Does someone falling into a black hole see the end of the universe? This question was prompted by Can matter really fall through an event horizon?. Notoriously, if you calculate the Schwarzschild coordinate time for anything, matter or light, to reach the event horizon the result is infinite. This implies that the universe ages by an infinite time before someone falling into the black hole reaches the event horizon, so could that person see the universe age by an infinite time?
To be more precise, suppose the observer starts falling from rest at time $t = 0$ and some initial distance $r > r_s$. If we wait for some time $T$ then shine a light ray at the falling observer. Will the light ray always reach the falling observer before they cross the event horizon? If not, what is the formula for the longest time $T$ that we can wait and still be sure the ray will catch the observer? If $T$ is not bounded it implies that observer could indeed see the end of the universe.
I can think of a qualitative argument for an upper limit on $T$, but I'm not sure how sound my argument is. The proper time for the observer to fall to the event horizon is finite - call this $\tau$. The proper time for the light ray to reach the horizon is zero, therefore the light ray will reach the observer before they cross the event horizon only if $T < \tau$. Hence $T$ is bounded and the observer won't see the end of the universe.
I think a more rigorous approach would be to determine the equations of motion (in the Schwarzschild coordinates) for the falling observer and the light ray, and then find the condition for the light to reach the falling observer at some distance $\epsilon$ from the event horizon. Then take the limit as $\epsilon \rightarrow 0$. In principle this seems straightforward, but in practice the algebra rapidly defeated me. Even for a light ray the radial distance:time equation isn't closed form (Wolfram claims it needs the $W$ function) and for the falling observer the calculation is even harder.
 A: I would recommend steering clear of Schwarzschild coordinates for these kind of questions. All the classical (i.e. firewall paradox aside) infinities having to do with the event horizon are due to poor coordinate choices. You want to use a coordinate system that is regular at the horizon, like Kruskal-Szekeres. Indeed, have a look at the Kruskal-Szekeres diagram:

(source: Wikipedia)
This is the maximally extended Schwarschild geometry, not a physical black hole forming from stellar collapse, but the differences shouldn't bother us for this question. Region I and III are asymptotically flat regions, II is the interior of the black hole and IV is a white hole. The bold hyperbolae in regions II and IV are the singularities. The diagonals through the origin are the event horizons. The origin (really a 2-sphere with angular coordinates suppressed) is the throat of a non-traversable wormhole joining the separate "universes" I and III. Radial light rays remain 45 degree diagonal lines on the Kruskal-Szekeres diagram. The dashed hyperbolae are lines of constant Schwarzschild $r$ coordinate, and the dashed radial rays are lines of constant $t$. You can see how the event horizon becomes a coordinate singularity where $r$ and $t$ switch roles.
Now if you draw a worldline from region I going into region II it becomes obvious that it crosses the horizon in finite proper time and, more importantly, the past light-cone of the event where it hits the singularity cannot possibly contain the whole spacetime. So the short answer to your question is no, someone falling into a black hole does not see the end of the universe. I don't know the formula you ask for for $T$, but in principle you can read it off from light rays on the diagram and just convert to whatever coordinate/proper time you want to use.
A: This is a rewrite of Michael Brown's answer to help me get my thoughts clear, and possibly to help everyone else who's interested to get their thoughts clear too :-) Michael presents a very simple answer to my question based on the geometry of spacetime around the black hole.
The key point is that the usual radius/time Schwarzschild coordinates are unhelpful because they obscure what's going on. To get round this we use a coordinate transformation to draw the spacetime around the black hole using the Kruskal-Szekeres coordinates $u$ and $v$. This what the result looks like:

The $u$ coordinate is horizontal and the $v$ coordinate is vertical.
The problem with these coordinates is that they are highly unintuitive. A displacement in $u$ or $v$ doesn't correspond to any simple physical quantity, unlike a displacement in the usual radial coordinate $r$ or time coordinate $t$. Nevertheless the KS coordinates simplify things drastically as follows:
In these coordinates constant $r$ is a hyperbola as shown by the dashed line. The event horizon is the solid 45° line. You can sort of think as $t$ increasing as you move up - it does, though not in a linear way. The singularity is the red hyperbola (this is a spacetime diagram remember, so the singularity is a curve not a point). The region I've labelled $I$ is the exterior of the black hole and the region I've labelled $II$ is the region inside the event horizon. Ignore the region of the diagram to the bottom left as it isn't relevant to my question.
