Computing a frequency shift
dpyikes said:
I’m afraid I continue to be perplexed about the way time dilation works inside the event horizon, particularly for the case of a collapsing star.
So-called "time dilation" concerns lightlike signals sent from one observer to another observer.
This notion has no meaning unless you specify an ordered pair of observers. In the above, I guess you mean an observer on the surface of a collapsing spherical dust ball (in an Oppenheimer-Snyder model) sending time signals once per second by his ideal clock to a very distant static observer.
dpyikes said:
What I am interested in is a sequence of light rays which had originated from flat space, and are striking the surface of the star.
OK, now I guess that you mean a distant static observer sending signals once per second (by his clock) to an observer on the surface of a collapsing spherical dust ball in an OS model (not the same thing at all!).
Everyone,
please, make the effort to try to specify which pair of observers you are talking about, or endless confusion will result! I'd also once again highly recommend the two popular books which I keep recommending (see link given in my immediately preceding posts in this thread above).
dpyikes said:
I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.
There is no flat
spacetime here. I doubt you were referring to the flat spatial hyperslices in the Painleve chart; rather, I guess that you were referring to
the fact that the exterior region is asymptotically flat, so that very far away from the collapsing dust ball, our OS spacetime is
approximately Minkowskian.
I don't understand what you are worried about, but I can help you compute the frequency shift in this scenario (assuming I understand what you have in mind).
To construct the OS model, we match a region filled with collapsing dust (modeled using a portion of an FRW dust model with E^3 hyperslices orthogonal to the world lines of the dust) across a collapsing sphere to an asymptotically flat vacuum exterior region (modeled using a portion of the Schwarzschild vacuum solution). A key observation is that since a dust is a pressureless perfect fluid, there are no forces acting on the dust particles, so their world lines will be timelike geodesics. In particular, the world lines of observers on the surface will be timelike geodesics.
Now, as you would expect, our dust ball must be momentarily at rest at some r=r_0, where r is the Schwarzschild radial coordinate of the surface in the exterior region. To keep things as simple as possible, we should take the limit where r_0 \rightarrow \infty, and then observers riding on the surface of the collapsing dust ball will be "Lemaitre observers" (the ones involved in the Painleve chart, called "free-fall chart" by Andrew Hamilton; see the figure just above the one you mentioned). If we took a finite dust ball, we would use "Novikov observers".
Let me outline how Hamilton came up with the picture he labels "free-fall spacetime diagram".
In the Painleve chart (1921), the metric tensor can be written
ds^2 = -dT^2 + \left( dr + \sqrt{m/r} \, dT \right)^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
= -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < T < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
The world lines of the Lemaitre observers are the integral curves of
\vec{e}_0 = \partial_T - \sqrt{2m/r}\, \partial_r
That is:
T - T_0 = \sqrt{\frac{2 \, (r-r_0)^3}{9m}}
The world lines of radially ingoing light rays are the integral curves of
\vec{k} = \partial_T - \left( 1 + \sqrt{2m/r} \right) \, \partial_r
That is:
T - T_0 = -\left( r-r_0 \right) + \sqrt{8 m} \, \left( r-r_0 \right) - 4 m \, \log \frac{\sqrt{r}+\sqrt{2m}}{\sqrt{r_0}+\sqrt{2m}}
So now you can draw a bunch of light signals sent from a very distant static observer (once per second, by his ideal clock) to an observer standing on the surface of the collapsing dust ball. You should get the same picture as Hamilton (see "Freely fall spacetime diagram", where the bright yellow curves depict the world lines of some radially infalling null geodesics, and where the light green curves depict the world lines of some Lemaitre observers.)
We can augment the timelike unit vector field \vec{e}_0 with three spacelike unit vector fields, all mutually orthogonal, in order to obtain the frame field we would use to describe the physical experience of our Lemaitre observers:
\vec{e}_1 = \partial_r, \; \vec{e}_2 = \frac{1}{r} \, \partial_{\theta}, \; \vec{e}_3 = \frac{1}{r \, \sin(\theta)} \, \partial_{\phi}
Then \vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3 is an example of an
inertial nonspinning frame field, which is as close as we can come in a curved spacetime to a "local Lorentz frame". As I said, the timelike congruence defining the Lemaitre observers is obtained from \vec{e}_0; note that the null congruence of radially ingoing null geodesics is \vec{k} = \vec{e}_0 - \vec{e}_1. Do you see why we can see at a glance that this is a null vector field, indeed a "radial" null vector field? Can you guess what would be the null vector field defining the null congruence of radially outgoing null geodesics?
I stress that the Painleve chart is defined down to the curvature singularity at r=0 and has two delightful properties:
1. the hyperslices T=T_0 are locally isometric to E^3, with a polar spherical chart in which the radial coordinate is just the Schwarzschild radial coordinate,
2. the difference of Painleve time coordinate T_2-T_1 for two events on the world line of one of our Lemaitre observers directly gives the elapsed proper time as measured by ideal clocks carried by this observer.
So we want to compute the frequency of the signals upon reception by our surface riding Lemaitre observer, as measured by his own ideal clock. But now that I've gotten you started, maybe I should give you an opportunity to try to finish the computation...
Let me just go back to something I said above: you probably already appreciate that if we compute the frequency shift for signals sent from the surface-riding observer back to our very distant static observer, this turns out to be a red shift which diverges as the falling observer passes the horizon. Before carrying out the computation, from looking at the diagram in Andrew Hamilton's website, can you guess what should happen in the case of signals sent from our very distant static observer to our surface riding observer?
My previous (post #52) was strictly aimed at Schwarzschild coordinates, where I tried to correct my poor definitions of observers outside the horizon.