[There has been much confusion and misinformation in this thread, as

well as a few sensible responses. I will answer the original question,

but cannot follow the many side issues raised.]

*Post by Ilya Kraskov*I've been trying to understand how gravity acts like a force.

Our best model of gravity is General Relativity. In GR, gravity is not a

force, and does not "act like a force". What we call "gravity" is an

aspect of the geometry of spacetime, as I'll discuss below.

*Post by Ilya Kraskov*On the net are many rubber trampoline examples [...]

These are very poor examples, as they attempt to "explain" gravity with

an analogy that requires gravity -- how else does a ball remain on the

rubber sheet, and fall down into a depression?

*Post by Ilya Kraskov*I realize EVERYTHING is moving [...]

For concepts like "moving", one must ALWAYS specify the coordinates one

is using. After all, every object can be described as at rest in some

set of coordinates (e.g. its instantaneously co-moving inertial frame).

Sloppy thinking like "everything is moving" leads to contradictions and

confusions. You MUST learn how to be more precise in thought and word,

or you will never be able to understand subtle concepts like those in

modern physics.

*Post by Ilya Kraskov*But if I think about gravity in the frame of reference from the

earth I'm not moving with respect to the earth. Am I? I'm standing

still with respect to the earth, aren't I?

Hmmm. Note that in physics, "frame" invariably means "inertial frame",

or "locally inertial frame"; the earth is rotating, and thus not at rest

in any such frame. You are not describing "motion" and "at rest" with

sufficient precision to understand the underlying concepts -- that is

the root of your lack of understanding.

[You are not alone in that around here.]

Let us start from the basics. Throughout this discussion I am speaking a

bit loosely, and using approximations that I don't always mention.

In GR, massive objects affect the geometry of spacetime, and that

geometry affects the motions of all objects. So let's start with the

simplest situation: the motion of a test particle. A test particle is a

pointlike object with nonzero mass, but a mass that is so small that its

effects on the geometry can be neglected [#]. For purposes of

discussion, consider a universe containing just a single massive object

that is spherical, with mass M and radius R, made of solid material

(i.e. our test particle cannot penetrate it); label it "M" for

discussion. Outside M, the entire universe is vacuum [@] (except for the

test particle).

[#] If the object in question has a mass large enough to

significantly affect the geometry, the analysis is far

too complicated for a discussion here. In practice this

means the mass of the test particle is much smaller than

the masses of other objects in the physical situation;

for instance a satellite or human compared to the earth.

[@] I have to make such approximations and assumptions

to keep this discussion reasonable in length.

In GR, given the above assumptions, the test particle follows a geodesic

path through spacetime. The geometry of spacetime for this case is

known, and outside M it is called "Schwarzschild spacetime" after the

man who first derived and described it. Applying GR, one can calculate

the geodesic paths of this geometry; they are all affected by M. Of

course which path our test particle follows depends upon the initial

conditions we assume for it. Let us consider three such paths, relative

to approximately-inertial coordinates in which M is at rest:

A) The particle is initially at rest in these coordinates, at a distance

r from M's center, with r > R. This geodesic remains on a radius of the

geometry, accelerating towards M; if M is the earth and r is less than a

few hundred km above the surface, the acceleration is 9.8 m/s^2. Note

this is the same path that Newtonian mechanics predicts, using its

gravitational force, but here the path is purely a consequence of the

geometry induced by M; no actual force is involved. (The particle will

eventually hit M, but we stop our analysis before then.)

B) The particle is initially located at radius r from M's center, with r

*Post by Ilya Kraskov*R, but has a sideways velocity such that the particle's path is a

circular orbit around M. This is obviously a different geodesic than in

(A). Again this is the same path that Newtonian mechanics predicts,

using its gravitational force, but here the path is purely a consequence

of the geometry induced by M; no actual force is involved. (Here the

particle orbits M forever.)

C) The particle is initially at rest on the surface of M. Since the

particle cannot penetrate M, the surface pushes upward on the particle

with precisely the right force to keep the particle at rest on the

surface. This is not a geodesic path, and the upward force is

proportional to the mass of the test particle -- it is precisely the

force required to divert the particle from its geodesic path of (A) and

remain at rest on the surface of M. The force on the particle is

determined by the geodesic path it would follow if it were not being

pushed upward by the surface, so the force depends on M, R, and the mass

of the test particle (just as in the gravitational force of Newtonian

mechanics).

In (A),(B),(C) I described the paths using approximately-inertial

coordinates in which M is at rest. I could have used locally-inertial

coordinates in which the test particle is at rest -- in these

coordinates "gravity disappears" because any other test particles nearby

move in uniform straight lines relative to these coordinates. In essence

such coordinates move with the test particle as it falls. So in GR

gravity is not a force, but it is similar to the "fictitious forces" of

"centrifugal and Coriolis forces" -- they all induce accelerations

relative to non-inertial coordinates, and they all disappear in

locally-inertial frames.

[Important note: coordinates at rest on the surface of

the earth are not inertial frames. Locally-inertial

frames at the surface are all accelerating downward at

9.8 m/s^2.]

So near earth, GR predicts that the behavior of objects much smaller

than the earth ("test particles") is essentially the same as predicted

by Newtonian mechanics. But GR's explanation is quite different:

spacetime geometry rather than Newton's gravitational force. As

Newtonian mechanics has been experimentally validated so thoroughly near

earth, if GR did not predict the same behavior it would have been

rejected long ago.

[Another note: I used "approximately-inertial" coordinates

above; how is that possible? Consider (B) above, with

M the earth and the ISS as the test particle. The ISS

path through spacetime is a helix with axis parallel

to the time coordinate, period 90 minutes, and radius

350km+6371km; this is a very long and thin helix with

period along its axis > 240,000 times longer than its

radius, which is very close to a straight line. (That

aspect ratio corresponds to a human hair about 6 meters

long.)]

General Relativity is a fascinating subject that is VASTLY more

complicated than I have alluded to above. To get a general understanding

of its basic concepts I recommend:

Geroch, _General_Relativity_from_A_to_B_.

This is a non-mathematical introduction to GR.

[Perhaps later I'll describe how "gravitational force" in GR is only

distantly related to "gravitational time dilation" -- they are simply

different manifestations of the same aspect of GR.]

Tom Roberts