Thanks for responding at such length, Zanket. We share much common ground of course, but I feel I'm still not quite communicating my "contradiction in terms" point. This is going to be a long post, but I'd like to start by taking a look at your first paragraph:

I agree with temur above. A frame can be inertial, and remain inertial, when it’s gravitationally accelerating. Only when it’s noninertially (nongravitationally) accelerating is it a noninertial frame. The frame of a thrusting rocket is an example of a noninertial frame.

OK, imagine you and I are in black boxes a million miles apart travelling through space at some steady velocity. You start to fall into a black hole, but you feel no acceleration. You remain in your inertial reference frame, and to you there are

*no* forces at work. There is no "force of gravity". But when I observe you, I see that you're accelerating away from me. To me you're in a non-inertial reference frame, and I claim that the force of gravity is accelerating you. Your paragraph above mixes these two viewpoints. That's the basis of the contradiction in terms.

Even in our gravity-endowed universe a frame can remain inertial indefinitely; I disagree that “a continuous gravitational acceleration continuously changes your inertial reference frame”. I think it is obvious from the supporting info in the OP that Einstein, Taylor, Thorne, and Wheeler are all in agreement with me.

I wouldn't be too sure about Einstein, but yes, I think the other people are in agreement with you. And I think they're missing the trick, especially Thorne when he talks about time travel. Going back to my scenario above, imagine that the black hole was suddenly removed and your freefall ceased. (Please can we ignore any accelerations or "gravity waves" caused by this). Thereafter both you and I would agree that you are back in an inertial reference frame, travelling at some steady but higher velocity. If the black hole reappears briefly to accelerate you some more, we would disagree about your inertial reference frame, then when it disappeared we would agree again. A freefall is like this but smooth and continuous. There's no acceleration in your eyes, no force, no action-at-a-distance, and no gravity. But there is in my eyes. To me your frame is constantly changing, to you it isn't. Not unless you look out the window. (You know, I consider myself an Einstein "fan", but I'm not a fan of reference frames because they don't actually exist).

Temur says: “So just drop a small ball from a bit above the horizon and the ball will be in free fall, and it will be in an inertial frame in a short time interval around the moment it crosses the horizon.” I say that a frame can in principle remain inertial for a year or more (on a clock affixed to the frame); you just need a sufficiently large black hole. The frame remains inertial for as long as the tidal force in the frame remains negligible for the purposes of your experiments. Taylor and Wheeler also talk about “small” regions and “short” times for an inertial frame, but these are arbitrary words whose precise magnitude depends on the experiment. In one of the examples in their book, they employ an inertial frame that is two million light years long and lasts for at least two million years. A frame in a stable circular orbit can remain inertial for billions of years because the tidal force in it does not increase.

I feel happier about circular orbits. Again "neglible" looks like a problem. If it's neglible and yet spans the horizon how can the horizon be inside the frame? You're saying

*there's no difference here* and yet the very presence of the horizon says

*there is a difference here*. How can I explain this? In your freefall inertial reference frame, there was no action at a distance force. The thing that was causing you to accelerate with respect to me

*has to be local to you* and every single atom of your body. Yes, the tidal force might be very small and your acceleration might be changing only slightly. But in my eyes you are definitely accelerating in a given direction. There is a definite non-uniformity and it is local to you, in what you consider to be your "uniform" local frame. The gravity

*is* the non-uniformity. Hence

**a uniform gravitational field is a uniform non-uniformity, and is contradiction in terms**.

In Pete Brown’s essay, a key point is this: “The main goal of this paper is not to present a new interpretation of gravity. For pedagogical purposes we review an old one, that of Albert Einstein’s”. The paper is just discussing an interpretation. Multiple interpretations can lead to the same predictions; e.g. both tidal force and spacetime curvature. The OP deals with GR’s predictions. It isn’t affected by an interpretation.

Noted. I'll reread the OP. Sorry if I've lost the plot somewhere along the line.

Brown says “A uniform gravitational field has, by definition, no tidal forces and thus no space-time curvature. Thus according to the interpretation of gravity as a curvature in spacetime a uniform gravitational field becomes a contradiction in terms (i.e. no tidal forces where there are tidal forces).” See that it’s only an apparent contradiction. Brown says “Given experimental limitations one can establish criteria in which tidal effects may be ignored.” That resolves the apparent contradiction. A uniform gravitational field is better defined as having a negligible tidal force, not no tidal force. A uniform gravitational field is better thought of as a negligibly nonuniform gravitational field. There’s a contradiction only when one rigidly sticks to definitions of a uniform gravitational field, inertial frame, or flat spacetime in which there are “no tidal forces and thus no space-time curvature”.

