Gravity-induced decay

Abstract

I suggest an object acted on by classical (i.e. non-coherent) force, in gravitational field, unavoidably decays. I estimate decay rates in some scenarios.

Full text

Gravity-induced decay
Sergei Viznyuk
Abstract
I suggest an object acted on by classical (i.e. non-coherent) force, in
gravitational field, unavoidably decays. I estimate decay rates in some scenarios.
Quantum phase shift between superposed states with different space geometries in
gravitational field inspired interest (Penrose, 1996) as well as controversy (Gao, 2013; Bonder,
Okon, & Sudarsky, 2016; Pikovski, Zych, Costa, & Brukner, 2015) since at least early 1960-ies
when (Feynman, 1962) first talked about role of gravity in breakdown of quantum mechanical
behavior. Despite the controversy, the nature of the phenomenon appears mundane: the states with
different space geometries in gravitational field experience phase shift with respect to each other.
If the shift is larger than the de-Broglie wave packet length the interference between packets with
different space geometries vanishes. It is the same effect as disappearance of interference pattern
in two-slit experiment when difference in optical lengths between two paths exceeds the coherence
length of the light source (Figger, Meschede, & Zimmermann, 2002). In such case, wave packets
(e.g. laser pulses) appear as if they travel down only one of the paths, as do classical particles.
However, the interference amplitude can be easily restored by introducing phase delay into one of
the optical paths. Perhaps one should not equate phase shift between wave packets or decrease in
interference amplitude with true de-coherence, which is an irreversible loss of phase relationships
between constituent states.
The observable effect of true de-coherence is disintegration of the object into wave packets
with no phase relationship between the packets. Such packets behave as separate objects.
True de-coherence arises if phases of constituent pure states no longer predictably relate to
each other, e.g. as a result of random dispersion. Random phase dispersion can be associated with:
1. Scattering (Uys, et al., 2010; Schlosshauer, 2007)
2. Brownian motion (Paz, 1994; Hornberger, 2009)
3. Dispersive media (Antonelli & al, 2011; Salemian & Mohammadnejad, 2011)
In this article I look into phase dispersion introduced by non-coherent coupling between external
non-gravitational fields and object’s constituent pure states, in the presence of gravitational field.
According to (Einstein, 1907) equivalence principle, the effect of the external non-gravitational
force on the object at rest in a [uniform] gravitational field is indistinguishable from the effect of
the same force on the object away from gravitating bodies, in object’s [accelerating] reference
frame. Thus, the influence of non-gravitational fields does not depend on presence of [uniform]
gravity. The inevitable conclusion is that any change to the state of the object, such as phase
dispersion between constituent pure states, can only be caused by non-gravitational fields:
It is clear, …, that, while some kind of decoherence might happen in specific
experimental situations, gravity, … is never the agent that causes the effect.
(Bonder, Okon, & Sudarsky, 2016)
Nevertheless, I suggest, gravity plays an important role, by establishing preferred basis in system’s
state vector space. Phase dispersion by coherent coupling between external field and pure states
retains coherence of the wave packet. If coupling is non-coherent, as e.g. with environment heat
bath, the phase dispersion includes random phase noise which, over sufficiently long time, results
in true de-coherence of the wave packet, and decay of the object is represents.

To demonstrate the principal feature, consider the object is in a stable state  if no external
fields. If object is free-falling in a gravitational field, with respect to lab frame the state  is a
superposition of eigenstates  of Hamiltonian  with field, i.e. in the preferred basis:


(1)
Thus in a lab,  is no longer a static state, as can be seen from solution to Schrodinger’s equation:

 


(2)
An observer performing measurement of object’s vertical coordinate  will find the expectation
value change according to:



, where

(3)
A two-level system is sufficient to demonstrate the main feature (Viznyuk, 2014):




(4)
From (4) the relation with acceleration of gravity  is:

 

