Helioseismology

Well-confirmed helioseismic data from several groups using various observational techniques at different sites have allowed us to determine the differential rotation in the outer half of the Sun's interior. The resulting rotation law is simple — the surface differential rotation persists through much of the convection zone with a transition toward solid body rotation beneath. To date there is no appealing evidence for a rapidly rotating core. There is however, weak evidence for a solar cycle dependence of the Sun's internal rotation.

Tones of the Oscillating Sun

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.

We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. All but high-latitude displacements align themselves with the observed surfaces of constant angular velocity. The tendency for the angular velocity to remain constant with depth in the bulk of the convective zone, together with other critical features of the rotation profile, emerge from little more than a visual inspection of the governing equation. In the absence of a background axial angular velocity gradient, displacements exhibit no poleward bias, suggesting that solar convection "plays-off" of prexisting shear rather than creates it. We argue that baroclinic vorticity of precisely the right order is generated at the radiative/convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces. If so, many features of the Sun's internal rotation become more clear, including: i) the general appearance of the tachocline; ii) the extension of differential rotation well into the radiative zone; iii) the abrupt change of morphology of convective zone isorotation surfaces; and iv) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions.

It is well known that the combination of barotropic rotation and radiative equilibrium are mutually incompatible in stars. The Sun's internal rotation is far from barotropic, however, which allows at least the theoretical possibility that the Sun's thermal balance is one of radiative equilibrium in the region of the tachocline near the outer boundary of the radiative zone. We show here that (i) the constraint of radiative equilibrium leads to a straightforward ordinary differential equation for the Sun's rotation profile, and (ii) solutions of this equation can be found that, to within current levels of accuracy, closely resemble the rotation profile deduced from helioseismology. More generally, we calculate how large a baroclinic deviation from uniform rotation is required to maintain radiative equilibrium without meridional circulation throughout the bulk of the radiative zone. Very little deviation is required, well below detectability. The feasibility of radiative equilibrium for the tachocline suggests that the issue of a spreading boundary layer may be less severe than previously thought.

We consider inversion techniques for investigating the structure and dynamics of the solar interior as functions of radius and latitude. In particular, we look at the problem of inferring the radial and latitudinal dependence of the Sun's internal rotation, using a fully two-dimensional least-squares inversion algorithm. Concepts such as averaging kernels, measures of resolution, and trade-off curves, which have previously been used in the one-dimensional case, are generalized to facilitate a comparison of two-dimensional methods. We investigate the weighting given to different modes and discuss the implications of this for observational strategies. As an illustration we use a mode set whose properties are similar to those expected for data from the GONG network.

We report inferences for the Sun's internal rotation from GONG months 4-10 averaged power spectra. In keeping with the international collaborative nature of the GONG project, the results presented here are based on the work of several groups around the world inverting the GONG data and sharing their results via the world-wide web. These groups are at the Observatoire de la Côte d'Azur, Nice (T. Corbard, G. Berthomieu, J. Provost); Theoretical Astrophysics Center, Aarhus (J. Christensen-Dalsgaard, F. Pijpers); Center for Astrophysics, Cambridge MA (A. Eff-Darwich, S. Korzennik); QMW, London (R. Howe, M. Thompson, in collaboration with J. Schou, Stanford); Institute of Astronomy, Cambridge (T. Sekii, D. Gough); University of Sydney (D. Burtonclay, Li Yan, P. Wilson); and Tata Institute of Fundamental Research, Bombay (H. Antia, S. Chitre).

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.

An introduction to current techniques to infer the Sun's internal rotation from observed acoustic oscillation modes is given, and some representative results are shown.

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.

To the best of my knowledge, it can safely be argued that the majority of recent studies on the foundational difficulties of quantum physics start from the assumptions that the conceptual foundations of classical physics and relativity theory are clear and unproblematic and the present problems should concern only quantum physics. Yet, a simple inquiry into the literature and especially into Einstein’s epistemological writings shows that, contrary to the view of a supposedly well-founded classical physics and relativity, important foundational problems in these sciences are still in need of further analysis.1 The belief that the only foundational difficulties belong to quantum physics (henceforth QP), as if they alone existed against an ideal unproblematic background of classical physics (henceforth CP) and relativity theories (RR), results in a limited approach in examining the historical documents.KeywordsClassical PhysicLocal Inertial FramePure MethodFoundational ProblemMethodological ViewThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The medium-l program of the Michelson Doppler Imager instrument on board SOHO provides continuous observations of oscillation modes of angular degree, l, from 0 to ∽ 300. The data for the program are partly processed on board because only about 3% of MDI observations can be transmitted continuously to the ground. The on-board data processing, the main component of which is Gaussian-weighted binning, has been optimized to reduce the negative influence of spatial aliasing of the high-degree oscillation modes. The data processing is completed in a data analysis pipeline at the SOI Stanford Support Center to determine the mean multiplet frequencies and splitting coefficients. The initial results show that the noise in the medium-l oscillation power spectrum is substantially lower than in ground-based measurements. This enables us to detect lower amplitude modes and, thus, to extend the range of measured mode frequencies. This is important for inferring the Sun's internal structure and rotation. The MDI observations also reveal the asymmetry of oscillation spectral lines. The line asymmetries agree with the theory of mode excitation by acoustic sources localized in the upper convective boundary layer. The sound-speed profile inferred from the mean frequencies gives evidence for a sharp variation at the edge of the energy-generating core. The results also confirm the previous finding by the GONG (Gough et al., 1996) that, in a thin layer just beneath the convection zone, helium appears to be less abundant than predicted by theory. Inverting the multiplet frequency splittings from MDI, we detect significant rotational shear in this thin layer. This layer is likely to be the place where the solar dynamo operates. In order to understand how the Sun works, it is extremely important to observe the evolution of this transition layer throughout the 11-year activity cycle.

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.

Motion is always relative to something. Is this thing a concrete body like the earth, is it an abstract space, or is it an imagined frame? Do the laws of physics depend on the choice of reference? Is there a choice for which the laws are simplest? Is this choice unique? Is there a physical cause for the choice made? These questions traverse the history of modern physics from Galileo to Einstein. The answers involved Galilean relativity, Newton’s absolute space, the purely relational concepts of Descartes, Leibniz, and Mach, and many forgotten uses of relativity principles in mechanics, optics, and electrodynamics―until the relativity theories of Poincaré, Einstein, Minkowski, and Laue radically redefined space and time to satisfy universal kinds of relativity. Accordingly, this book retraces the emergence of relativity principles in early modern mechanics, documents their constructive use in eighteenth- and nineteenth-century mechanics, optics, and electrodynamics, and gives a well-rooted account of the genesis of special and general relativity in the early twentieth century. As an exercise in long-term history, it demonstrates the connectivity of issues and approaches across several centuries, despite enormous changes in context and culture. As an account of the genesis of relativity theories, it brings unprecedented clarity and fullness by broadening the spectrum of resources on which the principal actors drew.

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.