The Concept of the Effective Mass Tensor in GR. The Gravitational Waves

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1 The Conept of the Effetive Mass Tensor in GR The Gravitational Waves Mirosław J. Kubiak Zespół Szkół Tehniznyh, Grudziądz, Poland Abstrat: In the paper [] we presented the onept of the effetive mass tensor (EMT) in General Relativity. In this paper we onsider the onept of the EMT but in the aspet of the gravitational waves. keywords: general theory of relativity; the effetive mass tensor; the gravitational waves I. Introdution In the Einstein s General Relativity (GR), gravitational waves are flutuations of the gravitational fields (or the ripples in the urvature of the spae-time) whih are propagate as a wave and they are generated mainly by moving massive bodies. A. Einstein predited these waves in 96 as a onsequene of the GR. Aording to him the gravitational waves should theoretially transport energy as a gravitational radiation. Soures of detetable gravitational waves ould possibly inlude binary star systems omposed of white dwarfs, neutron stars or blak holes []. Although gravitational radiation has not been diretly deteted, there is indiret evidene for its existene. For example, the 993 Nobel Prize in Physis (J. H. Taylor and R. A. Hulse) was awarded for measurements of the PSR B93+6 (also known as PSR J and PSR 93+6 and the Hulse -Taylor pulsar) system whih suggests gravitational waves are more than mathematial anomalies [3, 4]. Though many various gravitational wave detetors exist (a some examples of the gravitational wave detetors see the Table I) they still remain not disovers. In the Setion II we will present the gravitational waves in the GR, in the Setion III the onept of the effetive mass density tensor (EMDT) and the gravitational waves. In the Setion IV we will ompare a few the physial features onerning of the spae-time urvature with the oneption of the EMDT. Our onlusions we will present in the Setion V. II. The gravitational waves in the GR As we know the Einstein s field equation has the form: 8πG R g R = 4 T ()

2 where: R is the Rii urvature tensor, R is the Rii salar, g is the metri tensor, G is Newton s gravitational onstant, is the speed of light in the vauum, and T is the stress-energy tensor. Our onsiderations we will realize in the weak of the gravitational field, whih allows us to deompose the metri tensor g into the flat Minkowski metri plus a small perturbation h, g = + () η h where: h <<. We will restrit ourselves to oordinates in whih η takes its anonial form, η = diag(-, +, +, +). Well-known alulation (see to [5]) give the wave equation 6πG + h = τ 4 t (3) where: h = h η h and the gauge ondition h µ µ λ = 0. The tensor τ desribes the distribution of the matter, whih disturbs the gravitational field [5]. In the vauum the eq. (3) has the form: + h = 0 (4) Equations (3 and 4) we an interpret as the metri perturbations the flutuations of the spae-time urvature produed by disturbing the metri tensor h propagate at the speed as a wave in the free spae. The tensor wave eq. (4) has the solution with the form h [ ik ( z t = A e (5) whih represents a monohromati wave of the spae-time geometry propagating along the +z diretion with speed and frequeny k. III. The onept of the EMT and the gravitational waves Let s assume that in the gravitational field g = (6) where: is the effetive mass density tensor (EMDT), is the bare mass density. The metri we an express by ds g = gdx dx ( g ) ds ( ) µ ν µ ν where: ( ) and ( ) ds = (7) ds = dx dx [].

