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black hole gedanken


Norman Albers
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I want you to tell me about the eighth dimension, then see embarasement.

 

Dan, Norman seems to be getting the response he wants from the other posters in the thread, which he started with his question. Maybe it would be wisest for you not to interfere.

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WHAT, ME, WORRY??

 

I am a bit confused about units so here's how I proceed: [math] L=\int_{2m}^r dr/\sqrt{(1-2m/r) } =\int_{2m}^r dr \sqrt{r \over {r-2m} }[/math], where the upper limit is "far away" . Rescale according to: R=r/2m, and write: [math]L=2m\int_1^R dR \sqrt{R\over {R-1}} = 2m \int_1^R dR \frac{R}{ \sqrt{R^2-R}} [/math]. Add and subtract to make a perfect differential: [math] L=m \int_1^R dR \left( \frac {2R-1}{\sqrt{R^2-R}}+\frac{1} {\sqrt{R^2-R}}\right) = 2m\sqrt{R^2-R}|_1^R~~ +m\int_1^R {dR \over{\sqrt{R^2-R}} } [/math]. The latter integral from CRC tables, is:[math] m log(2\sqrt{R^2-R} +2R-1)|_1^R [/math]. We can see that at large R the expression tends to 2mR; at R=1, the event horizon, the first term is zero, and the argument of the logarithm becomes 1 and so the log is zero. This says the total distance covered by a string will be that extra 2m, compared with the external coordinate measure by which we are observing. This is a pretty fun result. I get messed up if I don't do the rescaling first, as I don't understand what goes into the log argument, but I trust this solution, as unitized above, better.

 

Satisfyingly, the answer to the scaling problem here came to me at 3am. It made me nervous that you come up with [math]log 2m [/math] if you don't unitize things by rescaling. HOWEVER, this only amounts to a constant of integration, so the problem goes away. Hand-waving and cheering. We'll see if BenTheMan buys this as a valid case (of hand-waving).

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  • 2 weeks later...

Yes Farsight, put that print through a 1/r inversion. Now I'm not ready to get sucked into interior physics; this makes turkeys of most of us. As I approach that study I will look at assumptions about proper-time physics, as it seems to me it might be looked at as a phase change in the vacuum fields.

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There is no interior physics, Norm. Put that print through a 1/r inversion and there is no interior. When your vacuum field is frozen because of a phase change, things change radically. If nothing can move there's no energy, no light, no time, no distance, no nothing. It isn't there any more. A black hole really is a hole in space. The event horizon is the end of events. And proper time isn't proper at all. Simple really.

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Looked at with my polarizable vacuum perspective, the interior modes of radiation in the transverse senses (tangent to a sphere) are strongly absorbed except near the center. This might imply a degenerate one-dimensional mode-space. Fred, yes but let's try to get clear on the relation of the "passage of time" in the different frames of reference. The ticking of clocks for the infalling observer really gets slower than the outer observer's. Yes one can speak of the experience of infalling but it takes asymptotically long by external clocks, and this will be mirrored by the apparent blueshift and time-lapse perception of the light coming in from the outer observer. Now maybe this becomes not relevant as the time constant is not that long, to where we get within quantum wavelengths, but you have to deal with the fact that the metric coefficient (1-2m/r) becomes zero. Then again, maybe the answer here lies in the question, "Given a BH and a shell of accreting matter, can we say this collapses to a BH of that much more mass?"

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We develop methods to extract resonance widths from finite volume spectra of (1+1)-dimensional quantum field theories. Our two methods are based on Lüscher's description of finite size corrections, and are dubbed the Breit–Wigner and the improved “mini-Hamiltonian” method, respectively. We establish a consistent framework for the finite volume description of sufficiently narrow resonances that takes into account the finite size corrections and mass shifts properly. Using predictions from form factor perturbation theory, we test the two methods against finite size data from truncated conformal space approach, and find excellent agreement which confirms both the theoretical framework and the numerical validity of the methods. Although our investigation is carried out in 1+1 dimensions, the extension to physical 3+1 space–time dimensions appears straightforward, given sufficiently accurate finite volume spectra.

Norman, would these modelling in (1+1) be of any interest, or what?

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Just wanting to know if anyone can explain what it means. What resonance is it talking about? I did experimental phys., and I've measured the DeBroglie electron. But I was more asking about quantum field theories in general. Here he's talking about mass resonance or something? Cheers

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