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If you have a test particle at location 2 radii from centre M (Earth) where the acceleration of gravity is (off top of head) in the region of 4.25m/s^2......where you see /s^2, what length of second^2 is that?The answer is:/standard second^2......where a standard second is a measure of time derived at ground level Earth.My suggestion of a 3rd time dilation of the g-field - that affects the time directly for m=0 only, but has an indirect affect upon m by affecting m's motions in the m=0 space of the g-field - has longer seconds at r=2radii.Instead of calculating acceleration held relative to the standard second, if one were to calculate the acceleration at r=2radii held relative to the 3rd time dilation length of second at r=2radii then the measure of acceleration held relative to that longer second would be in the region of 9.807m/s^2, as opposed to holding the acceleration at r=2radii relative to the standard second where the acceleration would be 4.25m/s^2.The acceleration at any r from M (Earth), as measured held relative to the length of second at that r, will always be in the region of 9.807m/s^2.So - when one takes the measure of acceleration at each r from M as per held relative to the standard second, where near Earth is 9.807m/s^2, and by r=2radii the acceleration has decreased to 4.25m/s^2, this can tell one by how much a 3rd time dilation second has become longer at each r.

Consider a sine wave. Nothing to do with light or gravity. Forget those. If the wave length (second length) is constant we can move (at constant speed) along the wave (metres of distance) marking it off at regular intervals. Everything will be constant and cyclic. Now if we start again but this time continuously vary the intervals (seconds) at which we mark off the wave (metres of distance) using a function (3rd time dilation) to determine the increase or decrease in the steps (length of seconds) we can see how this can make it appear that something (acceleration) has changed. If we were blissfully unaware that our function (3rd time dilation) existed then we may come to the conclusion that it was the wave (metres of distance) that was changing.

Why not?It's not as though the concept is difficult Jeff...If m is travelling metres of space towards M where at every radius seconds are becoming progressively shorter, m will experience acceleration.All I have outlined is a means of converting the physics measurement of m/s^2 acceleration of the g-field into a time dilation related phenomenon.The only difficulty you will no doubt encounter within this simple mathematical conversion - 'cos let's face it, the maths of that conversation, although beyond my capabilities, are not difficult - is related to your pre-conceived understanding of time dilation....where taking on board new ideas is more difficult for some than others...

Ok let's take you at face value. How does your vector space operate with respect to the gravitational field? Even simple equations will do.

Consider this. There cannot be quanta in the gravitational field if it is simply a case of mass curving spacetime that causes acceleration. This would imply a continuum.

Tell that to NIST who have conducted tests and found that SR effects are observed at speeds of less than 30mph.And GR effects at 1 metre altitude.As said you'd be advised to watch the program I suggest and listen to what the professionals say before proceeding with this line of discussion.

Is there a vector space that can be used with linear combinations that is representative of a non-linear space such as that of the gravitational field? If this exists can it be formulated as an energy vector space?