Only if one is a mathematician.
I didn't mean a strict definition vs intuitive one. I meant that one needs to say what he means when there is more than one possibility even if it is vague and intuitive definition.Only if one is a mathematician.
Now seriously. No, because it seemed that the source of disagreement between PeroK and me was not in different definitions.
Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.
Anyone teaching SR would immediately be able to give you the precise definition of (coordinate) length as the distance between simultaneous measurements of the two ends of an object (in the given reference frame - usually assumed to be inertial).
Sure, but here we are talking about GR, so precise definition is not so immediate.
There is none. That's the point. It's not a core concept in GR. It only appears in more obscure scenarios.
Can you give any definition, not necessarily precise?
I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer [1]. One can then compare distances as seen by different observers in relative motion with respect to each other, and this gives a GR generalization of length contraction. Sure, it's not a core concept in GR, but it does make some sense.
See my post above.
No, that is a local situation.
I'm not sure what do you mean by that. If you mean that Eq. (84.6) in LL is local, one can always integrate it ##l=\int dl## to get a global distance. The integral is taken along a geodesic with respect to the spatial metric (84.7).
Where are you getting this from? Do you have a reference?
Reference?
Reference?
More than that, one should first read subsequent posts to see what the OP was actually asking about. See in particular post #25 and my responses to it.
No, that is still local.
The object actually defines a worldtube in spacetime. The reference frame in which it is at rest should have as "grid lines of constant spatial coordinates" the worldlines in the congruence defined from the object's worldtube.
It shows some of the issues involved if you try to give a well-defined meaning to "gravitational length contraction".
Is that a problem?
Yes, if the stick is not near you how do you assign length to it? But even if that worked, the main point since post #2 is that whatever conventions there are in the question, they need to be explicitly made. Otherwise the question is ill posed.
The reference in post #43 says how.
It's explicit in the reference in post #43.
No, it is not there. At least I cannot find it. Can you quote it or point out exactly where to look?
Page 234, "Suppose a light signal is directed ...". It describes an experimental procedure for measuring distances.
Btw, even though the spacetime is static you must evaluate the integral in the appropriate coordinate chart -- i.e. in the adapted coordinate chart in which the metric does not depends on timelike coordinate.
Which continues "...from some point B to an infinetesimally close point A..."
And later says that it can be integrated to finite distances (provided that metric is time-independent).
We are going in circles. This is local. An observer can do that and say what the length of a stick is, if the stick is right next to him. What about a second observe, who is far away? What is he going to do to measure the stick's length? All this requires some conventions to be specified. It seems that the OP is unaware of that.
https://ui.adsabs.harvard.edu/abs/1959PhRv..116.1041T/abstract
Well, my understanding of a Rindler-horizon is that it's a horizon were motion of stuff appears to freeze, as seen from far above the horizon. Now, if a ruler's one end near said horizon appears to be quite motionless, while the other end is still appears to be moving a little bit, then the length of the ruler appears to be decreasing.
That depends a bit what you mean by "seen". Get a 1m long strip lamp and a 1m long piece of photo paper facing each other a centimetre or so apart and have a rod of rest length 1m pass between at 0.866c. Blink the light as the rod passes through and the shadow on the photo paper will be 0.5m long. However, if you stand and watch the rod or use a regular camera that we can model as pointlike then the light coming from the far end of the rod is older than the light from the near end, so you don't see the whole rod as it is at one time (per an Einstein frame) and this can sometimes counter the length contraction effect.