Ashtekar's complex variables in general relativity and its teleparallelism equivalent

The simplest variant of gauge gravitation theory in Riemann-Cartan spacetime leading to the solution of the problem of cosmological singularity and dark energy problem is investigated. It is shown that this theory by certain restrictions on indefinite parameters of gravitational Lagrangian in the case of usual gravitating systems leads to Einstein gravitational equations with effective cosmological constant.

The theory of special and general relativity causes a “schizophrenic” dilemma in physics. It undeniably provides mathematically correct values, but it is undeniably epistemologically wrong in many respects. Including the relativistic explanation of the gravitational “time dilation” and the curvature of light beams at the surfaces of large masses, the author demonstrates the illogical character of relativistic physics. When one thinks the relativistic explanations of gravitational time dilatation and of the curvature of light rays by masses through to the end, they lead to absurd and contradictory logical conclusions.

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\omega)$ is defined on $e$ and $\omega$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$.

Two radically different views about time are possible. According to the first, the universe is three dimensional. It has a past and a future, but that does not mean it is spread out in time as it is spread out in the three dimensions of space. This view requires that there is an unambiguous, absolute, cosmic-wide “now” at each instant. According to the second view about time, the universe is four dimensional. It is spread out in both space and time—in space-time in short. Special and general relativity rule out the first view. There is, according to relativity theory, no such thing as an unambiguous, absolute cosmic-wide “now” at each instant. However, we have every reason to hold that both special and general relativity are false. Not only does the historical record tell us that physics advances from one false theory to another. Furthermore, elsewhere I have shown that we must interpret physics as having established physicalism—in so far as physics can ever establish anything theoretical. Physicalism, here, is to be interpreted as the thesis that the universe is such that some unified “theory of everything” is true. Granted physicalism, it follows immediately that any physical theory that is about a restricted range of phenomena only, cannot be true, whatever its empirical success may be. It follows that both special and general relativity are false. This does not mean of course that the implication of these two theories that there is no unambiguous cosmic-wide “now” at each instant is false. It still may be the case that the first view of time, indicated at the outset, is false. Are there grounds for holding that an unambiguous cosmic-wide “now” does exist, despite special and general relativity, both of which imply that it does not exist? There are such grounds. Elsewhere I have argued that, in order to solve the quantum wave/particle problem and make sense of the quantum domain we need to interpret quantum theory as a fundamentally probabilistic theory, a theory which specifies how quantum entities—electrons, photons, atoms—interact with one another probabilistically. It is conceivable that this is correct, and the ultimate laws of the universe are probabilistic in character. If so, probabilistic transitions could define unambiguous, absolute cosmic-wide “nows” at each instant. It is entirely unsurprising that special and general relativity have nothing to say about the matter. Both theories are pre-quantum mechanical, classical theories, and general relativity in particular is deterministic. The universe may indeed be three dimensional, with a past and a future, but not spread out in four dimensional space-time, despite the fact that relativity theories appear to rule this out. These considerations, finally, have implications for views about the arrow of time and free will.

A RECENT paper by Einstein and Bargmann1 declared that "Ever since the theory of general relativity has been developed there has existed the problem of finding a unified theory of the physical field by some generalization of the relativistic theory of gravitation ... a decisive modification of the fundamental concepts is unavoidable". Schrödinger2 has also explained the need for a generalization of Einstein's original postulates in order to unify the theories of gravitation, electromagnetism, and the mesonic field responsible for binding the nucleus. There is no danger of the work of Einstein and Schrödinger being overlooked, but there is great danger that what is apparently an investigation of great importance, namely, "The Theory of Indeterminate Space-Time", by F. R. Saxby, may be missed by physicists, as it appears not in any of the usual scientific journals, but in the Bulletin of the Research Laboratories of the National Cash Register Company (pp. 13–72, September 1943), of which laboratories he is mathematics staff engineer.

