What are the equations for a massless particle's worldline?

[NeedsHelpBadly]

Hello,

I've been trying for a while to calculate the paths of massless particles around a Schwarzschild black hole, and basically I've failed, but I really need the answer for writing a simulator.

Knowing the initial conditions of the particles, I need to know the equations for:

$r(\lambda)$
$\theta(\lambda)$
$\phi(\lambda)$
$t(\lambda)$

or equations that I can integrate easily.

 

[Markus Hanke]

What have you tried so far to tackle this problem ?
Basically, the way to do it is to plug the Schwarzschild metric into the geodesic equations, and, given initial conditions, obtain parametrized curves from the differential equations. I should warn you though that the integrals you will be getting aren't exactly elementary.

 

[Nugatory]

Google for "schwarzschild equation of motion" will find these equations fairly quickly.

 

[NeedsHelpBadly]

I've tried plugging the Christoffel symbols into the geodesic equation, but I just don't understand how to integrate the equations I get. They have terms which are velocities along other coordinate axes, and which depend on the function which I would find by integrating. A worked example, maybe of the radial velocity with respect to time would be helpful. Also, I've been told that I can use specific energy and specific angular momentum as conserved quantities, but I don't know how to use them.

 

[NeedsHelpBadly]

Google for "schwarzschild equation of motion" will find these equations fairly quickly.

Could you actually give a link to the pages that contain them, because I can't see them.

 

[Markus Hanke]

maybe of the radial velocity with respect to time would be helpful.

You don't need the geodesic equations for this; total energy and momentum is conserved, so the radial velocity relation must be of the form

$$\displaystyle{\frac{\dot{r}}{2}-\frac{GM}{r}=const.}$$

The dot refers to differentiation with respect to proper time. The constant must be chosen so that it matches with your initial conditions. What this basically says is that the sum of kinetic and potential energy of the infalling body remains that same at all times.

Bear in mind though that this gives you radial velocity only; to get the full trajectory you need to consider the complete set of equations.

How's this for starters :

http://physicspages.com/tag/geodesic-equation/

 

[Bill_K]

If you're doing it in terms of Christoffel symbols, you're doing it the hard way. Take advantage of the symmetries of Schwarzschild, which imply the existence of several first integrals.

First of all, assume the orbit is in the equatorial plane, so θ is out of the picture. The three first integrals are:

L = r²(dφ/ds)
Γ = (1 - r_s/r) (dt/ds)
F = (1 - r_s/r)^-1(dr/ds)² + r²(dφ/ds)² - (1 - r_s/r)(dt/ds)²

where r_s is the Schwarzwschild radius and s is an affine parameter along the curve. L represents angular momentum, Γ represents energy, and F represents the norm of the velocity 4-vector. These quantities are conserved (constant) for the orbit of any test particle, whether it be massless or not.

For a particle coming in from infinity, choose s such that s = t at r = ∞. Then Γ = 1. Also for a particle which is massless, F = 0. Putting it all together gives

(dr/ds)² = 1 - (1 - r_s/r)L²/r²

This is the only equation you have to solve.

It's convenient to let b be the perihelion distance, i.e. r = b where dr/ds = 0
This determines L:

L² = b²/(1 - r_s/b).

 

[Markus Hanke]

If you're doing it in terms of Christoffel symbols, you're doing it the hard way. Take advantage of the symmetries of Schwarzschild, which imply the existence of several first integrals.

Good point :)

 

[George Jones]

There is also this thread:

https://www.physicsforums.com/showthread.php?t=347915

 

[NeedsHelpBadly]

There is also this thread:

https://www.physicsforums.com/showthread.php?t=347915

That's perfect for me, except they never finished finding the equation for the radial component of velocity.

 

[Bill_K]

That's perfect for me, except they never finished finding the equation for the radial component of velocity.

See post #7 above.

 

[NeedsHelpBadly]

See post #7 above.

Yes, I saw that, but I still have no idea how to integrate that, as it contains terms with r.

 

[Bill_K]

Well that IS the equation. If you're writing a simulator, I assume you're planning to do the integration numerically.

 

[NeedsHelpBadly]

Yes, but what do I put as the value of r, if the point of integrating the r component of velocity is to find r?

 

[Bill_K]

There's many methods of integrating differential equations numerically. They all share the feature that you start with an assumed value at an initial point, then use the DE to calculate the value at the next point, and so on repeatedly.

For example here's a video that explains Euler's method, which is one of the simplest.

 

[pervect]

Yes, but what do I put as the value of r, if the point of integrating the r component of velocity is to find r?

I'm not sure what your question is. Conceptually, you have some function r(s), where s is an affine parameter, and you have a differential equation for r(s) that says:

dr/ds = f(r)

See Bill K's post for f(r)

A very low order numerical approach would be to say that given r(s), r(s+delta) = r(s)+ delta*f(r). This comes from the definition of the derivative:

dr/ds = f(r) = [r(s+delta) - r(s) ] / delta

You know r(s), you assume a value for delta, and you solve this to find r(s+delta)

This wouldn't be recommendable in practice, the error would be too large. You'd want to use a more sophisticated and faster converging algorithm in practice. I hesitate to recommend a particular numerical algorithm at this point for several reasons.

An analytical approach would be to say that $\int \frac{dr}{f(r)} = \int ds = s$