This issue is often referred to as the "initial velocity problem" or the "problem of motion." It arises because the curvature of spacetime can cause the direction of a particle's velocity to deviate from what would be expected based solely on its initial conditions. In other words, even if a particle is initially following a geodesic, the curvature of spacetime can cause its trajectory to deviate over time.
Geodesic motion
In GR, geodesic motion describes the path that a particle follows in the presence of a gravitational field. A geodesic is the equivalent of a straight line in curved spacetime.
The issue of initial velocity
In the context of the initial velocity issue in the deviation of geodesic motion, we consider the motion of a test particle in the vicinity of a massive object, such as a star or a black hole. According to general relativity, the presence of this massive object curves the surrounding spacetime, which in turn affects the motion of nearby particles.
When discussing the initial velocity issue, we are concerned with the conditions that must be satisfied at a given point for a particle to follow a particular geodesic. In other words, if we want a particle to follow a specific trajectory (geodesic) near a massive object, we need to determine the appropriate initial conditions, including the particle's initial position and velocity.
The challenges
The initial challenge was to get the initial velocity dependence out of the equations so the predictions match reality in the theory i.e the standard geodesics of 4D and not the ever changing ones via the initial velocity parameter. The sution is the time additive path or d=ct Einstein introduced (Wheeler).
The other challenge arises because the curvature of spacetime affects the motion of the particle in a nontrivial way. Unlike in Newtonian physics, where the initial position and velocity fully determine the subsequent motion of a particle, in general relativity, the initial conditions alone are not sufficient to uniquely determine the particle's trajectory.
The opening statement is, simply, wrong. Initial conditions are defined at a point in spacetime, not on any curve.
The geodesic equation has a unique solution, defined by the initial conditions, so the statement that it doesn't is, simply, wrong.
What happens in black hole spacetimes, with a space-like singularity, is that the spacetime isn't complete. So there exist solutions to the geodesic equation, that hit the singularity in finite proper time-these define the interior of the black hole; and solutions that don't-these define the exterior of the black hole.
Stam Nichols,
I' m not talking about an application of GR to a specific case - i' m talking about the derivation of GR. At one stage, Einstein faces the so-called "initial velocity issue" in the formulation of his theory
This is where I'm referring
Once more: There's no such problem. In any given curved spacetime, without a singularity, the geodesic equation has a unique solution. The only case where it might not is if the geometry admits closed time-like curves.
Einstein didn't realize that general relativity-like any theory, in fact-is completely defined by its symmetries (which for general relativity is the imvariance under diffeomorphisms) and the requirement that the equations of motion can't contain derivatives of order higher than two. These results were deduced by Emmy Noether in 1918, though it did take time for them to be fully understood. There's no excuse now to confuse the history of physics with physics.
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