I The Shapiro delay and falling into a black hole

[Zan24C]

There's a difference between "never sees it happen" and "never happens". Your experiment concerns the former - you're just sending somebody down close to the horizon to do the observing and waiting for him to report back.

But according to my thought experiment, there is always a time for the distant observer which corresponds with the infalling person being able to reverse course and come back.
So, by logic, as far as the distant observer is concerned it never happens because there is never a time when the infalling observer cannot come back. Except at infinity. Yet in his own perspective the infalling person could activate his rocket a fraction of a second too late and cross the event horizon. That fraction of a second has somehow bridged the gap between all of the observer's future time and infinity.

Just to illustrate the change in speed of light near a black hole, I have found this graphic posted by John Rennie as a reply in http://physics.stackexchange.com/questions/77227/speed-of-light-in-a-gravitational-field

The variation of the velocity of light with distance from the black hole looks like:

calculated for a light ray heading directly towards a black hole in the reference frame of a distant observer, where v is the speed of light as a fraction of c and r is the distance from the centre of the black hole as a fraction of the radius of the event horizon.

 

[PeterDonis]

it's not the subject of this thread and I would like to stop talking about it now.

Fair enough, but you should be aware that we have no experimental evidence that extremely small probabilities like 1 / googol have any physical meaning. It is one thing to say that non-relativistic quantum mechanics mathematically includes such probabilities (and even there the "non-relativistic" is key--as has already been pointed out, when you include relativity you are using quantum field theory, which works differently); it is quite another to insist that such probabilities are real. You can open a new thread in the QM forum if you want to ask about the limitations of this viewpoint.

I truly appreciate that you took time to reply to this thread and are trying to help me.

Thanks for the appreciation. I understand that this is a difficult topic, and what GR says in this case is very counterintuitive. I'll limit my response to your previous posts to pointing out a few specific statements that seem to me to pinpoint where the counterintuitiveness is concentrated, so to speak.

Ignore the reference frame of the observer. AGAIN.

There is a very important point to be made here, which I think is made in the Insights series but perhaps I need to go back and stress it more. Every observer's "reference frame", strictly speaking, is local. In other words, every observer can construct a "reference frame" that describes events in his local vicinity. And in his local vicinity, he has a valid argument for treating the description of his observations in terms of his local reference frame as "real".

But any attempt to extend that beyond the local vicinity of that observer no longer can be justified by such an argument in the general case. In flat spacetime, there happens to be a set of reference frames--the inertial frames--that can be so justified; but that is a particular feature of flat spacetime and inertial frames and cannot be extended beyond that. For example, even in flat spacetime, non-inertial frames will not, in general, cover the entire spacetime. (Rindler coordinates, for example, which are the natural "reference frame" for an observer with uniform proper acceleration, have a horizon that has a number of similarities with the event horizon in Schwarzschild spacetime.) And in curved spacetime, even inertial frames (i.e., frames in which an object at rest is in free fall--note that Schwarzschild coordinates are not such a frame, btw) are only local and cannot cover an extended region of spacetime, let alone the entire spacetime.

So when you say that "the reference frame of the distant observer says the falling observer never crosses the horizon", you are trying to extrapolate the distant observer's frame beyond the region where it is valid. You simply can't treat that reference frame, or any reference frame, in curved spacetime the way you treat an inertial frame in flat spacetime. This goes for the falling observer's frame too, btw: strictly speaking, the falling observer's frame, in which it is simple to show that he reaches the horizon in finite time, only covers his local region and cannot be extrapolated to cover the distant observer. So he can't assume, for example, that the coordinate time in his reference frame (which is basically Painleve coordinates) will be meaningful in the distant observer's vicinity.

The upshot of all this is that there is no "reference frame" that you can use to describe the entire geometry of a curved spacetime like Schwarzschild spacetime. The only way to do it is to use coordinate charts, which in general will not correspond everywhere to any observer's "reference frame". And there is no single chart that can represent the entire geometry without distortion, so coordinate charts have limitations too if you're trying to visualize what's going on. We don't have to confront this in everyday life because we are used to limited domains, like the vicinity of the Earth, in which we can find single "reference frames" or coordinate charts that cover everything we are interested in. But that's because we live our everyday lives in a limited domain, not because it's always possible.

