Geodesics general relativity Definition and 31 Threads

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.

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  1. H

    I Geodesics in Schwarzschild: Reparametrizing the Equations

    Hi all, I am working through Sean Carroll's Textbook, particularly Chapter 5 regarding the Schwarzschild Solution. In this chapter, Energy and Angular Momentum are defined as follows: $$ \begin{align} E &= (1-\frac{2GM}{r})\frac{dt}{d\lambda} \Rightarrow \frac{dt}{d\lambda} = (1-...
  2. cianfa72

    I Constants of motion symmetric spacetimes

    As discussed in this thread, for a symmetric spacetime (i.e. with a KVF) there are conserved quantities as constants of motion. For instance in Schwarzschild spacetime there is a timelike KVF, hence for example the contraction of a geodesic tangent vector (4-velocity) and the timelike KVF...
  3. M

    I Einstein brother-in-law elevator

    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...
  4. G

    A How to Express the Schwarzschild Metric in Fermi Normal Coordinates?

    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...
  5. cianfa72

    I General Relativity and the precession of the perihelion of Mercury

    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.
  6. cianfa72

    I Wald synchronous reference frame proof

    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...
  7. Onyx

    B Find Geodesics in Dynamic Ellis Orbits Metric

    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.
  8. cianfa72

    B Basic introduction to gravitation as curved spacetime

    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...
  9. cianfa72

    I About global inertial frames in GR

    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...
  10. M

    I Equivalence principle question

    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)...
  11. M

    A How to Integrate the geodesic equations numerically?

    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...
  12. A

    B Euler-Lagrange equation for calculating geodesics

    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?
  13. A

    I Solving Geodesic Equations with Killing Vectors: Is There a General Solution?

    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!
  14. D

    Deriving geodesic equation using variational principle

    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} =...
  15. U

    Conditions on Christoffel Symbols?

    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...
  16. U

    What Are the Conditions on Christoffel Symbols for Given Geodesics?

    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...
  17. U

    How is Frequency Redshift Related to Sphere's Proper Area and Flux Ratio?

    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)...
  18. U

    Frequency of Photon in Schwarzschild Metric?

    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...
  19. U

    Satellite orbiting around Earth - Spacetime Metric

    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...
  20. U

    Light-like Geodesic - What are the limits of integration?

    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 =...
  21. U

    How do I find the scale factor of cosmological constant?

    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...
  22. U

    What is the Geodesic Equation for FRW Metric's Time Component?

    Taken from Hobson's book: Metric is given by ds^2 = c^2 dt^2 - R^2(t) \left[ d\chi^2 + S^2(\chi) (d\theta^2 + sin^2\theta d\phi^2) \right] Thus, ##g_{00} = c^2, g_{11} = -R^2(t), g_{22} = -R^2(t) S^2(\chi), g_{33} = -R^2(t) S^2(\chi) sin^2 \theta##. Geodesic equation is given by: \dot...
  23. U

    Lowering Indices: Tensor Calculus Basics

    At low speeds and assuming pressure ##P=0##, T^{\alpha \beta} = \rho U^\alpha U^\beta g_{\alpha \mu} g_{\gamma \beta} T^{\alpha \beta} = \rho g_{\alpha \mu} g_{\gamma \beta} U^\alpha U^\beta T_{\gamma \mu} = \rho U_\mu U^\beta g_{\gamma \beta} Setting ##\gamma = \mu = 0##: T_{00} = \rho...
  24. U

    Geodesic Deviation Equation Solved

    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...
  25. U

    Einstein Tensor - Particle at rest?

    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 +...
  26. U

    Flat Space - Christoffel symbols and Ricci = 0?

    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} =...
  27. U

    General Relativity - Deflection of light

    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 -...
  28. U

    General Relativity - Circular Orbit around Earth

    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 =...
  29. U

    Index Notation: Understanding LHS = RHS

    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...
  30. DOTDO

    By what method did Einstein derived his gravitational field equation?

    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...
  31. RCopernicus

    Geodesics - Some help, please.

    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...
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