Download On quantization of gravitational waves

On quantization of gravitational waves
Giorgio Torrieri
Universidade Estadual de Campinas - Instituto de Fisica ”Gleb Wataghin”
Rua Srgio Buarque de Holanda, 777
CEP 13083-859 - Campinas SP
[email protected]
,
[email protected]
Essay written for the Gravity Research Foundation 2016 Awards for Essays on Gravitation
Date of submission: 1/4/2016. Country of submission: Brazil
We point out that well-known heuristic arguments justifying quantum mechanics on the basis of idealized experiments (real and Gedanken), such Heisenberg’s
microscope, the photoelectric effect, the double slit experiment and so on, lead to
conceptual problems when, instead of electromagnetic radiation, the experiment uses
gravitational waves. This casts doubt on the feasibility of “quantizing gravity” in
the canonical way. We illustrate the difficulties and also discuss these issues in light
of the relationship between gravity and thermodynamics
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The search for a quantum theory of gravity [1, 2] has been the pre-eminent problem of
theoretical physics in the last half of a century, and yet, despite thousands of papers, many
conceptual issues, such as the black hole information paradox or the nature of fundamental
quantum-gravitational degrees of freedom, are as mysterious as decades ago.
A deceptively simple quantization, known since the 30s, is to perturb around Minkowski
space via a quantum “graviton” [3, 4] hµν . The metric is
gµν = Diag [1, −1, −1, −1] + ĥµν
(1)
and the graviton is quantized in the same way as the photon, via creation and annihilation
operators.
ĥµν =
Z
d3 k
√
ǫµν âk eik + ǫ∗µν â+
e−ik
k
k0
(2)
For photons, this quantization (with field ĥµν → µ and polarization ǫµν → ǫµ ) provides a
formal basis for Einstein’s photoelectric effect ansatz that the amplitude of the wave gives
the expected density of photons, while the frequency gives the energy of photons. Particles
are defined in momentum space and uncertainty relations can be derived from the width
of propagators in configuration and momentum space. Non-probe Interactions are handled
perturbatively.
The only difference is that Gravitons, unlike photons, are spin 2 particles, but for a
free theory this distinction is technical. General relativity’s equivalence principle acquires
the same role as Gauge theory in reducing the five polarization states of ĥ to two. This
requirement uniquely fixes the linearized lagrangian to a function of the Ricci scalar, and
the source to the conserved energy-momentum tensor [1, 4].
D E D E
In the appropriate gauge, the linearized equation of motion for the average ĥµν , µ
both reduce to the wave equation, sourced respectively by the energy-momentum tensor and
electromagnetic charge
D E  D E

ĥµν   G T̂µν 
∂
2
D
E
D E
−∇
=
 µ   e Jˆµ 
∂t2
2
(3)
The classical limit of such electromagnetic waves is of course textbook material [5], and
gravitational waves obeying Eq. 3 are also experimentally discovered [6].
This success motivates the hope that “quantum gravity” is a technical problem of consistently including the backreaction of the gravitational degrees of freedom on matter, the
3
Low wavelength
gravity wave
FIG. 1. Einstein’s elevator with a gravitational wave experiment
“loops” in Feynman diagrams [4]. Since the 70s it became clear this problem is much more
difficult than originally thought [1], with the resolution or even nature of this [7, 8] difficulty
still controversial [1].
In this work, we point out that there is another relevant distinction between electromagnetic and gravitational waves: Electromagnetic waves of wavenumber k require a detector
of size not ≪ k −1 to be measured. Beyond this limitation, one can measure either the position or the momentum of each photon to any degree of precision limited by the uncertainty
principle.
The limitations the equivalence principle imposes on gravity waves detection are much
more stringent. The equivalence principle ensures that the graviton cannot be detected by
an apparatus much smaller than its wavelength, since all rods and clocks of the detector
would contract and expand as the wave passes through it. For the detector to register tidal
differences in contraction and expansion, an apparatus of size L ∼ k −2 is necessary (note
that by size we mean both space and “time”, the experiments duration). This is simply an
extension of Einstein’s elevator Gedankenexperiment (Fig. 1) [2], where the elevator of size
∼ k −1 passes through a wavefront. Unless the elevator feels tidal forces, all laboratory equipment in it will shift in the same way ( comparing the shift to measurements of a stationary
laboratory requires a distance or time separation between the two). Note that Gauge invariance rather than quantum mechanics ensures frequency, rather than amplitude is relevant
here, since large (compared to the detector) wavelengths can be brought arbitrarily close
to inertial frames, locally to the detector, by a gauge redefinition. Furthermore, assuming
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gravity is quantized the gravitational wave has to be distinguishable from quantum noise
due to the detector’s position and the equivalence principle. This implies a detector size
∼ (Gk)−1 . Putting everything together we get L ∼ G−1 k −3 estimate for the detector size,
of course amply obeyed in [6].
