Re: [Phys-L] gravitational waves

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding JD's quarrel with the trampoline model/analogy for GR.

> I'm not convinced.
>
> It seems to me that the trampoline model qualitatively
> represents some sort of field that mediates an interaction
> between the particles.  This field is a real physical thing with
> its own dynamics.  So far so good.
>
> However, things go downhill rapidly after that.

I think I'm mostly on JD's side in quarrel.  Over and above the problems JD already mentioned in his post there are other serious ones as well.  For instance, often diagrams of the trampoline model show gravitating bodies sitting on top of the trampoline and having them locally distort the fabric there.  Such diagrams can be very confusing to the uninitiated who don't already know what's going on.  Such diagrams make it look like there is some sort of background gravitational field orthogonal to the spatial fabric pulling the gravitating bodies down into the fabric and causing the distortions.  Also, putting such 3-d objects outside of the 2-d fabric into the 3rd embedding dimension is a good way to confuse people who are already uncertain as to which dimensions are which in the analogy, i.e. what is a 2-d model for space or spacetime & what is the significance of the orthogonal embedding dimension in the analogy.

But my biggest quibble with the analogy/model is that it makes it look like *Newtonian* gravitational forces are something that fall out of the *spatial* curvature caused distortions of space (when presented as a distorted map in flat space).  In fact, in the Newtonian limit, the curvature of space, per se, is quite irrelevant because it is so tiny.  Rather the curvature in the way the *time-like* dimension of space-time is incorporated into the manifold in a spatially locally differential manner is what is responsible for Newtonian gravitation.  IOW, the Newtonian gravitational forces come from locally varying amounts of gravitational *time* dilation.  But the model doesn't even describe or address the issue of the gravitational time dilation at all as it completely suppresses the time-like dimension.  Admittedly the absolute amount of distortion due to curvature in the time-like dimension (i.e. gravitational time-dilation) is comparable in intrinsic magnitude to the spatial distortions due to purely spatial curvature, but those distortions appear to higher order in 1/c^2 in the slow motion Newtonian limit where gravitating massive bodies never move at speeds approaching an appreciable fraction of c.  So it is only the effect of gravitational time dilation that is responsible for Newtonian gravitational effects.

BTW, a nice way to extract Newtonian gravitational effects on a freely falling object (with non-zero mass) is to realize that the Newtonian Lagrangian functional for it is simply -m*c^2 times the differential amount of local relative gravitational time dilation (from the sqrt of the coefficient of the time-like coordinate in the GR line element for the square proper time for the object) relative to time kept by a clock at spatial infinity (assuming all gravitating masses are spatially localized). The factor of c^2 in the unit conversion factor between relative dilation and the Lagrangian's energy units is what amplifies the tiny amounts of time dilation into something that has significant Newtonian forces in that limit.  Since the spatial contribution to the line element is of higher order in 1/c^2 than the temporal one the spatial distortion/curvature effects tend to drop out in the slow motion regime.

Note if one wants to see how GR makes a *light ray* deflect as it passes a gravitating body then *both* the local time dilation along the path *and* the spatial distortions show up with equal contributions to the overall effect.  This is because the light ray is sampling equal amounts of space and time as it moves on a null geodesic at 45 deg.  But the slow speed Newtonian bodies are hardly sampling any space for the amount of time their world lines cover as they are directed almost completely parallel to the time axis, and they are far more sensitive to just the temporal distortions.  If one only included the temporal effects on the light ray path one would find a predicted deflection of 1/2 of the actual amount predicted by GR and the experimentally observed results.

David Bowman