Sunday, January 02, 2005

Just Info's About The Development of Our Humble Research Problem

This (result from long weeks of doing the physics and mathematics of our present selected research problem) might get wrong but I'm shameless enough to blog this here. Besides today, I'm very much dismayed, the only ferryboat bound for jagna is already closed for additional passengers. So, I will not be able to attend a conference in theoretical physics about path integrals to be lectured by some of the world's authorities in such field, coming from the USA, Japan, Germany. Exhausted, I just visit an internet cafe to relieve myself from such disappointment but it was my fault anyway.

So may I shamelessly blog it here the title, abstract and introduction of our supposed (much just for the sake of getting the needed mastery not totally of having to get a discovery to publish) new paper.

Please don't laugh if you find something funny, as I said I'm disappointed today :)

TITLE: Schwarzschild Metric Fixes The Klein-Gordon Field To Act As Cosmological Constant

ABSTRACT: We consider the field equations of General Relativity with the Klein-Gordon (scalar) field acting as the source in the stress-energy tensor. In this paper, we set the spacetime to be that of the spherically symmetric and static Schwarzschild metric, and take the scalar field to be static. Final results show that the Schwarzschild solutions fix the static Klein-Gordon field to be a non-zero constant with the vanishing of the space-like derivatives in the non-Minkowskian equations of motion of the field. The fixing leads to an associated result that the scalar field acts as a cosmological constant.

INTRODUCTION: The most well known simplified solution of the field equation of General Relativity in the non-cosmological context is the Schwarschild metric. This metric corresponds to a sphericall symmetric and static setting of space-time, which could either be devoid of energy density, or is pervaded with a kind of energy[2, 3, 4, 6, 8, 9] that is constant in time. These basic features, being static and spherically symmetric, are also the basic features of other metric solutions such as the Reissner-Nordstrom metric for the field equations of gravity with an electromagnetic tensor[3]. Both in the Schwarzschild metric and Reissner-Nordstrom metric, the time-time (g00) and radial-radial (grr) components of the metric tensor are mutually reciprocal functions. Karl Schwarzschild [10] originally obtained these basic metric features in considering a completely zero content of the stress-energy tensor for the gravitational field equations. His assumptions are discussed in [10] and are also presented in [7] with reference to [2, 3, 4, 10].

The Schwarzschild metric is extended as a solution to the gravitational field equations that are augmented with the cosmological constant. These field equations are presented and expounded in [2, 8, 9, 10] and the corresponding Schwarzschild solutions are discussed in [2] and are also presented in [6] with chief reference to [2].

There had been a re-emergence of theoretical interest to consider gravitational field equations augmented by the cosmological constant, more significantly in the cosmological context. This had been motivated by the astrophysical observations that the universe is in the phase of accelerating expansion. (For the accelerating expansion phase of the universe see for example [11, 12], and for an introductory and brief discussion on the cosmological implications of the cosmological constant see for example [2, 8, 9, 13]. Alongside with this is the emergence of theoretical alternatives to the cosmological constant as for example, there is the "120-ORDER-OF-MAGNITUDE DISCREPANCY", (see [8, 9, 13]).

There are alternatives that pose as promising, known as scalar-tensor theories. In these theories, the field equations of gravity are modified in such a way that Newton's Gravitational Constant is considered to vary with a scalar field, and all other relevant terms represent partial derivatives of the scalar field, and in addition, the existence of potential, which is a function of the scalar field. Originally, scalar-tensor theory was invented by Brans and Dicke upon the motivation founded on a suggestion by Dirac that Newton's Gravitational Constant changes with time, (brief account[3]). In the cosmological context, a scalar field theory is held as the originator of negative pressure needed in the inflation phase of the BIGBANG[1].

A single component scalar-field equation in relativistic quantum mechanics was discovered by Schroedinger before he arrived at his famous non-relativistic quantum equation. This single component scalar field equation is of the 2nd order both in time and space derivatives and describes the spin-zero particles[2, 5]. This equation is the Klein-Gordon equation.

It is the Klein-Gordon equation of relativistic quantum mechanics that we adapt (in this paper) in the generally curved space of General Relativity, having to treat it as a classical field. We investigate how this field adapts to classical curved metric solutions. We seek the spherically symmetric and static metric solutions that solve the field equations of General Relativity with the stress-energy tensor whose content is the Klein-Gordon field that we take as static. We find that the mathematical simplicity of the spherically symmetric and static metric (that of the Schwarzschild space) will lead to the vanishing of space derivatives of the scalar field, fixing it as a non-zero constant. Satisfying the constraints imposed by the component field equations of gravity, this fixed scalar field is identifiable with the cosmological constant in the metric solutions.



Belated merry x'mass and happy new year!!!

Pero dili ta magsobra og lipay, silingan nato nga mga nasud sa asya giigo sa TSUNAMI. PRAY TO YOUR OWN GOD FOR THE INNOCENT SOULS. Me... hmmm... hunahuna sa ko ho is my God . . . :)
Basta, polpol kaayo ko, lako kalarga to Jagna, kita unta ko Physicist nga Amerikano... hehe...



3 comments:

jazen said...

looks familiar (the theories you mentioned)... lecture namo sa relativity sa una under doctor tee.

try daw si doctor tee ug approach, jo. basin d i willing sya mutabang.

apprentice01 said...

Hello Mam,

Tnx 4 d gud advice, testingan nako aproach him after mafinis na write-up ani...

Bydwy, txt me f daun ka dz sund...

K...TC!

apprentice01 said...

References

[1]Peebles, P. J. E., Principles of Physical Cosmology, Princeton University Press, Princeton, New Jersey, U. S. A., copyright 1993

[2]Ohanian, H. C., Gravitation and Spacetime, W. W. Norton and Company, New York, copyright 1976

[3]Carroll, S. M., Lecture Notes On General Relativity, INSTITUTE OF THEORETICAL PHYSICS UNIVERSITY OF CALIFORNIA SANTA BARBARA, C93106

[4]http://itp.ucsb.edu/~carroll/notes/

[5]D. J., Griffiths, Introduction To Elementary Particles, John Wiley & Sons, Inc., USA, copyright 1987

[6]Roa, F. P., Yacapin, J. P., Bantaculo, R. V., A Theoretical Try-out On Source-Free Einstein’s Field Equations With The Cosmological Constant, 4th NATIONAL PHYSICS CONFERENCE & WORKSHOP – SAMAHANG PISIKA NG VISAYAS AT MINDANAO, proceedings of extended abstract 2002

[7]Roa, F. P., dela Torre, I., A Brief Introduction on Maximally Symmetric Spaces, Birkhoff’s Theorem and Spherically Symmetric Vacuum Static Solutions, 6th NATIONAL PHYSICS CONFERENCE & WORKSHOP – SAMAHANG PISIKA NG VISAYAS AT MINDANAO, proceedings of extended abstracts 2004

[8]http://www.livingreviews.org/Articles/Volume4/2001-1carroll/node3.html

[9]http://www.livingreviews.org/Articles/Volume4/2001-1carroll/node4.html

[10]Foster, J., Nightingale, J. D., A Short Course In General Relativity, 2nd edition, springer-verlag, New York, Inc., copyright 1995

[11]http://cfa-www.harvard.edu/cfa/oir/Research/supernova/HighZ.html

[12]http://www-supernova.lbl.gov/public/

[13]http://www.astro.ucla.edu/~wright/cosmo_constant.html