Maximizing survival time falling into a black hole

[George Jones]

Cusp == Geraint

Sorry, I should have read post #6 more closely. I only scanned it, and I got confused by alternating passages in quotes and not in quotes. This shines a very ironic light on my comments in post #7 .

I hope your interesting paper becomes well known. I read it when v2 came out on the arxiv, and I, too, thought that there was a problem with one of the problems in Misner, Thorne, and Wheeler. After looking more closely at the problem in MTW, I decided there was no problem.

 

[MeJennifer]

Yes, that's what is claimed

So then it is presumed that different observers can reach the singularity at different coordinate times?

 

[Cusp]

So then it is presumed that different observers can reach the singularity at different coordinate times?

Yes - the figures in the paper show this, although it's not presumed, it's calculated.

 

[MeJennifer]

Yes - the figures in the paper show this, although it's not presumed, it's calculated.

Yes, I read the paper, observers reaching the singularity at different coordinate times.

So perhaps, in general, for the good understanding, which charts of the Schwartzschild metric show an observer reach the singularity in finite T?

 

[George Jones]

So then it is presumed that different observers can reach the singularity at different coordinate times?

Yes, different observers can reach the singularity at values of the coordinate $t$. See Figure 4. in the arXiv paper (link given in post #2).

I have bit of a beef with the terminology used by both you and the paper. As with Schwarzschild coordinates, $t$ is a space coordinate inside the event horizon, since

$$g \left( \frac{ \partial}{\partial t} , \frac{ \partial}{\partial t} \right) = - \left( 1 - \frac{2m}{r} \right),$$

which is positive inside the horizon.

It is interesting to note that, unlike Schwarzschild coordinates, $r$ is spacelike outside the event horizon, on the event horizon, and inside the event horizon, since

$$g \left( \frac{ \partial}{\partial r} , \frac{ \partial}{\partial r} \right) = \left( 1 + \frac{2m}{r} \right)$$

is always positive.

This is an example of what Penrose calls Woodhouse's Second (or is it first?) Fundamental Confusion of Calculus.

Let $(t,r)$ be Schwarzschild coodinates. Define Eddington-Finkelstein coordinates by

$$T(t,r) = t + 2m ln \left| \frac{r}{2m} - 1 \right|$$
$$R(t,r) = r.$$

Even though $R = r,$ $\partial / \partial R$ is not the same as $\partial / \partial r$ because lines of constant $(T,\theta,\phi)$ are not the same as lines of constant $(t,\theta,\phi).$

 

[MeJennifer]

As with Schwarzschild coordinates, $t$ is a space coordinate inside the event horizon, since

$$g \left( \frac{ \partial}{\partial t} , \frac{ \partial}{\partial t} \right) = - \left( 1 - \frac{2m}{r} \right),$$

which is positive inside the horizon.

A short OT reply here:
Looking at it from a complex Euclidean CE⁴ space with both a + - - - and a - + + + section, I am not at all convinced that we can equate a sign change in t with space.

 

[pervect]

A short OT reply here:
Looking at it from a complex Euclidean CE⁴ space with both a + - - - and a - + + + section, I am not at all convinced that we can equate a sign change in t with space.

I don't understand your difficulty. Which of the following are you having difficulty with?

1)A vector u can be either space-like or time-like, depending on the sign of
$$g_{ij} u^i u^u$$.

2) Given a coordinate system, (t, etc), a vector pointing in the direction of increasing t is $$\frac{\partial}{\partial t}$$ . Similarly, a vector pointing in the direction of x is $$\frac{\partial}{\partial x}$$

3) Given 1 & 2, the only remaining issue is that George has made some sort of math error, but I don't think that's your claim.

 

[pervect]

This is an example of what Penrose calls Woodhouse's Second (or is it first?) Fundamental Confusion of Calculus.

It's "second fundamental confusion of calculus" in my edition of "The Road to Reality" - pg 190, fig 10.7.

I ran into some similar issues not very long ago.

 

[MeJennifer]

I don't understand your difficulty.

It is not related to some difficulty but a matter of interpretation of the reality condition.
I remember we had a discussion about this before where everybody claimed not to understand my "difficulty" so there does not seem to be a point to repeat this. It appears I can only have a serious discussion about this with myself.

1)A vector u can be either space-like or time-like, depending on the sign of
$$g_{ij} u^i u^u$$.

That´s like saying a number can be either positive or negative. There happen to be imaginary numbers as well.

Anyway, we really should get back on topic. So let´s just say that I seriously misunderstand things here and move on.

So the argument, if I understand the paper correctly, is that since free falling observers can arrive at the singularity at different coordinate times it must be possible for some of those observers to accelerate in such a way that the coordinate time of arrival changes, with a result that the total proper time is increased with respect to the case in which that particular observer would not have accelerated. Is that right?

 

[pervect]

Travelling a geodesic path means that the proper time between any two fixed points is maximized.

To apply the analysis, the points must be fixed. The problem is that "hitting the singularity" isn't such a fixed point - it can be and usually is a different event with different starting velociteis.

However, everyone seems to agree that the limiting case is an observer who falls in such a way that (dr/dtau) -> 0 at the event horizon, r=2M. This isn't just a local maximum, it's a global maximum. (At least that's what I gather, I haven't looked at this terribly closely). It's really is not too surprising that such a global maximum is also a local maximum (i.e. a geodesic).

Strictly speaking, however, this limiting-case observer isn't actually crossing the event horizon.

So for someone who actually crosses the event horizon, they have to accelerate, as quickly as possible, to achieve the same energy-at-infinity (zero) as the observer above, one who starts out with dr/dtau = 0 at 2M. After they have done this, their best bet is to "coast" the rest of the way in.

