What is Geodesics general relativity: Definition and 29 Discussions
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.
In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space.
As personal curiosity, I want to calculate which is the difference in "travelled height" between a photon that goes across the width of an elevator - which is more or less 2[m] in my country - and a tiny mass particle that free-falls starting at the same "height" as the photon origin, and is...
I have been learning a bit about Fermi normal coordinates in Eric Poisson's "A Relativist's Toolkit". Problem 1.10 in this book is to express the Schwarzschild metric in Fermi normal coordinates about a radially infalling, timelike geodesic.
I know that in the Fermi normal coordinates (denoted...
Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury.
My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.
Hi, on Wald's book on GR there is a claim at pag. 43 about the construction of synchronous reference frame (i.e. Gaussian coordinate chart) in a finite region of any spacetime. In particular he says: $$n^b\nabla_b (n_aX^a)=n_aX^b\nabla_b \, n^a$$Then he claims from Leibnitz rule the above equals...
Does anyone see a way I can find geodesics in the metric ##ds^2=-dt^2+dp^2+(5p^2+4t^2)d\phi^2## (ones with nonzero angular momentum)? I'm hoping it can be done analytically, but that may be wishful thinking. FYI, this is the metric listed at the bottom of the Wikipedia article about Ellis Wormholes.
Hi,
my daughter saw my MTW copy on the desk and she asked me about the picture with the apple in front. To introduce her to the idea of gravitation as curved spacetime I answered like this:
Consider you (A) and a your friend (B) at two different spots on a garden each with a firecracker. Take...
Hi,
starting from this thread Principle of relativity for proper accelerating frame of reference I'm convincing myself of some misunderstanding about what a global inertial frame should actually be.
In GR we take as definition of inertial frame (aka inertial coordinate system or inertial...
The equivalence principle states that a person stood on Earth would experience “gravity” the same as if he was in an elevator in space traveling at 1g. I get this. but when Einstein was first exploring this, I read he came to the realisation that a person free falling on Earth (if in a vacuum)...
Hello there, I've been considering the geodesic equations of motion for a test particle in Schwarzschild geometry for some time now. Similar to what we can do with the Kepler problem I would like to be able to numerically integrate the equations of motion. I'm quite interested to see how...
Hello I am little bit confused about lagrange approximation to geodesic equation:
So we have lagrange equal to L=gμνd/dxμd/dxν
And we have Euler-Lagrange equation:∂L/∂xμ-d/dt ∂/∂x(dot)μ=0
And x(dot)μ=dxμ/dτ. How do I find the value of x(dot)μ?
Hello I am concered about way of solving geodesic equation. Is there a general solution to geodesic equation? How to calculate the Cristoffel symbol at the right side of the equation?
Thanks for helping me out!
I am trying to derive the geodesic equation using variational principle.
My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$
Using the Euler-Lagrange equation, I have got this.
$$ \frac{d^2 x^u}{dt^2} + \Gamma^u_{mk} \frac{dx^m}{dt} \frac{dx^k}{dt} =...
Homework Statement
Write down the geodesic equation. For ##x^0 = c\tau## and ##x^i = constant##, find the condition on the christoffel symbols ##\Gamma^\mu~_{\alpha \beta}##. Show these conditions always work when the metric is of the form ##ds^2 = -c^2dt^2 +g_{ij}dx^idx^j##.Homework...
Homework Statement
Using the geodesic equation, find the conditions on christoffel symbols for ##x^\mu(\tau)## geodesics where ##x^0 = c\tau, x^i = constant##.
Show the metric is of the form ##ds^2 = -c^2 d\tau^2 + g_{ij}dx^i dx^j##.
Homework EquationsThe Attempt at a Solution
The geodesic...
Homework Statement
(a) Show the relation between frequency received and emitted
(b) Find the proper area of sphere
(c) Find ratio of fluxes
Homework EquationsThe Attempt at a Solution
Part (a)
Metric is ##ds^2 = -c^2dt^2 + a(t)^2 \left( \frac{dr^2}{1-kr^2}+ r^2(d\theta^2 + \sin^2\theta)...
Homework Statement
The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##.
(a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...
Homework Statement
The metric near Earth is ##ds^2 = -c^2 \left(1-\frac{2GM}{rc^2} \right)dt^2 + \left(1+\frac{2GM}{rc^2} \right)\left( dx^2+dy^2+dz^2\right)##.
(a) Find all non-zero christoffel symbols for this metric.
(b) Find satellite's period.
(c) Why does ##R^i_{0j0} \simeq \partial_j...
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 =...
Homework Statement
(a)Sketch how the contributions change with time
(b)For no cosmological constant, how long will this universe exist?
(c)How far would a photon travel in this metric?
(d)Find particular density ##\rho_E## and scale factor
(e)How would this universe evolve?[/B]
Homework...
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...
Homework Statement
(a)Find Christoffel symbols
(b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors
(c) Find zeroth component of Einstein Tensor
Homework EquationsThe Attempt at a Solution
Part (a)[/B]
Let lagrangian be:
-c^2 \left( \frac{dt}{d\tau}\right)^2 +...
Homework Statement
[/B]
(a) Find christoffel symbols and ricci tensor
(b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##.
Homework EquationsThe Attempt at a Solution
Part(a)
[/B]
I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} = g_{tx} =...
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 -...
Homework Statement
(a) Find the proper time in the rest frame of particle
(b) Find the proper time in the laboratory frame
(c) Find the proper time in a photon that travels from A to B in time P
Homework EquationsThe Attempt at a Solution
Part(a)
[/B]
The metric is given by:
ds^2 =...
I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read:
Summing them up,
\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij}
I'm trying to understand how LHS = RHS...
Hi.
In class, the professor has tried to derive the equation by using the principle of least-action. (But not yet completed. Maybe next class...)
However I heard this method is used by Hilbert, who had derived the equation 5 days before Einstein derived it.
Then, what method did Einstein use...
We've all seen an image similar to this one:
This is displaying the projection of GR Geodesics onto 3-D space (well, 2D in the picture). I'm still working my way through the General Relativity texts, so I'm not yet able to do the calculation on my own. Can anyone give me a formula that I can...