After a thorogh discussion about black holes and the Schwarzschild metric, we will now look at how objects move around a black hole, or any spherical, uniform mass.
How might we obtain an equation of motion?
In a system without any external force, objects follow the "least" path. On a flat surface, this is simply just a straight segment connecting two points. In the 4-dimensional spacetime, this least path (or the geodesics) describes the motion of the object. In Minkowski space, the geodesics is a line, with a time dimension.
What about curved space?
Consider a spherical surface--a curved 2-dimensional space. The shortest path now isn't a straight line. It is in fact a great circle connecting the points, that is, a circle with its center at the center of the sphere, and its radius equal to that of the sphere. In a more complicated curved space, the geodesics is what makes the spacetime interval (as described by the metric) take its extreme value. Sounds like Fermat's principle, eh?
A great circle
For any metric given to us, we can calculate the geodesics using a method known as calculus of variations. In fact, we could write the interval as:
L is any function, and λ is a parameter factored out of L. (It is named the affine parameter). Note that the dot on x, y, z signifies derivative with respect to λ. To give the extremal interval, it can be shown that x,y,z satisfies the Euler-Lagrange equations, or:
(The Euler-Lagrange equations are commonly used in classical mechanics with the notion of "Lagrangian")
Now with all these tools, we can finally look at the metric!
We take the affine parameter in this case to be the proper time, or the time as measured in the object's own frame. Recall the definition of proper time in the schwarzschild metric:
that is, the spacetime interval when dr = 0. This means we could write:
and hence (with L = 1):
taking the negative square root of ds, we obtain the following information of L:
L is now dependent on the 4 variables: t, r, φ and θ. All that is left is for us to plug in the equations, and Voila! so easy~~
The 4 variables give us the for equations. For t:
The lack of dependence on t in L shows that there is in fact a conserved quantity：
Where E/m is the energy per mass. Expanding, we have:
Why do we call it E/m? The argument is that, for large r, this reduces down to the Minkowski metric, and for the Minkowski metric,
for φ，θ, the same calculations holds. We define the following:
to be the angular momentum per mass. this is simpler to understand than E/m. (Simply recall the definition of θ in a spherical system)
The existence of the sinθ term means there isn't a conserved quantity for θ. We write:
r's equations of motion are quite difficult to obtain using the Euler-Lagrange equations. Instead, we use the definition of L that L = L^2 = 1：
When looking orbits in a single plane, θ=π/2, so that it reduces to:
Now we plug in the conserved quantities, the energy per mass and angular momentum per mass:
Basic algbera to separate the variables give:
Finally, the art of guessing allows us to plug back all the c's (also known as dimensional analysis), so that finally we obtain the equation:
And this is our equation of motion.
No aspects of the black hole are actually discussed during our work--these equations fit any spherically uniform mass. To solve explicitly for r is very difficult indeed, so we use approximations...
Nope. Sadly, no approximation exists. We will instead use the method of effective potential to discuss all of the possible orbits. Finally, an interesting case to note is the light orbit. Newtonian physics gives no light orbits. Light goes in a straight line. But in general relativity, light in fact is curved (and red-shifted) near masses. Such facts has already been confirmed with gravitational lensing, but what's more interesting is the existence of a light circular orbit.
...And all of that is for next time!