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'.
Return to the Special & General Relativity Questions and Answers page.
All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the NASA Astronomy Cafe, part of the NASA Education and Public Outreach program.