Of course gravitational waves have an additional experimental barrier: a gravitational
wave of frequency G−1/2 would interact “strongly” with any probe but would also have the
capacity to collapse this probe into a black hole [7], rendering it unobservable. As a corollary,
“diffraction” and absorption of gravitational waves would only be possible by a black hole
like object which would in turn inevitably result in a wave’s redshift.
To appreciate the significance of these differences, one must remember that quantum mechanics has both a theoretical and phenomenological dimension. The formalism of operators
and Hilbert spaces is the most elegant way we know of accounting for both a large set of
experimental data (level quantization, photoelectric effect, double slit experiments, SternGerlach experiments and so on) and Gedanken setups extrapolated from this experimental
data. The latter played a fundamental role in getting quantum mechanics accepted [9, 10].
The quantization of gravity, or a modification of quantum mechanics in the presence of
gravity, are commonly motivated by Gedankenexperiments showing the paradox of having
non-quantized gravity interacting with quantized matter [11, 12]. However, if gravity is to
be quantized Gedankenexperiments demanding quantization must also work when the most
elementary gravitational dynamical degrees of freedom, linearized gravitational waves, are
substituted to the photons and atoms used in the historical generation of experiments. In
the following examples (Fig 2), it is clear these experiments are profoundly altered because
of the limitations on gravitational wave detectability given by the equivalence principle.
Heisenberg’s microscope and Einstein’s box Heisenberg’s microscope’s relationship
to the mathematical structure of quantum mechanics is dubious [10], but it is the first
heuristic explanation for the Uncertainty principle. The essential idea is to understand the
behavior of a pointlike particle bombarded with light and observed via a standard microscope
given Planck’s law. To observe the position precisely, the light required has to have a small
wavenumber since the uncertainty in position is in turn determined by the microscope’s
resolving power, which is always proportional to the photon’s wavelength divided by lens
size. The collision of the particle with the photon will result in a scattering whose momentum
exchange depends on the photon’s momentum, proportional to wavenumber, and whose
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FIG. 2. Experiments, real and Gedanken, motivating the key concepts of quantum mechanics,
respectively uncertainty relations and wave-particle duality. From the left to the right we have
Heisenberg’s microscope, the photoelectric effect, and the double-slit experiment. The blue-dashed
frames indicate the part of the experiment where, if a gravitational wave were to be substituted for
a photon, the effect would either be unobservable in principle or lead to the formation of a black
hole. Image on the left and right courtesy of Wikipedia
direction is random. The resulting uncertainty in momentum is therefore proportional to
wavenumber, times the lens size for the photon to be captured. With some elementary
algebra, the lens size factors out and Heisenberg’s uncertainty principle relating momentum
and position uncertainty follows.
Let us now try to substitute gravitational waves for photons. Since we are dealing with
point-like particles, strictly speaking the G−1 k −3 requirement already precludes the analogy
(the locality criterion used in the conclusion of [13], the Gravitational Compton wavelength,
would ensure a black hole is created after every scattering). Let us however assume that
individual Gravitons propagate and scatter as spin 2 quantum particles. Even given this
assumption a consistent replication of Heisenberg’s thought experiment is problematic due
to the behavior of the microscope. In general relativity all matter interacts with the gravitational wave in a manner proportional to its energy-momentum tensor. The “microscope” in
this case is a gravitational wave detector of large enough size as to resolve the gravitational
wave, whose resolving power has no relation to the frequency of the wave beyond the G−1 k −3
limit. We could place a gravitational lens in front of the detector, but any mass distribution
large enough to be a lens will also redshift the wave, amplifying the position uncertainty.
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Hence, “Heisenberg’s gravitational microscope” will never be able to resolve the position of
a single graviton, independently of the graviton momentum exchange.