 

[MeJennifer]

The problem is that "hitting the singularity" isn't such a fixed point - it can be and usually is a different event with different starting velocities.

Oh?

Frankly, I think that this does not make a lot of sense (and it looks like I am the only one again), perhaps you could provide me with some specific references to what you say, so I can study this.

Hopefully the paper we are discussing is not debunking a myth by affirming another one, namely that "the" singularity is some local entity.

 

[pervect]

The best references (IMO) have already been give - MTW, $25.5 and pgs 824-832 and exercise 31.4 on pg 836 and the preprint.

For the preprint http://www.arxiv.org/abs/0705.1029 see figure 4

For the red line, the faller fires their rocket all the way to the singularity, while the dark blue, light blue and green turn off their rocket when e = 0.3, e = −0.3 and e = 0 respectively. An examination of the proper time in the left-hand panel reveals that it the path that settles on e = 0 that possesses the longest proper time.

also

In dropping from rest at the event horizon, the firing of a
rocket does not extend the time left, it only diminishes it

The only question is what is meant by "at rest at the event horizon" - I assume that this means that dr/dtau is zero at the horizon. The observer "at rest" can't be one with a constant r coordinate - such observers, usually called stationary observers, are timelike only outside the event horizon. The null geodesic of an outgoing light ray is stationary at the event horizon, but the path of an observer must be timelike, not lightlike or spacelike.

In any given coordinate system, two events are the same if and only if ALL coordinates for both events are the same. If any coordinate is different, the events are different.

As far as computing geodesics go, I'd strongly recommend reading a textbook or the preprint section on conserved quantities, aka "Killing vectors". For geodesic radial motion, the conserved energy parameter (which I call energy-at-infinity as that's what MTW calls it) gives dr/dtau as a function of r. (For general motion, you also need the angular momentum, but for radial motion, the angular momentum is zero and you only need the energy parameter).

This doesn't work for non-geodesics, but it should be enough to illustrate how energy influences geodesic path. Note that you'll have to pick the correct formula for the conserved quantity depending on what coordinates you use, but all coordinate systems will have some way of expressing the numerical value of this conserved quantity.

Exercise: pick two radial geodesic paths, which terminate on the same event. Trace them backwards in time. Do they ever cross?

 

[MeJennifer]

I was simply inquiring about this notion of observers hitting "the" singularity at different times.

If "the" singularity is supposedly something that is a collection of separate events then there is no point in talking about hitting it. Also if we time reverse the situation (something that is perfectly valid in relativity) and look at it as a white hole, I am led to believe that geodesics start from this singularity at different times? What is supposedly the meaning of time and space at the singularity, to me is seems it is uttter nonsense to claim that different observers hit "the" singularity at different times.

Everybody else seems to claim the contrary, hence my request for references.

 

[pervect]

I was simply inquiring about this notion of observers hitting "the" singularity at different times.

If "the" singularity is supposedly something that is a collection of separate events then there is no point in talking about hitting it.

Huh?

Look at a Penrose diagram, or a Kruskal-Szerkes diagram. I think you have MTW? If so look at figure 31.4.

If not, try

http://casa.colorado.edu/~ajsh/schwp.html#penrose
http://casa.colorado.edu/~ajsh/schwp.html#kruskal

Is the singularity a point on these diagrams (for the lurkers, NO!), or is it a line (for the lurkers, YES!).

For extra credit. Consider two nearby points on set of points (the line) that forms the singularity. These two points occur at the same Schwarzschild coordinate r, and at different Schwarzschild coordinate t.

Is the Lorentz interval between these points timelike, or spacelike?

Also if we time reverse the situation (something that is perfectly valid in relativity) and look at it as a white hole, I am led to believe that geodesics start from this singularity at different times? What is supposedly the meaning of time and space at the singularity, to me is seems it is uttter nonsense to claim that different observers hit "the" singularity at different times.

Everybody else seems to claim the contrary, hence my request for references.

I don't understand where you got your ideas, much less why you think "everyone else claims the contrary". If you think there are contrary claims, please cite them.

 

[MeJennifer]

Yes, I have MTW and most other published GR books.

Is the singularity a point on these diagrams (for the lurkers, NO!), or is it a line (for the lurkers, YES!).

I see, so are you saying the singularity is a line?

I don't understand where you got your ideas, much less why you think "everyone else claims the contrary". If you think there are contrary claims, please cite them.

You misunderstand me, I was asking for references that help me to show that I am supposedly wrong or misguided in thinking that the notion of hitting the singularity at different times does not make any physical sense.
So far none has been given.

For extra credit. Consider two nearby points on set of points (the line) that forms the singularity. These two points occur at the same Schwarzschild coordinate r, and at different Schwarzschild coordinate t.

Ok for extra credit my answer:
The notion above that the singularity is at charted points on the manifold is false. The singularity is not on the manifold.

 

[pervect]

The notion that geodesics hit the singularity at different "times" is indeed suspect. However, if you will go back over the thread, you should notice that I said "different events", and not "different times". The reason I phrased it this way is that while the topology of the set of events that form the singularity is that of a line, the separation between nearby points of that line is not timelike, but spacelike.

I was hoping to lead you to this conclusion on your own, but you are resisting. A large part of the problem seems to be your rejection of the classification of 4-vectors as space-like or time-like. I really don't understand what your problem is here, but I think I've done about all I can to explain, the rest is up to you.

I think I've given enough references - what in my opinion needs to happen next is that you have to start reading them. And if you do have a disagreement, it's time (past time) for you to start citing specific references.

I'm going to lock this thread to give you some more time to read, before you shoot your mouth off again. I intend to unlock in again in a few days when things have hopefully calmed down (and if they don't calm down, I'll relock it).