Are Hawking's Equations for Black Hole Energy and Evaporation Time Flawed?

[liometopum]

Hawking's equations for black hole energy, power, and evaporation time are unattractive. http://en.wikipedia.org/wiki/Hawking_radiation

The energy equation, with its denominator of 8 pi GM, isn't too bad, but it has already been challenged as being off by a factor of two. See arxiv, "Black Hole Thermodynamics and the Factor of 2 Problem"

The power equation is just plain ugly, with "15360 pi " in the denominator. I don't think there is any other cosmological equation that looks so wrong.
The evaporation time appears ugly too, with a "5120 pi" in its denominator.

Given that the energy equation is possibly wrong, simply transferring that correction through the power and evaporation equation reduces their awkwardness, but the problem remains that these equations are anything but elegant, and elegance appears to be a rule in physics and cosmology. Consider how simple and elegant the Planck relation is.

Do any of you have ideas or opinions on this? Do you people believe these equations are correct? There is no experimental confirmation for them. And if the energy equation is wrong, the others are too.

 

[Simon Bridge]

Well, "unattractive", "ugly", "awkward", are not normally considered scientific reasons for considering a proposed relation invalid.
I realize that the truth is poetically famous for it's aesthetic appeal, however, how it looks written down as math is not usually part of that.
Math is infamous for it's lack of appeal isn't it?

One thing does spring out from the account though:
In the absence of experimental evidence - how do the authors know it's out by a factor of 2?

My take: Looks like business as usual to me.

These are theories - lots of things get proposed all the time.
Almost all of them turn out to be wrong, and almost all of the rest need to be modified.
You get used to it.

Your reference is:
Pilling T. Black hole thermodynamics and the factor of 2 problem
Physics Letters B; Volume 660, Issue 4, 28 February 2008, Pages 402–406
... but it's $40 for the full text ... a (possibly) earlier version (v4. Jan 2008) is found in
http://arxiv.org/abs/0709.1624

The author explicitly assumes that Hawking got black-hole thermodynamics right (p2) and go on to show that the tunneling relation depends on the definition used for entropy.

In particular, if the radiation temperature were different from that found by Hawking, the entropy-area law would change accordingly and this would change the resulting tunneling formula
-- Pilling 2008 (p3)​

Since 2008 - there are a lot of papers referencing this one (63 in scopus - it's hardly news). A quick trawl reveals that, in 2010, there have been claims that Hawking radiation has been observed in lab conditions. I have not reviewed the rest of the papers - it is likely that there are a variety of models being proposed, with different characteristics.

BTW: the factor-2 problem comes from two different approximations for the temperature.
I think the references cited are:
Emil T. Akhmedov, Valeria Akhmedova, Douglas Singleton, and Terry Pilling, Int. J. Mod. Phys. A 22, 1 (2007);
Emil T. Akhmedov, Valeria Akhmedova, and Douglas Singleton, Phys. Lett. B 642, 124 (2006).

You'll faind a lot of papers about the characteristics of the different approximation methods before and since. We have yet to hear the last word by a long shot.

... I'm wondering how the subject comes to your attention?

 

[Vanadium 50]

The muon lifetime has an ugly 192pi^3 in it. Does that make muons stable?

 

[liometopum]

E=mc2, E=h-bar x frequency, delta E=h-bar/ (2 delta T), are all core quantum energy-related equations. They are simple. They do not contain pi. Even rotational kinetic energy has no pi.

Black hole emission should be a quantum value and E= (h-bar c3 )/ ( 8 pi G M ) is out of character in relation to the presence of pi in the equation.

Shouldn't we be able to equate the energy of a photon emited by a black hole with other photons? You cannot, with a pi in one and not in the others.

From Pillings' paper: "Our present demonstration shows that the paradox can be resolved if the black hole temperature really is a factor of two higher than that originally given by Hawking."

I agree with the factor of two part, but the pi problem remains. The pi in the Hawking energy equation just does not mesh with other quantum energy equations.

The muon equation is too far removed from the core concept to be of much relevance here. But it is ugly! Digressing some, maybe ugly is an emergent property and the probablility of ugly increases with distance from the origin.

 

[Vanadium 50]

1) The muon lifetime has an ugly 192pi^3 in it. Does that make muons stable?

2) h-bar does not contain pi? It surely does: it's h/2pi.

 

[Simon Bridge]

I'm more used to peers, particularly the religiously inclined for some reason, complaining about all the ugly empirical constants like h and asserting that a True theory of everything won't have any.

I'd suspect liometopum meant to use ##E=h\nu## rather than ##E=\hbar\omega## as an example but for that ##E=\hbar/2\Delta T## (?!) thing.

It's overall puzzling - i.e.

Even rotational kinetic energy has no pi

##K_{rot}=\frac{1}{2}I\omega^2 = 2\pi^2I\nu^2## ... and there's the pi - hiding behind the units for frequency.

The muon equation is too far removed from the core concept to be of much relevance here. But it is ugly! Digressing some, maybe ugly is an emergent property and the probablility of ugly increases with distance from the origin.

... or, maybe, beauty, like ugliness, is in the eye of the beholder? Come up with an objective definition of ugliness and then we'll have something to talk about. The rest is pork-pi's ;)

Meantime - I do suspect that we are off topic.
Isn't there a philosophy forum for this sort of thing?

 

[liometopum]

Maybe an example will make it clear.

In the attachment we have looked only at the Hawking equation. If, for example, we solve for what mass gives its own energy, we get an answer that is clearly a mutant.

Remove the pi, and the Hawking equation is cleaner. Reduce by 2 pi and it is right-on.

 

[Simon Bridge]

we get an answer that is clearly a mutant ... Remove the pi, and the Hawking equation is cleaner.

... what objective measure are you using for "mutant" and "clearly" and "cleaner"? It looks like you just don't like the look of the equation as it is written down on the page and you seem to think that everyone else will share your aesthetics. Clearly the Universe doesn't since almost all of your other examples of clean, non-ugly, relations, have included the thing you find so ugly and the only way you've been able to remove it is by hiding it in the units.

Why do you find plank's constant OK but pi is ugly?

Like I said: define your terms and then we can talk.
And the place to talk about this is the philosophy forum.

 

[Jorriss]

This would fundamentally change physics, as dimensional analysis would be rendered invalid. And then further, we can't define different unit systems. h bar can not be set equal to one as they are not equally beautiful (I think, I haven't tried to prove their equivalence on beautiful spaces).

 

[Simon Bridge]

I haven't tried to prove their equivalence on beautiful spaces.

:)
Maybe this is the tip of an iceburg - could we get funding to research laying out equations according to different aesthetic systems - feng-shui for example?

 

[micromass]

Nature doesn't care about aesthetics. Something can look really beautiful, but be wrong. And something can be really ugly, but be correct.

Since this thread is only about the aesthetics of an equation and not about actual science, I'm locking it.