Riemannian Geometry with Maintaining the Notion of Distant Parallelism (Albert Einstein, 1928):
Riemannian Geometry has led to a physical description of the gravitational field in the theory of general relativity, but it did not provide concepts that can be attributed to the electromagnetic field. Therefore, theoreticians aim to find natural generalizations or extensions of riemannian geometry that are richer in concepts, hoping to arrive at a logical construction that unifies all physical field concepts under one single leading point. Such endeavors have brought be to a theory which should be communicated even without attempting any physical interpretation, because it can claim a certain interest just because of the naturality of the concepts introduced thererin.
Riemannian geometry is characterized by an Euclidean metric in an infinitesmal neighborhood of any point P. Furthermore, the absolute values of the line elements which belong to the neighborhood of two points P and Q of finite distance can be compared. However, the notion of parallelism of such line elements is missing; a concept of direction does not exist for the finite case. The theory outlined In the following is characterized by introducing - beyond the Riemannian metric - the concept of 'direction', 'equality of directions' or parallelism for finite distances. Therefore, new invariants and tensors will arise besides those known in Riemannian geometry.
http://www.alexander-unzicker.de/rep1.pdf
Torsion Gravity: a Reappraisal
"The role played by torsion in gravitation is critically reviewed. After a description of the problems and controversies involving the physics of torsion, a comprehensive presentation of the teleparallel equivalent of general relativity is made. According to this theory, curvature and torsion are alternative ways of describing the gravitational field, and consequently related to the same degrees of freedom of gravity. However, more general gravity theories, like for example Einstein-Cartan and gauge theories for the Poincare and the affine groups, consider curvature and torsion as representing independent degrees of freedom. By using an active version of the strong equivalence principle, a possible solution to this conceptual question is reviewed. This solution favors ultimately the teleparallel point of view, and consequently the completeness of general relativity. A discussion of the consequences for gravitation is presented."
http://arxiv.org/abs/gr-qc/0501017
"In theoretical physics, teleparallelism is an alternative theory to general relativity. The setting is a (locally) flat Minkowski space (i.e. no curvature), but having torsion. Gravitational effects are due to this torsion and not the curvature of spacetime.
Current experiments still cannot distinguish between general relativity and teleparallelism."
http://www.wordiq.com/definition/Teleparallelism
Generalizations of teleparallel gravity and local Lorentz symmetry
"We analyze the relation between teleparallelism and local Lorentz invariance. We show that generic modifications of the teleparallel equivalent to general relativity will not respect local Lorentz symmetry. We clarify the reasons for this and explain why the situation is different in general relativity. We give a prescription for constructing teleparallel equivalents for known theories. We also explicitly consider a recently proposed class of generalized teleparallel theories, called f(T) theories of gravity, and show why restoring local Lorentz symmetry in such theories cannot lead to sensible dynamics, even if one gives up teleparallelism."
http://arxiv.org/abs/1012.4039
"In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance is a related concept, covariance being a measure of how much two variables change together.
Lorentz covariance (from Hendrik Lorentz) is a key property of spacetime that follows from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings:
A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a scalar (e.g. the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz invariant (i.e. they transform under the trivial representation).
An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term invariant here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame (this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame). This condition is a requirement according to the principle of relativity, i.e. all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference."
http://en.wikipedia.org/wiki/Lorentz_covariance
Lorentz invariance and loop quantum gravity
"Loop quantum gravity (LQG) is a quantization of a classical Lagrangian field theory. It is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just like it is broken in general relativity (unless one is dealing with
Minkowski spacetime, which is one particular solution of the Einstein field equations). On the other hand, there has been much talk about possible local and global violations of Lorentz invariance beyond those expected in straightforward general relativity. Of interest in this connection would be to see whether the LQG analogue of Minkowski spacetime breaks or preserves global lorentz invariance, and
Carlo Rovelli and coworkers have recently been investigating the Minkowski state of LQG using spin-foam techniques.
These questions will all remain open as long as the classical limits of various LQG models (see below for the sources of variation) cannot be calculated.
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact SO(3) or SU(2) gauge group as opposed to the non-compact SO(3,1) or SL(2.C). The compactness of the Lie group
avoids some thus-far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The infamous Immirzi parameter is necessary to resolve an ambiguity in the process of complexification. These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through.
It should be pointed out that the reasons why one can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level is that the respective Lie algebras are the same. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (these subtleties are usually ignored in elementary particle physics). The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.
To make matters more complicated, it can be shown that a positive cosmological constant can be realized in LQG by replacing the
Lorentz group with the corresponding quantum group. At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated (in the case of the rotation group, instead of having representations labelled by all half-integral spins, one is left with all representations with total spin j less than some constant). It is entirely possible to formulate LQG in terms of q-deformed Lie algebras instead of ordinary Lie algebras, and in the case of the Lorentz group the result would, again, be indistinguishable for sufficiently small velocity paramenters.
In the spin-foam formalism the Barrett-Crane model, which was for a while the most promising state-sum model of 4D Lorentzian quantum gravity, was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks.
These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Lee Smolin and Joao Magueijo then went on to study doubly-special relativity, in which not only there is a constant velocity c but also a constant distance l. They showed that there are nonlinear representations of the Lorentz lie algebra with these properties (the usual Lorentz group being obtained from a linear representation). Doubly-special relativity predicts deviations from the special relativity dispersion relation at large energies (corresponding to small wavelengths of the order of the constant length l in the doubly-special theory).
Giovanni Amelino-Camelia then proposed that the mystery of ultra-high-energy cosmic rays might be solved by assuming such violations of the special-relativity dispertion relation for photons.
Phenomenological (hence, not specific to LQG) constraints on anomalous dispersion relations can be obtained by considering a variety of astrophysical experimental data, of which high-energy cosmic rays are but one part. Current observations are already able to place exceedingly stringent constraints on these phenomenological parameters."
http://www.wordiq.com/definition/Lorentz_invariance
No comments:
Post a Comment