Ultimately, the physics comes down to invariants--quantities that do not depend on your choice of reference frames or coordinate charts. When I say that the falling observer reaches the horizon in finite time, that is an invariant: the invariant length of a particular timelike curve in the spacetime geometry. It's no different from saying that the great circle distance from New York to London is an invariant on the Earth's surface, and doesn't depend on whether you use spherical coordiates, Mercator coordinates, or stereographic coordinates, or even if you choose a reference frame or coordinates that include New York but don't include London. London is there in the geometry whether your coordinates include it or not; you can compute geometric invariants that show that. Similarly, the horizon and the region inside it are there in the Schwarzschild geometry, whether your coordinates include it or not; you can compute geometric invariants that show that.

Because you are now factoring in a change in reference frame from local to global, where there is a change in spacetime curvature.

No, I'm not. The signal delay is an invariant, as I said. Invariants are independent of any choice of reference frame or coordinates. See above.

Also, the spacetime curvature is itself an invariant: it also doesn't depend on your choice of reference frame or coordinates. So saying that changing reference frames changes the spacetime curvature is wrong.

If the person traveling around the world took an infinite amount of time from the perspective of the person
going south on the interstate - which is the more appropriate analogy by the way

No, it isn't. In your thought experiment about an observer going down close to the horizon, staying there for a while, then coming back up to the distant observer, both of their elapsed times are finite.

If, OTOH, the falling observer actually reaches the horizon, then he can never get back out to the distant observer again, so there is no way for them to compare clock readings. And since both of their reference frames are local (see above), neither one can extrapolate his own clock readings to the other. In other words, there is no way to make the comparison you are trying to make unless the two observers can come back together, and if the falling observer reaches the horizon, they can't.

 

[PAllen]

I'm not an expert on the Alcubierre drive but I believe all the problems you mentioned have been previously debunked on this very forum. Regardless, I only introduced the Alcubierre drive as a thought experiment. Theoretically the distant observer could also quantum tunnel to and from the in falling object extremely rapidly, again requiring both reference frames to reconcile.

It is rigorously proven that two alcublerre drives can be used to construct closed time like curves, allowing general time travel. Similarly, it is established that a traversible wormhole, if it exists, can be converted to wormhole back in time.

 

[PeterDonis]

according to my thought experiment, there is always a time for the distant observer which corresponds with the infalling person being able to reverse course and come back.

Yes, but that's because all of the distant observer's "times" correspond to the falling observer being somewhere above the event horizon. And as long as the falling observer is above the horizon, then yes, he can reverse course and come back.

But once the falling observer reaches the horizon, he can no longer reverse course and come back, and there is also no longer any "time" according to the distant observer that can be extrapolated to his location. See my previous post.

The variation of the velocity of light with distance from the black hole

The variation of the coordinate velocity of light, in Schwarzschild coordinates. But no observer ever measures light moving at this velocity locally. Any observer measuring the velocity of a light beam passing him will measure it to be $c$.

 

[PeterDonis]

@Zan24C I am closing this thread because if it continues as it has been you are going to get a misinformation warning. Your posts have been addressed, and I don't see the point of continuing to repeat the same responses to the same statements by you. If you have something new to suggest that has not been already discussed in this thread, PM me and I will consider it.

 

[pervect]

Any particle or system with non-zero rest mass follows a time-like path through space-time. If you imagine the system being a stop-watch, the difference between the readings on the stop watch as it follows the time-like path would be the proper time associated with that section of the path. So we can think of proper time as what would be measured by a very small stop-watch, ideally a point-like stopwatch.

This observer-independent quantity, the change in reading on the stop-watch, can be computed by some formulae from observations made in a different frame in which the stop-watch may be (and usually is) moving. So we'd assign coordinates (usually t,x,y,z) to every point along the clock's path through space-time, and we can relate the changes in the clock reading to the changes in the coordinates via some mathematical formula (I'll avoid the details for now) involving the changes in coordinates.

Wiki has a writeup of proper time and the necessary formulas, <<link>> but I doubt it's the best place to start learning about the concept.

If you grasp the principle of computing the proper time in SR, you can read about the relatively minor modifications (involving the metric coefficients) that are needed to compute proper time in GR. If you take these formula as a given, using the Schwarzschild metric coefficients, you can compute the proper time interval that experienced by a stop-watch falling into a black hole - unfortunately, you need also to either know or be able to figure out which path a free-falling clock takes. You can find some paths that are not free-fall that do take an infinite time, so it's not a trivial question.

Most GR textbooks will do some version of this problem, though, and you can fact-check the result that the time on the clock is finite.