Graviton Fields Under Helicity Rotations. Thought of the Day 156.0

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Einstein described gravity as equivalent to curves in space and time, but physicists have long searched for a theory of gravitons, its putative quantum-scale source. Though gravitons are individually too weak to detect, most physicists believe the particles roam the quantum realm in droves, and that their behavior somehow collectively gives rise to the macroscopic force of gravity, just as light is a macroscopic effect of particles called photons. But every proposed theory of how gravity particles might behave faces the same problem: upon close inspection, it doesn’t make mathematical sense. Calculations of graviton interactions might seem to work at first, but when physicists attempt to make them more exact, they yield gibberish – an answer of “infinity.” This is the disease of quantized gravity. With regard to the exchange particles concept in the quantum electrodynamics theory and the existence of graviton, let’s consider a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the gravitational field. But when we define the graviton relative to the photon, it is necessary to explain the properties and behavior of photon in the gravitational field. The fields around a “ray of light” are electromagnetic waves, not static fields. The electromagnetic field generated by a photon is much stronger than the associated gravitational field. When a photon is falling in the gravitational field, it goes from a low layer to a higher layer density of gravitons. We should assume that the graviton is not a solid sphere without any considerable effect. Graviton carries gravity force, so it is absorbable by other gravitons; in general; gravitons absorb each other and combine. This new view on graviton shows, identities of graviton changes, in fact it has mass with changeable spin.

When we derive various supermultiplets of states, at the noninteracting level, these states can easily be described in terms of local fields. But, at the interacting level, there are certain ambiguities that withdraw as a result of different field representations describing the same massless free states. So the proper choice of the field representation may be subtle. The supermultiplets can then be converted into supersymmetric actions, quadratic in the fields. For selfdual tensor fields, the action must be augmented by a duality constraint on the corresponding field strength. For the graviton field,

The linearized Einstein equation for gμν = ημν + κhμν implies that (for D ≥ 3)

Rμν ∝ ∂2hμν + ∂μνh – ∂μρhνρ – ∂νρhρμ = 0 —– (1)

where h ≡ hμμ and Rμν is the Ricci tensor. To analyze the number of states implied by this equation, one may count the number of plane-wave solutions with given momentum qμ. It then turns out that there are D arbitrary solutions, corresponding to the linearized gauge invariance hμν → hμν + ∂μξν + ∂νξμ, which can be discarded. Many other components vanish and the only nonvanishing ones require the momentum to be lightlike. Thee reside in the fields hij, where the components i, j are in the transverse (D-2) dimensional subspace. In addition, the trace of hij must be zero. Hence, the relevant plane-wave solutions are massless and have polarizations (helicities) characterized by a symmetric traceless 2-rank tensor. This tensor comprises 1/2D(D-3), which transform irreducibly under the SO(D-2) helicity group of transverse rotations. For the special case of D = 6 spacetime dimensions, the helicity group is SO(4), which factorizes into two SU(2) groups. The symmetric traceless representation then transforms as a doublet under each of the SU(2) factors and it is thus denoted by (2,2). As for D = 3, there are obviously no dynamic degrees of freedom associated with the gravitational field. When D = 2 there are again no dynamic degrees of freedom, but here (1) should be replaced by Rμν = 1/2gμνR.

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.

Typicality. Cosmological Constant and Boltzmann Brains. Note Quote.

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In a multiverse we would expect there to be relatively many universe domains with large values of the cosmological constant, but none of these allow gravitationally bound structures (such as our galaxy) to occur, so the likelihood of observing ourselves to be in one is essentially zero.

The cosmological constant has negative pressure, but positive energy.  The negative pressure ensures that as the volume expands then matter loses energy (photons get red shifted, particles slow down); this loss of energy by matter causes the expansion to slow down – but the increase in energy of the increased volume is more important .  The increase of energy associated with the extra space the cosmological constant fills has to be balanced by a decrease in the gravitational energy of the expansion – and this expansion energy is negative, allowing the universe to carry on expanding.  If you put all the terms on one side in the Friedmann equation – which is just an energy balancing equation – (with the other side equal to zero) you will see that the expansion energy is negative, whereas the cosmological constant and matter (including dark matter) all have positive energy.

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However, as the cosmological constant is decreased, we eventually reach a transition point where it becomes just small enough for gravitational structures to occur. Reduce it a bit further still, and you now get universes resembling ours. Given the increased likelihood of observing such a universe, the chances of our universe being one of these will be near its peak. Theoretical physicist Steven Weinberg used this reasoning to correctly predict the order of magnitude of the cosmological constant well before the acceleration of our universe was even measured.