We introduced the preceding chapter by mentioning the two great physical theories that changed not only the Newtonian and Max-wellian laws of nature, but also the Galilean-Newtonian concepts of space and time. We described the first of these, Planck’s quantum theory, which introduced into physics the corpuscle-wave concept and led Bohr to the correct nuclear-planetary model of the atom. We now present the second of these two revolutionary theories, the theory of relativity, without which modern astrophysics and cosmology could not have been developed. Albert Einstein presented the theory of relativity in two steps: the special theory of relativity first appeared in 1905 in one of Einstein’s three famous papers that were published that year in Annalen der physik. The great importance of these three papers may be gauged by their evaluation, many years later, by the Nobel laureate Max Born, who stated that each of these papers opened up a “whole new branch of physics.” The second step in the presentation of the theory of relativity appeared in 1915 as the general theory of relativity.

The confrontation between Einstein's gravitation theory and experimental results, notably binary pulsar data, is summarized and its significance discussed. Experiment and theory agree at the 10^{-3} level. All the basic structures of Einstein's theory (coupling of gravity to matter; propagation and self-interaction of the gravitational field, including in strong-field conditions) have been verified. However, some recent theoretical findings (cosmological relaxation toward zero scalar couplings) suggest that the present agreement between Einstein's theory and experiment might be naturally compatible with the existence of a long-range scalar contribution to gravity (such as the dilaton, or a moduli field of string theory). This provides a new theoretical paradigm, and new motivations for improving the experimental tests of gravity. Ultra-high precision tests of the Equivalence Principle appear as the most sensitive way to look for possible long-range deviations from General Relativity: they might open a low-energy window on string-scale physics.

Sound waves generated in the solar convection zone propagate through the Sun and cause it to vibrate. Resonant vibrations are detected by measuring wavelike motion that they produce on the solar surface; by analyzing them we can learn about conditions in the solar interior. At present we have only limited data but already we have drawn an important conclusion. We have inferred that outside the core of the Sun, and beneath the subsurface convection zone, the abundance of helium is about 25% by mass. It is generally believed that the intermediate region in the Sun is uncontaminated by the products of the nuclear reactions in the core and by material that might have fallen into the Sun from interstellar space. If that is indeed the case, we have measured the helium abundance of the gas cloud from which the Sun condensed some 4.7 thousand million years ago. This information is relevant to our understanding of the early stages of the evolution of the Universe. It is exciting that there are hints in the vibration data that might even enable us to determine whether the Sun really has remained unmixed, and hence whether we have a faithful record of the primordial mixture. There is also evidence in the data concerning the Sun's internal rotation. It appears that the core of the Sun rotate faster than the surface, but by how much is a subject of some controversy. It is very important to settle the matter, because the result is an essential step in one of the most important tests of Einstein's General Theory of Relativity. What evidence we have at present does not contradict Einstein's theory.

It is shown that the general theory of relativity is not satisfactory physical theory, since in it there are no laws of conservation for the matter and gravitational field taken together and it does not satisfy the principle of correspondence with Newton's theory. In the present paper, we construct a new theory of gravitation which possesses conservation laws, can describe all the existing gravitational experiments, satisfies the correspondence principle, and predicts a number of fundamental consequences.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Gravity Probe B is an orbital test of Einstein’s general theory of relativity using gyroscopes. The precession of the gyroscopes will measure both the geodetic effect (6.6 arc sec/yr) through the curved space-time surrounding the Earth and the motional effect (0.042 arc sec/yr) due to the rotating Earth dragging space-time around with it. To achieve the extraordinary accuracies needed to measure these small precessions, it is necessary to have the gyroscopes operating in the following environments: a vacuum of < 10−10 torr; an acceleration level of < 10−10 g’s; a magnetic field of < 10−7 gauss; and a temperature near 2 K. This paper discusses designs that allow scientific instruments to be installed into a dewar at 4.2 K. Methods for structurally supporting the instruments, transferring heat across joints at low temperature in vacuum and excluding air during the insertion process are discussed. The structural support method is designed for Shuttle launch loads. The separable heat transfer joints are needed for cooling the instruments through the four cool vapor cooled shields and structural support in the dewar. Both structural and thermal resistance test data for the joints are presented. A design of a helium purged air lock is discussed that allows the insertion or removal of the instruments into or out of the dewar. The design permits multiple insertions and removals without refurbishment.

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc. In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz. In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time. In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements. For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.

