<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>[mechanic]: Our dispute here seems to be whether one can state that "momentum has velocity" to mean anything other than the velocity a mass is moving at. Momentum is mass times velocity. I'd like to know how you define the "velocity of momentum".<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
There is one and only one velocity associated with momentum. The "velocity of momentum" is the velocity in the expression "momentum equals mass times velocity".
In physics (which usually has a different character than math), consider a particle ejected from a source mass and traveling to a target body. (Other scenarios are possible. The traveling entity might be a wave instead of a particle. I'm just using one example.)
Let the particle have mass m and velocity v. Then the momentum being transferred from the source to the target is mv, and propagates at speed v. The target body may already have a mass M and a velocity V. When the particle impacts the target and deposits its total momentum there, the target acquires the momentum of the particle and ends up with a momentum that is the vector sum of MV+mv.
This is all elementary mechanics, mechanic. (Sorry, I had to say that.)
<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Obviously, momentum is velocity scaled by mass. The time rate of change of momentum is force. How do you define the "speed of force"?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
If there is a single particle, the target feels no force, but only an impulse. If there is a series of momentum-carrying particles, then there is a continual change in the momentum of the target body. As you say, the time rate of change is called "force". Obviously, only the momentum carriers have a speed. It is not even dimensionally consistent to speak of the speed of force. Nonetheless, in common parlance in physics, the speed of the momentum carriers or the speed of the force generators is said to be the speed of the force.
<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Force is mass times acceleration. According to your example, force "has acceleration". Force is acceleration scaled by mass. What is then the "speed of acceleration"? The simplest answer is to say that the "speed of a force" is the speed resulting in a given acceleration.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
Divide force by target mass and you get target acceleration. The "speed of acceleration" is the same as the "speed of force", which is the same as the "speed of momentum", all of which are mass-independent. These are common conventions, especially in celestial mechanics. You can see these concepts in operation in, for example, <i>Fundamentals of Celestial Mechanics</i> by J.M.A. Danby.
<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>In this context, in pushing gravity the "speed of gravity" is the speed of gravitons. In GR geometric interpretation, there are no forces and "speed of gravity" is the speed at which geodesic paths are formed due to space-time curving and that's equal to the speed of light. Do you agree Tom?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
In a highly technical sense, there are no forces in gravitation. But there are 3-space accelerations, so one can invoke Newton's second law (F = m*a) and speak of a "force" without ambiguity, even if it is a purely mathematical concept. I don't care for that use of "force" myself because it implies that the "force" of the Earth on a dust grain and the "force" of the dust grain back on the Earth are equal -- somehting that is anti-intuitive to say the least. By sticking to accelerations, everything behaves in accord with our intuitions. And in astrophysics, only accelerations are observable, but never forces.
The geometric interpretation has 3-space acceleration without any ambiguity whatever. That is what an orbit is. It is what the equations of motion describe. And we could not compare GR to observations (which are all made in 3-space) if we could not convert solutions to the Einstein equations into 3-space equations.
In GR, the accelerations of bodies depend on the true, instantaneous positions of source masses. No aspect of gravitational force or 3-space acceleration -- <b>none whatever</b> -- propagates as slowly as lightspeed. In GR, it is all instantaneous interactions.
After orbits of bodies are established, their accelerations disturb the "space-time medium" in which they are embedded every so slightly. The space-time medium, the light-carrying medium, the gravitational potential field, and the "elysium" of Pushing Gravity are all one and the same. These small disturbances are gravitational waves, and propagate through the medium at its natural wave speed, the speed of light. These waves are so weak that they cannot accelerate any other body. But they act like a slight drag force on the body whose acceleration created them.
Does that help clear up the physical relationships involved here? -|Tom|-