Finally, the key feature that makes it possible to answer my question is that all radial ingoing light rays are straight 45° lines running from bottom right to top left. I've drawn several such light rays as magenta lines.
Now we can answer my question. We start with a rocket hovering at a constant distance away from the black hole, which is represented by the black dashed hyperbola of constant $r$ (as I mentioned above you can sort of think about time increasing as you move up). At time $t_0$ our observer leaves the rocket and starts falling towards the black hole. The blue line shows the trajectory followed by the observer. The observer hits the singularity at the point where the blue and red lines meet.
At time $t_1$ the rocket shines a light ray at the infalling observer. The light ray, travelling at 45°, reaches the observer before they cross the event horizon - so far so good. At time $t_2$ the rocket shines a second light ray at the observer, and this light ray reaches the observer just as they hit the singularity. At time $t_3$ the rocket shines a third light ray into the black hole, but this doesn't reach the observer because the observer has already hit the singularity and no longer exists. That means the observer never sees the light ray released at time $t_3$. The observer sees any light ray released between $t_0$ and $t_2$, but doesn't see any light ray released after $t_2$. So the dashed magenta line marks the boundary between light rays the observer can see and ones they can't.
And there is the answer to my question. The observer does not see the end of the universe because the last light ray they see is the one released at time $t_2$.
This doesn't give me an easy way to calculate the value of $t_2$, because I'd have to derive an expression for the trajectory of the infalling observer (blue line) and that's hard. Nevertheless is shows that $t_2$ is finite so, using the notation in my question, $T$ is bounded.
A: To add to the excellent answers above, here is a spacetime diagram in Gullstrand-Painleve or "rain" coordinates. This is from the superb and accessible book Exploring Black Holes (2000) by Taylor & Wheeler, $\S B.6$. Their metaphor "rain" means a test particle with mass, which initially fell from rest far away from the black hole. Think of them as astronauts / observers, for this problem.
$t_\textrm{rain}$ is the proper time of a raindrop, which is used as a coordinate. $r$ is the usual curvature coordinate as in Schwarzschild[-Droste] coordinates, and $M$ is the mass of the black hole. The diagram shows that most "light pulses" never catch up to a given "rain plunger"; in particular they will not see the end of the universe.

A: I agree that for a spacetime that is exactly Schwarzschild, the infalling observer does not see the entire history of the universe.  However, this turns out not to be the generic case you would expect for an astrophysical black hole, which formed from collapse of some approximately spherical distribution of matter.  This topic is actually being actively researched, and there are some very interesting results about what the inside of a black hole actually looks like.  See, for example, this recent paper.
The reason that in Schwarzschild the infalling observer does not see the entire history of the universe is that the singularity is spacelike.  This means that there is a range of points where the infalling observer can hit the singularity, and each point can only see part of the universe in its causal past.  
But people have known of other kinds of black holes for a long time that do not share this behavior.  The best know examples are the Reissner-Nordstrom solution for a charged, spherically symmetric black hole, and the Kerr solution for a spinning black hole.  These both have timelike singularities, and hence the situation is quite different.  Here is a causal diagram of a Reissner-Norstrom black hole:

The vertical jagged lines represent the timelike singularities for this black hole. In this case it is possible to avoid the singularity and emerge into a new universe that you could attach to the top of this picture.  In this case, when you cross the inner horizon, you should be able to look back and see the entire history or the universe.  
This brings up a problematic point however.  The observer passes the inner horizon in finite proper time, yet he is able to see all the light that enters the black hole from the entire infinite history of the universe.  Since light has energy, you might think that this pile up of radiation from the outside universe should lead to a great deal of curvature, and indeed it does.  This is known as a mass inflation instability of the black hole.  Kerr black holes share this feature, although the structure of the singularity in that case is more complicated.
