Search or add a thesis

Advanced Search (Beta)
Home > Use of Approximate Symmetry Methods to Define Energy of Gravitational Waves

Use of Approximate Symmetry Methods to Define Energy of Gravitational Waves

Thesis Info

Access Option

External Link

Author

Hussain, Ibrar

Program

PhD

Institute

National University of Sciences & Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2009

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2491/1/2568S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727644180

Similar


In this thesis approximate Lie symmetry methods for differential equations are used to investigate the problem of energy in general relativity and in particular in gravitational waves. For this purpose second-order approximate symmetries of the system of geodesic equations for the Reissner-Nordström (RN) spacetime are studied. It is shown that in the second-order approximation, energy must be rescaled for the RN spacetime. Then the approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime are investigated. Taking the Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional algebra of conformal isometries of the Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter first-order approximate symmetries of the Kerr spacetime as a first perturbation of the Schwarzschild spacetime are considered. We then investigate the second-order approximate symmetries of the Kerr spacetime as a second perturbation of the Minkowski spacetime. Next, second-order approximate symmetries of the system of geodesic equations for the charged-Kerr spacetime are investigated. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr spacetime has to be rescaled. Since gravitational wave spacetimes are time-varying vacuum solutions of Einstein''s field equations, there is no unambiguous means to define their energy content. Here a definition, using slightly broken Noether symmetries is proposed. A problem is noted with the use of the proposal for plane-fronted gravitational waves. To attain a better understanding of the implications of this proposal we also use an artificially constructed time-varying non-vacuum plane symmetric metric and evaluate its Weyl and stress-energy tensors so as to obtain the gravitational and matter components separately and compare them with the energy content obtained by our proposal. The procedure is also used for cylindrical gravitational wave solutions. The usefulness of the definition is demonstrated by the fact that it leads to a result on whether gravitational waves suffer self-damping.
Loading...
Loading...

Similar Books

Loading...

Similar Chapters

Loading...

Similar News

Loading...

Similar Articles

Loading...

Similar Article Headings

Loading...

بانگ درا

3 ستمبر 1924 ء کو منظر عام پر آنے والا یہ اردو کا پہلا شعری مجموعہ ہے۔ اس میں تقریباً چوبیس سال تک کا کلام موجود ہے۔ اقبال اردو کلام کی اشاعت کے حق میں نہ تھے۔ ایک صاحب نے شوق میں یہ کام کر بھی دیا تو اقبال نے قانونی طور پر پابندی عائد کروادی۔ اس کے بعد خود توجہ دی۔ بہت سا حصہ حذف کر دیا۔ اقبال کے تمام مجموعوں میں بانگ در اسب سے بڑا مجموعہ ہے۔ اس میں 1901ء سے 1905 ء کا کلام پہلے حصے میں 1905ء سے 1908ء کا کلام دوسرے حصہ میں اور یورپ سے واپسی کے بعد سے لے کر 1924 ء تک کا کلام تیسرے حصے میں شامل ہے۔ اس طرح بانگ درا کے کل تین حصے ہیں۔ اس میں143 نظمیں اور 28 غزلیں شامل ہیں۔عمدہ افکار و اسالیب پر مبنی مرثیے اس تصنیف میں موجود ہیں۔ زیادہ تر طویل نظمیں اسی تصنیف میں موجود ہیں۔

عدالتی خلع اورپاکستانی عدالتوں کے فیصلوں کی روشنی میں اس کی قانونی بنیادیں Judicial Khul’ and its Legal Grounds in the light of Decisions of Pakistani Courts

The doctrine of Khul’ has, within the course of last few years, assumed a great deal of importance in Pakistan because literacy rate in women is increasing rapidly and their dependency on men is decreasing. Now, women can easily make their own decisions with free consent due to their some awareness about their rights, especially regarding dissolution of marriage. Majority of women is still ignorant about dissolution of marriage on the ground of Khul’. Therefore, it is very necessary to explain all different aspects of “doctrine of Khul’” for ensuring justice in our society. The present research has mainly explored the grounds of judicial Khul’ and other relevant incidents in the light of Pakistani Case Law based on Islamic family Law. The research is based primarily on the decisions of superior courts of Pakistan. The decisions of family courts of Pakistan have been included in the discussion. The relevant provisions of the Dissolution of Muslim Marriages Act, 1939 and the Family Courts Act, 1964 have also been debated. The difference between Khul’ and other modes of dissolution of marriage have also been elaborated briefly.

