Space-time Description Research Articles

Newton’s absolute time and space in general relativity

I describe a reference system in a spherically symmetric gravitational field that is built around times recorded by radially moving geodesic clocks. The geodesic time coordinate t and the curvature spatial radial coordinate R result in spacetime descriptions of the motion of the geodesic clocks that are exactly identical with equations following from Newton’s absolute time and space used with his inverse square law. I show how to use the resulting Newtonian/general-relativistic equations for geodesic clocks to generate exact relativistic metric forms in terms of the coordinates (R,t). Newtonian theory does not describe light. However, the motion of light can be determined from the (R,t) general-relativistic metric forms obtained from Newtonian theory by setting ds2(R,t)=0. In this sense, a theory of light can be related to absolute time and space of Newtonian gravitational theory. I illustrate the (R,t) methodology by first solving the equations that result from a Newtonian picture and then examining the exact metric forms for the general-relativistic problems of the Schwarzschild field, gravitational collapse and expansion of a zero-pressure perfect fluid, and zero-pressure big-bang cosmology. I also briefly describe other applications of the Newtonian/general-relativistic formulation to: embedding a Schwarzschild mass into cosmology; continuously following an expanding universe from radiation to matter domination; Dirac’s Large Numbers hypothesis; the incompleteness of Kruskal–Szekeres spacetime; double valuedness in cosmology; and the de Sitter universe.

Effective noise in a stochastic description of inflation

Stochastic description of inflationary spacetimes emulates the growth of vacuum fluctuations by an effective stochastic ``noise field'' which drives the dynamics of the volume-smoothed inflaton. We investigate statistical properties of this field and find its correlator to be a function of distance measured in units of the smoothing length. Our results apply for a wide class of smoothing window functions and are different from previous calculations by Starobinsky and others who used a sharp momentum cutoff. We also discuss the applicability of some approximate noise descriptions to simulations of stochastic inflation.

A space-time structure determination of human CD2 reveals the CD58-binding mode.

We describe a procedure for a space-time description of protein structures. The method is capable of determining populations of conformational substates, and amplitudes and directions of internal protein motions. This is achieved by fitting static and dynamic NMR data. The approach is based on the jumping-among-minima concept. First, a wide conformational space compatible with structural NMR data is sampled to find a large set of substates. Subsequently, intrasubstate motions are sampled by using molecular dynamics calculations with force field energy terms. Next, the populations of substates are fitted to NMR relaxation data. By diagonalizing a second moment matrix, directions and amplitudes of motions are identified. The method was applied to the adhesion domain of human CD2. We found that very few substates can account for most of the experimental data. Furthermore, only two types of collective motions have high amplitudes. They represent transitions between a concave (closed) and flat (open) binding face and resemble the change upon counter-receptor (CD58) binding.

The spatial behavior of nonclassical light

Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.

D-branes in the Green-Schwarz formalism

We give a basic account of supersymmetric open strings and D-branes using the Green-Schwarz formalism, obtaining a manifestly spacetime supersymmetric description of their spectrum. In addition we discuss a mechanism whereby some of the D-brane states are projected out and which can lead to chiral quantum field theories on the brane.

Event horizon of a Schwarzschild black hole: Magnifying glass for Planck length physics

An attempt is made to describe the `thermodynamics' of semiclassical spacetime without specifying the detailed `molecular structure' of the quantum spacetime, using the known properties of blackholes. I give detailed arguments, essentially based on the behaviour of quantum systems near the event horizon, which suggest that event horizon acts as a magnifying glass to probe Planck length physics even in those contexts in which the spacetime curvature is arbitrarily low. The quantum state describing a blackhole, in any microscopic description of spacetime, has to possess certain universal form of density of states which can be ascertained from general considerations. Since a blackhole can be formed from the collapse of any physical system with a low energy Hamiltonian H, it is suggested that when such a system collapses to form a blackhole, it should be described by a modified Hamiltonian of the form $H^2_{\rm mod} =A^2 \ln (1+ H^2/A^2)$ where $A^2 \propto E_P^2$.I also show that it is possible to construct several physical systems which have the blackhole density of states and hence will be indistinguishable from a blackhole as far as thermodynamic interactions are concerned. In particular, blackholes can be thought of as one-particle excitations of a class of {\it nonlocal} field theories with the thermodynamics of blackholes arising essentially from the asymptotic form of the dispersion relation satisfied by these excitations. These field theoretic models have correlation functions with a universal short distance behaviour, which translates into the generic behaviour of semiclassical blackholes. Several implications of this paradigm are discussed.