So for generic black holes that are not exactly Schwarzschild, a different behavior is expected.  The perturbations tend to change the singularity from being spacelike to behaving like a null surface, i.e. following the trajectories of light.  A picture from the above paper shows this situation:

The outside universe lives in the bottom right triangle of this picture.  The lines labeled $\mathcal{CH}^+$ are the null singularities.  The paper found that this situation resulted from perturbing the Schwarzschild solution with scalar field matter.  In this case if you fell into the black hole from the outside universe, you would run into the null singularities, and assuming you hit the one on the right, you will see everything the entire history of the universe, in the sense that all that you will have access to light that enters the black hole from arbitrarily late times of the universe's history.  
A: (Michael Brown's answer is the correct answer and this is merely to amplify via an added diagram.)
Below is figure 31.4 from page 835 of Gravitation (MTW).

Both diagrams are of the Schwarzschild geometry.  Note that in the Kruskal-Szekeres coordinates, light cones appear as they do in Minkowski spacetime.
As Michael points out, lightlike radial geodesics are 45 degree lines as can be seen by looking at geodesic B.
Clearly, there are lightlike worldlines that cross the horizon after some timelike worldlines so the worldline of an astronaut falling radially towards the hole does not intersect all the lightlike radial worldlines before crossing the horizon.
Also, it is clear that there are lightlike worldlines that end on the singularity after some timelike worldlines.
Thus, the astronaut does not see the infinite future before crossing the horizon or encountering the singularity.
Also, and this is just an interesting side note to consider,  the Schwarzschild solution is the spherically symmetric static (well, outside the horizon at least) solution to Einstein's equations.  In other words, there is no "end of the universe" in this solution.    
A: No. The black hole will completely evaporate in finite time, so by the end of the universe it will no longer exist.
A: Your question is due to some confusion with the spacetime concept of a black hole. You must distinguish between your coordinate system and what you see. Both are different concepts: One simple example is a Minkowski space: If a Minkowski diagram represents your coordinates, you get a fourdimensional view of the whole spacetime. In contrast, what you  s e e  are elements which are located on your lightcone which is showing into the past.
Near a black hole we must apply the same distinction of this twofold concept which can be shown in the following Kruskal diagram, with one infalling particle A and one particle remaining outside B:

The time coordinates of a far-away observer are indicated by the lines going through the center: t = 0, t = 1, t = 2, limited by the event horizon where t = $\infty$. According to these time coordinates, the infalling particle will never reach the horizon. And conversely, when A is approaching the horizon, the clock of an outside observer will approach the end of time.
Perhaps this is the reason why you asked your question. But your question is not asking: what is the position with respect to the coordinates of an outside observer, but: what does the infalling particle see, and for this question you must refer (as shown in other answers) to the small 45°-arrows between the communicating particles A and B. The 3 diagonal arrows from the bottom to the left are showing that B is at a certain point when A touches the event horizon.
A: Inspired by a similar question, I've been working on this topic at the same time as Rob Jeffries. Irritatingly, he beat me to it; but since I use a slightly different approach and since I don't want my efforts to be in vain, I'll post my own derivation. If nothing else, it serves as a confirmation of his fantastic answer :)
Let us start by stating the (region I) Kruskal–Szekeres coordinates
$$u = f(r)\cosh\left(\frac{ct}{2r_\text{s}}\right), \qquad v = f(r)\sinh\left(\frac{ct}{2r_\text{s}}\right),\\
f(r) = \left(\frac{r}{r_\text{s}}-1\right)^{\!1/2}\,\text{e}^{r/2r_\text{s}}.$$
As is well known, in these coordinates the geodesics of radially infalling light rays are straight lines at $-45^\circ$ angles. Indeed, if we plug $u + v= k$ into the equations, with $k$ a constant, then from $u^2 - v^2 = f(r)^2$ we find $k(u -v) = f(r)^2$, so that
$$
k\exp\left(-\frac{ct}{2r_\text{s}}\right) = f(r) =  \left(\frac{r}{r_\text{s}}-1\right)^{\!1/2}\,\exp\left(\frac{r}{2r_\text{s}}\right),
$$
or
$$
\frac{ct}{r_\text{s}} = \ln(k^2)-\frac{r}{r_\text{s}} - \ln\left(\frac{r}{r_\text{s}}-1\right),
$$
which are indeed the geodesics of a radially infalling photon, with $k = f(r_{0,\gamma})$ and $r_{0,\gamma}$ the initial position of the photon at $t=0$.
