As pervect says, a star directly "above" an observer that freely falls into a black hole will be infinitely redshifted - I give another derivation below. But this is not true for the light from all stars in the universe!
The shadow (black region of the sky) of the black hole grows bigger as the observer falls, but, even at the event horizon take up less that half the observer's field of view. "At" the singularity, the shadow takes up half the field of view.
Consider far way stars that are distributed roughly equally in all (Schwarzschild coordinate) directions. Images of stars that are not directly "above" the observer move towards the shadow region because of stellar aberration. Also, depending on image location, some of the stars are blue shifted and some are red shifted.
Just before the singularity, all the images (except for stars directly above) are concentrated in a brilliant ring around the shadow region, i.e, in a ring perpendicular to "direction" of travel.
All of this can be derived fairly easily without tensors. As an example, consider the image of a star directly above.
The proper time of an observer hovering with constant r, \theta, and \phi above a black hole is related to Schwarzschild coordinate time by
<br />
d\tau_S = \left(1 - \frac{2M}{r} \right)^{\frac{1}{2}} dt,<br />
and coordinate time is proper time for a far observer. Frequency is the inverse of time, so if light with frequency f starts far from the black hole with frequency and fall radially towards the hole, the hovering observer will see frequency
<br />
f_S = \left(1 - \frac{2M}{r} \right)^{-\frac{1}{2}} f.<br />
Suppose an observer freely falling from rest far away passes the hovering observer at the instant the light is observed. Since the two observers are coincident, the special relativistic Doppler effect relates the frequencies seen by the two observers. The relative speed between the two observers is given by
<br />
v = \sqrt{\frac{2M}{r}}.<br />
For a derivation, see
https://www.physicsforums.com/showpost.php?p=621802&postcount=32.
Consequently, the freely falling observer sees a frequency
<br />
\begin{equation*}<br />
\begin{split}<br />
f' &= f_S \sqrt{\frac{1 - v}{1 + v}} \\<br />
&= f \left(1 - \frac{2M}{r} \right)^{-\frac{1}{2}} \sqrt{\frac{1 - \sqrt{\frac{2M}{r}}}{1 + \sqrt{\frac{2M}{r}}}} \\<br />
&= \frac{f}{1 + \sqrt{\frac{2M}{r}}}<br />
\end{split}<br />
\end{equation*}<br />
As pervect calculated, the frequency seen by the freely falling observer as he crosses the event horizon will be half the frequency that the light had far away.
This result can also be calculated using Lorentz transformations.
Even though this derivation is only valid outside the event horizon, I think (but haven't checked) that the result is valid inside the horizon, where is r a timelike coordinate.
Regards,
George
i really don't get all this math and stuff, but besides that what do u all mean by being in a black hole? i thoght u become speghtee once u got sucked in.
