Equations of motion for the Schwarzschild metric (nonlinear PDE)

In summary, the person is working on general relativity and trying to solve for equations of motion using the Schwarzschild Metric. They are new to nonlinear pde and are unsure of what methods to try. They have 2 out of 3 equations for t and r, and have solutions for dct/dlambda and dphi/dlambda. They believe the constant A is related to total energy and B is related to angular momentum in the phi direction. They have tried rearranging their equations, but are unsure how to integrate a random function of r with respect to lambda. They are stuck and would appreciate any help or tips.
  • #1
BiGyElLoWhAt
Gold Member
1,622
131
TL;DR Summary
Solution to equation of the form d^2r/dl^2 + f(r)(dt/dl)^2 + g(r)(dphi/dl)^2 where l is an arbitrary parameter. Working in spherical coordinates
I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance, most of the things I know to try from ODE don't look like they'll work.

##
\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s(r-r_s)}{r^3}(\frac{dct}{d\lambda})^2 +1/2 \frac{r_s}{r(r_s-r)}(\frac{dr}{d\lambda})^2 +(r_s-r)(\frac{d\phi}{d\lambda})^2 =0
##
Any pointers, solution methods, etc would be greatly appreciated.**crosses fingers for good latex render because preview is broken**Edit**
Having stepped away and played a video game for 45 or so minutes, I'm fairly certain that my other solutions are relevant here, as I effectively have solutions for ##\frac{dct}{d\lambda}## and ##\frac{d\phi}{d\lambda}##

I have:
##\frac{dct}{d\lambda} = \frac{Ar}{r-r_s}##
and
##\frac{d\phi}{d\lambda} = \frac{B}{r^2}##
I'm pretty sure the constant A has to to with total energy and B has to do with angular momentum in the phi direction, beyond that I'm not really sure what those 2 are at the moment.
Plugging that into what I have for my dr equations:

##
\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s(r-r_s)}{r^3}[\frac{Ar}{r-r_s}]^2 +1/2 \frac{r_s}{r(r_s-r)}(\frac{dr}{d\lambda})^2 +(r_s-r)[\frac{B}{r^2}]^2 =0
##
I haven't gotten it down on paper, just did some copy paste. I'm going to continue to stare at this paper and see if anything pops out at me, but still (obviously) feel free to jump in at any time and give me some tips.

Edit** ctd.
So I cleaned it up a bit, but am not really sure what to do now. This is what I have:
##\frac{d^2r}{d\lambda^2} + 1/2 \frac{r_s}{r(r_s-r)} (\frac{dr}{d\lambda})^2 = -A^2/2 \frac{r_s}{r(r-r_s)} - B^2\frac{r_s - r}{r^4} ##
So for the other 2 equations, I had homogeneous equations, and was basically able to do something like ##\frac{d}{d\lambda}[f(r)\frac{dct}{d\lambda}]## or with dphi for the phi equations. This almost looks like I should be able to do that, but I can't easily rearrange to get a condition on f(r) (before i had f'/f = the function attached to the first order term), and 2nd, I'm not sure how to integrate a random function of r w.r.t. lambda (rhs) where r is a function of lambda.

--this really reminds me of the drag equation (nonlinear drag)
 
Last edited:
Physics news on Phys.org
  • #2
where you have a drag term on the rhs, and I think it's related to the "coupling" terms in the equation of motion.Anyways, I'm still stuck and figured I'd update this post with more information. Any help is appreciated!
 

Related to Equations of motion for the Schwarzschild metric (nonlinear PDE)

1. What is the Schwarzschild metric?

The Schwarzschild metric is a mathematical equation that describes the curvature of space and time around a non-rotating, spherically symmetric mass. It is a fundamental solution to Einstein's field equations in general relativity.

2. What are the equations of motion for the Schwarzschild metric?

The equations of motion for the Schwarzschild metric are a set of nonlinear partial differential equations (PDEs) that describe how objects move in the curved spacetime around a massive object. They take into account the effects of gravity and the curvature of spacetime on the motion of objects.

3. How are the equations of motion derived for the Schwarzschild metric?

The equations of motion for the Schwarzschild metric are derived from Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy. By solving these equations for a spherically symmetric mass, the equations of motion for the Schwarzschild metric can be obtained.

4. What are the applications of the equations of motion for the Schwarzschild metric?

The equations of motion for the Schwarzschild metric have numerous applications in astrophysics and cosmology. They are used to study the motion of objects in the vicinity of massive objects such as black holes, as well as to understand the behavior of light and other forms of radiation in curved spacetime.

5. Are there any limitations or assumptions in the equations of motion for the Schwarzschild metric?

Yes, there are some limitations and assumptions in the equations of motion for the Schwarzschild metric. They are based on the assumption of a non-rotating, spherically symmetric mass and do not take into account the effects of other forces, such as electromagnetic or nuclear forces. Additionally, these equations are only valid in the weak gravitational field limit and break down in extreme cases such as near the event horizon of a black hole.

Similar threads

  • Special and General Relativity
Replies
9
Views
1K
  • Special and General Relativity
Replies
11
Views
524
  • Differential Equations
Replies
3
Views
2K
  • Special and General Relativity
2
Replies
43
Views
2K
  • Differential Equations
Replies
1
Views
1K
  • Special and General Relativity
Replies
6
Views
1K
Replies
10
Views
1K
  • Differential Equations
Replies
1
Views
2K
  • Special and General Relativity
2
Replies
42
Views
3K
  • Advanced Physics Homework Help
Replies
5
Views
1K
Back
Top