re - If I am on the gravitating mass will I experience an increase in weight when the object moves vs when it is stationary? - Yes. Suppose you are standing on a large sheet of mass iin the frame of reference in which the sheet of mass is at rest. This "sheet is at rest" is a valid statement. If you now hop in a car and drive very fast then you will weigh more than when the car was stationary. The "speed" is the relative velocity of you and the rest frame of the sheet. However there are other situations such as a a sheet above you and a sheet below you which are moving with respect to each other. In this case there is no frame in which each sheet is at rest. However there is a frame in which the total momentum is zero. I haven't calculated this one but I've seen some comments on it. Weird motion of test particles in the field!! I'll dig it out.

re - But again, moving with respect to what? : Maybe I can ask it this way; Can there ever be a : situation where I feel an increased in gravitational : pull because of the velocity of the gravitating : object I am on?

Here's something to think about which may help you understand.

Dopper shift (no gravity here) - The frequency of an electromagnetic wave depends on your frame of reference. In different frames of reference which are in relative motion you will reckon different frequencies.

Suppose I say "In frame O there is a beam if light, frequency = F, moving in the x-direction. You are in frame O' moving in the +x direction. What is the frequency F' in your frame?"

Questions to ask yourself

(1) Do I need to know about what the source is doing to answer this question?

(2) Does the frequency depend on speed?

(3)Speed with respect to what?

Cheers

Pete