Ooooh boy! I hate these questions!!!
The particular story evidently has a spaceship approaching a neutron star on whose surface live some creatures. Apparently, because of time dilation effects, the spaceship seems to approach their world in some kind of slow motion and for long periods of time seems to remain frozen in space. This apparently gives them a long time to decide upon what to do about the spacecraft.
It is true that the gravitational field of the neutron star causes clocks on the surface to run slow compared to distant clocks. Example, for a neutron star with a typical radius of 20 kilometers and a mass of say 1.0 times the Sun, the dilation factor is:
1 Gamma = -------------------- 1/2 ( 1 - E) 2 G M where E = ------- c^2 R <\pre> The relationship between the time of the distant observer (T) and the surface dwellers (t) is then just t = T x gamma. For M = 2 x 10^33 grams, G = 6.6 x 10^-8, c = 3 x 10^10 and R = 20 x 10^5 all in the appropriate 'centimeters-grams-seconds' units, you get E = 0.146 so that Gamma = 1.08. The 'black hole' radius for this neutron star is 2.7 kilometers so its surface is well outside this radius. The gamma factor means that distant observers will gauge that the clocks on the surface of the neutron star are running about 1.08 times SLOWER than their clocks which follows from the principle that clocks inside strong gravitational fields run slow. Alternately, as viewed by the surface dwellers, they will assume their clocks (t) are running just fine, and see that the clocks on the spacecraft (T) are running 1.08 times FASTER because the 1 second the surface dwellers are measuring is bigger by a factor of 1.08 than the 1 second measured on the distant clock. If I have gotten the general relativity correct, the surface dwellers ought to see events on the spacecraft speeded-up slightly, not slowed down.
If the neutron star were only 1 kilometer bigger than its black hole radius of 2.7 kilometers, the gamma factor would increase to 1.9 which is still rather modest, and such an object being so close to its black hole size would probably be massively unstable anyway.
Not being a relativist, my first impression is that the story does not properly take into consideration that, unlike the relative motion in special relativity, the symmetry between the surface dwellers and the space ship inhabitants is broken by the fact that the surface dwellers are in a much stronger gravitational field. This means that what each party sees will not be symmetrical with the other parties clocks running slower. The surface dweller's clocks run slower in an absolute sense because it is they that are in the strong gravitational field relative to distant observers either in the spacecraft or at 'infinity'.
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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the NASA Astronomy Cafe, part of the NASA Education and Public Outreach program.