Now, let us assume that we have a radially infalling object, which starts at rest at a position $r_0$ at $t=0$. Which radially infalling photons will reach the object before it crosses the event horizon? To answer this, we will try to derive the geodesic of a radially infalling photon such that it catches up with the object right at the event horizon.
The geodesic of a radially infalling object can be written in the form (Misner, Thorne & Wheeler Eq. (31.10), Pag. 824) 
$$\begin{align}
r &= \frac{r_0}{2}(1+\cos\eta) = r_0\cos^2\eta/2,\\
\frac{c\tau}{r_\text{s}} &= \frac{1}{2}\left(\frac{r_0}{r_\text{s}}\right)^{\!3/2}(\eta + \sin\eta),\\
\frac{ct}{r_\text{s}} &= \ln\left(\frac{\sqrt{r_0/r_\text{s} -1} + \tan\eta/2}{\sqrt{r_0/r_\text{s} -1} - \tan\eta/2}\right) + \left(\frac{r_0}{r_\text{s}}-1\right)^{\!1/2}\left(\eta + \frac{r_0}{2r_\text{s}}(\eta + \sin\eta)\right).
\end{align}
$$
It's also instructive to introduce the (dimensionless) total energy of the object
$$
E = \frac{\mathcal{E}}{mc^2} = \left(1- \frac{r_\text{s}}{r}\right)\frac{\text{d}t}{\text{d}\tau}.
$$
Radial orbits satisfy the equation
$$
\left(\frac{\text{d}r}{c\,\text{d}\tau}\right)^2 = E^2 - \left(1- \frac{r_\text{s}}{r}\right),
$$
so if the object is at rest at position $r_0$ at $t = \tau = 0$, then
$$
E = \sqrt{1- \frac{r_\text{s}}{r_0}}.
$$
The equation for $t(\eta)$ can thus be rewritten as
$$
\frac{ct}{r_\text{s}} = \ln\left(\frac{E + \sqrt{1-E^2}\tan\eta/2}{E - \sqrt{1-E^2}\tan\eta/2}\right) + \frac{E}{\sqrt{1-E^2}}\left(\eta + \frac{\eta + \sin\eta}{2(1-E^2)}\right).
$$ 
Next, I will follow this article (which contains a few errors) to derive how this equation behaves as $r$ approaches the event horizon. We write
$$
r = r_\text{s}(1+\varepsilon),\qquad \varepsilon\rightarrow 0.
$$
Near the event horizon we can ignore higher-order terms in $\varepsilon$, so that
$$\begin{align}
\cos^2\eta/2 &= (1+\varepsilon)\frac{r_\text{s}}{r_0} = (1+\varepsilon)(1-E^2) = 1 - E^2 + \varepsilon(1-E^2),\\
\sin^2\eta/2 &= E^2 - \varepsilon(1-E^2),
\end{align}
$$
and
$$\begin{align}
(1-E^2)\tan^2\eta/2 &= \frac{E^2 - \varepsilon(1-E^2)}{(1+\varepsilon)} \\ &\approx \left[E^2 - \varepsilon(1-E^2)\right](1-\varepsilon)\\
&\approx E^2 - \varepsilon(1-E^2) - \varepsilon)E^2 = E^2 - \varepsilon.
