Online research in philosophy

We take for granted that Gravitational Waves (GWs) exist, but examine critically the possibility for their direct observation with ground and space-based laser interferometers. It is argued that the detection of GWs can, at least theoretically, be achieved iff three requirements are met en bloc. Alternatively, a hypothetical case related to the so-called dark energy would render the task impossible in principle. The discussion is kept at conceptual level, to make it accessible to the general audience.

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96..

Four empirically equivalent versions of general relativity, namely standard GR, Lorentz-invariant gravitational theory, and the gravitational gauge theories of the Lorentz and translation groups, are investigated in the form of a case study for theory underdetermination. The various ontological indeterminacies (both underdetermination and inscrutability of reference) inherent in gravitational theories are analyzed in a detailed comparative study. The concept of practical underdetermination is proposed, followed by a discussion of its adequacy to describe scientific progress.

Four empirically equivalent versions of general relativity, namely standard GR, Lorentz-invariant gravitational theory,and the gravitational gauge theories of the Lorentz and translation groups, are investigated in the form of a case study for theory underdetermination. The various ontological indeterminacies (both underdetermination and inscrutability of reference) inherent in gravitational theories are analyzed in a detailed comparative study. The concept of practical underdetermination is proposed, followed by a discussion of its adequacy to describe scientific progress.

In 1907, Einstein set out to fully relativize all motion, no matter whether uniform or accelerated. After five failed attempts between 1907 and 1918, he finally threw in the towel around 1920, setting himself a new goal. For the rest of his life he searched for a classical field theory unifying gravity and electromagnetism. As he struggled to relativize motion, Einstein had to readjust both his approach and his objectives at almost every step along the way; he got himself hopelessly confused at times; he fooled himself with fallacious arguments and sloppy calculations; and he committed what he later allegedly called the biggest blunder of his career: he introduced the cosmological constant. There is a very uplifting moral to this somber tale. Although Einstein never reached his original destination, the harvest of his thirteen-year odyssey is quite impressive. First of all, what is left of absolute motion in general relativity is far more palatable than absolute motion in special relativity or Newtonian theory. And general relativity does seem to eliminate absolute space. More importantly, from a modern physics point of view, Einstein produced a spectacular new theory of gravity based on what he called the equivalence principle. This principle says that inertial and gravitational effects are due to one and the same structure, the inertio-gravitational field, which in Einstein’s theory is represented by a metric tensor field. In addition to laying the foundations of this theory, Einstein, among other things, launched relativistic cosmology, suggested the possibility of gravitational waves, gave the first sensible definition of a space-time singularity, and caught on to the intimate connection between general covariance and energy-momentum conservation, an example of the general connection between symmetries and conservation laws of Noether’s theorems. These results more than make up for the—at least by the standards of modern philosophy of science—rather opportunistic way in which they were obtained..

On Mass Problem in Relativistic Mechanics and Gravitational Physics Anatoli Vankov (dated 12.16.2003, e-mail: anatolivankov@hotmail.com) The proper mass of a test particle in General Relativity Theory (GRT) is a rest mass, so it is considered principally constant, just as in Kinematics of Special Relativity Theory (SRT). One may think that the same is true in SRT Mechanics (Dynamics). We found that a proper mass change occurs under a force action that is, during a transition from one inertial reference frame to another. The proper mass constancy in SRT Mechanics is, in fact, a weak field approximation leading to the Newtonian limit. We show that a variability of the proper mass is a fundamental physical phenomenon. It becomes especially important under strong field conditions, therefore, for understanding of the so-called self-energy divergence. The problem was seemingly overcome with help of the known renormalization procedure in Electrodynamics but not in gravitational field theory. GRT was shown to be nonrenormalizable. Our analysis of the SRT mass-energy concept showed that, after the proper mass variation was taken into account in SRT Mechanics equations, arguments for an exclusion of the gravity phenomenon from the SRT domain fell away. Moreover, this approach resulted in principal elimination of the gravitational divergence problem. Another new result concerned the speed of light. The conclusion was that the speed of light is not a fundamental physical constant: it is a physical quantity determined by a gravitational potential and has a cosmological meaning. In spite of radically different physical interpretation, the alternative approach to the gravitational problem gives an adequate description of “weak-field” gravitational experiments as GRT does: a numerical difference from GRT predictions is not meaningful. However, the difference in predictions progressively rises with field strength and an energy increase. One particular result concerns a behavior of a massive particle being in free fall in a gravitational field. In GRT, both a free particle and a photon, when approaching a gravitational center, tend to slow down, the particle speed being always less then the photon speed. In the SRT approach, the photon similarly slows down but not the particle. If so, superluminal particles exist. This is a new physical phenomenon, which may be called a gravitational refraction. We propose the experiment on the detection of superluminal particles in high-energy cosmic rays. It should be considered a new relativistic test having a falsifying power in a strong-field domain. This work is mainly conceptual. The purpose is to present in a simple form for a wide physical community some results of our study of Relativistic Mechanics, in which a source of a gravitational field is the proper mass. The main conclusion is that the development of the SRT-based divergence-free gravitation field theory is possible. PACS 04.80.Cc Key words: 1. General relativity

The problem of finding a covariant expression for the distribution and conservation of gravitational energy-momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann's realization that there are infinitely many gravitational energy-momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of pseudotensors of a given type, such as the Einstein pseudotensor, in _every_ coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau-Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether's theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many.

I describe how relativistic field theory generalizes the paradigm property of material systems, the possession of mass, to the requirement that they have a mass–energy–momentum density tensor T µ associated with them. I argue that T µ does not represent an intrinsic property of matter. For it will become evident that the definition of T µ depends on the metric field g µ in a variety of ways. Accordingly, since g µ represents the geometry of spacetime itself, the properties of mass, stress, energy, and momentum should not be seen as intrinsic properties of matter, but as relational properties that material systems have only in virtue of their relation to spacetime structure.

The determination of inertia by matter is looked at in general relativity, where inertia can be represented by affine or projective structure. The matter tensor T seems to underdetermine affine structure by ten degrees of freedom, eight of which can be eliminated by gauge choices, leaving two. Their physical meaning---which is bound up with that of gravitational waves and the pseudotensor t, and with the conservation of energy-momentum---is considered, along with the dependence of reality on invariance and of causal explanation on conservation.

The topics of gravitational field energy and energy-momentum conservation in General Relativity theory have been unjustly neglected by philosophers. If the gravitational field in space free of ordinary matter, as represented by the metric g ab itself, can be said to carry genuine energy and momentum, this is a powerful argument for adopting the substantivalist view of spacetime.This paper explores the standard textbook account of gravitational field energy and argues that (a) so-called stress-energy of the gravitational field is well-defined neither locally nor globally; and (b) there is no general principle of energy-momentum conservation to be found in General Relativity. I discuss the nature and justification of the zero-divergence law for ordinary stress-energy, and its possible connection with the failure of General Relativity to realise Mach's principle.