Black holes, string theory and quantum coherence

On the basis of recently discovered connections between D-branes and black holes, I show how the information puzzle is solved by superstring theory as the fundamental theory of quantum gravity. The picture that emerges is that a well-defined quantum state does not give rise to a black hole even if the apparent distribution of energy, momenta, charges, etc. would predict one on classical grounds. Indeed, geometry – general relativistic space time description – is unwarranted at the quantum microstate level. It is the decoherence leading to macrostates (average over degenerate microstates) that provides – on the same token – the loss of quantum coherence, the emergence of a space time description with causal properties and, thus, the formation of a black hole and its Hawking evaporation.

More comments on string theory on AdS3

We clarify a number of issues regarding the worldsheet and spacetime descriptions of string propagation on AdS_3. We construct the vertex operators of spacetime current algebra and spacetime (super) Virasoro generators in the full interacting SL(2) WZW theory and study their Ward identities. We also explain the relation between the analysis in this note and some recent work on this subject.

Beyond strings, multiple times and gauge theories of area-scalings relativistic transformations

Nottale's special scale-relativity principle was proposed earlier by the author as a plausible geometrical origin to string theory and extended objects. Scale relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations in nature is taken to be the Planck scale, which is not the same as the string scale. Starting with ordinary actions for strings and other extended objects, we show that gauge theories of volume-resolutions scale-relativistic symmetries, of the world volume measure associated with the extended “fuzzy” objects, are a natural and viable way to formulate the geometrical principle underlying the theory of all extended objects. Gauge invariance can only be implemented if the extendon actions in D target dimensions are embedded in D + 1 dimensions with an extra temporal variable corresponding to the scaling dimension of the original string coordinates. This is achieved upon viewing the extendon coordinates, from the fuzzy world volume point of view, as noncommuting matrices valued in the Lie algebra of Lorentz-scale relativistic transformations. Preliminary steps are taken to merge motion relativity with scale relativity by introducing the gauge field that gauges the Lorentz-scale symmetries in the same vain that the spin connection gauges ordinary Lorentz transformations and, in this fashion, one may go beyond string theory to construct the sought-after General Theory of Scale-Motion Relativity. Such theory requires the introduction of the scale-graviton (in addition to the ordinary graviton) which is the field that gauges the symmetry which converts motion dynamics into scaling-resolutions dynamics and vice versa (the analog of the gravitino that gauges supersymmetry). To go beyond the quantum string geometry most probably would require a curved fractal spacetime description (curved from both scaling and motion points of views) with a curvilinear fractal coordinate system. Non-Archimedean geometry and p-adic numbers are essential ingredients comprising the geometrical arena of such extensions of quantum string geometry.

Bulk versus boundary dynamics in anti–de Sitter spacetime

We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti--de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates. For ${\mathrm{AdS}}_{3},$ the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\mathrm{SL}(2,R{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SL}(2,R{)}_{R}$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincar\'e coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.

