Breakdown of Lorentz Invariance: The Order of Quantum Gravity Phenomenology. Thought of the Day 132.0

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The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, MPlanck = 1.2 × 1019 GeV/c2, (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω2 = ω02 + (1 + η2) c2k2 + η4(ħ/MLorentz violation)2 + k4 + ….

where the coefficients ηn are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and MLorentz violation is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of MPlanck.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η2, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η2 coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ MPlanck (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η2, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η2 ∝ (μ/MPlanck)σ where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(MPlanckc) ≪ (μ/MPlanck)σ the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(MPlanckc) ≫ (μ/MPlanck)σ the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/MPlanck. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/MPlanck would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients ηn, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

Black Holes. Thought of the Day 23.0

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The formation of black holes can be understood, at least partially, within the context of general relativity. According to general relativity the gravitational collapse leads to a spacetime singularity. But this spacetime singularity can not be adequately described within general relativity, because the equivalence principle of general relativity is not valid for spacetime singularities; therefore, general relativity does not give a complete description of black holes. The same problem exists with regard to the postulated initial singularity of the expanding cosmos. In these cases, quantum mechanics and quantum field theory also reach their limit; they are not applicable for highly curved spacetimes. For a certain curving parameter (the famous Planck scale), gravity has the same strength as the other interactions; then it is not possible to ignore gravity in the context of a quantum field theoretical description. So, there exists no theory which would be able to describe gravitational collapses or which could explain, why (although they are predicted by general relativity) they don’t happen, or why there is no spacetime singularity. And the real problems start, if one brings general relativity and quantum field theory together to describe black holes. Then it comes to rather strange forms of contradictions, and the mutual conceptual incompatibility of general relativity and quantum field theory becomes very clear:

Black holes are according to general relativity surrounded by an event horizon. Material objects and radiation can enter the black hole, but nothing inside its event horizon can leave this region, because the gravitational pull is strong enough to hold back even radiation; the escape velocity is greater than the speed of light. Not even photons can leave a black hole. Black holes have a mass; in the case of the Schwarzschild metrics, they have exclusively a mass. In the case of the Reissner-Nordström metrics, they have a mass and an electric charge; in case of the Kerr metrics, they have a mass and an angular momentum; and in case of the Kerr-Newman metrics, they have mass, electric charge and angular momentum. These are, according to the no-hair theorem, all the characteristics a black hole has at its disposal. Let’s restrict the argument in the following to the Reissner-Nordström metrics in which a black hole has only mass and electric charge. In the classical picture, the electric charge of a black hole becomes noticeable in form of a force exerted on an electrically charged probe outside its event horizon. In the quantum field theoretical picture, interactions are the result of the exchange of virtual interaction bosons, in case of an electric charge: virtual photons. But how can photons be exchanged between an electrically charged black hole and an electrically charged probe outside its event horizon, if no photon can leave a black hole – which can be considered a definition of a black hole? One could think, that virtual photons, mediating electrical interaction, are possibly able (in contrast to real photons, representing radiation) to leave the black hole. But why? There is no good reason and no good answer for that within our present theoretical framework. The same problem exists for the gravitational interaction, for the gravitational pull of the black hole exerted on massive objects outside its event horizon, if the gravitational force is understood as an exchange of gravitons between massive objects, as the quantum field theoretical picture in its extrapolation to gravity suggests. How could (virtual) gravitons leave a black hole at all?

There are three possible scenarios resulting from the incompatibility of our assumptions about the characteristics of a black hole, based on general relativity, and on the picture quantum field theory draws with regard to interactions:

(i) Black holes don’t exist in nature. They are a theoretical artifact, demonstrating the asymptotic inadequacy of Einstein’s general theory of relativity. Only a quantum theory of gravity will explain where the general relativistic predictions fail, and why.

(ii) Black holes exist, as predicted by general relativity, and they have a mass and, in some cases, an electric charge, both leading to physical effects outside the event horizon. Then, we would have to explain, how these effects are realized physically. The quantum field theoretical picture of interactions is either fundamentally wrong, or we would have to explain, why virtual photons behave completely different, with regard to black holes, from real radiation photons. Or the features of a black hole – mass, electric charge and angular momentum – would be features imprinted during its formation onto the spacetime surrounding the black hole or onto its event horizon. Then, interactions between a black hole and its environment would rather be interactions between the environment and the event horizon or even interactions within the environmental spacetime.

(iii) Black holes exist as the product of gravitational collapses, but they do not exert any effects on their environment. This is the craziest of all scenarios. For this scenario, general relativity would have to be fundamentally wrong. In contrast to the picture given by general relativity, black holes would have no physically effective features at all: no mass, no electric charge, no angular momentum, nothing. And after the formation of a black hole, there would be no spacetime curvature, because there remains no mass. (Or, the spacetime curvature has to result from other effects.) The mass and the electric charge of objects falling (casually) into a black hole would be irretrievably lost. They would simply disappear from the universe, when they pass the event horizon. Black holes would not exert any forces on massive or electrically charged objects in their environment. They would not pull any massive objects into their event horizon and increase thereby their mass. Moreover, their event horizon would mark a region causally disconnected with our universe: a region outside of our universe. Everything falling casually into the black hole, or thrown intentionally into this region, would disappear from the universe.