3 + In the weak gravitational field we an deompose of the EMDT of the body to the simple form: = bare bare +, where: = η = diag(, +, +, ) we will all the bare mass densi- ty tensor, η is the Minkowski tensor, h << is a small EMDT perturbation. Note = that in the absene of the gravitational field the EMDT beomes the bare mass density tensor and bare. The wave equation (eq. 3) has now the form 6πG + = τ (0) 4 µ where: = η and the gauge ondition µ σ = 0. In the vauum the eq. (0) has the form: + = Equations (0) and () we an interpret as the small perturbation of the effetive mass density, whih propagates with the speed as a wave. The tensor wave eq. () has the solution with the form 0 () [ ik ( z t = A e () whih represents a monohromati wave in the effetive mass density propagating along the +z diretion with speed and frequeny k. In the GR the gravitational waves are the ripples in the urvature of the spae-time that propagate as a wave. The onept of the EMDT predit that the gravitational waves are perturbation in the effetive mass density that propagate as a wave. And although in both ases the gravitational waves are propagated with the speed of light, it the way of their deteting should be different. (see to Table I). IV. The spae-time urvature vs. the effetive mass density tensor Let s ompare a few physial features onerning of the spae-time urvature with the physial features of the EMDT disussed in this paper. The results of this omparison are presented in Table I below. Table. I. The spae-time urvature vs. the effetive mass density tensor. The metri tensor The spae-time urvature g The effetive mass density tensor The effetive mass density tensor = g 3

4 The weak field approximation g + = η h The weakness of the gravitational field is expressed as ability to deompose the metri tensor into the flat Minkowski metri tensor plus a small perturbation tensor, h <<. The gravitational waves in the weak field The weak field approximation bare ( + h ) = = + η The weakness of the gravitational field is expressed as ability to deompose the EMDT to the bare mass density tensor plus a small perturbation of the EMDT tensor h <<. = The gravitational waves in the weak field + h 6πG = τ 4 6πG + = τ 4 The solution The solution = The tensor wave eq. (4) has the solution with the [ ik ( z t form h A e whih represents a monohromati wave of spae-time geometry propagating along the +z diretion with speed and frequeny k. The physial interpretation of the wave equation The gravitational waves are ripples in the urvature of the spae-time that propagate as a wave with the speed as a wave. What we measure in the detetor? Contemporary detetor an measure the dimensionless amplitude h =, whih is generated L L by the gravitational waves. Some examples of the gravitational wave detetors:. Ground based: GEO 600, LIGO, Virgo [6].. Spae-based: LISA [7]. 3. Pulsar Timing Arrays [8]. The results of searh Though many various gravitational wave detetors exist the gravitational waves still remain not disover. The tensor wave eq. () has the solution with the [ ik ( z t form = A e whih represents a monohromati wave of the effetive mass density propagating along the +z diretion with speed and frequeny k. The physial interpretation of the wave equation The perturbation in the effetive mass density, whih propagates through the spae-time with the speed as a wave. What should we measure in the detetor? Designed detetor should measure the dimension- less amplitude =, whih is generated by the gravitational waves. The gravitational wave detetor The results of searh Waiting for disovery.? 4

5 V. Conlusion In the GR the gravitational waves are the ripples in the urvature of the spae-time that propagate as a wave. The onept of the EMDT predit that the gravitational waves are the perturbation in the effetive mass density that propagate as a wave. And although in both ases the gravitational waves are propagated with the speed of light, it the way of their deteting should be different, what open a new ways to searhes of the gravitational waves. Referene []. M. J. Kubiak, The Conept of the Effetive Mass Tensor in the General Relativity, []. B. C. Barish, The Detetion of Gravitational Waves with LIGO, [3]. R. A. Hulse, J. H. Taylor, Disovery of a pulsar in a binary system. ApJ 95, L5 L53 (975). [4]. J. H. Taylor, J. M. Weisberg, A new test of general relativity: Gravitational radiation and the binary pulsar PSR ApJ 53, (98). [5]. S. M. Carroll, Leture Notes on General Relativity, [6]. Ground based gravitational wave detetors: GEO 600 (The British-German GEO 600 projet whih aims at the diret detetion of gravitational waves by means of a laser interferometer of 600m arm length), LIGO (The Laser Interferometer Gravitational-Wave Observatory is a faility dediated to the detetion of osmi gravitational waves and the measurement of these waves for sientifi researh), Virgo (Virgo is a 3km detetor loated in Casina near Pisa, whih ommened siene runs in 007) - [7]. National Aeronautis and Spae Administration, Gravitational-Wave Mission Conept Study Final Report, [8]. G. Hobbs, Pulsars as gravitational wave detetors, 5