Orbits & Trajectories
A trajectory can be described as the path taken by an object, which has both radial [vR] and orbital [vO] components of velocity. In part, the previous discussion has reviewed some of the implications associated with a mass [m] free-falling towards mass [M] along a radial path based on the Schwarzschild metric. What is observed in this situation depended on the position of the observer, which we aligned to our basic model:
- A distant observer [A] far from any gravitational mass [M].
- A shell observer [B] at a stationary distance from the gravitational mass [M].
- A free-falling observer [C] accelerating towards the gravitational mass [M].
While this model appears to be generally useful in respect to the free-fall case, it needs to be modify in order to discuss orbital paths and composite trajectories involving radial and orbital components. So, we might begin by considering a small mass [m] orbiting a much larger mass [M], such that we may assume the orbit to be circular around the larger mass [M].
We can label the frames of reference in a similar fashion to the basic model introduced earlier, although orbital paths were not really represented. Therefore, it should be noted that [C] is no longer free falling towards mass [M] under gravity, but rather orbiting it at some coordinate radius [rC]. Simply to maintain a constant perspective between [A] and [C], we might placed observer [A] centrally above the rotating orbit of [C]. Observer [B] is positioned at the same radius [r] as the orbit of [C], so as to represent an equivalent position within the gravity well, but remembering that [C] will also have to account for the effects of its orbital velocity [vO]. As such, we now have a basic framework in which to discuss the effects of gravity in the context of general relativity, where the mass [M] has curved spacetime such that the shortest geodesic path through spacetime becomes a circular orbit. However, this does not mean that we cannot continue to use the visualization of a gravitational force [F] between two masses [M] and [m], as defined by Newtonian physics:
In the context of , a stable orbit between objects of mass [M] and [m] requires a balance between the inward force of gravitation to that of an outward `centrifugal force`. While this configuration typically leads to elliptical orbits, as defined by Kepler’s laws, we can ignore this complexity by making the assumption that mass [M] is very much larger than mass [m]. As such, any circular orbit must balance the force of gravitational attraction defined in , which leads to the definition of the centrifugal force:
If we re-arrange , we see that a given orbital velocity [vo] will maintain a stable orbit of radius [r] around mass [M]:
However, the diagram on the right suggests that we are still conceptually dealing with radial [vR] and orbital [vO] components of velocities, even though the radial component is being held in check to maintain a stable circular obit. We can also see from the diagram that the orbital velocity [vO] can be described as the instantaneous tangential velocity at any point, which is ‘forced’ to follow a curved path in Newtonian space, although relativity would describe this as the shortest geodesic path in curved spacetime. However, if we wish to continue to cross-reference Newtonian concepts, we need to translate the orbital tangential velocity [vO] into its angular velocity [ω]:
In , we see both the conversion between the angular and linear forms of velocity, which is then linked to the idea of a rate of change of an angular displacement [dφ] with time [dt], as used in the Schwarzschild metric. Now, at this point, it might also be useful to express the total energy [Et] of a closed system in terms of the kinetic energy connected to the velocity of mass [m] in respect to the potential energy connected with the gravitation of mass [M]:
Of course,  is predicated on Newtonian concepts, which we might guess are subject to modification when reviewed in the context of the Schwarzschild metric, although it might not be obvious as to how energy is associated with this metric, at this stage. Therefore, let us address this issue by considering an equatorial orbital, where [θ=π/2], which allows us to reduce the Schwarzschild metric to the form:
If we divide through by [dτ2], we then need to consider the solution in terms of the rates of change of [dr] and [dφ] with respect to the proper time [dτ]. The subscripts attached to the various variables in  are again used to simply identify the distant [A] and orbiting [C] frames:
At this point, we can make some substitutions to highlight that  is making reference to both the radial and orbital components of a trajectory in spacetime affected by mass [M].
If we now substitute the forms in  back into :
The next steps are primarily re-arranging the form of  to separate the various components of velocity, i.e. [vR, vO] or [c] onto one side:
Now remember, we started the process of modifying the Schwarzschild metric in  so that we might develop some form of energy equation that could be compared to the Newtonian form in . We might finally do this by multiplying the result in  through by the factor [1/2m]:
What we might first recognise is that  is now an energy equation based on the Schwarzschild metric associated with the timeframe [dτ], which we might initially associate with the onboard observer [C]. We might also wish to have an adjacent comparison with the previous classical energy equation:
By comparing  and , we can see that there are comparable kinetic energy [Ek] terms associated with the radial [vR] and orbital [vO] components of the trajectory in both  and . There is also a term that looks like the potential energy [Ep] associated with mass [M], although it now appears to be linked to some relativistic factor. However, there are also some other expressions which we might initially assume align to the total energy [Et].
So how do we solve  for a given radius?
Before we can really address this question we need to consider some
basic implications of the conservation of energy and momentum associated
with a given trajectory and then outline the concept of effective potential,
both for the classical and relativistic systems.