Would it be possible to measure the distance to a black hole of known mass by measuring the radius of the most redshifted radiation that is detectable? The idea is that since the point where that radiation vanishes is related to the gravitational potential difference between the observer and the point of emission of that radiation, we should be able to get an estimate of how far we are by observing how far into the black hole we can see (assuming that for a given mass and an observer at infinity that radius is fixed).
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$\begingroup$ What do you mean by distance to a black hole? The distance to its Schwarzschild radius or the distance to its center? $\endgroup$– danCommented May 8, 2017 at 19:40
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$\begingroup$ @danielAzuelos Measuring the distance to the SR would not be as useful I think as that would depend on the distance to the observer $\endgroup$– oarsomeCommented May 10, 2017 at 11:30
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Yes. If you could spatially resolve a black hole and measure the redshift of radiation at a given angular radius, then that would yield the distance to the black hole.
This works because the amount of redshift depends on the ratio of radius to Schwarzschild radius, but the latter is known if the mass is known. Thus the redshift yields the radius. The distance is then given by the radius divided by the apparent angular radius.
There will be complications due to gravitational lensing and any orbital velocity around the black hole, but this does not stop it working in principle.
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1$\begingroup$ Isn't the sharp variation of redshift close to a BH a major impediment and source of errors in practice? Normally you want a function that varies more smoothly to perform such measurements. $\endgroup$ Commented May 8, 2017 at 19:12
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1$\begingroup$ @FlorinAndrei If you are simply arguing that this is a difficult measurement; well actually it is impossible at the moment. Doppler shift measurements accurate to 1 part in a million are routine these days., so measurements "close" to the black hole are not required. It is just getting an accurate angular measurement that is difficult. For example, we can easily measure M/R for a white dwarf from GR redshift, but the angular radii of white dwarfs are unreachable. The principle is exactly the same for a black hole. $\endgroup$– ProfRobCommented May 8, 2017 at 21:14
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$\begingroup$ @RobJeffries Would you not be able to estimate the radius based on that variation? $\endgroup$– oarsomeCommented May 10, 2017 at 11:32
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$\begingroup$ @tarzan You need both the redshift and a measurement of angular size. The latter cannot be done at present. $\endgroup$– ProfRobCommented May 10, 2017 at 13:22
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$\begingroup$ @RobJeffries I understand that the angular resolution of current instruments don't allow for such measurements, but if I understand correctly spectrum line widths are used for velocity dispersion measurements. In this case there is no "line" or velocity but I was wondering whether there is a spectral feature (e.g. the slope at a certain wavelength) that could be used to estimate how much of the radiation is emitted from the near side. I gather from your answer that there is no such technique, but is that theoretically possible? $\endgroup$– oarsomeCommented May 11, 2017 at 17:37