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.
Let us start with a simple two-dimensional space that is asymptotically flat, like e.g. sheet of paper on a desk. Geodesics in this space are straight lines in the traditional sense.
Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort...
Question: The idea of a continuum breaks down for a singularity when a geodesic become incomplete (the breaking of the idea that there was a continuous succession, where no part could be distinguished from neighboring parts, except by arbitrary division), and so with that does this indicate a...
Does anybody know to which "Gauss embedding theorem" the speaker in this video talk at minute 14 (point 5. in the displayed notes) is refering too? Sounds to be a standard result in differential geometry, but after detailed googling I found nothing to which the speaker may refering too in the...
Hi, reading this old thread I'd like a clarification about the following:
Fermi Normal hypersurface at an event on a comoving FLRW worldline is defined by the collection of spacetime orthogonal geodesics. Such geodesics should be spacelike since they are orthogonal to the timelike comoving...
I think I have a slight misconception maybe, but I was wondering about this question.
Usually when we say that the vectors are parallel, we say that it means that there's an equation ##k = \alpha l##, for the vectors ##k## and ##l## and some scalar ##\alpha##. In the context of differential...
Hi All,
I was just watching the video on the Veritassium channel about scientific dissemination of general relativity. And I could see once again that when you try to use animations and/or computer graphics to show the transition from free space geodesics to geodesics around a planet, what you...
Hello friends of the Forum. I want to ask you why the inertial acceleration in free fall in the relativistic geodesic equations is assumed equal to zero in free fall and equal to 9.8 m/s at rest on the earth's surface. On the other hand, assuming that zero acceleration in free fall, what would...
I have the following question to solve:Use the metric:
$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$
Test bodies are arranged in a circle on the metric at rest at $$t=0$$.
The circle define as $$x^2 +y^2 \leq R^2$$
The bodies start to move on geodesic when we have $$a(0)=0$$
a. we have to...
Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
1 Does the connection vanish along a affine geodesic?
2 In《Introducing Einstein's Relativity》Ed2 on page 96"It can be shown that the result(the connection vanishes at P) can be extended to obtain a coordinate system in which the connection vanishes along a curve,but not in general to a...
Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not?
My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)##...
I've got that length of a curve on the surface is:
$$L=\int_{-\infty}^{\infty}\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}dx$$
So the function to extremise is:
$$f(\rho,\rho')=\sqrt{1+\frac{4\pi^{2}}{h^{2}}\rho^{2}+\left(\frac{d\rho}{dz}\right)^{2}}$$
Where...
So. It was late night, the limpid sky a near cloudless darkness, somewhat lightened by the waxing moon. I being somewhat stoned and looking at the stars and constellations as I sometimes do, and it came to seem to me, in my imagination, that In the spaces between the stars, I was observing (in...
Suppose there is a three dimensional graph (such as z=x^2+y^2).
Suppose there is a point on the surface of the 3 dimensional graph, for example at (x,y,z)=(1,1,2).
Suppose the point is moving along the surface (along a geodesic) according to a unit vector, such as <0,1,0>.
Is there a...
Hi,
I would like to ask for some clarification about the physics involved in the gravitational waves detection using interferometers.
Starting from this thread Light speed and the LIGO experiment I'm aware of the two ends of an arm of the interferometer (e.g. LIGO) can be taken as the...
~ Shower Thoughts ~
Twin A is in a spaceship, Twin B is in a spaceship. Both in 'deep space'.
B follows a highly elliptical geodesic which goes around a planet (or black hole) with strong gravity, very far away.
When they meet again, who is younger and why?
I genuinely don't know what this...
If I'm standing on Earth, is my time dilation actually greater than if I was in a rocket accelerating at 9.8m/s^2 in deep space due to me being in a gravitational field on top of the acceleration? Geodesics experience time dilation in gravitational fields, so it seems like there is an additive...
Hello, there. I am learning the chapter, The Schwarzschild Solution, in Spacetime and geometry by Caroll. I could not grasp the idea of circular orbits.
It starts from the equations for ##r##, $$\frac 1 2 (\frac {dr}{d \lambda})^2 +V(r) =\mathcal E$$ where $$V(r)=\frac {L^2}{2r^2}-\frac...
Computing timelike geodesics in the Schwarzschild geometry is pretty straightforward using conserved quantities. You can treat the problem as a variational problem with an effective Lagrangian of
##\mathcal{L} = \frac{1}{2} (Q \frac{dt}{d\tau}^2 - \frac{1}{Q} \frac{dr}{d\tau}^2 - r^2...
Homework Statement:: i) If ##\bar{g} = \Omega^2 g## for some positive function ##\Omega##, show that ##\bar{g}## and ##g## have the same null geodesics.
ii) Let ##\psi## solve ##g^{ab} \nabla_a \nabla_b \psi + \xi R \psi = 0##. Determine ##\xi## such that ##\bar{\psi} = \Omega^p \psi## for...
Let us consider a sphere of a unit radius . Therefore, by choosing the canonical spherical coordinates ##\theta## and ##\phi## we have, for the differential length element:
$$dl = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2} $$
In order to find the geodesic we need to extremize the...
