# Does curved spacetime arise from inhomogeneity of gravitational field?

In general relativity textbooks such as Sean Carroll's Spacetime and Geometry, there is often a line of reasoning that goes like this:

1. Strong equivalence principle states that free falling frames are equivalent everywhere and are locally inertial.

2. Thus if gravitational field were homogeneous (say, there is a constant gravitational field of g pointing in a particular direction, everywhere in space), we can treat the free falling frame as a global inertial frame.

3. However, since gravitational field is not homogeneous (e.g. near the earth, gravitational field is roughly radial), we cannot construct a global inertial frame. Thus we have to patch local inertial frames together to form a curved spacetime.

I am very confused by this reasoning. In the case of homogeneous gravitational field, the free falling frame would be defined by observers who undergo the same proper acceleration everywhere. The frame defined by these moving observers are clearly not inertial because their relative proper distance changes due to special relativistic kinematic effects. In fact, I believe that the only way to set up observers with constant (but not homogeneous) proper acceleration AND constant relative proper distance is to construct Rindler observers.

So are the textbooks wrong? Or am I missing something?

Does curved spacetime arise from inhomogeneity of gravitational field?

Not quite. It actually arises from the inhomogeneity of space, see Einstein talking about it here. This reduces with distance from the gravitating body. See the Riemann-curvature depiction of curved spacetime. Where the plot is tilted, space is inhomogeneous. In that location, light curves and objects fall down:

CCASA image by Johnstone, see Wikipedia

As you move away from the gravitating body the tilt reduces along with the inhomogeneity of space and the force of gravity. Because the force of gravity reduces you could say it's inhomogeneous, but that's because there's a reduction in the inhomogeneity of space. Have a look at this paper on how curved spacetime relates to inhomogeneous space.

In general relativity textbooks such as Sean Carroll's Spacetime and Geometry, there is often a line of reasoning that goes like this: Strong equivalence principle states that free falling frames are equivalent everywhere and are locally inertial. Thus if gravitational field were homogeneous (say, there is a constant gravitational field of g pointing in a particular direction, everywhere in space), we can treat the free falling frame as a global inertial frame. However, since gravitational field is not homogeneous (e.g. near the earth, gravitational field is roughly radial), we cannot construct a global inertial frame. Thus we have to patch local inertial frames together to form a curved spacetime.

This reasoning is incorrect. You can "zoom in" to the Earth's gravitational field to avoid being confused by the shape of the Earth. Then the relative size of the squares depicts the inhomogeneity of space. It's a bit like looking at a flat version of the picture above.

I am very confused by this reasoning. In the case of homogeneous gravitational field, the free falling frame would be defined by observers who undergo the same proper acceleration everywhere.

It's confusing because a homogeneous gravitational field is something of a contradiction in terms. It's where space is inhomogeneous in a uniform fashion. If you had to depict it like the first picture above, you'd be drawing an inverted cone.

The frame defined by these moving observers are clearly not inertial because their relative proper distance changes due to special relativistic kinematic effects. In fact, I believe that the only way to set up observers with constant (but not homogeneous) proper acceleration AND constant relative proper distance is to construct Rindler observers.

Avoid Rindler observers. Accelerating through homogeneous space is not the same as standing on a planet in inhomogeneous space. This is why Einstein said a real gravitational field cannot be transformed away. See section 20 of Relativity: the Special and General Theory, Einstein said "special form" where we would nowadays say "real":

“We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes”.

So are the textbooks wrong? Or am I missing something?

If I had to hazard a guess, I'd say Sean Carroll is wrong. Note that the equivalence principle only applies to a region of infinitesimal extent, so the idea of a "strong" equivalence principle doesn't quite work. See this and note the "nowhere precisely realized in the real world".