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So I take particle A and place it in space, then I place particle B 1,000,000 light years away from particle A. I strike a line between the two, and measure that line as having a length of 1,000,000 light years. I verify this measure by taking out my handy cosmic calipers and confirm that there is precisely 1,000,000 light years of distance between the two particles.

Now I drop a black hole halfway along the length of the line. While my tape measure gets sucked into the singularity halfway along the line, and I can no longer "see" particle B from particle A's locale, and vice versa, as any reflected or emitted light from either particle gets sucked down the black hole, my handy cosmic calipers seem to still show a distance of 1,000,000 light years between the particles.

My question... Has the distance between the two particles changed by the addition of the black hole along the line between the two?

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    $\begingroup$ There is no single distance between the two particles. Even in flat space the distance depends on the path one takes between A and B. What you are talking about is the shortest distance, which in flat space would be a straight line. That shortest distance still exists, it just doesn't go straight trough the black hole but somewhere around it. How do you get that shortest distance physically? By shooting light rays from A to B under different angles. The one that goes right trough B is the one with the shortest path. $\endgroup$ – CuriousOne Oct 28 '15 at 4:24
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    $\begingroup$ Why would the initial shortest line not disappear into the singularity, geometrically it doesn't change direction, it just runs into a bottomless pit? If a "new" shortest path is then created "bending" around the black hole, do my cosmic calipers reflect that change, or do they still measure the same distance between points? $\endgroup$ – Hep Oct 28 '15 at 6:04
  • $\begingroup$ Depends. What is the frame of reference you consider for your cosmic calipers? $\endgroup$ – Tamoghna Chowdhury Oct 28 '15 at 6:11
  • $\begingroup$ Big. Really, really big. I can contain both particles between the jaws while standing, a super-being in space, observing the whole of the experiment. $\endgroup$ – Hep Oct 28 '15 at 6:59
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So I take particle A and place it in space, then I place particle B 1,000,000 light years away from particle A.

Alright.
But, just to be sure:
In astronomy, and (prehaps somewhat more recently) in cosmology and in physics in general, we understand this measure of "having been apart" as chronometric mutual separation. In your example this means specificly that

  • A and B should have conducted a trial which took A 4,000,000 years (at least) and which took B 4,000,000 years (at least) where

  • for (at least) each signal indication which A stated during the first 2,000,000 years of the trial A observed the corresponding echo indication of B exactly 2,000,000 years later, and

  • for (at least) each signal indication which B stated during the first 2,000,000 years of the trial B observed the corresponding echo indication of A exactly 2,000,000 years later, and

  • all durations of A which A calls "$t$ years" (for some real number $t$) and all durations of B which B calls "$t$ years" (for the same real number $t$) were confirmed equal by means of "our handy cosmic chronometeres" (Marzke-Wheeler clocks); see also MTW Box 16.4. The region containing A and B may be thought of as being (sufficiently densely) filled with such Marzke-Wheeler clocks.

I strike a line between the two, and measure that line as having a length of 1,000,000 light years.

Do you mean that additional particles would have to be found, throughout the trial(s) by A and B described above,
namely one particle P for each real number $0 \lt p \lt 1.000.000$, such that

  • P was and remained "placed $p$ light years away from particle A", and

  • P was and remained "placed $(1.000.000 - p)$ light years away from particle B"

??

Well, if that's your requirement for your setup prescription, so be it. But, just to be sure:
That's not necessarily required for plainly attributing to A and B some particular value of (their) chronometric mutual separation; such as "1,000,000 light years".

I verify this measure by taking out my handy cosmic calipers [... or tape measure] and confirm that there is precisely 1,000,000 light years of distance between the two particles.

Here you're apparently kidding.
Anyways, there's no mentioning of a "line" in this mildly hysterical description of yours; and the actual (chronometric) method is of course rather as laid out above.

Now I drop a black hole halfway along the length of the line.

This seems as if you had the idea that such a black hole had not been "in place" already during the trial(s) of A and B described above. But of course: conditions which are not explicitly forbidden by a setup prescription (including the comprehensive methodology how to determine them) had apparently already been admitted from the outset.

I can no longer "see" particle B from particle A's locale, and vice versa

But sure you can, unless the presumed additional black hole is too large such that A and (by symmetry) B were both inside its horizon.

Has the distance between the two particles changed by the addition of the black hole along the line between the two?

Surely the trial(s) described above can succeed even while the region containing A and B is (significantly and varyingly) curved; e.g. by (not too large) black holes being "dropped", and "added". Accordingly, a value of mutual separation would be attributed to A and B (in either case, with or without black holes being "dropped"); and A and B are accordingly called chronometrically rigid with respect to each other.

But, strictly, a value of "distance" is attributed only to a pair (of "ends") which was and remained chronometrically rigid with respect to each other if, moreover, the region containing this pair had been flat throughout their trials; such that these two ends could (also) be called having been and remained at rest to each other, as two members of the same inertial system.

This is of course not satisfied in case black holes were "dropped" (or already "present") during the trial(s) of A and B. According to this setup prescription, A and B would be attributed a value of their mutual separation (if their trial(s) succeded, as described above), but not a value of (their mutual) distance.

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  • $\begingroup$ So what you're saying is that we can measure their separation, but not their distance from one another, and in measuring the former, the black hole would be irrelevant? $\endgroup$ – Hep Oct 28 '15 at 7:19
  • $\begingroup$ @Hep: "So what you're saying is that we can measure their separation [...]" -- If A and B found (separately) constant and (mutually) equal ping durations of 2.000.000 years each then they're attributed a value of their mutual separation: 1.000.000 light years. "but [we can] not [measure] their distance from one another" -- If A and B found their ping durations as described, and are accordingly being attributed a mutual separation of 1.000.000 light years, but if the region containing them was not flat (due to whatever), then we don't call this result a "distance" value. $\endgroup$ – user12262 Oct 28 '15 at 7:26
  • $\begingroup$ Ah, now I get it. Its semantics. Distance is not a relevant term. $\endgroup$ – Hep Oct 28 '15 at 7:40

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