intuitively true that all inertial frams should be equivalent (*), but there seems to be no philosophical justification that light should travel at constant velocity to all inertial frames (+). Could someone show...