In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.
The noun "geodesic" and the adjective "geodetic" come from geodesy, the science of measuring the size and shape of Earth, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.
Hi,
very basic question. Take an object like a rock or the Earth itself. If we consider their internal constituents, there will be electromagnetic forces acting between them (Newton's 3th law pairs).
From a global perspective if the rock is free from external non-gravitational forces, then it...
I start with the 2D plane. Suppose y(x) is the curve that connects these two points. Its length is given by:
$$S=\int_1^2 \, ds=\int_1^2 (1+y'^2)^{\frac {1}{2}} \, dx$$ Applying Euler's equation we get:$$\frac {\partial f} {\partial y'}=A$$$$\dfrac {y'}{(1+y'^2)^{\frac {1}{2}}}=A$$...
If one stands on a large planetary body, like the moon, and throws a large object, like a rock straight up, the object will leave with some velocity, slow down to a stop, and then come back down with the same velocity once it returns to its origin. In Newtonian mechanics, the understanding is...
Using EL equation, $$L=\left(\frac{t^2}{\alpha}\dot{x}^2-\frac{c^2t^2}{\alpha}\dot{t}^2\right)^{0.5} \Longrightarrow \mathrm{constant} =\left(\dot{x}^2 -c^2 \dot{t}^2\right)^{-0.5} \left(\frac{t^2}{\alpha}\right)^{0.5} \dot{x}$$.
Get another equation from the metric: $$ds^2=-\frac{c^2t^2}\alpha...
The object takes a step [x, y] in 2 dimensional space. This is represented the change in coordinate ##x \vec e_x + y \vec e_y## where ##e_x## and ##e_y## are basis vectors in this space.
Suppose we define a non-linear / parametric transformation of this ##\vec e_x## and ##\vec e_y## basis...
I'm reading《Introducing Einstein's Relativity_ A Deeper Understanding Ed 2》on page 180,it says:
since we are interested in the Newtonian limit,we restrict our attention to the spatial part of the geodesic equation,i.e.when a=##\alpha####\quad ##,and we obtain,by using...
On pages 106-107 of Spacetime & Geometry, Carroll derives the geodesic equation by extremizing the proper time functional. He writes:
What I am unclear on is the step in 3.47. I understand that the four velocity is normalized to -1 for timelike paths, but if the value of f is fixed, how can we...
The metric is $$ds^2 = \frac{dr^2 + r^2 d\theta ^2}{r^2-a^2} - \frac{r^2 dr^2}{(r^2-a^2)^2}$$
I need to prove the geodesic is: $$a^2 (\frac{dr}{d \theta})^2 + a^2 r^2 = K r^4$$
My method was to variate the action ##\int\frac{(\frac{dr}{d\theta})^2 + r^2 }{r^2-a^2} - \frac{r^2...
I was following David tongs notes on GR, right after deriving the Euler Lagrange equation, he jumps into writing the Lagrangian of a free particle and then applying the EL equation to it, he mentions curved spaces by specifying the infinitesimal distance between any two points, ##x^i##and ##x^i...
This is my attempt to re-write the geodesic deviation equation in the special case of 3 dimensions and +++ signature in matrix notation.
We start with assuming an orthonormal basis. Matrix notation allows one to express vectors as column vectors, and dual vectors as row vectors, but by...
Hey guys, as you may remember, I have posted a question about plotting the orbit of timelike and null-like particles for a given metric.
I think this discussion might be helpful for me and some other people in future studies. I have found an article, and in that article, the authors are using...
Since the EFE describes the shape of spacetime, it describes the way black holes, for example, evolve. Can one derive the geodesic equation from it in some limit ?
Hello,
I need the natural parametrization or a geodesic curve contained in the surface z=x^2+y^2, that goes through the origin, with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with "a" constant, expressed as a function of the arc length, i.e., I need r(s)=r(x(s),y(s)).
Thank...
When parallel transporting a vector along a straight line on flat space, does the connection (when calculating the covariant derivative) always equal zero? Do things change at all when using an arbitrary connection, rather than Christoffel symbols?
Hi,
My question can result a bit odd.
Consider flat spacetime. We know that inertial motions are defined by 'zero proper acceleration'. Suppose there exist just one free body in the context of SR flat spacetime (an accelerometer attached to it reads zero). We know that 'zero proper...
In the formula : ##\frac{d^2 x^\mu}{d\tau^2}=-\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}##
How is the ##x^\mu## understood : a 4-vector or the ##\mu##-st component simply ?
If it is a vector, how to write it in spherical coordinate with extra time dimension ?
Btw...
I'm still on section 5.4 of Carroll's book on Schwarzschild geodesics
Carroll says "In addition, we always have another constant of the motion for geodesics: the geodesic equation (together with metric compatibility) implies that the quantity $$...
I'm on to section 5.4 of Carroll's book on Schwarzschild geodesics and he says stuff in it which, I think, enlightens me on the use of Killing vectors. I had to go back to section 3.8 on Symmetries and Killing vectors. I now understand the following:
A Killing vector satisfies $$...
I have no idea if this is an “A” level question, but I will put that down.
From the Schwarzschild metric, and with the help of the Maxima program, one of the geodesic equations is:
(I will have to attach a pdf for the equations...)
I believe this integrates to the following, with ...
Consider a free particle with rest mass ##m## moving along a geodesic in some curved spacetime with metric ##g_{\mu\nu}##:
$$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$...
I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$
And I apply the worm hole space...
Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it.
There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...
Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...
I have been working with the Godel metric (- + + + signature). I wanted to derive the geodesics for the metric, so I took to the geodesic equation:
(d2xm/ds2) + Γmab(dxa/ds)(dxb/ds) = 0
In the case of the Godel metric, the geodesic equations that I was able to derive after deriving the...
I hope I'm asking this in the right place! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...
I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error:
I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative
\begin{align}...
From Thomas Moore A General Relativity Workbook I have the geodesic equation as,
$$ 0=\frac{d}{d \tau} (g_{\alpha \beta} \frac{dx^\beta}{d \tau}) - \frac{1}{2} \partial_\alpha g_{\mu\nu} \frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau} $$
as well as
$$ 0= \frac{d^2x^\gamma}{d \tau^2} +...
The Born frame field (see ref below) describes a rotating system and the proper acceleration ##\vec{a}=\nabla _{{{\vec {p}}_{0}}}\,{\vec {p}}_{0}={\frac {-\omega ^{2}\,r}{1-\omega ^{2}\,r^{2}}}\,{\vec {p}}_{2}##. If ##\omega## depends on coordinate ##r## then...
The geodesic for 2-D, 3-D are straight lines.
For a 4-D spacetime (x1,x2,x3,t) what would be it's geodesic.??
The tangent vector components are ##V^0=\frac{∂t}{∂λ} , V^i=\frac{∂x^i}{∂λ},i=1,2,3## & ##(\nabla_V V)^\mu=(V^\nu \nabla_\nu V)^\mu=0,(\nu,\mu=0,1,2,3)##
For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve).
At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##.
The non-zero christoffel symbol are Γ122 and Γ212.
From covariant derivative...
I have tried twice now to calculate acceleration of gravity using the general relativistic equation of geodesic deviation and both times my solution is twice the correct answer. What am I doing wrong? As an example here is one problem: Calculate the acceleration of gravity g at the earth’s...
Homework Statement
We are asked to show that:
## \frac{d^2x_\mu}{d\tau^2}= \frac{1}{2} \frac{dx^\nu}{d\tau} \frac{dx^{\rho}}{d\tau} \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} ##
( please ignore the image in this section i cannot remove it for some reason )
Homework Equations
The...
From the invariance of space time interval the metric dΓ2=dt2-dx2-dy2-dz2
dΓ2=gμνdxμdxν
dΓ=√(gμνvμvμ)dt
dΓ=proper time.
Can someone please help me in sort out why the term √(gμνdxμdxν) is taken as the Lagrangian,as geodesic equation is solved by taking this to be the Lagrangian.
Homework Statement
Let ##U## be a plane given by ##\frac{x^2}{2}-z=0##
Find the curve with the shortest path on ##U## between the points ##A(-1,0,\frac{1}{2})## and ##B(1,1,\frac{1}{2})##
I have a question regarding the answer we got in class.
Homework Equations
Euler-Lagrange
##L(y)=\int...
I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
Hi,
If I have a massive particle constrained to the surface of a Riemannian manifold (the metric tensor is positive definite) with kinetic energy $$T=\dfrac 12mg_{\mu\nu} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}$$ then I believe I should be able to derive the geodesic...
Hello! I read a derivation for the geodesic deviation: you have 2 nearby geodesics and define a vector connecting points of equal proper time and calculate the second covariant derivative of this vector. I understand the derivation but I am a bit confused about the actual definition of this...
Can a person inside a spaceship falling freely on a geodesic path, experience the same just like a person inside a car experience a force on a turn on Earth i.e when the geodesic path is no more straight near a huge planet.
Thanks.
I have a question which asks show that a null geodesic to get to r> R , r some constant, given the space time metric etc, takes infinite coordinate time but finite proper time. ( It may be vice versa ).
I just want to confirm that, ofc there is no affine parameter for a null geodesic and so you...
Homework Statement
Question attached:
Homework Equations
see below
The Attempt at a Solution
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my main question really is
1) what is meant by 'abstract tensors' as I have this for my definition:
to part a)
##V^u\nabla_uV^a=0##
but you do say that ##V^u=/dot{x^u}## ; x^u is a...
Hello,
i know that relativists like to extend solutions of einstein equations so that they are geodesicly maximal (i.e. geodesics end only in singularity or infinite value of affine parameter). But why only geodesicly? Thus this geodesic maximality imply, that if i take any timelike or...
... via plugging in the Fundamental theorem of Riemmanian Geometry :
##\Gamma^u_{ab}=\frac{1}{2}g^{uc}(\partial_ag_{bc}+\partial_bg_{ca}-\partial_cg_{ab})##
Expanding out the covariant definition gives the geodesic equation as:
(1) ##\ddot{x^u}+\Gamma^u_{ab} x^a x^b =0 ##
(2) Lagrangian is...
some questions I have seen tend to word as show that some quantity/tensor/scalar (e.g let this be ##K##) is constant along an affinely parameterised geodesic, others ask show covariantly constant.
the definiton of covariantly constant/ parallel transport is:
## V^a\nabla_u K = 0 ##for the...
Let us assume a "toy-metric" of the form
$$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$
where ##J## is the angular-momentum vector of the source.
Consider the curve
$$ \gamma(\tau)=(x^\mu...
Hi,
So the geodesic equation is saying in my frame of reference I may see acceleration and then in your frame of reference you may see gravity? So by just changing coordinates you can create a "force" ?
And also is this relevant to the Minkowski space or do I need to be in GR to be able to get...