Time, Phenomenologically.

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Phenomenological philosophy claims that time is not the place, the scene, the container or the medium for events (changes), nor a dimension along which everything flows. According to Jean-Toussaint Desanti, a French scholar of Husserl, we should forget “the ordinary meaning of the preposition “in” we spontaneously use when we talk about our experience of time. It is even this use, so ancient that should be the subject of our review. Really it would be strange that what we have learned to call “time” can contain anything. And yet we say without anxiety: “It is time that everything goes.”

But what is happening “in” time does not remain as a place. In fact, this is the major objection of Bergson against Einstein’s Special Relativity, that he has dimensioned time, something immeasurable in the same way as space, which is, of course (in everyday life), measurable. This kind of reasoning in phenomenology is not that far from the one in modern physics.

As Smolin says,

There is a deeper problem, perhaps going back to the origin of physics… time is frozen as if it were another dimension of space. Motion is frozen, and a whole history of constant motion and change is presented to us as something static and unchanging… We have to find a way to unfreeze time — to represent time without turning it into space.

In the words of Carlos Rovelli,

Today, the novelty that comes from quantum gravity is that space does not exist. … But combining this idea with relativity, one must conclude that the non-existence of space also implies the non-existence of time. Indeed, this is exactly what happens in quantum gravity: the variable t does not appear in the Wheeler-DeWitt equation, or elsewhere in the basic structure of the theory. … Time does not exist.

The claim about the imaginary, surreal, even exotic nature of time is not new in philosophy and physics. Of course, there have always been, too, physicists defending the real existence of time, even so real to define such a quantum variable as the chronon with the idea in mind to reconcile special and general relativity with quantum field theory. This “atom” of time was supposed to be the duration for light to travel the distance of the classical (non-quantum) radius of an electron. This model implies a lowest level of actuality, as asserted in the Planck scale.

In his book “Time Reborn” Smolin argues that physicists have inappropriately banned the reality of time because they confuse their timeless mathematical models with reality. His claim was that time is both real (which means external to him) and fundamental, hypothesizing that the very laws of physics are not fixed, but evolve over time. This stance is not really a new one. But it means again an absolute external reference axis and a direction for placing events in a sequence, which phenomenologists decline as the only option. Some of them, partly inspired by the late works of Heidegger and Merleau-Ponty, approach time neither from the standpoint of simultaneity alone, nor from that of succession. For instance, the dualism of these two concepts is surpassed in favor of a temporal dialectic in which simultaneity and succession are entwined, without denying their separate meanings. Heidegger’s concept of “true time” speaks to this approach to phenomenology.

Unruh Radiation, Black Holes and Partial Waves. Note Quote.

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It is well known that Hawking radiation from an asymptotically flat Schwarzschild black hole is dominated by low angular momentum modes. This is a consequence of the fact that a black hole of Hawking temperaure TH and Schwarzschild radius rs has TH rs ∼ 1, so that high angular momentum modes of energy TH are trapped behind a large barrier in the effective radial potential. Since a local observer is unlikely to encounter such quanta, one might then conclude that a (much-weakened) version of postulate “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” might still hold in which the suppression is replaced by a fixed (1/area) power law. In addition, one would need to propose a mechanism through which these quanta would arise from the infalling perspective. This would appear to require that the infalling observer experience violations of local quantum field theory at this (power-law-suppressed) level.

This would already be a striking result: these quanta must appear quite close to the horizon and so violate the standard wisdom that the horizon is not a distinguished location. And they are not rare in the sense that their number is of the same order as the number of actual Hawking quanta.

As noted long ago by Unruh and Wald, it is possible to ‘mine’ energy from the modes trapped behind the effective potential. The basic procedure is to lower some object below the potential barrier, let the object absorb the trapped modes, and then raise the object back above the barrier. Unruh and Wald thought of the object as a box that could be opened to collect ambient radiation and then closed to keep the radiation from escaping. One may also visualize the object as a particle detector, though the two are equivalent at the level discussed here.

In the context of such a mining operation, one need only consider the internal state of the mining equipment to be part of the late-time Hawking radiation. In particular, postulate “outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations”, can be used to evolve the mode to be mined backward in time and to conclude for an old black hole that, even before the mining process takes place, the mode must be fully entangled with the early-time radiation. “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” is then violated for these modes as well, suggesting that the infalling observer encounters a Planck density of Planck scale radiation and burns up. One might say that the black hole is protected by a Planck-scale firewall.

Note that this firewall need not be visible to any observer that remains outside the horizon. All that we have argued is that the infalling observer does not experience a pure state. There remains considerable freedom in the possible reduced density matrices that could describe a few localized degrees of freedom outside the black hole, so that this matrix might still agree perfectly with that predicted by Hawking. In this case any local signal that an external observer might hope to ascribe to the firewall at distance 1/ω cannot be disentangled from the Unruh radiation that results from probing this scale without falling into the black hole.