arXiv subjects

Lau Loi So

Publications and source records attributed to Lau Loi So.

At least 19 recordsLinked to original sources

arXiv2024

The gravitational angular momentum for the super-energy Bel-Robinson tensor

Although the super-energy Bel-Robinson tensor gives a desirable gravitational energy-momentum in a small sphere region, the angular-momentum is vanishing. Intuitively, it should be non-zero. Our present work shows that indeed the angular momentum is non-vanishing under the continuity equation requirement. Meanwhile this angular momentum can be converted as a ``Poynting" vector. In addition, by the analog, we also constructed the four ``Maxwell" equations in general relativity.

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arXiv2023

The gravitational energy-momentum for the super-energy Bel-Robinson tensor

Describing the gravitational energy-momentum, the super-energy Bel-Robinson tensor is the best candidate. In the past, people seems only explore the lowest order: the electric part $E_{ab}$ and magnetic part $B_{ab}$ for the Riemann tensor. These two components are related with the static case, however, for the energy transfer situation, one may need to consider the time varying $\dot{E}_{ab}$ and $\dot{B}_{ab}$. Here we use $(\dot{E}_{ab},\dot{B}_{ab}$) to study the energy-momentum for the Bel-Robinson tensor in a small sphere limit. Meanwhile, our result illustrates how the gravitational field carries the 4-momentum including this extra information.

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arXiv2022

The Bel-Robinson tensor and the classical pseudotensors

Describing the gravitational energy and momentum, the Bel-Robinson tensor is the best tensor. However, the classical pseudotensors can also manage the the same job. As Deser mentioned in 1999, a certain linear combination between Einstein pseudotensor and Landau-Lifshitz pseudoetnsor give a pure Bel-Robinson tensor. Here we used the same idea but adapted the harmonic gauge, we found that all the classical pseudotensor alone cannot give a multiple of the Bel-Robinson tensor. But under a modification, all of them contribute the same energy-momentum and stress as the Bel-Robinson tensor does.

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arXiv2022

Quasilocal energy-momentum for tensors B and 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 a 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. In addition, we also demonstrate the tidal heating, rate of change of momentum and spin angular momentum flux by using these two tensors.

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arXiv2020

General relativistic tidal heating for the M$ø$ller 58 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 this tidal heating must be independent of how one localizes gravitational energy and is unambiguously given by that expression. Favata calculated the tidal heating for a number of classical gravitational pseudotensors including that of M$ø$ller, and obtained the result that all of them produced the same (Newtonian) value. After a re-examination of the calculation using the M$ø$ller pseudotensor we find that indeed this pseudotensor gives the desired result under the condition that the mass $M$ is a constant, while Favata considered $M$ as being time dependent, which is illegitimate since it violates the harmonic gauge condition, which he used. Moreover, we carry on to consider this M$ø$ller pseudtensor even in a black hole situation, i.e., beyond Newtonian physics.

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arXiv2019

Black hole's tidal heating and angular momentum

In 1985 Thorne and Hartle used the Landau-Lifshitz pseudotensor to demonstrate the tidal heating and angular momentum flux for a black hole. Later in 2004, Poisson used the gravitational perturbation method to study a black hole and obtained the same result. Poisson proposed a new idea, that the mass quardupole moment and current quadrupole moment can be written as the rate of change of the tidal gravitational field. Inspired by these two papers, we use the method of Thorne and Hartle to study other classical pseudotensors: Einstein, Bergmann-Thomson, Papapetrou and Weinberg. Moreover, we also constructed a general expression pseudotensor. We find that for (i) tidal heating: other classical pseudotensors give the same result as the Landau-Lifshitz contribution. (ii) angular momentum flux: except for the Einstein pseudotensor, all of them give the same value as the Landau-Lifshitz pseudotensor.

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arXiv2018

The modification of the Bel-Robinson energy-momentum

For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson `momentum' in a quasi-local small sphere limit, the Bel-Robinson tensor $B$ is desirable. However, we found this Bel-Robinson `momentum' can be modified such that it still satisfy the non-spacelike and future pointing requirement. These particular energy-momentum properties can be obtained from a linear combination between $B$ with other tensor $S$ in a small sphere limit. This implies that the Landau-Lifshitz pseudo-tensor is no longer disqualified for this non-spacelike and future pointing requirement. Moreover, we constructed a certain linear combination using tensors $B,S,T$ that gives the dominate energy condition in a small sphere region.

