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If I make two rods with 1 meter length here on the surface of earth, and send one of them near a black hole that is at rest relative to earth, would I see the rod close to the black hole with a length shorter than 1 meter, (would the black hole rod be perceived as shorter than the rod here on earth)?

Then, if I go close to the rod near the black hole, would I see it with its original length, while the rod on earth would appear longer than 1 meter?

Below are some assumptions before we can discuss the problem:

  • Let assume that the bar is made with a very high youngs modulus and tensile strength material, so that the stretching/rupture due to tidal forces near the black hole be negligible when compared to the total length of 1 meter.

  • For calculus purposes, one should consider that the rod is "stationary", because the measurement will be made "instantly" at a given moment. (but we know that the rod will be circularly orbiting the black hole.)

  • The distance between the observer and the rod that is far away is known at any given moment, so that at first glance, the lenght will be measured using apparent angular size from a side view of the rod, not considering gavitational lensing (naive aproach).

  • The mass of the black hole and the distance between the rod that its near it and the black hole center is also known, so that in a second time, the length will still be measured using apparent angular size from a side view of the rod, but, considering the influences of the gravitational lensing.

  • The capacity to measure meter-scale lengths over cosmical distances should not be considered a problem, given that this question must be seen as a fundamental problem instead of a engineering one.

So, in short, my question is: Does matter (with known dimensions) in curved spacetime appears to have its dimensions distorted when viewed from a region with flat spacetime?

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  • $\begingroup$ Things don’t look distorted locally. The problem here is that “locally” means in an area much smaller than the distance to the horizon. This is commonly misunderstood. So when the rod’s distance to the horizon is comparable to its length, the rod cannot be local and looks distorted to any observer. $\endgroup$ – safesphere May 1 '20 at 7:10
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So, in short, my question is: Does matter (with known dimensions) in curved spacetime appears to have its dimensions distorted when viewed from a region with flat spacetime?

In short, no it does not. Any distortion is a result of the way in which you choose coordinates to describe curved spacetime, just as distortions are different in different maps of the surface of the Earth drawn on flat paper. These scaling distortions are not real. To find real physical quantities, we must use the metric, which undoes the scaling distortions of maps. Once you apply the metric, you will find that a metre rod is still one metre long.

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  • $\begingroup$ The OP is asking if objects near a black hole appear to a Schwarzschild observer as contracted while appearing normal to themselves. The answer is yes, but in the radial direction only, as given by the Schwarzschild metric. $\endgroup$ – safesphere Apr 25 '20 at 4:46
  • $\begingroup$ @The OP did not specify coordinates, although he may have implicitly assumed them. The answer is the in any case the same. Any distortion is due to the property of the map. $\endgroup$ – Charles Francis Apr 25 '20 at 5:01
  • $\begingroup$ "Once you apply the metric, you will find that a metre rod is still one metre long." This doesn't answer the question. The OP asks under which angular size the rod would be perceived on Earth. $\endgroup$ – timm Apr 26 '20 at 9:17
  • $\begingroup$ The angular size is not a dimension of the rod. The answer to the question is that apparent distortions depend an arbitrary, and ultimately meaningless, choice of coordinates. If the observer wants to correctly measure the length of the rod, then he must use the metric, as I said. $\endgroup$ – Charles Francis Apr 26 '20 at 9:25
  • $\begingroup$ @Charles Francis You say the proper length of the rod is unchanged to any observer when the rod is placed in the gravitational field near a BH. Is the proper length also unchanged in the presence of a GW? Kip Thorne's derivation as to how LIGO works says the proper length of LIGO's arms is changed by a GW. $\endgroup$ – Gary Godfrey May 4 '20 at 23:56

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