General relativistic tidal work for Papapetrou, Weinberg and Goldberg pseudotensors

In 1998 Thorne claimed that all pseudotensors give the same tidal work as the Newtonian theory. In 1999, Purdue used the Landau-Lifshitz pseudotensor to calculate the tidal heating and the result matched with the Newtonian gravity. Soon after in 2001, Favata employed the same method to examine the Einstein, Bergmann-Thomson and M{\o}ller pseudotensors, all of them give the same result as Purdue did. Inspired by the work of Purdue and Favata, for the completeness, here we manipulate the tidal work for Papapetrou, Weinberg and Goldberg pseudotensors. We obtained the same tidal work as Purdue achieved. In addition, we emphasize that a suitable gravitational energy-momentum pseudotensor requires fulfill the inside matter condition and all of the classical pseudotensors pass this test except M\oller. Moreover, we constructed a general pseudotesnor which is modified by 13 linear artificial higher order terms combination with Einstein pseudotensor. We find that the result agrees with Thorne's prediction, i.e., relativistic tidal work is pseudotensor independent.Comment: 6 page

General relativistic tidal heating for the Moller pseudotensor

In his study of tidal stabilization of fully relativistic neutron stars Thorne showed that the fully relativistic expression for tidal heating is the same as in non-relativistic Newtonian theory. Furthermore, Thorne also noted that the tidal heating must be independent of how one localizes gravitational energy and is unambiguously given by that expression. Purdue and Favata calculated the tidal heating for a number of classical gravitational pseudotensors including that of Moller, and obtained the result that all of them produced the same (Newtonian) value. However, in a re-examination of the calculation using the Moller pseudotensor we find that there is no tidal heating. This leads us to the conclusion that Thorne's assertion needs a minor modification: the relativistic tidal heating is pseudotensor independent only if the pseudotensor is derived from a Freud type superpotential.Comment: 10 pages, a major revision of arXiv:1509.0920

General relativistic tidal heating for Moller pseudotensor

Thorne elucidated that the relativistic tidal heating is the same as the Newtonian theory. Moreover, Thorne also claimed that the tidal heating is independent of how one localizes gravitational energy and is unambiguously given by a certain formula. Purdue and Favata calculated the tidal heating for different classical pseudotensors including Moller and obtained the results all matched with the Newtonian perspective. After re-examined this Moller pseudotensor, we find that there does not exist any tidal heating value. Thus we claim that the relativistic tidal heating is pseudotensor independent under the condition that if the peusdotensor is a Freud typed superpotential.Comment: 4 page

Gravitational energy in small regions for the quasilocal expressions in orthonormal frames

The M\oller tetrad gravitational energy-momentum expression was recently evaluated for a small vacuum region using orthonormal frames adapted to Riemann normal coordinates. However the result was not proportional to the Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$. Treating a modified quasilocal expressions in a similar way, we found one unique combination that gives a multiple of $B_{\alpha\beta\mu\nu}$ which provides a non-negative gravitational energy-momentum in the small sphere approximation. Moreover, in addition to $B_{\alpha\beta\mu\nu}$, we found a certain tensor $S_{\alpha\beta\mu\nu}+K_{\alpha\beta\mu\nu}$ which gives the same "energy-momentum" density in vacuum. Using this tensor combination, we obtained an infinite set of solutions that provides a positive gravitational energy within the same limit.Comment:

Application of the 3-space approach to the Bianchi II cosmological model

Einstein used 4-dimensional space time geometry to explain gravity. However, in 1962, Baierlein, Sharp and Wheeler proposed a Jacobi type timeless Lagrangian based on the 3-dimensional geometry of space to reproduce the same physics. In 2002, Barbour $et$. $al$. further extended this idea and they call it 3-space approach. Here we use Bianchi II cosmological model to demonstrate the 3-space idea. Indeed, we find that this theory is more fundamental and the manipulation is more practical. We recover the known and find a new solutions.Comment: 5 page

The modification of the Einstein and Landau-Lifshitz pseudotensrs

Deser et al. proposed a combination of the Einstein and Landau-Lifshitz pseudotensors such that the second derivatives in vacuum are proportional to the Bel-Robinson tensor. Stimulated by their work, the present paper discuss the gravitational energy-momentum expression which has the same desirable Bel-Robinson tensor property. We find modifications of the Einstein and Landau-Lifshitz pseudotensors that both give the same coefficient of the Bel-Robinson tensor in vacuum in holonomic frames.Comment: 10 page

Modification of the Bel-Robinson type energy-momentum

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson type energy-momentum in a quasilocal 2-surface, both the Bel-Robinson tensor $B$ and tensor $V$ are suitable. We found that this Bel-Robinson type energy-momentum can be modified such that it satisfies the Lorentz covariant, future pointing and non-spacelike properties. We find that these particular energy-momentum properties can be obtained from (i): $B$ or $V$ plus a tensor $S$ in a quasilocal small cube limit, or (ii): directly evaluating the energy-momentum of $B$ or $V$ in a quasilocal small box region.Comment: 6 page

Relativistic tidal heating of Hamiltonian quasi-local boundary expressions

Purdue and Favata calculate the tidal heating used certain classical pseudotensors. Booth and Creighton employed the quasi-local mass formalism of Brown and York to demonstrate the same subject. All of them give the result matched with the Newtonian theory. Here we present another Hamiltonian quasi-local boundary expressions and all give the same desired value. This indicates that the tidal heating is unique as Thorne predicted. Moreover, we discovered that the pseudo-tensor method and quasi-local method are fundamentally different.Comment: 6 page

Quasilocal energy-momentum for tensor V in small regions

The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal energy-momentum in a small sphere. Using a pseudotensor approach to evaluate the energy-momentum in a half-cylinder, we find that $B$ and $V$ have different values, not proportional to the `Bel-Robinson energy-momentum'. Furthermore, even if we arrange things so that we do get the same `Bel-Robinson energy-momentum' value, the angular momentum gives different values using $B$ and $V$ in half-cylinder. In addition, we find that $B$ and $V$ have a different number of independent components. The fully trace free property of $B$ and $V$ implies conservation of pure `Bel-Robinson energy-momentum' in small regions, and vice versa.Comment: 8 pages, a major revision of arXiv:1006.527

K\"ahler immersions of K\"ahler manifolds into complex space forms

The study of K\"ahler immersions of a given real analytic K\"ahler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally K\"ahler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a K\"ahler immersion into another. Although, a complete classification of K\"ahler manifolds admitting a K\"ahler immersion into complex space forms is not known, not even when the K\"ahler manifolds involved are of great interest, e.g. when they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.Comment: 116 page