Geodesic Definition and 256 Threads

Homework Statement Consider the following geodesic of a massless particle where ##\alpha## is a constant: \dot r = \frac{\alpha}{a(t)^2} c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)} Homework EquationsThe Attempt at a Solution Part (a) c \frac{dt}{d\lambda} = \frac{\alpha}{a} a dt =...

I'm utterly confused by co-moving distance, transverse comoving distance and proper distance. Is comoving distance = proper distance? Then what is transverse comoving distance? Here's what I know so far: The FRW metric can either be expressed as ds^2 = c^2dt^2 - a^2(t) \left[ \frac{dr^2}{1-kr^2}...

Taken from Hobson's book: How did they get this form? \dot u^{\mu} = - \Gamma_{v\sigma}^\mu u^v u^\sigma \dot u^{\mu} g_{\mu \beta} \delta_\mu ^\beta = - g_{\mu \beta} \delta_\mu ^\beta \Gamma_{v\sigma}^\mu u^v u^\sigma \dot u_{\mu} = - \frac{1}{2} g_{\mu \beta} \delta_\mu ^\beta...

My book says in the slow motion approx, so ## v << c ##, ##v=\frac{dx^{i}}{dt}=O(\epsilon) ## It then states: i) ##\frac{dx^{i}}{ds}=\frac{dt}{ds}\frac{dx^{i}}{dt}=O(\epsilon) ## ii) ## \frac{dx^{0}}{ds}=\frac{dt}{ds}=1+O(\epsilon) ## The geodesic equation reduces from...

Hi according to the text I am reading a curve is geodesic if these conditions are met ##\frac{d}{ds}(2g_{mi} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{m}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0##, where ##m=1,...,N## a curve is a null geodesic if exactly the same conditions are...

Homework Statement (a) Show that the equations satisfy FRW equations. (b) Show the metric when ##\eta## is taken as time Homework EquationsThe Attempt at a Solution [/B] The FRW equation is: 3 \left( \frac{\dot a}{a} \right)^2 = 8\pi G \rho Using ##\frac{da}{dt} = \frac{da}{d\eta}...

What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.

Taken from my lecturer's notes on GR: I'm trying to understand what goes on from 2nd to 3rd line: N^\beta \nabla_\beta (T^\mu \nabla_\mu T^\alpha) - N^\beta \nabla_\beta T^\mu \nabla_\mu T^\alpha = -T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha Using commutator relation ## T^v \nabla_v...

What do they mean by contracting ##\mu## with ##\alpha## ?

Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...

Starting with the geodesic equation with non-relativistic approximation: \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma_{00}^{\mu} \left( \frac{dx^0}{d\tau} \right)^2 = 0 I know that ## \Gamma_{\alpha \beta}^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\lambda}} \frac{\partial^2 y^{\lambda}}{\partial...

Homework Statement Find the deflection of light given this metric, along null geodesics. Homework EquationsThe Attempt at a Solution [/B] Conserved quantities are: e \equiv -\zeta \cdot u = \left( 1 - \frac{2GM}{c^2r} \right) c \frac{dt}{d\lambda} l \equiv \eta \cdot u = r^2 \left( 1 -...

In Dirac's book on relativity, he begins and ends his section on proving the stationary property of geodesics with references to "null geodesics". His last sentence is: "Thus we may use the stationary condition as the definition of a geodesic, except in the case of a null geodesic." What is a...

I am trying to follow a proof that given a Kiling vector ##V^{u}##, the quantity ##V_{u}U^{u} ## is conserved along a geodesic. I am given the Killiing Equation: ## \bigtriangledown_{(v}U_{u)}=0 ## [1] Below ## U^{u} ## is tangent vector ## U^{u} = \frac{dx^{u}}{d\lambda} ## The proof...

Hi all, I was looking through this proof and have no idea where the "u" comes from., any help apreciated. http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0 http://s0.wp.com/latex.php?latex=%5Cdisplaystyle++&bg=eedbbd&fg=000000&s=0...

