How to calculate redshift from the schwartzchild metric

[BiGyElLoWhAt]

Homework Statement

I'm doing a project on the redshift from a star system (I chose a binomial system because why not). I might be going a little overboard using topology to calculate redshift, but whatever. First off, can I just treat a binomial system as the superposition of 2 sources which result in the schwartzchild metric? By that I mean one star isolated from the other would have the sc metric, as would the other, and the resulting system would have a similar metric, but there would be a path along 2 metrics to consider. So the topology from metric 1 + the metric from topology 2 along a path. I think this would work, but I'm not sure. I'm also not 100% on how to put this into a computer program.

Homework Equations

Schwartzchild metric.
$c^2d\tau^2= (1-\frac{r_s}{r})c^2dt^2 - (1-\frac{r_s}{r})^{-1}dr^2 -r^2(d\theta^2 + sin(\theta)d\phi^2)$

The Attempt at a Solution

I mean... I'm not sure what to put here.
First off, the left hand side is the proper time (of a photon), a differential element of which should be zero, I believe. Are the differentials on the RHS for an observer? So dr, dt, etc would be traced from the surface of emission to my observer? I'm tempted to assume that dt is a function of dr, or vice versa, along with theta and phi (zero, since I'm working in 2 spatial dimensions).
So correct me, please, but what I think I'm working with is something to the effect of:
$0 = (1-\frac{r_s}{r})c^2(dt(r,\theta))^2 - (1-\frac{r_s}{r})^{-1}dr^2 -r^2d\theta^2$
Or is this not useful (or even correct)?
Would it be better to solve for dt?

The reason I want to do it this way, is the only equation I've been able to find for redshift is either the Newtonian limit, or the limit as r-> inf. I want the redshift over a finite spatial distance.
I'm probably missing some things. So feel free to point them out.

**Edit
Ok, I suppose the RHS wouldn't be the coordinates for the observer in that manner, explicitly. However, the coordinates of the observer would be the end point of the path.

 

[andrewkirk]

I think that, unfortunately, you can't use superposed Schwarzschild metrics for this calculation. As I recall, a key step in the development of the Swarzschild metric is the assumption that the spacetime is static and rotationally symmetric around the centre of the star, and neither of those conditions will hold in a binary system, either in relation to the centre of mass of the system, or the centre of either of the stars.

might be going a little overboard using topology to calculate redshift

I don't understand what you mean by using topology. Do you mean using the gravitational equation - which is differential geometry, rather than topology? So far as I know, that is the only way to derive the formula for gravitational redshift.

 

[BiGyElLoWhAt]

Yes, sometimes I mix up the two terms and use them interchangeably. Sorry. Is there a metric that can be used for this type of problem?

 

[andrewkirk]

Is there a metric that can be used for this type of problem?

Not an exact one. Follow this link to an earlier question on this. pervect gives some links to papers that use approximate metrics. But it sounds like even those are horribly complicated. Perhaps numerical solutions are the most accessible, practical way to estimate geodesics for a binary system.