Curvature without tidal forces

[Ratzinger]

How is spacetime curved if the present gravity field is completely uniform and there are no tidal forces. Clocks at a same height would tick the same, at different heights (to the gravity source) would tick differently. But what about space? How is space curved in the absence of tidal forces?

Often curvature is introduced with falling elevators without tidal forces. The observer in a falling elevator sees a light ray going from on side of the elevator wall to the other as a straight line. An outside observer sees a bended line. Thus gravity bends spacetime they say. Later then tidal forces and the non-uniformity of gravity fields is mentioned and made responsible for curvature.

So again my question: how would space be bent in a complete uniform gravity space?

thank you

 

[JesseM]

In a completely uniform gravitational field, it's my understanding that spacetime would be flat. For example, this page on the twin paradox in flat spacetime mentions that you can understand things from the point of view of the accelerated twin by introducing a uniform gravitational field during the period of acceleration...but the curvature of spacetime is supposed to be independent of your coordinate system, so if the inertial twin sees spacetime as flat throughout the journey, then the accelerating twin should see the same thing.

 

[pervect]

How is spacetime curved if the present gravity field is completely uniform and there are no tidal forces. Clocks at a same height would tick the same, at different heights (to the gravity source) would tick differently. But what about space? How is space curved in the absence of tidal forces?

Often curvature is introduced with falling elevators without tidal forces. The observer in a falling elevator sees a light ray going from on side of the elevator wall to the other as a straight line. An outside observer sees a bended line. Thus gravity bends spacetime they say. Later then tidal forces and the non-uniformity of gravity fields is mentioned and made responsible for curvature.

So again my question: how would space be bent in a complete uniform gravity space?

thank you

If you have a flat Minkowski space-time, the curvature tensor is zero, and this will be true regardles of the coordinate system used.

However, if you adopt non-inertial coordinates to describe a flat minkowskian space-time, like the coordinate system of an accelerated observer, you can make the Chirsotffel symbols non-zero, even though you can never make the curvature tensor non-zero.

Non-zero Christoffel symbols can cause, for instance, the opposite sides of a parallelogram to have different lengths. This is somtimes called "curvature", but that's really speaking very losely. It's quite commonly done, though, including in many textbooks.

 

[pmb_phy]

How is spacetime curved if the present gravity field is completely uniform and there are no tidal forces. Clocks at a same height would tick the same, at different heights (to the gravity source) would tick differently. But what about space? How is space curved in the absence of tidal forces?

Often curvature is introduced with falling elevators without tidal forces. The observer in a falling elevator sees a light ray going from on side of the elevator wall to the other as a straight line. An outside observer sees a bended line. Thus gravity bends spacetime they say. Later then tidal forces and the non-uniformity of gravity fields is mentioned and made responsible for curvature.

So again my question: how would space be bent in a complete uniform gravity space?

thank you

At one time (before I learned GR) I wondered about this too so after I learned GR I wrote up the answers to your question in this article

http://xxx.lanl.gov/abs/physics/0204044

Pete

 

[Stingray]

As already mentioned, there is no spacetime curvature with a uniform gravitational field. The modern interpretation is that things "fall" in such fields due to the observer's nonzero (4-)acceleration (the magnitude of which is an invariant), and that it really isn't a gravitational effect.

Of course other interpretations are possible - and are occasionally convenient - but this one tends to be the most foolproof. Relying on nonvanishing Christoffel symbols is a great way to confuse yourself. Even the physicists working on GR were confused by it (and similar ideas) back when such statements were still popular. Hardly anyone made any progress in the field until the old coordinate-dependent notions were removed from everyones' minds. That took about 40 years.