Synthesis, Characterization of Silver Nanoparticles Using Plant Extract and Their Application in Environmental Remediation

Nanotechnology has endorsed enormous development in material science to formulate innovative products by manipulating matter at nano-scale. Due to certain limitations associated with conventional physico-chemical synthesis protocols, novel techniques are still being pursued for fabrication of nanomaterials. Among these protocols, biological synthesis of nanoparticles using plant extract has been considered comparatively, eco-friendly, efficient and cost-effective. This thesis presents study on the investigation of different parts of Eucalyptus camaldulensis and Eucalyptus citriodora for their ability to synthesize silver nanoparticles. Further, work on the nature and scope of nanoparticles in different applications has been reported. A green synthesis route for the production of silver nanoparticles using aqueous extracts of leaves stems and fruit of E. camaldulensis and E. citriodora was explored. Five solutions with varying concentration of plant material were prepared for each of three parts i.e. leaves, stems and fruit of both plant. Among these thirty (30) collected samples, six were selected for further analysis and applications depending upon the information collected from their UV-Vis spectra. Three samples from each plant including one sample for each part were collected. 1 Mm silver nitrate solution was mixed with plant extract at 45 °C under basic medium with stirring time 30 min and formation of silver nanoparticles was observed through color change from light yellow to dark brown Thus synthesized silver nanoparticles were characterized by UV-Vis spectrophotometer, having a surface Plasmon resonance (SPR) band centered at 406 nm, Dispersity and morphology was characterized by scanning electron microscope (SEM). Crystalline nature and purity of synthesized silver nanoparticles were revealed by X-ray diffraction (XRD) and energy dispersive X-ray spectroscopy (EDX). FTIR spectrum was examined to identify the effective functional molecules responsible for the reduction and stabilization of silver nanoparticles synthesized by extracts of parts of plants. For the selected six samples, the effect of time and temperature variation was also observed. The results showed that time of reaction, temperature and concentration of extract could accelerate the reduction rate of Ag+ and also affect the size and shape of Ag NPs. The nanoparticles were found to be about 50 nm to 100 nm in size, mono-dispersed in nature and spherical in shape. xi The investigated applications of Ag NPs include catalytic degradation of different dyes and antimicrobial activity. The work emphasizes the effect of the size of silver nanoparticles on the degradation rate of hazardous dyes like methyl orange, methylene blue and eosin Y by NaBH4. The efficiency of silver nanoparticles as a promising candidate for the catalysis of degradation of organic dyes by NaBH4 through the electron transfer process is established in the present study. Catalytic degradation of methyl orange, methylene blue and Eosin Y was measured spectrophotometrically under visible light illumination. These processes were studied by monitoring the simultaneous decrease in the height of absorbance peak of dye solution and increase in the height or shifting of plasmon peak corresponding to silver nanoparticles. The results showed that silver nanoparticles are an efficient catalyst for degradation of dyes and their efficiency depends on morphology of nanoparticles. Biosynthesized samples of silver nanoparticles were found to be impressive in degrading methyl orange. The finalized six samples were also used to evaluate their antimicrobial activity against two bacterial strains, one from gram-positive and one from gram-negative. The zone inhibition results were found to be in comparison of reference standard of Streptomycin. Among the prepared samples of both plants, six were forwarded for further evaluation and their sizes and shapes were related to time, pH, temperature and concentration of plant extract. After complete characterization, these were evaluated for degradation of dyes and antimicrobial behavior. Both of the applications were assisted through notably prominent results.