Talk:Schwarzschild geodesics/Archive 1

Archive 1

Adaptation from Two-body problem in general relativity

I have changed the scope of this article from being a redirect (to the Schwarzschild metric) to a description of the metric and the geodesic orbits of particles moving in that metric - I hope that's OK! :)

I wrote much of this material for the Two-body problem in general relativity (originally called the Kepler problem in general relativity). However, it was wisely pointed out to me that the Schwarzschild solution pertains to more than just the motion of two bodies. Rather, it's more closely akin to the classical problem of n particles moving in a static central potential. Certainly there are applications of the Schwarzschild solution that pertain to more than two bodies. Therefore, I have split off the Schwarzschild portion of that article and would like to develop it separately here.

I hope that this change is agreeable to everyone. I look forward to everyone's contributions! :) Willow (talk) 19:56, 13 July 2010 (UTC)

Name change?

The title of this article is less than ideal, since the Schwarzschild metric is a solution of the Einstein field equations, and could be construed as the "Schwarzschild solution". The scope of this article is to solve for the geodesics of the Schwarzschild metric, and thereby connect with many of the experimental tests of general relativity.

Should we rename this article to "Geodesics of the Schwarzschild metric" or "Schwarzschild geodesics" or some such? I'll consult my sources to see what's customary in the literature and report back, and I hope that others here will do likewise. Thanks for your help! :) Willow (talk) 05:13, 22 July 2010 (UTC)

Sampling of sources

Adler, Bazin, & Schiffer: "General relativistic Kepler problem" for massive test particles (p. 177), else "trajectory of a light ray in a Schwarzschild field" (p. 188)

Misner, Thorne, and Wheeler: "Particle motion in Schwarzschild geometry" (running title of Chapter 25); "orbit of a particle in the Schwarzschild geometry" (p. 659, 672)

Synge: "Orbits and rays in the [Schwarzschild] field" (p. 289); Synge refers to the "geodesic hypothesis"

Weinberg: "Motion of a freely falling material particle or photon in a static, isotropic, gravitational field" (p. 185)

Landau and Lifshitz: "Motion in a centrally symmetric gravitational field" (p. 306)

Weyl: "Problem of one body in a radially symmetric, static field" (p. 252)

Pauli: "Paths of point masses and light rays in the [Schwarzschild] field" (pp. 166–167)

Rindler: "Rays and orbits in Schwarzschild space" (p. 143)

Wald: "Geodesics of Schwarzschild" (p. 136)

Is the energy formula correct?

In case anyone has any doubts about whether the formula,

$E=m\,c^{2}\left(1-{\frac {r_{s}}{r}}\right){\frac {dt}{d\tau }}\,,$

given for the energy of the particle of mass m is correct, I will justify it here. Firstly, the article shows that it is a conserved quantity which it should be from Noether's theorem and the static character of the solution. Secondly, if we consider the special case where $r_{s}=0\,$ we get

$E=m\,c^{2}{\frac {dt}{d\tau }}={\frac {m\,c^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,$

which is the formula for energy in special relativity as it should be. Thirdly, if we consider the special case where the particle is instantaneously at rest, we get

${\frac {d\tau }{dt}}={\sqrt {1-{\frac {r_{s}}{r}}}}\,$

from which we can derive

$E=m\,c^{2}{\sqrt {1-{\frac {r_{s}}{r}}}}\approx m\,c^{2}\left(1-{\frac {r_{s}}{2r}}\right)=m\,c^{2}-{\frac {G\,M\,m}{r}}\,$

which is the rest energy plus the (negative) Newtonian gravitational potential energy as it should be. JRSpriggs (talk) 14:13, 23 July 2010 (UTC)

Consider an observer who is momentarily co-located with the particle, but at rest relative to the star. Suppose that he constructs a local Cartesian inertial frame of reference. Let v be the velocity of the particle as measured in that frame. Then one can show that, in general,

${\frac {d\tau }{dt}}={\frac {dt_{\text{local}}}{dt}}{\frac {d\tau }{dt_{\text{local}}}}={\sqrt {1-{\frac {r_{s}}{r}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\,.$

Consequently,

$E=m\,c^{2}{\frac {\sqrt {1-{\frac {r_{s}}{r}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\,$

holds in any case. And thus

$E=E_{\text{local}}{\sqrt {1-{\frac {r_{s}}{r}}}}\,.$

JRSpriggs (talk) 05:18, 28 July 2010 (UTC)

More concepts please, less formula cranking, please!