The Close Limit of Colliding Black Holes: An Update

The collision of binary black holes is one of the primary expected sources of gravitational waves to be detected by the broadband interferometric gravitational wave telescopes currently under construction, like the LIGO project in the US, the British/German GEO project, the TAMA project in Japan and the French/Italian VIRGO project. A collision of two binary black holes can be divided into three distinct regimes. Initially, the black holes spiral around each other in quasi-Newtonian orbits. The radius of the orbits decrease due to the emission of gravitational radiation. Let us call this period the “inspiral” phase. The gravitational waves produced during this phase are well described by the post-Newtonian approximation. Notice that such an approximation does not provide a good description of the whole spacetime, since it breaks down close to each hole (to first approximation, the holes are singular point particles), but as long as the holes are far apart, this is not expected to be relevant from the point of view of the waveforms observed at infinity.∗) The gravitational waves from this phase of the collision correspond to a quasi-regular sinusoid whose frequency and amplitude increases with time as the holes start getting closer to each other, known as the “chirp”. Good descriptions of this approximation applied to the binary black hole case and appropriate references can be found in Blanchet et al. 2) When the separation of the holes is around 10 to 12 times the mass of each individual holes, it is expected that the post-Newtonian approximation breaks down. It is not completely clear what is the extent of the breakdown, since the post-Newtonian approximation leads to an asymptotic perturbation series. In fact, attempts are currently being made to extend the validity of the domain of the approximation using Pade approximants. 3) In any event, there will be a limiting separation of the holes such that if they are any closer, one cannot use the post-Newtonian approximation. The domain that starts at that point and continues up to the point in which the

A Pure Geometric Approach to Cosmology

It is well known that standard Big-Bang cosmology suffers from certain problems, e.g. singularity, horizon, flatness, … In the present work it is claimed that the appearence of some of these problems is due to two main assumptions. The first is the assumption that the 4-dimentional Riemannian (RIE)-geometry gives a complete description of the cosmic space-time. The second is the assumption that the material distribution in the universe is described by a phenomenological (PH)-matter tensor. It is shown that, by relaxing these two assumptions, some of the problems of the standard Big-Bang cosmology could be avoided. The following table summarises some results in favour of the above claim. The absolute parallelism (AP)-geometry is used to construct some of the theories mentioned in the table.

Localisation of Large Scale Structures in a Supersonic Mixing Layer: A New Method and First Analysis

The space-time description of the organised structures in a supersonic mixing layer with a convective Mach number of 0.6 is given. Fourier analysis gives some evidence of the existence of large scales nearly periodic in time, but gives only global information on length and velocity scales. A new method based on the wavelet transform is then proposed to perform a space-time analysis localised in time. It is used to detect and to analyse such structures, in particular to determine their convection velocity. Characteristic time and length scales are given and compared with the scales deduced from the spectral analysis. The results are used to extract the contribution of the large scales to the turbulence signal. The flow under examination has a convective Mach number of 0.6. Therefore, in contrast with subsonic mixing layers, there are probably significant three-dimensional effects. Results on spacing between large eddies are discussed in terms of merging processes and of three-dimensional effects. A conclusion is that diffusion and/or entrainment are modified rather than the type and frequency of merging.

VACUUM DECAY VIA LORENTZIAN WORMHOLES

We speculate about the space–time description due to the presence of Lorentzian worm-holes (handles in space–time joining two distant regions or other universes) in quantum gravity. The semiclassical rate of production of these Lorentzian wormholes in Reissner–Nordström space–times is calculated as a result of the spontaneous decay of vacuum due to a real tunneling configuration. In the magnetic case it only depends on the value of the field theoretical fine structure constant. We predict that the quantum probability corresponding to the nucleation of such geodesically complete space–times should be acutally negligible in our physical Universe.