(5)
, where  ought to be taken as a distance (radius) to the center of gravity at  ,
and  
.
Now consider the object is acted on by an external non-gravitational forces keeping the
object around fixed position in a lab frame. If external fields couple coherently, the evolution of
object’s state  is still described by (2), where  are eigenstates of Hamiltonian with all fields
included. With non-coherent coupling, the phases  of -states no longer relate to each other as
 

(6)
Instead, non-coherent coupling can be modelled as a random phase walk, with phase
difference  changing by   during single act of interaction with external
field. Here  has a meaning of mean free time between interactions, with    
 being a
scattering rate. During time  there is    scattering events, each with probability 
 
 to increment phase difference  by  in positive or negative direction. After 
scattering events phase difference  will be within standard deviation  of its coherent
value, where 
 is the variance of binomial distribution 
 . So, instead of
(6), for non-coherent coupling I have:
  
 
(7)
Once random dispersion (7) grows to   the , -states can be considered de-coherent
with respect to each other. From here I estimate time  for the object to decay, considering it a
simple two-level system as in (4), (5):




, and decay rate
 

 

(8)

Eqs. (7-8) has been derived in approximation of large number of scattering events per rotation
period  
, i.e. in approximation . In the opposite  case I expect:  as  .
The inverse proportionality of decay rate  on de-coherent scattering rate  in (8) can be viewed
as manifestation of quantum Zeno effect (Misra B., 1977). Each scattering event constitutes the
measurement by the environment. The more frequent are the measurements performed on the
object, the more likely the object will be found in its initial state after fixed elapsed time.
Furthermore, the effect may not even be so “quantum” in nature. I can simulate it using the
following experiment: take a [fragile] wine glass and try to keep it vertically around the same
height with respect to lab floor by applying periodic kicks from below using some hard-surface
object, e.g. a wood board. In the mean free time between the kicks the wine glass will be in free
fall. The more frequent are the kicks, the lesser distance the wine glass will fall in between the
kicks, the lesser is the force of each kick required to return the wine glass to its initial position. In
the limit of infinite frequency of kicks, the wine glass will be simply standing steady on hard
surface. It can stay in such position forever. However, if the frequency of kicks is decreased, in
the mean free time the wine glass would fall greater distance. Each kick would have to be stronger
to return wine glass to its initial position; and with stronger kick it is more likely the wine glass
would eventually break into smaller pieces. As I expect, there is also a second limit, when ;
; corresponding to the case when wine glass is allowed to fall indefinitely without
experiencing kicks, and that would preserve wine glass too.
It is conceivable (8) could be applicable to any object, including elementary particles which
are otherwise considered stable, such as electrons and protons. I shall estimate  from (8) in some
scenarios, using input parameters from the cited sources:
Scenario
Distance to the
center of gravity ()
Acceleration  of
gravity ()
Scattering rate 
(
Decay time 
electron in Sun’s
photosphere
 [21]

 [10]

electron in Sun’s
corona
 [21]

 [10]
24 years
proton in Sun’s core
at the maximum
gravity location
 [18]
2342
 [9]
 s
With solar mass distribution (Stix, 2004), the gravity reaches maximum of  at about
 of solar radius from the center of the Sun. The proton collision rate in Sun’s core is estimated
from  (Knapp, 2011) to  (Mullan, 2009). Thus from (8) the decay rate of
protons in Sun’s core could range from  to . The claimed hydrogen
fusion rate of  (Knapp, 2011; Mullan, 2009) in Sun’s core lays right in that
range. If the presented reasoning is valid, the thermonuclear fusion might not be the primary factor
behind Sun’s radiative output. Indeed, as radiative modes have higher entropy than heavy nuclei
the Second Law of Thermodynamics favors Sun evaporating into thermal radiation rather than
forming iron core and turning into a white dwarf or neutron star, a relatively low entropy states.
Also, the presented estimates show the proposed particle decay rates in Sun’s corona are many
orders of magnitude higher than in Sun’s photosphere, which may explain why the temperature of
Sun’s corona is , and up to  in the hottest regions, in comparison with
photosphere which is about .

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