Hello, I am quite new to GR and I have a question regarding the construction of the action to find the geodesic equation. In pretty much every book, you'll find: ##S=-m ∫ dS## using: ## dS=dS\frac{d\tau}{d\tau}=\sqrt{g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau## with...

Homework Statement Write down the Newtonian approximation to the geodesic deviation equation for a family of geodesics. ##V^\mu## is the particle 4-velocity and ##Y^\mu## is the deviation vector. Homework Equations D = V^\mu\nabla_\mu \\ V^\mu\approx(1,0,0,0) \\...

Hi All, I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So...

Hi guys So I am having trouble reparameterizing the geodesic equation in terms of coordinate time. Normally you have: \frac{d^2 x^{\alpha}}{d \tau^2} + \Gamma_{nm}^{\alpha} \frac{d x^{n}}{d \tau}\frac{d x^{m}}{d \tau} = 0 Where \tau is the proper time. I class we were told to express the...

Homework Statement Hi all. For some reason I have been having a lot of difficulty with this problem in Peter Petersen's text. The problem is Prove: ##d(exp_p(tv), exp_p(tw)) = |t||v-w| + O(t^2 )## Homework Equations The exponential map is the usual geodesic exponential map. And ##d(p,q)## is...

On the surface of a unit sphere two cars are on the equator moving north with velocity v. Their initial separation on the equator is d. I've used the equation of geodesic deviation...

Hi, How can null geodesics be defined? Obviously the concept of parallel-transport, of the tangent to the curve, applies equally well to null curves as to time/space-like curves. Technically this is only the definition for an "auto-parallel", not for a "geodesic". For example in...

Dear all, I was reading "Nature of space and time" By Penrose and Hawking pg.13, > If $$\rho=\rho_0$$ at $$\nu=\nu_0$$, then the RNP equation > > $$\frac{d\rho}{d\nu} = \rho^2 + \sigma^{ij}\sigma_{ij} + \frac{1}{n} R_{\mu\nu} l^\mu l^\nu$$ implies that the convergence $$\rho$$ will become...

Definition/Summary Where two particles very close together have the same velocities, their two geodesics are parallel, though only instantaneously, and so the gap (a 4-vector, of time and distance) between them has zero rate of increase, but has non-zero acceleration. The acceleration (a...

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G} where s is the arc length parameter and E, G are the coefficents of the first fundamental form. Can you...

Greetings, I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have some knowledge (from an introductory differential geometry class in my...

I've been meaning to ask this for some time, and now I've plucked up the courage! It is puzzling to me that many fundamental relationships in GR are explained in terms of euclidean space. Taking for example the geodesic deviation equation, it occurs to me that if defined in 3+1 spacetime there...

I've been thinking about this quite a bit. So it is clear that one can determine the Christoffel symbols from the first fundamental form. Is it possible to derive the geodesics of a surface from the Christoffel symbols?

Kevin Brown, in his excellent book "Reflections on Relativity" p. 409, "immediately" integrates 2 geodesic equations: \frac{d^{2}t}{ds^{2}}=-\frac{2m}{r(r-2m)}\frac{dr}{ds}\frac{dt}{ds} \frac{d^{2}\phi}{ds^{2}}=-\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds} to get...

Hi, I've found geodesic equations for the metric: \begin{equation} ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2} \end{equation} where \begin{equation} \alpha = 1 - \frac{r^{2}}{r_{s}^{2}} \end{equation} I have found that for a light ray: \begin{equation}...

Hello all, In Carroll's on page 109 it is pointed out that for derivation of the geodesic equation, 3.44, a "hidden" assumption is that we have used an affine parameter. Some few lines below we see that "any other parametrization" could be used, called alpha, but in that case the general...

I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field...