Background
While watching Does time cause gravity? from PBS Spacetime, i wondered if its possible to "derive" the geodesic equation
not from GR alone, but by assuming each particle is described by an extended wave function and the time evolution
of this wave is not constant but the rate varies...
*Moving this thread from 'General Math Forum' to 'General Relativity Forum' in order to generate more discussion.*
Any object will move through spacetime along its geodesic. Since mass bends spacetime, an object initially at rest near the mass will move towards the mass along a geodesic. It...
Hello there.Curvature can be informally defined as the deviation from a straight line in the context of curves, a circle in R^2 has curvature, then if we get higher dimensions than three we can't see the manifolds because it is their nature and the nature of our eyes that it is bounded by the...
I would like to determine how a point (xo,yo,zo) moves along a geodesic on a three dimensional graph when it initially starts moving in a direction according to a unit vector <vxo,vyo,vzo>. So, if I start at that point, after a very small amount of time, what is its new coordinate (x1,y1,z1) and...
1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities
$$Q^t :=...
I am currently reading Foster and Nightingale and when it comes to the concept of parallel transport, the authors don't go very deep in explaining it except just stating that if a vector is subject to parallel transport along a parameterized curve, there is no change in its length or direction...
I'm trying to evaluate the arc length between two points on a 2-sphere.
The geodesic equation of a 2-sphere is:
$$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$
According to this article:http://vixra.org/pdf/1404.0016v1.pdfthe arc length parameterization of the 2-sphere...
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}##...
I am working from Sean Carroll's Spacetime and Geometry : An Introduction to General Relativity and have got to the geodesic equation. I wanted to test it on the surface of a sphere where I know that great circles are geodesics and is about the simplest non-trivial case I can think of.
Carroll...
The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is
\begin{equation}
T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2}
\end{equation}
The covariant...
I will start with an example.
Consider components of metric tensor g' in a coordinate system
$$ g'=
\begin{pmatrix}
xy & 1 \\
1 & xy \\
\end{pmatrix}
$$
We can find a transformation rule which brings g' to euclidean metric g=\begin{pmatrix}
1 & 0 \\
0 & 1\\
\end{pmatrix}, namely...
My project for obtaining my master's degree in computer science involved ray tracing in Schwarzschild spacetime in order to render images of black holes. These light rays had to be computed numerically using the geodesic equation. However, I ran into a problem. The geodesic equation is given as...
Given ##ds^2 = y^{-2}(dx^2 + dy^2)##, I am trying to prove that a demicircle centred on the x-axis, written parametrically as ##x=r\cos\theta + x_0 ## and ##y= r \sin \theta ## are geodesics. Where ##r## is constant and ##\theta \in (0,\pi)##
I have already found the general form of the...
Hi, I'm the given the following line element:
ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And I'm asked to calculate the null geodesics.
I know that in order to do that I have to solve the Euler-Lagrange equations. For...
I was wondering where does the 1/2 factor come from in the Euler-Lagrange equation, that is:
L = \sqrt{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu}
implies that \partial_\mu L = \pm \frac{1}{2} (\partial_\mu g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu )
I'm not sure I entirely understand where it comes...
In the following link,
https://en.wikipedia.org/wiki/Geodesics_in_general_relativity
they write:"the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space."
Is there anything wrong with the following circular...
In general, is the number of distinct geodesics between two fixed points purely a feature of the topology of the manifold? i.e. Can there ever exist two topologically equivalent (I can’t remember the proper word right now- is homeomorphic the right one?) manifolds, ##\mathcal M## and ##\mathcal...
Homework Statement
I am unsure of Q3 but have posted my solutions to other parts
Homework EquationsThe Attempt at a Solution
3)
ok so it is clear that because the metric components are independent of ##x^i## each ##x^i## has an associated conserved quantity ##d/ds (\dot{x^i})=0##. (1)
The...
Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
I am reading a book of General Relativity and I am stuck on a demonstration. If I consider the FLRW metric as :
##\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta\text{d}\phi^2)\bigg]##
with ##g_{tt}=1##, ##\quad...
I am using from the following Euler equations :
$$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$
with function ##f## is equal to :
$$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$
and we have...
The ##t## in the title states the time evolution of our universe.
How does one show that Relativity doesn't require geodesics extending indefinetely in the past? That is, how does one show that the theory allows for geodesics to have a starting point in the past? Is it hard to show?
Homework Statement
Show that any geodesic with constant ##\theta## lies on the equator of a sphere, with the north pole being on the ##\theta = 0## line. Hence explain why all geodesics on a sphere will be arcs of a great circle.
Homework EquationsThe Attempt at a Solution
I've had a go at it...
Consider a non-radial timelike geodesic outside the event horizon. Will it nevertheless cross the horizon radially or are non-radial geodesics also possible inside? I couldn't find any reference regarding a possible angle dependence in this respect.
Homework Statement
The Lagrange Function corresponding to a geodesic is $$\mathcal{L}(x^\mu,\dot{x}^\nu)=\frac{1}{2}g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$
Calculate the Euler-Lagrange equations
Homework Equations
The Euler Lagrange equations are $$\frac{\mathrm{d}}{\mathrm{d}s}...