I agree with this. I say gravity is not "curved spacetime", but instead is a gradient in c across your local frame that you don't notice but I do.

It seems to me that what happens is this: People first learn SR, which is an idealized environment where there is no tidal force. When they learn GR they stick to SR’s definitions, causing contradictions to be seen. But there’s no requirement that SR’s definitions carry over to GR unchanged. Einstein effectively changed them with the equivalence principle, to allow a negligible tidal force in an inertial frame. That makes sense, since otherwise SR could not even be experimentally confirmed. We might not be having this discussion if it weren’t for the fact that terms like “negligibly nonuniform” and “negligibly curved” get tedious, resulting in the shorthand terms “uniform” and “flat”.

Agreed. The frames can become quite confusing. The whole subject can. If it didn't I think we'd have bottomed out gravity a long long time ago.

*Farsight: “Because that's what the gravity is. It's equivalent to a continuously changing inertial reference frame, and if there is no continuously changing frame there is no gravity.”* I disagree. To me and Taylor and Wheeler, gravity is indicated by the tidal force, synonymous with spacetime curvature. A negligible tidal force exists in any inertial frame.

See my paragraphs above regarding viewpoint. If you stick to your reference frame, there's no force acting on you, no "force of gravity",

*no gravity*. If we shift to my viewpoint, everything changes.

*Farsight: But if you apply that "neglible" to your calculus analogy, you'd be doing away with the very curve itself. You equate the curve to a perfectly flat line, and you're left with no area beneath the curve, and no route to understanding what gravity is. The neglible is axiomatic. There's no justification for it. There can be no curve if the small section of it is absolutely the same as that flat line.* Using negligibly wide rectangles under the curve does not equate the curve to a single perfectly flat line; it equates it to a series of negligibly wide segments of perfectly flat lines, the area under which approximates the area under the curve. The approximation can be as precise as desired. The narrower the rectangles, the more precise the approximation.

We're talking at cross purposes here. At right angles in fact. I was looking at sections of the curve and saying they can't be flat, because if they were you'd have no curve.

Some of the letters on your screen now look like they have curves right? Well, if you could zoom in on them you’d see that they comprise only square blocks, which, by virtue of being negligibly large (to your eyes), can be used to approximate a curve.

No problem. But if you say they

*are* flat, all the letters end up looking like this _.

If the crew of the International Space Station (ISS) let some object float across the station, it will seem to follow a straight path relative to the station even if the time required is the time it takes for them to complete an orbit around the Earth. The object’s path curves relative to the Earth, but relative to the ISS it is straight.

No problem. It's straight

*and* it's curved. Ain't relativity fun!

*Farsight: So you just can't have an inertial reference frame crossing an event horizon.* It wouldn’t matter to the OP even if you were right, because GR predicts that an inertial frame can fall through a horizon, as the supporting info in the OP shows. You’d be arguing against GR, not the OP.

I've said elsewhere that I consider this to be a hypothetical situation. You cannot in reality fall through the horizon, because at this point time dilation becomes infinite. My rule of thumb says that infinities don't happen in reality, so we should look to see if the mathematics needs some refinement. Hence I say c goes to zero at the horizon. This means a collapsing star takes forever to collapse, and none have finished collapsing yet. Hence they're frozen stars. The event horizon becomes a wall of eventless "solid space". An eventful and "uniform" local frame simply cannot span it.

*Farsight: As I've said before, I think you do have a point that is generally misunderstood. But IMHO it doesn't invalidate GR. It just challenges the current interpretation, which I believe is not quite the same as the one Einstein eventually held.* I disagree. The OP does invalidate GR, by showing that it blatantly violates the relativity principle (one way to put it). The OP does not challenge an interpretation. It shows that what GR claims is the same cannot be the same by any stretch of the imagination.

Noted. I've been thinking a shift in interpretation might fix GR singularity infinities and resolve your issue, but like I said I'll reread the OP again. If you'd prefer to agree to differ don't hesitate to say so and I'll leave it at that.