Schwarzschild Metric & Satellite Orbits: A Question

[TrueBlue1990]

Thanks in advance - this problem has been bothering me for a while!

I'm working with an unpowered spaceship orbiting a large mass M. The orbit is circular and it is following the geodesic freely. It has an orbit radies of r = R.

My question is this. The metric of the space-time curvature is the Schwarzschild metric:

ds^2 = - (1 - 2GM/(r*c^2) )*(c^2)dt^2 + [(1 - 2GM/(r*c^2) )^-1]*dr^2 + (r^2)dθ^2 + (r^2)*(sinθ^2)d∅^2 -(1)

But I keep seeing references to ds^2 = - (c^2)*d(tau)^2 -(2)

I understand the angular terms and dr disappear as the d(tau) means we are observing the orbit from the reference frame stationary with the satellite. But I can only how equation (1) = equation (2) in a Schwarzschild curved space if r (distance between mass and satellite) tends to infinity. But, in the case of the satellite's reference frame r=R.

I will be eternally grateful to anyone that can shed light on my error of understanding. Many thanks.

 

[Mentz114]

I would like to answer but I have no idea what you mean. The circular orbit has

$$\frac{dt}{d\tau}=\frac{\sqrt{r}}{\sqrt{r-3\,m}},\ \ \frac{d\phi}{d\tau}=\frac{\sqrt{m}}{r\,\sqrt{r-3\,m}}$$ in case that helps.

Remember that the proper-length s in the metric is also cτ, so ds²= c²dτ²

[Edit]As Bill_K points out below this is true up to a sign, depending on the metric signature.

 

[Passionflower]

Also r > 6m for orbits to be stable.

 

[Bill_K]

But I keep seeing references to ds^2 = - (c^2)*d(tau)^2

This is just a matter of notation. ds² is called the spacetime interval. It's positive for a spacelike distance and negative for a timelike one. For a timelike interval it's more convenient to use in place of ds² the proper time. So you pull out a minus sign, and pull out a factor of c² to give it the right dimensions. Otherwise ds² and dτ² are interchangeable.