Modeling Path of Object in General Relativistic/Riemannian Geom

  • Thread starter Philosophaie
  • Start date
  • Tags
    Path
In summary, the conversation discusses using the Earth and/or Sun to model a General Relativistic or Riemannian Geometric path of an object with an initial velocity in a gravitational field. The Schwarzschid metric for a spherically symmetric, nonrotating body is suggested as a starting point, and the geodesic equations are derived from it. The equations can be analyzed using similar techniques as classical equations of motion, and there is a solution for all values of theta. However, there are no known analytical solutions except for special cases. The conversation also mentions the possibility of using perturbation theory to approximate the perihelion precession of a planet.
  • #1
Philosophaie
462
0
I want to use the Earth and/or Sun to model a General Relativistic or Riemannian Geometric path of an object with an initial velocity coming into a gravitational field until it comes into contact with the ground or skips off in another direction. Does anyone have any suggestion where to start?
 
Physics news on Phys.org
  • #2
Start by writing down the Schwarzschid metric for a spherically symmetric, nonrotating body:
[tex]
ds^2 = -\left( 1 - \frac{2GM}{r} \right) \textrm{d} t^2 + \left( 1 - \frac{2GM}{r} \right)^{-1} \textrm{d} r^2 + r^2 ( \textrm{d} \theta^2 + \sin^2(\theta) \textrm{d} \varphi^2 ) \textrm{.}
[/tex]
To get the geodesic equations from this, you can either calculate all the Christoffel symbols (long and tedious, but straightforward) or use the calculus of variations (my favorite method). In either case, I'd suggest setting [tex] e^{f(r)} = \left( 1 - \frac{2GM}{r} \right) [/tex] for ease of notation. You'll end up with four very ugly-looking equations (one for each of [tex] t, r, \theta [/tex], and [tex] \varphi [/tex]). By symmetry, you can set [tex] \theta = \frac{\pi}{2} [/tex] (i.e., [tex] \theta' = 0 [/tex]). The [tex] t [/tex] equation will then give you [tex] e^{f(r)} t' = E [/tex], where [tex] E [/tex] is a constant of motion. The [tex] \varphi [/tex] equation will give [tex] r^2 \varphi' = L [/tex], where [tex] L [/tex] is another constant of motion ("angular momentum"). The [tex] r [/tex] equation gives
[tex]
e^{-f(r)} (r')^2 + \frac{L^2}{r^2} - e^{-f(r)} E^2 = \epsilon \textrm{,}
[/tex]
where [tex] \epsilon [/tex] is another consant of motion. For massive particles, you can set [tex] \epsilon = -1 [/tex] without loss of generality (why?). Then you can rearrange the above equation to get
[tex]
\frac{1}{2} (r')^2 + V(r) = \mathcal{E} \textrm{,}
[/tex]
where [tex] V(r) = \frac{1}{2} \left(1 + \frac{L^2}{r^2} \right) e^{f(r)} [/tex] is a "potential energy" and [tex] \mathcal{E} = \frac{1}{2} E^2 [/tex] is a "total energy." This looks exactly like the form of a classical equation of motion, and you can analyze it using similar techniques. In particular, determining values for [tex] E [/tex], [tex] L [/tex], and [tex] r_{0} [/tex] that lead to bounded or unbounded paths is straightforward.
 
Last edited:
  • #3
The Schwarzschild metric is:

[tex]ds^{2}=-(1-2GM/r)dt^{2}+(1-2GM/r)^{-1}dr^{2}[/tex]...

I already calculated Affinity (Torsion=0), Riemann Tensor and the Ricci Tensor from the metric. I got the four ugly looking equations from the diagonal of the Geodesic Equation. This is where I get stuck. I would like to have more than just [tex]\theta=\pi/2[/tex]. Is there a solution or a set of solution to these four Geodesic Equations no matter how difficult for all [tex]\theta[/tex]?
 
  • #4
The Schwarzschild metric is:

LaTeX Code: ds^{2}=-(1-2GM/r)dt^{2}+(1-2GM/r)^{-1}dr^{2} ...

Yeah...sorry about that. \facepalm

Due to spherical symmetry, you can always change coordinates so that [tex] \theta = \frac{\pi}{2} [/tex], [tex] \theta' = 0 [/tex]. Briefly, this is because, since we have invariance under rotations, the "direction of angular momentum" is conserved, hence the motion of a particle must be planar. (More rigorously, to each Killing vector on a manifold we can associate a constant of geodesic motion. The timelike Killing vector of Schwarzschild yields conservation of "energy," while the three spherical Killing vectors yield conservation of the three components of the "angular momentum.")

I don't know of any analytical solutions to these equations, except for very special cases. Like I said, however, they're (fairly) straightforward to analyze nonetheless; for example, you can derive, without too much work, the rate of precession of the perihelion of a planet using perturbation theory (to a first approximation). You should get [tex] \Delta \varphi = \frac{6 \pi G^2 M^2}{L^2} [/tex], where [tex] \Delta \varphi [/tex] indicates the amount the perihelion advances per orbital period.
 
  • #5
Thanks for the info!
 
Last edited:

What is general relativity and how does it relate to modeling the path of objects?

General relativity is a theory of gravity developed by Albert Einstein in the early 20th century. It explains how massive objects distort the fabric of space-time, causing objects to move along curved paths. This theory is essential in modeling the path of objects in a Riemannian geometry, as it takes into account the curvature of space-time in the presence of massive objects.

What is a Riemannian geometry and how does it differ from Euclidean geometry?

Riemannian geometry is a branch of mathematics that deals with curved spaces and their properties. In contrast to Euclidean geometry, which describes flat spaces, Riemannian geometry allows for the measurement of distances and angles in curved spaces. It is used in general relativity to model the curvature of space-time caused by massive objects.

How is the path of an object modeled in general relativity?

In general relativity, the path of an object is modeled using the equations of motion derived from the theory. These equations take into account the curvature of space-time caused by massive objects, as well as the mass and velocity of the object itself. Solving these equations allows us to determine the path of the object in a given space-time geometry.

Can general relativity accurately model the path of objects in extreme environments, such as near a black hole?

Yes, general relativity has been extensively tested and proven to accurately model the path of objects in extreme environments, including near black holes. In fact, the predictions of general relativity have been confirmed by numerous observations and experiments, making it one of the most successful theories in physics.

What are some potential applications of modeling the path of objects in general relativity?

Modeling the path of objects in general relativity has various applications, including predicting the orbits of planets and other celestial bodies, understanding the behavior of stars and galaxies, and even aiding in the development of technology such as GPS systems. It also has implications for our understanding of the universe and the fundamental laws of physics.

Similar threads

  • Special and General Relativity
2
Replies
50
Views
2K
  • Special and General Relativity
2
Replies
35
Views
387
Replies
14
Views
975
  • Special and General Relativity
Replies
27
Views
4K
  • Special and General Relativity
Replies
6
Views
2K
  • Special and General Relativity
Replies
23
Views
2K
  • Special and General Relativity
Replies
24
Views
2K
  • Special and General Relativity
Replies
4
Views
867
Replies
32
Views
852
  • Special and General Relativity
Replies
6
Views
863
Back
Top