It is already giving you one, but it's not along the normal directions you can feel: you are not moving at your normal free-fall speed that you "should" have in the Earth's curved spacetime. It's like a friction force, it's not moving you, it's actively stopping you from moving.
You're asking good questions. The reason standing on Earth gives you an upward push is that in fact this is a semantics game. What's really happening is that the upward push is what we call a "normal force", and it is entirely the result of interactions between the electrons and protons of the matter we and the Earth are made up of. The accelerations measured by accelerometers (including the ones in our ears) and by the pressure sensors in our skin are strictly electromagnetic in nature, but ultimately these are caused by gravity being a force indeed that is pulling us down, and it is the normal forces that stop us from accelerating further into the planet.
A lot of people are confused by GR. But really, gravity can be seen as any of these things:
- an effect that curves spacetime,
yielding accelerations when that
curved spacetime is mapped to
flat spacetime (think of a Mercator-
like projection of curved spacetime
to flat spacetime)
- a force that, when applied to waves
and matter (which... is standing
waves anyways) causes them to be
accelerated in such ways as to
follow paths indistinguishable from
the ones they would follow in
curved spacetime
It's really a lot simpler than some people make it out to be.
Consider the Schwarzschild metric \[ds^2 = -(1 - \frac{2M}{r}) dt^2 + (1 - \frac{2M}{r})^{-1} dr^2 + r^2 d\Omega^2\] -- its terms show you the distortions of space and time (independently, though related by the distance r to the center of the massive object) relative to flat spacetime. The first term shows you the distortion of time, and the second and third show you the distortion of space. I've yet to see a physicist put it that way, but that is exactly how it is. (This metric is a second derivative of spacetime, so it's not immediately obvious what the actual mapping between flat and curved spacetime is -- you have to do a double integral for that. However, what you get is the normal time dilation factor we know \[\sqrt{1-\frac{2M}{rc^2}] as the distortion of time, and... the same as the radial space expansion (that is, space expands, but only in the direction to/away from the massive body), + a three-dimentional angular (two angles) displacement.
It is often said that you can't tease out the space and the time curvature from spacetime curvature because they are intimately intertwined. But if you look at the Schwarzschild metric this is not really quite true. Yes, the first two terms have an r in them, so space and time curvature are very much interrelated, but they are still terms in an addition, with one term for time and one for space, so they can in fact be described separately, as indeed the Schwarzschild metric does.
How long until this upward a gives me a little upward v? What's the dt here?