This article is very heavy on the crank, and very light on the concepts and abstraction. In particular, it gets to the end with essentially no mention of tangent bundles, geodesic flows, integrability conditions. The treatment of hamilton-jacobi and lagrangian formulas are also so close to the ground as to make them almost impenetrable. Hamilton-Jacobi and Lagrangian formulations, when properly formulated, open good conceptual insight; here its been used to kill that insight. It would be much easier to read and understand if this was written in a far more abstract fashion: write down some differential forms, some covariant derivatives, whatever, some lie brackets, whatever it takes, use the language of dynamical systems, demonstrate the basins of attraction. Write down the caustics. Cause right now, its just an avalanche of dull, deadly formulas, with no intuition or insight at all into what is going on, and why these formulas are the way they are. I mean, its great if I had a homework problem and had to come up with a number; but, for understanding the concepts, understanding what is really going on in here, its very very opaque. 67.198.37.16 (talk) 06:46, 24 February 2016 (UTC)

While the formulas in the article may be difficult for you to understand, the abstractions for which you are asking are equally difficult for me to understand. If you want to add something more abstract, you may do so, but please leave the formulas alone. JRSpriggs (talk) 10:17, 24 February 2016 (UTC)

Its not that the formulas are "difficult to understand", its the opposite: they are "too easy" to understand, are lacking in content, and, as a result, a bit boring. They fail to be illustrative. Its like asking how an automobile works, and being told that "the motor turns and the wheels go round" -- child talk. OK, so this article isn't quite child-talk, but its very nearly at that level. And worse: it takes a plug-n-chug approach to the subject. (A very interesting post on conceptual understanding vs plug-n-chug, quoting Eric Mazur & the Force Concept Inventory: https://plus.google.com/+ChrisReeveOnlineScientificDiscourseIsBroken/posts/HV5W3xBr1AQ )

I'm not quite sure how to make it better. Maybe look at congruence (general relativity), which has a nice section on kinematics, and work out the tensors for the various cases. Clearly, vorticity vanishes; I can guess at the others, but simply seeing expressions for those quantities would increase intuitive understanding.

Some ideas:

• The 'almost SO(4)' symmetry of Runge-Lenz is broken. Aside from precession, can anything more be said?
• The horocycles (well, horospheres) and if possible, the hypercircles. They hypercircle would come from integrating the tidal tensor. Horocycles are traditinally used in dynamical systems to illustrate the unstable and stable bundles (gack. lets try stable set wikilink -- the stable bundle article is off in another unwanted direction) e.g hyperbolic set. There's no chance that any of this is Anosov, I guess; I've never heard that term applied before to GR, but my readings have been minor, but it is another "obvious" question.

67.198.37.16 (talk) 16:53, 24 February 2016 (UTC)

dr/dt vs dr/dτ

@B wik, your edits are wrong, I'll revert them all. If it were dr/dt (the observed radial coordinate velocity in the system of the bookkeeper) instead of dr/dτ (the celerity) the far away bookkeeper would observe a much higher velocity than the local velocity, which is clearly nonsense. --Yukterez (talk) 19:46, 23 September 2018 (UTC)

The definition of v comes from the book by Landau and Lifshitz. The formulas for dr/dt should therefore also be taken from this source. --B wik (talk) 21:17, 23 September 2018 (UTC)

Since you can not provide a link for your source I can only assume that the letter v in your source is used for a different variable. In this article v is used for the local velocity, as measured by the test particle itself. This v must be c for a transverse orbit at the photons sphere, and c for the radial infall velocity at the horizon. You always have to check what a letter stands for, you can not quote "a²+b²=c²" with c being the speed of light, or "e^(iπ)-1=0" where e is not Euler's number but the energy. The same goes for letters like v, especially if you don't have a link for your source. --Yukterez (talk) 22:39, 23 September 2018 (UTC)

The definition of v does not come from your book, it is the normal v everybody knows from the Lorentzfactor 1/√(1-v²/c²) where c is the limit for v. Also I bet that your reference does NOT give your dubious definition for dr/dt, since I (and everybody else who knows a little thing about relativity) knows for sure that this is dr/dτ. Even if we were to use the letter v for the shapirodelayed velocity instead of the local one, the formula for dr/dτ would not become your formula for dr/dt. To transform from one into the other you can not just replace the v in the Lorentz factor by the shapirodelayed velocity! --Yukterez (talk) 23:00, 23 September 2018 (UTC)

The formulas for $v_{\parallel }$ and $v_{\perp }$ from the section "Local and delayed velocitites" are correct, but only valid for test bodys with mass. With the stated formula for the time dilation dtau/dt, one could also calculate the more general and simpler version in dependence from dr/dt or dphi/dt, which describe also the movement of massless particles. --B wik (talk) 03:04, 2 October 2018 (UTC)

I don't see how this helps your case to just simply replace all the dτ's with dt's. --Yukterez (talk) 09:10, 2 October 2018 (UTC)

At least you are correct in one point: only massive test particles have a proper time, so I'll mention that it takes the role of an affine parameter when applied to photons, while the equations stay the same. --Yukterez (talk) 09:50, 2 October 2018 (UTC)