\end{align}
$$
Therefore,
$$\begin{align}
E + \sqrt{1-E^2}\tan\eta/2&\approx E\left(1 + \sqrt{1 - \varepsilon/E^2}\right)\\
&\approx 2E - \frac{\varepsilon}{2E}\approx 2E,
\end{align}
$$
and
$$\begin{align}
E - \sqrt{1-E^2}\tan\eta/2&\approx E\left(1 - \sqrt{1 - \varepsilon/E^2}\right)\\
&\approx \frac{\varepsilon}{2E},
\end{align}
$$
So that finally, as $r\rightarrow r_\text{s}$, 
$$
\frac{ct_\text{s}}{r_\text{s}} \approx \ln\left(\frac{4E^2}{\varepsilon}\right) + \frac{E}{\sqrt{1-E^2}}\left(\eta_\text{s} + \frac{\eta_\text{s} + \sin\eta_\text{s}}{2(1-E^2)}\right),
$$ 
with $\eta_\text{s}$ the value of $\eta$ at the event horizon. Since
$\cosh(x) = \sinh(x) \rightarrow \text{e}^x/2$ as $x\rightarrow\infty$, the Kruskal–Szekeres coordinates of the object at the event horizon become (since $t\rightarrow\infty$)
$$\begin{align}
u_\text{s}^2 = v_\text{s}^2 &= \frac{1}{4}f(r_\text{s})^2\,\exp\left(\frac{ct_\text{s}}{r_\text{s}}\right)
= \frac{\varepsilon\text{e}}{4}\exp\left(\frac{ct_\text{s}}{r_\text{s}}\right)\\
&= \text{e}E^2
\exp\left[\frac{E}{\sqrt{1-E^2}}\left(\eta_\text{s} + \frac{\eta_\text{s} + \sin\eta_\text{s}}{2(1-E^2)}\right)\right],
\end{align}
$$
or
$$\begin{align}
u_\text{s} = v_\text{s} &= \sqrt{\text{e}}E
\exp\left[\frac{E}{2\sqrt{1-E^2}}\left(\eta_\text{s} + \frac{\eta_\text{s} + \sin\eta_\text{s}}{2(1-E^2)}\right)\right]\\
&= \sqrt{\text{e}}\sqrt{\frac{r_\text{s}}{r_0}}\left(\frac{r_0}{r_\text{s}}- 1\right)^{\!1/2}
\exp\left[\frac{1}{2}\left(\frac{r_0}{r_\text{s}}-1\right)^{\!1/2}\left(\eta_\text{s} + \frac{r_0}{2r_\text{s}}(\eta_\text{s} + \sin\eta_\text{s})\right)\right].
\end{align}
$$
The corresponding coordinates for a radially infalling photon satisfy
$u_\text{s} + v_\text{s} = k_\text{b}$, for some boundary value $k_\text{b}$.
Therefore $k_\text{b} = 2u_\text{s}$ and we find the corresponding null geodesic
$$
\frac{ct}{r_\text{s}} = \ln(k_\text{b}^2)-\frac{r}{r_\text{s}} - \ln\left(\frac{r}{r_\text{s}}-1\right).
$$
We could solve this for $r$ at $t=0$, which yields a boundary radius $r_\text{b}$ beyond which radially infalling photons cannot catch up with the object before it crosses the event horizon. Alternatively, we could plug in $r=r_0$ and ask what the maximum time $\Delta t$ is such that photons sent out at $r_0$ before $t=\Delta t$ can still catch up with the object. We find
$$\begin{align}
\frac{c\Delta t}{r_\text{s}} &= \ln(k_\text{b}^2)-\frac{r_0}{r_\text{s}} - \ln\left(\frac{r_0}{r_\text{s}}-1\right)\\
&= 1 + \ln\left(\frac{4r_\text{s}}{r_0}\right) + \left[\left(\frac{r_0}{r_\text{s}}-1\right)^{\!1/2}\left(\eta_\text{s} + \frac{r_0}{2r_\text{s}}(\eta_\text{s} + \sin\eta_\text{s})\right)\right]-\frac{r_0}{r_\text{s}},
\end{align}
$$
which is exactly the same result as given by Rob Jeffries.
I made a plot to visualize the results, in Schwarzschild and Kruskal–Szekeres coordinates:

The blue curve is the geodesic of an object, at rest at $t=0$ (here, $r_0 = 2r_\text{s}$). The orange curve is the geodesic of a photon which is at position $r_0$ at $t=0$. The red curve is the geodesic that I derived in this post. It starts at position $r_\text{b}$ at $t=0$ and catches up with the object right at the event horizon. Geodesics of photons that lie between the orange and red curves (I've plotted two, the dashed-lined curves) will catch up with the object, geodesics beyond the red curve will not.