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arXiv2017

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.

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arXiv2016

Dominant property for the Bel-Robinson tensor and tensor S

The Bel-Robinson tensor contains many nice mathematical properties and its dominant energy condition is desirable for describing the positive gravitational energy. The dominant property is a basic requirement for the quasi-local mass, i.e., in small sphere limit. We claim that there exists another option, a linear combination between the Bel-Robinson tensor $B$ and tensor $S$, which contributes the same dominant property. Moreover, using the 5 Petrov types as the verification, we found that this dominant property justification for the Bel-Robinson tensor can be simplified as examining $B_{0000}\geq|B_{αβ00}|$ and $B_{0000}\geq|B_{0123}|$, instead of $B_{0000}\geq|B_{αβλσ}|$ for all $α,β,λ,σ=0,1,2,3$.

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arXiv2015

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.

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arXiv2015

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.

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arXiv2015

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 Møller 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$ø$ller. 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.

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arXiv2014

The 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 quasi-local 2-surface, both the Bel-Robinson tensor $B$ and tensor $V$ are suitable. We have 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 quasi-local energy-momentum properties can be obtained from (i): $B$ or $V$ plus a tensor $S$ in a small sphere limit, or (ii): directly evaluating the energy-momentum of $B$ or $V$ in a small ellipsoid region. (iii): calculate the total energy using the Landau-Lifshitz pseudotensor in a small ellipsoid, from Jupiter's tidal force to Io in Schwarzchild spacetime, in an elliptic orbit.

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arXiv2013

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.

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arXiv2013

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.

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arXiv2010

A simple tensorial proof for the completely symmetric property of the Bel-Robinson tensor

The Bel-Robinson tensor $T_{αβμν}$ was proposed in 1958. The main application of this tensor is for describing gravitational energy. It is known that $T_{αβμν}$ has many nice properties such as being completely symmetric. It is easy to prove this property using spinors as shown in Penrose's book. The present paper provides an alternative way, using the tensorial method to prove that $T_{αβμν}$ is indeed totally symmetric. Moreover, we also found that the well known identity in vacuum, $R_{αλστ}R_β{}^{λστ} \equiv 1/4g_{αβ}R_{ρλστ} R^{ρλστ}$, which can be proven by the same tensorial method.

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arXiv2010

An unique alternative non-negative gravitational energy tensor to the Bel-Robinson tensor in the quasilocal small sphere limit

The Bel-Robinson tensor $B_{αβμν}$ gives a positive definite gravitational energy in the quasilocal small sphere limit approximation. However, there is an alternative tensor $V_{αβμν}$ that was proposed recently that offers the same positivity as $B_{αβμν}$ does. We have found that $V_{αβμν}$ is the unique alternative tensor with $B_{αβμν}$ which implies that these two tensors are a basis for expressions that have the desirable non-negative gravitational energy in the small sphere limit. In other words, the `energy-momentum' density according to $B_{αβμν}$ and $V_{αβμν}$ are on equal footing at the same limit.

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arXiv2010

Algebraic Rainich conditions for the tensor V

Algebraic conditions on the Ricci tensor in the Rainich-Misner-Wheeler unified field theory are known as the Rainich conditions. Penrose and more recently Bergqvist and Lankinen made an analogy from the Ricci tensor to the Bel-Robinson tensor $B_{αβμν}$, a certain fourth rank tensor quadratic in the Weyl curvature, which also satisfies algebraic Rainich-like conditions. However, we found that not only does the tensor $B_{αβμν}$ fulfill these conditions, but so also does our recently proposed tensor $V_{αβμν}$, which has many of the desirable properties of $B_{αβμν}$. For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors $B_{αβμν}$ and $V_{αβμν}$ which form a basis for good energy expressions. Both of them have the completely trace free and causal properties, these two form necessary and sufficient conditions. Surprisingly either completely traceless or causal is enough to fulfill the algebraic Rainich conditions. Furthermore, relaxing the quasilocal restriction and considering the general fourth rank tensor, we found two remarkable results: (i) without any symmetry requirement, the algebraic Rainich conditions only require totally trace free; (ii) with a symmetry requirement, we recovered the same result as in the quasilocal small sphere limit.

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