Hawking radiation versus special relativity - equation incompatibility

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Hawking radiation versus special relativity - equation incompatibility

Classical Hawking radiation seems to be impossible under an SR-based system of physics, due to the characteristics of the SR equations.

Infinite horizon blueshift

Suppose that a rocket-ship is magically transported to the event horizon of a black hole and released there, so that it is stationary with respect to the hole at the moment of release. it can then either freefall into the hole's core, or fire up its engines and attempt to escape. We'll try to assume that the ship's engines are built using some future technology that represents the ultimate propulsion system (such as a warpdrive).

No matter how clever the spaceship's engines, or how many tricks the engine designer has used that we currently don't understand (perhaps even able to transport the shift faster than background lightspeed), the fundamental obstacle preventing the ship's escape seemsot be the SR Doppler shift equations.

ror simpel recession, the SR relativistic Doppler relationships is:[ref]

$\frac{frequency'}{frequency} = \sqrt{\frac{c-v}{c+v}}$ ,

or,

$\frac{frequency'}{frequency} = \frac{c}{c+v} × ({ 1 - \frac{v^2}{c^2}})^{0.5}$

If we put v=c then we have the shift prediction for a body recedingn from us at lightspeed (f'/f=zero), and if we put v=(-c), we have the prediction for the shift on the body approaching us as lightspeed (f'/f = infinity).

The surface escape velocity of a black hole is the speed of light, so if the velocity differential of a body falling "downhill" from a great distance to the horizon surface is also the speed of light, then the blueshift on the object that we see just before it hits us will, according to the equation (with v=(-c) ) be infinite.

If the same equation also tells us the gravitational blueshift of light infalling to our position, then this would seem to be infinite, too. Setting aside the small matter of our being instantly vaporised by the incoming infinite-energy light, and pushed inwards by its associate infinite light-momentum, if we did somehow manage to survive and fire up our fancy engines, the fact that we see the outside universe to be aging infinitely rapidly would mean that if we did manage to escape, it would only be after an infinite amount of outsider-time had elapsed – by which time there might not be a universe left to escape to. Certainly, nobody in our current epoch would see us escaping.

If our engines are already firing when we are placed in position, then things are even worse - the geeforces of our acceleration make the blackhole's gravity feel even stronger, and should make the infinite blushift greater-than-infinite.

Under GR1960, none of these issues are problems – to hover or escape the horizon is supposed to be just as impossible as travelling at or faster then the spee dof light, so if we assume that we can hover or escape, it's perhaps only right that we find ourself=ves dealign with physical infinities. It's meant to be an impossible and illegal exercise.

Reconciling with QM

other systems of physics (such as an C18yth dark star or a current-day cosmological horizon do seem to be able ot support the concept of indirect radition though a horizonand mesh with QM. But if we require a generla theory to be modified ot repriduce this bahviour, it appears that th emodification must have to involve modifying special relativity.

This woud mean that our hoped-for solution to the black hole informaiton paradox – of somehow constructing a larger theory of quantum gravity that includes both QM and GR in theor current states – is probably impossible. The only way of reconciling QM and GR woudl be by changing the GR equaitons so that they no longer corresponded exactly to special relativity.

Changing the equations

Referrign back to the relativistic ellispe exercise,we find that the principel fo relativity suggests a rangeof possible solutions identified with thgeneralised equation

$\frac{frequency'}{frequency} = \frac{c}{c+v} × ({ 1 - \frac{v^2}{c^2}})^{x}$

, where the exponent "x" has a value between 0.5 and 1.

In order to recreate indirect particle escape, we require the inward blueshift at 2=2M to have a finite positive valie soemwher ebetween zero and infinity, and the only value of x that gives a finite nonzero result is x=1.

Exploring x=1

if we change the Lorentzlike term to x=1, weget a "knife-edge solution where the inward blueshift is f'/f = 2. any greater than x=1 gives a zero result, and any less gives an infinity. This suggests that any resolutio not the BHIP that involves making GR QM-compatibleinvolves changign the basic equaitons to a set that are redder than those of SR by exactly one additiona Lorentz factor.

this has the effect of simplifying the doppler relationships, turnign the recession redshift prediction to just

$\frac{frequency'}{frequency} = \frac{c-v}{c}$

, which is the Newtoinian relationship that SR was supposed ot replace.

If we now look at our two known case sof classical Hawkign radiation, the Newtonian Dark star obviously uses the Newtonian relaitonships of x=1, and the Hubble shift relationship also diverges frim the Sr predictiuons. Accordign to MTW[ref], the Hubble shift law uses a simple Doppler relationship ratehr than the SR version. since the only two findamental relationships are x=0 and x=1, and x=0 does not genreate a horizon, then it seems that both expmples that we have in relativity theory that do conform to QM behaviour both, coincidentally, use the same equation that we have just identified.

Put another way, we do not know of any astronomiocal cases that create classical Hawkign radiation that do NOT use x=1.

Our current inability to reconcile QM and Gr maightbe due to an ill-advised adoption fo x=0.5, and the subsequent stadardisation on only x=0.5 theories that tookplac ein 1960.

Notes

We hose a set fo equations for our curved-spacetime model usign flat-spacetime reasoning ,ad without checkign how those equaitons operated in curved spacetime. In the common vernacular, "a cock-up"