Note that this “small apparatus” objection easily translates into extensions of Heisenberg’s microscope to more complicated systems, including the “Einstein box” thought experiment [9]. An Einstein Box full of gravity waves would either fail to contain Gravitons,
and hence have an infinite time uncertainty, or collapse into a black hole.
The photoelectric effect The phenomenology of the photoelectric effect [10] has been
crucial to establish the wave-particle duality of photons. If one fires electromagnetic waves
at atoms, below a certain threshold in frequency no ionization occurs independently of the
amplitude of the waves. Once the critical frequency is reached, ionization starts occurring,
at a rate which does linearly depend on amplitude. The correct explanation seems so obvious
today we forget its revolutionary dimension: Atoms interact with individual photons, the
photon energy is given by the frequency via Planck’s law, the amplitude gives the density
of photons via Born’s rule.
The equivalent setup with gravitons is unfortunately firmly in the Gedankenexperiment
regime, no realistic setup is remotely feasible. However, our knowledge of classical general
relativity is enough to understand that the result of this experiment would be wildly different
from the traditional photon one. Bounded objects are characterized by a radius (the Bohr
radius for central force problems) determined by masses, the ionizing potential and quantum
mechanics. For a binding energy ∆E, in the non-relativistic limit elementary considerations
give a radius ∆R ∼ (m∆E)−1/2 . Putting the G−1 k −3 ∼ ∆R requirement means that these
gravitational waves will create black holes out of the atoms, so no ionization ever occurs.
The double slit experiment If the photoelectric effect was the affirmation of waveparticle duality on a phenomenological level, the double slit experiment is the exploration
of its implications.
One scatters particles on a wall with narrow slits, and measures the number of impacts
on a screen behind the slits. An interference pattern forms, in accordance to the wave
interpretation of light. However, if one performs this experiment with O (1) single quanta at
a time, the screen will record O (1) single readings, distributed in a probability distribution
that looks like an interference pattern. This validates directly Einstein’s explanation for the
photoelectric effect via wave-particle duality. If one puts a photon detector by one of the
slits, of resolution comparable to the slits size, the interference pattern disappears, clarifying
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the connection between wave-particle duality and the uncertainty principle.
Here, once again, the equivalence principle immediately precludes, in principle, the possibility of a “wall” with slits. Any wall capable of stopping gravitons would, by its nature,
be a black ”brane”’. Even assuming that stable slits can be maintained on such a brane (the
no-hair theorem, and recent work on brane “hydrodynamics” [14], makes this assumption
dubious) propagation through them susceptible to gravitational lensing effects and redshifts.
The most obvious of these effects would be to stretch any interference patterns to the extent
that they will be undetectable by a screen at infinity.
The considerations made in this work need an update once horizon thermodynamics is
taken in into account. In a universe with a time-like horizon (our universe seems have one),
an observer of arbitrarily small size can detect a gravitational wave of arbitrary frequency
by “observing” the horizon (Fig. 3). The gravitational wave will momentarily wobble the
curvature of the horizon, and locally increase the Hawking temperature by ∆T ∼ Gk. This
detection method does not evade the issues discussed in the first part of this work, since the
cosmological horizon has very low temperature, on which the emission frequency ω depends
(ω ∼ T ∼ GR, where R is the size of the region within the horizon). But this implies that
while gravitational waves are stable in Minkowski space (where there is no time-like horizon
to perturb) they should decay in “cosmological Hawking radiation”, of wavenumber ≪ k, in
a universe with a time-like horizon. This is an intrinsically non-local and non-perturbative
effect, since momentum conservation, phase space and Gauge symmetry will generally forbid
the decay of massless particles into any number of other massless particles at any order in
perturbation theory [15]. The UV/IR mixing at work here clearly shows that gravitational
waves of arbitrary wavenumber are sensitive to the global topology of space, something that
is obviously in tension with an intrinsically local equivalence principle.
In conclusion, we discussed the application of gravitational waves to the standard
Gedankenexperiments which motivate the formalism of quantization and its consequences.
We find that such experiments would give different results when gravitational waves are
substituted for photons. If one thinks such heuristic experiments motivate quantization, it
follows that gravity is not to be quantized like other forces.
The author acknowledges support from FAPESP proc. 2014/13120-7 and CNPQ bolsa
8
Low wavelength
gravity wave
Observer
Hawking
radiation
Horizon
FIG. 3. Hawking radiation from a horizon perturbation
de produtividade 301996/2014-8.
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