Unfortunately this argument runs into conceptually murky water. The multiverse is infinite and it is not clear whether we can calculate the odds for anything to happen in an infinite volume of space- time. All we have is the single case of our apparently small but positive value of the cosmological constant, so it’s hard to see how we could ever test whether or not Weinberg’s prediction was a lucky coincidence. Such questions concerning infinity, and what one can reasonably infer from a single data point, are just the tip of the philosophical iceberg that cosmologists face.

Another conundrum is where the laws of physics come from. Even if these laws vary across the multiverse, there must be, so it seems, meta-laws that dictate the manner in which they are distributed. How can we, inhabitants on a planet in a solar system in a galaxy, meaningfully debate the origin of the laws of physics as well as the origins of something, the very universe, that we are part of? What about the parts of space-time we can never see? These regions could infinitely outnumber our visible patch. The laws of physics could differ there, for all we know.

We cannot settle any of these questions by experiment, and this is where philosophers enter the debate. Central to this is the so-called observational-selection effect, whereby an observation is influenced by the observer’s “telescope”, whatever form that may take. But what exactly is it to be an observer, or more specifically a “typical” observer, in a system where every possible sort of observer will come about infinitely many times? The same basic question, centred on the role of observers, is as fundamental to the science of the indefinitely large (cosmology) as it is to that of the infinitesimally small (quantum theory).

This key issue of typicality also confronted Austrian physicist and philosopher Ludwig Boltzmann. In 1897 he posited an infinite space-time as a means to explain how extraordinarily well-ordered the universe is compared with the state of high entropy (or disorder) predicted by thermodynamics. Given such an arena, where every conceivable combination of particle position and momenta would exist somewhere, he suggested that the orderliness around us might be that of an incredibly rare fluctuation within an infinite space-time.

But Boltzmann’s reasoning was undermined by another, more absurd, conclusion. Rare fluctuations could also give rise to single momentary brains – self aware entities that spontaneously arises through random collisions of particles. Such “Boltzmann brains”, the argument goes, are far more likely to arise than the entire visible universe or even the solar system. Ludwig Boltzmann reasoned that brains and other complex, orderly objects on Earth were the result of random fluctuations. But why, then, do we see billions of other complex, orderly objects all around us? Why aren’t we like the lone being in the sea of nonsense?Boltzmann theorized that if random fluctuations create brains like ours, there should be Boltzmann brains floating around in space or sitting alone on uninhabited planets untold lightyears away. And in fact, those Boltzmann brains should be incredibly more common than the herds of complex, orderly objects we see here on Earth. So we have another paradox. If the only requirement of consciousness is a brain like the one in your head, why aren’t you a Boltzmann brain? If you were assigned to experience a random consciousness, you should almost certainly find yourself alone in the depths of space rather than surrounded by similar consciousnesses. The easy answers seem to all require a touch of magic. Perhaps consciousness doesn’t arise naturally from a brain like yours but requires some metaphysical endowment. Or maybe we’re not random fluctuations in a thermodynamic soup, and we were put here by an intelligent being. An infinity of space would therefore contain an infinitude of such disembodied brains, which would then be the “typical observer”, not us. OR. Starting at the very beginning: entropy must always stay the same or increase over time, according to the second law of thermodynamics. However, Boltzmann (the Ludwig one, not the brain one) formulated a version of the law of entropy that was statistical. What this means for what you’re asking is that while entropy almost always increases or stays the same, over billions of billions of billions of billions of billions…you get the idea years, entropy might go down a bit. This is called a fluctuation. So backing up a tad, if entropy always increases/stays the same, what is surprising for cosmologists is that the universe started in such a low-entropy state. So to (try) to explain this, Boltzmann said, hey, what if there’s a bigger universe that our universe is in, and it is in a state of the most possible entropy, or thermal equilibrium. Then, let’s say it exists for a long long time, those billions we talked about earlier. There’ll be statistical fluctuations, right? And those statistical fluctuations might be represented by the birth of universes. Ahem, our universe is one of them. So now, we get into the brains. Our universe must be a HUGE statistical fluctuation comparatively to other fluctuations. I mean, think about it. If it is so nuts for entropy to decrease by just a little tiny bit, how nuts would it be for it to decrease enough for the birth of a universe to happen!? So the question is, why aren’t we just brains? That is, why aren’t we a statistical fluctuation just big enough for intelligent life to develop, look around, see it exists, and melt back into goop. And it is this goopy-not-long-existing intelligent life that is a Boltzmann brain. This is a huge challenge to the Boltzmann (Ludwig) theory.

Can this bizarre vision possibly be real, or does it indicate something fundamentally wrong with our notion of “typicality”? Or is our notion of “the observer” flawed – can thermodynamic fluctuations that give rise to Boltzmann’s brains really suffice? Or could a futuristic supercomputer even play the Matrix-like role of a multitude of observers?