H_A_Landman said:
It's really disheartening to see people still getting this exactly wrong and being loudly insistent that they are right. And other people upvoting them.
It's really disheartening to see someone talking about one particular solution of the Einstein Field Equation, described in one particular coordinate chart, as though it were "GR" and "gravity" without qualification.
H_A_Landman said:
In the Newtonian (weak-field, low-speed) limit of the Schwarzschild metric of GR
See the bolded addition I made above. It makes a huge difference. The Schwarzschild metric is just
one solution of the Einstein Field Equation. There are many others. Most of them do not even have a simple concept of "gravity" as arising from a Newtonian potential.
H_A_Landman said:
the reduced metric in Schwarzschild coordinates
Again, see the bolded addition I made above. It makes a huge difference.
H_A_Landman said:
consists of flat Minkowski spacetime (which causes no gravity by definition)
Depends on how you define "gravity". By at least one fairly common definition (the one that underlies the equivalence principle), there is "gravity" in the non-inertial rest frame of a rocket accelerating in a straight line in flat spacetime.
H_A_Landman said:
plus a time-curvature term, proportional to the Newtonian gravitational potential, that represents the time dilation field.
The ##- 2M / r## term in the ##g_{tt}## metric coefficient in Schwarzschild coordinates, which is what you are referring to here, is
not a "time curvature" term. Curvature is expressed by the Riemann curvature tensor, not the metric. This term can be thought of, in this particular solution, in these particular coordinates, in this particular approximation, as arising from the Newtonian gravitational potential. But that just means that spacetime curvature, in its manifestation in this particular solution, in these particular coordinates, in this particular approximation as Newtonian gravitational potential, causes both time dilation for stationary observers (note that qualifier, it also makes a big difference--"time dilation" is observer-dependent)
and "gravity" (more precisely, the fact that radial geodesics converge towards the central mass). It does
not mean that time dilation causes gravity.
H_A_Landman said:
If you back out of taking the low-speed limit (so it's only weak-field), then there is a space curvature term that ranges from zero (at zero speed) to equal in size to the time-curvature term (at the speed of light).
This claim, with appropriate caveats (just as for ##g_{tt}## above, the appearance of a term proportional to ##2M / r## in the spatial terms in the metric is not "space curvature", since that, as noted, is expressed by the Riemann curvature tensor, not the metric), is also only true for this particular solution, in these particular coordinates, in this particular approximation (weak field).