QCD space-time analysis of quarkonium formation and evolution in hadronic collisions

The production of heavy quarkonium as $Q\overline{Q}$ bound states in hadron-hadron collisions is considered within the framework of a space-time description, combining parton-cascade evolution with a coalescence model for bound-state formation. The ``hard'' production of the initial $Q\overline{Q}$, directly or via gluon fragmentation and including both color-singlet and color-octet contributions, is calculated from the perturbative QCD cross sections. The subsequent development of the $Q\overline{Q}$ system is described within a space-time generalization of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi parton-evolution formalism in position and momentum space. The actual formation of the bound states is accomplished through overlap of the $Q\overline{Q}$ pair and a spectrum of quarkonium wave functions. This coalescence can only occur after sufficient gluon radiation reduces the $Q\overline{Q}$ relative velocity to a value commensurate with the nonrelativistic kinematics of these bound systems. The presence of gluon participants in the cascade then is both necessary and leads to the natural inclusion of both color-singlet and color-octet mechanisms. The application of this approach to $\mathrm{pp}$ $(p\overline{p})$ collisions from $\sqrt{s}=$ 30 GeV--14 TeV reveals very decent agreement with available data from the CERN Intersecting Storage Rings and Fermilab Tevatron---without the necessity of introducing fit parameters. Moreover, production probabilities are calculated for a complete spectrum of charmonium and bottomonium states, with the relative significance compared to open charm (bottom) production. An analysis of the space-time development is carried through which sheds light on the relevance of gluon radiation and color structure, suggesting a corresponding experimental investigation.

Towards a quantum pregeometric description of space-time

The ubiquity of singularities in solutions of the field equations of general relativity, and the non-separability that is central to quantum theory cast serious doubt on the validity of continuum pictures of space-time. Furthermore, attempts to construct a quantum theory of gravity indicate that a continuum picture must break down on small scales. To overcome these problems, it has been suggested that space-time should be considered as an emergent property of an underlying structure, or ‘pregeometry’. Here, the problems posed for continuum space-time by general relativity and quantum theory are discussed, and an approach to pregeometry based on the notion of quantum process is outlined.

Quantum Space-Time and Tetrads

The description of space-time in aquantum-theoretic framework must be considered as afundamental problem in physics. Most attempts start withan already given classical space-time, then thequantization is done. In contrast to this, the centralassumption in this paper is not to start withspace-time, but to derive it from some more abstractpresuppositions as done in Von Weizsacker's quantumtheory of ur-alternatives. Mathematically, the transitionfrom a manifold with spin structure to a manifold withfour real space-time coordinates has to be considered.The suggestion is made that this transition can be well described by using a tetradialformalism which appears to be the most naturalconnection between ur-spinors and realfour-vectors.

First massive state of the superstring in superspace

Using the manifestly spacetime supersymmetric description of the four-dimensional open superstring, we construct the vertex operator in superspace for the first massive state. This construction provides an N=1 D=4 superspace representation of the massive spin-2 multiplet.

Brisure de symétrie spontanée et géométrie du point de vue spectral

We present a new notion of geometric space, which, abandoning the central role played by points in standard geometry, allows a greater freedom in description of space-time at short scale. The proposed framework is sufficiently general to describe discrete spaces, Riemannian spaces, configuration spaces of quantum field theory and duals of discrete groups not necessarily commutative.

Space–Time Description of Nonstationary Trapped Lee Waves Using ST Radars, Aircraft, and Constant Volume Balloons during the PYREX Experiment

Abstract The third intensive observation period (IOP3) of PYREX was a case of strong lee waves generated by a southerly wind crossing the Pyrenees chain. Upstream radiosounds and measurements obtained by aircraft along the chain transect and by constant volume balloons launched near the crest provided spatial characteristics of the lee wave at different times and heights. Values ranging from 7 to 14 km for the horizontal wavelength, and from 3 to 5 m s−1 for the maximum amplitude of the air vertical velocity, were observed. The lee wave horizontal extent, measured from the crest line, reached 30 to 55 km. In addition, two very high frequency stratosphere-troposphere (VHF ST) radars, one on the mountain mean axis and another downstream in the lee wave field, observed the temporal variations of the vertical profiles of the vertical velocity. The analysis of those observed variations and their vertical distribution allowed the stationarity of the wave to be studied. The lee wave was found to be far from stat...