The topic of this chapter is the debate between adherents to respectively relationalism and substantivalism in the settings of relativity theory. In special relativity theory one may consistently hold that spacetime intervals are mere attributes of pairs of material objects. Even though spacetime often is treated as an object in relativity theory, there is no reason to think that it is a physical object; we may very well accept it as a mathematical object without concluding that this mathematical object has a counterpart in the physical world. By the same argument one may conclude that even if string theory is a nearly correct theory, the fact that there are 10 (or 11) spacetime dimensions in this theory doesn’t say anything about the number of spacetime dimensions in the real world.When we go to general relativity we have no obvious way to distinguish between object and attribute. Einstein’s equation, the fundamental principle in general relativity, tells us that a function of the spacetime metric is equal to the matter-energy tensor. The equation merely relates two quantities to each other. One is then inclined to ask which object are these quantities attributes of and the answer is that general relativity doesn’t make any distinction between object and attribute. One may say that either we can treat matter-energy distribution as a description of what exists and the function of the spacetime metric an attribute of that object, or reverse and hold that spacetime with its metric is what exists. The structure of general relativity transcends the structure of our common sense thinking in terms of objects with attributes.

As soon as the theory of general relativity was propounded, astronomers and physicists made ready to test its observational predictions. There was of course the excess motion of the perihelion of the orbit of Mercury, which the theory had been designed in part to explain. There was the gravitational bending of light beams (verified at the eclipse of 1919), the gravitational redshifting of light and the gravitational retarding of light (both verified in recent years). Light rays and the planet Mercury would exist without the theory of general relativity. Its effects on their behavior are small. But there is a class of objects whose existence is predicted by general relativity and which do not exist if the theory in some form is not true. These are the black holes, objects that have collapsed under the influence of their own gravitation until they are so dense and have such strong gravitational fields that neither matter nor radiation can escape from their surfaces. To a general relativist a black hole is an example of extremely curved space. In and around it the effects of general relativity are not minor; they are dominant, and they determine the nature of the phenomena that occur. The discovery of an actual black hole would provide an astrophysical working place for general relativity, a region where the effects of space curvature could be studied readily rather than having to be sorted out as tiny corrections to the nearly flat space that surrounds ordinary objects. As Remo J. Ruffini of Princeton University puts it, the discovery of a black hole would be for general relativity what the discovery of the photon (also, by the way, associated with Einstein) was for quantum theory and the physics of the microcosm. Ruffini believes, and has believed for more than a year, that at least one black hole is currently under observation (SN: 1/13/73, p. 28). Controversy rages, and to the discussion Ruffini now brings theoretical arguments and observational evidence, some of which was barely analyzed in time for him to use it at the meeting of the American Physical Society two weeks ago in Chicago. Ruffini's candidate for a black hole is Cygnus X-1, one of a newly discovered class of celestial X-ray sources, X-ray binaries. Up to, a point there is general agreement as to what these binary sources are: They are systems in which a more or less normal star is gravitationally bound to a highly condensed dark companion. The usual explanation as to how the companion got condensed is that it is the last stage of stellar evolution, what is left over after a supernova explosion. Matter streams out from the normal star and falls onto the dark companion. The streaming matter forms a rotating disk around the dark companion in which pieces of matter gradually spiral inward until they fall on the dark companion. As they do so they convert energy they gain through their fall to X-rays. (The matter in the disks is electrically charged ionized plasma, not neutral gas, so it has means to do this.) The argument starts over the nature of the condensed body. The main candidates are black holes and neutron stars-though a few observers may hold out for white dwarfs. To make the case for black holes or for a black hole one needs a way of distinguishing observationally between the two. Ruffini provides a way. It has to do with magnetic fields. Stars have magnetic fields. When they collapse, even through the violence of a supernova explosion, they should retain those fields. Thus neutron stars and black holes can have magnetic fields. But in the configuration of those fields lies the difference. A neutron star

Einstein’s theory of gravity — the general theory of relativity — is based on the principle that all objects accelerate identically in an external gravitational field. A triple-star system provides a stringent test of this principle. Einstein’s theory of gravity — the general theory of relativity — is based on the principle that all objects accelerate identically in an external gravitational field. A triple-star system provides a stringent test of this principle.