publius

2006-Aug-10, 05:31 AM

Spurred on by the question of gravitational lensing, and how, according to Maxwell in gravitational field as derived by Landau & Lifsh-itz, one can see the gravitational field as a "medium" of increasing permeability, I arrived at what to me is a rather strange conclusion.

For a an object that starts a free-fall far away, but at relativistic initial velocity, *it will appear to slow down*, not accelerate from a stationary frame watching it*. From that frame, the coordinate speed of light decreases with depth in the field. But nothing can appear to be going faster than the local speed of light. So if light appears to slow down, then a relativistic free-faller would have to slow down as well to keep its speed below the coordinate speed of light.

We all know that an object free-falling toward a BH will appear to stop at the event horizon. I had never thought about the consequences of this too much, but at the event horizon, the coordinate speed of light goes to zero. Hence everything has to appear to stop.

So, for a free faller starting out at zero velocity at infinity, it will have accelerate, but at some distance it will have to appear to start slowing down. But this has to depend on the initial velocity, as I just realized. If it starts out at nearly light speed, it cannot accelerate much before it would exceed the coordinate speed of light.

Above, I'm thinking about a straight line radial free fall, L = 0. If we have angular momentum for a relativistic initial velocity, gravity still appears attractive in the directional sense, pulling the object toward it, yet its (scalar)speed must decrease as it falls into the gravity well!

And that's my sanity check, here. Is this correct or am I messing up big time somewhere?

-Richard

For a an object that starts a free-fall far away, but at relativistic initial velocity, *it will appear to slow down*, not accelerate from a stationary frame watching it*. From that frame, the coordinate speed of light decreases with depth in the field. But nothing can appear to be going faster than the local speed of light. So if light appears to slow down, then a relativistic free-faller would have to slow down as well to keep its speed below the coordinate speed of light.

We all know that an object free-falling toward a BH will appear to stop at the event horizon. I had never thought about the consequences of this too much, but at the event horizon, the coordinate speed of light goes to zero. Hence everything has to appear to stop.

So, for a free faller starting out at zero velocity at infinity, it will have accelerate, but at some distance it will have to appear to start slowing down. But this has to depend on the initial velocity, as I just realized. If it starts out at nearly light speed, it cannot accelerate much before it would exceed the coordinate speed of light.

Above, I'm thinking about a straight line radial free fall, L = 0. If we have angular momentum for a relativistic initial velocity, gravity still appears attractive in the directional sense, pulling the object toward it, yet its (scalar)speed must decrease as it falls into the gravity well!

And that's my sanity check, here. Is this correct or am I messing up big time somewhere?

-Richard