Time dilation when falling into a black hole

[PeterDonis]

Then, how would you fit this in?:

PAllen gave a good answer. The only thing I would add is that, using the convention that A.T. was using, it "takes forever" for you to reach the *horizon*--this convention simply can't comprehend *at all* anything that happens to you at or below the horizon. So assuming that you are still alive when you cross the horizon, it would take "more than forever", so to speak, for you to die from the viewpoint of the people on Earth. But as PAllen said, this is just a convention, and not the only possible one that the people on Earth could adopt.

 

[PAllen]

I thought it might be useful to make concrete the point about the conventionality of simultaneity in both SR and GR.

For non-inertial observers in SR, two common conventions are:

- Fermi-Normal coordinates (using the simultaneity of an instantaneously comoving inertial observer).
- Radar simultaneity - using round trip light signals to define simultaneity.

These become arbitrarily close to each other in a sufficiently small region of spacetime around the observer (as would be desired of any 'physically plausible' simultaneity). However, they are not only radically different 'far away', but they cover different regions of spacetime!. The Fermi-Normal convention typically covers only part of spacetime with a valid coordinate chart, while radar (in SR) will cover all of spacetime.

Similarly, I propose a physically plausible alternate simultaneity for SC geometry for a distant static observer (it closely matches SC coordinates over a small region of spacetime for the distant static observer). The convention is as follows:

- consider launching test bodies on radial geodesics at .9c from the distant static observer. For every SC (t,r) coordinate (both interior and exterior), there is an event on the the observe'rs world line that could reach that (t,r) with a test body probe. Compute the proper time along the test body world line, multiply by gamma(.9c), and declare this simultaneous with this amount past the launch event, on the observer's world line.

Having charted (t,r), the angular coordinates are the same as SC due spherical symmetry. This chart smoothly covers one exterior and one black hole interior section of the complete Kruskal geometry in physcially motivated way. You can now talk about 'when' any event inside the horizon occurs in relation to the distant observer, and even compute proper distances along these simultaneity surfaces to events inside the horizon.

 

[PeterDonis]

Similarly, I propose a physically plausible alternate simultaneity for SC geometry for a distant static observer (it closely matches SC coordinates over a small region of spacetime for the distant static observer).

I have an even easier proposal, which basically amounts to using Painleve coordinate time as the standard instead of Schwarzschild coordinate time. Since at very large r, the two are almost the same, the idea is this: just use proper time along any infalling geodesic as the standard of simultaneity. That is, if it takes me one year of my proper time to fall from a faraway space station to the horizon of a BH, then the event of my crossing the horizon is deemed to be simultaneous with an event on the station's worldline one year, by the station clock, after I leave the station.

One key thing I would want to look at, for both these proposals, is how they match up with light travel times. That is, when we combine the suggested simultaneity convention with the travel times of light, do we get something reasonable? For example, if the faraway space station sends a laser message after me, at what time by the station's clock would it have to be sent in order to reach me just as I cross the horizon?

In the case of your suggested convention, I would want to formulate a similar comparison: the "time" by the station clock at which I am deemed to cross the horizon is determined by the time at which a 0.9c test probe would need to be sent after me in order to reach me just as I cross the horizon, plus the elapsed proper time for the probe times gamma. At what time, by the station clock, would a laser message have to be sent in order to reach me just as the test probe does?

(One thing that bothers me a bit about your proposal is that the comparison I just described depends partly on gamma--i.e., on the velocity of the test probe--as well as on when I leave the station. In my version, that's not the case.)

 

[PAllen]

I have an even easier proposal, which basically amounts to using Painleve coordinate time as the standard instead of Schwarzschild coordinate time. Since at very large r, the two are almost the same, the idea is this: just use proper time along any infalling geodesic as the standard of simultaneity. That is, if it takes me one year of my proper time to fall from a faraway space station to the horizon of a BH, then the event of my crossing the horizon is deemed to be simultaneous with an event on the station's worldline one year, by the station clock, after I leave the station.

One key thing I would want to look at, for both these proposals, is how they match up with light travel times. That is, when we combine the suggested simultaneity convention with the travel times of light, do we get something reasonable? For example, if the faraway space station sends a laser message after me, at what time by the station's clock would it have to be sent in order to reach me just as I cross the horizon?

In the case of your suggested convention, I would want to formulate a similar comparison: the "time" by the station clock at which I am deemed to cross the horizon is determined by the time at which a 0.9c test probe would need to be sent after me in order to reach me just as I cross the horizon, plus the elapsed proper time for the probe times gamma. At what time, by the station clock, would a laser message have to be sent in order to reach me just as the test probe does?

(One thing that bothers me a bit about your proposal is that the comparison I just described depends partly on gamma--i.e., on the velocity of the test probe--as well as on when I leave the station. In my version, that's not the case.)

Yours is effectively the gamma=1 case of mine, I think. I am really proposing a family of possible coordinate charts, that depend on two parameters: reference r value, and gamma. I think you do need a specific reference r value since proper time on a radial geodesic from infinity is infinite. My motivation for using a significant gamma was simply to make it 'look' more like half of the radar simultaneity convention. It would be interesting to compute how gamma affects the chart, but I am not motivated to do that. I suspect the effect may be subtle, because multiplication by gamma removes a large part of the difference.

Also, I believe these charts lose the 'manifest' static representation of SC coordinates. Some metric components would seem to depend on the new t. I suspect this is an inevitable tradeoff.

[edit: one can even conceive of taking the gamma=infinity limit of these charts. Not sure how that would work out.]

[edit2: Maybe you are thinking of the family of free fall from infinity geodesics. In which case this becomes the (gamma,r)=(1,infinity) member of my family of coordinates. ]

 

[guss]

Thanks a lot guys, I think I have a good understanding of it now (or at least a good understanding of what I currently understand and what I don't :p). That previous post by A.T. was sort of what was throwing me off the whole time, because I was misunderstanding what was being said.