I am trying to derive the geodesic equation by extremising the integral $$ \ell = \int d\tau $$ Now after applying Euler-Lagrange equation, I finally get the following: $$ \ddot{x}^\tau + \Gamma^\tau_{\mu \nu} \dot{x}^\mu \dot{x}^\nu = \frac{1}{2} \dot{x}^\tau \frac{d}{ds} \ln \left|...

Hello all, I have a geodesic equation from extremizing the action which is second order. I am curious as to what the significance is of having 2 independent geodesic equations is. Also I was wondering what the best way to deal with this is.

Geodesic equation: m_{0}\frac{du^{\alpha}}{d\tau}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}= qF^{\alpha\beta}u_{\beta} Weak-field: ds^{2}= - (1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2}) Magnetic field, B is set to be zero. I want to find electric field, E, but don't know where to start, so...

The geodesic equation for a path X^\mu(s) is: \frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0 where U^\mu = \frac{d}{ds} X^\mu But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter s must be...

I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so...

Homework Statement My question is about a step in the lecture notes [http://arxiv.org/abs/hep-th/0307101] on page 6, and it is probably quite trivial: I want to see why a lightlike particle in AdS_5\times S^5 sees the metric as plane wave background. The metric is ds^2=R^2(-dt^2...

I understand why the geodesic equation works in flat space. It just basically gives a set of differential equations to solve for a path as a function of a single variable s where the output is the coordinates of whichever parameterization of the space you are using. But the derivation I know and...

Let's say I have my pen on my desk; does it describe a geodesic.? Or not because there is the normalforce working on it.

The geodesic equation follows from vanishing variation ##\delta S = 0## with ##S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}## In many cases one uses the energy functional with ##\delta E = 0## instead: ##E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}## Can...

Homework Statement 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 are given by u^2=v^2+av+b where a,b are constants. Homework Equations...

I'm trying to derive the Geodesic equation, \ddot{x}^{α} + {Γ}^{α}_{βγ} \dot{x}^{β} \dot{x}^{γ} = 0. However, when I take the Lagrangian to be {L} = {g}_{γβ} \dot{x}^{γ} \dot{x}^{β}, and I'm taking \frac{\partial {L}}{\partial \dot{x}^{α}}, I don't understand why the partial derivative of...

Homework Statement "The shortest path between two point on a curved surface, such as the surface of a sphere is called a geodesic. To find a geodesic, one has to first set up an integral that gives the length of a path on the surface in question. This will always be similar to the integral...

In General Relativity (GR), we have the _geodesic deviation equation_ (GDE) $$\tag{1}\frac{D^2\xi^{\alpha}}{d\tau^2}=R^{\alpha}_{\beta\gamma\delta}\frac{dx^{\beta}}{d\tau}\xi^{\gamma}\frac{dx^{\delta}}{d\tau}, $$ see e.g...

Hello Everybody, Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead. So take for eg. Carroll, he looks at the killing equation and extracts the equation K_\mu...

Hello Everybody, Carroll introduces in page 106 of his book "Spacetime and Geometry" the variational method to derive the geodesic equation. I have a couple of questions regarding his derivation. First, he writes:" it makes things easier to specify the parameter to be the proper time τ...

Homework Statement Consider the parametrization of a torus: \tau(u,v)=((2+cos(v)cos(u),(2+cosv)sinu,sinv) The distance from the origin to the center of the tube of the torus is 2 and he radius of the tube is 1. Let the coordinates on \mathbb R^3 be (x,y,z) . If p = \tau(u,v) then u is...

Stated loosely, the geodesic hypothesis says that test particles follow geodesics, where "test particle" means it has to be small in some sense (size, mass, ...), and there is an ambiguity in the word "geodesics" because we want to talk about geodesics of the spacetime that would have existed...

Homework Statement Use the result (6.41) of Problem 6.1 to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(ψ,ψ',θ) in (6.41) is independent of ψ, so the Euler-Lagrange equation reduces to ∂f/ψ' = c, a constant. This gives...