The Concept of the Effective Mass Tensor in GR. Clocks and Rods

Transcription

1 The Concept of the Effective Mass Tenso in GR Clocks and Rods Miosław J. Kubiak Zespół Szkół Technicznych, Gudziądz, Poland Abstact: In the pape [] we pesented the concept of the effective ass tenso (EMT) in the Geneal Relativity (GR). Accoding to this concept unde the influence of the gavitational field the bae ass tenso becoes the EMT. The concept of the EMT is a new physical intepetation of GR, whee the cuvatue of space-tie has been eplaced by the EMT. In this pape we conside the concept of the EMT in GR but in the aspect of the clocks and ods. keywods: the effective ass tenso, clocks and ods I. Intoduction In the pape [] we pesented the concept of the effective ass tenso (EMT) in the Geneal Relativity (GR). Accoding to this concept unde the influence of the gavitational field the bae ass tenso becoes the EMT. In this pape we again conside the concept of the EMT in the GR but in the aspect of the clocks and ods. Clocks easue the tie and the ods easue the length. In the GR clocks and ods ae assless. The ain question is sounds: whethe the clocks and ods with the effective ass will easue the sae ties and the sae lengths as clocks and ods with the bae ass? Ou atheatical consideations we will ealize in the Schwazschild etic. In the pape [] we postulated that in the paticula case the EMT can iics the etic tenso g and whee coponents µ, ν,,, 3. Theefoe the etic ds g gdx dx ds g () ( g ) ds ( ) () ds dx dx µ ν µ ν whee: ( ) and ( ). II. Clocks and ods in the Schwazschild etic The line eleent in the Schwazschild etic has the fo

2 GM GM dτ c dt d + dθ sin θdφ + c + c c (3) whee: τ is a pope tie, c is the speed of light, t is the tie coodinate, M is the ass of sta, distance fo the sta, G is the gavitational constant, θ is the colatitude (angle fo Noth, in units of adians), φ is the longitude (also in adians). The sae line eleent with the effective ass has the fo [] c dτ c dt + d + dθ + sin θdφ (4) If we take a slice given by t const we obtain a tee-diensional anifold with line eleent ds d + dθ + sin θdφ (5) Equation (5) we obtained puing dt in equation (4). Thus, the linea eleent ds i dx dx j (6) whee coponents i, j,, 3, (x, x θ, x 3 φ). We see that the space-tie and is positive-define EMT on this 3-anifold, so the slice is a space athe than a sin θ (7) Coponents of the the fo does not depends on the tie t. If we assue that then the EMT has (8) sin θ Note that the 3-diensional EMT has also an inteesting fo. Coponent of the, coponent of the The θθ, whee is the oent of inetia. Coponent of the and does not depends on diection in the space, while the othe coponents yes. φφ sin θ.

3 Let's go back to the equation (5). If we assue that θ const and φ const then we have the infinitesial adial distance dr in the fo dr d d (9) If we assue that then ods with the effective ass will easue diffeent length than the ods with the bae ass and dr > d. But if we assue that then ods with the effective ass and the ods with the bae ass will easue the sae length dr d. The easueent esults ae consistent with GR but thei physical eaning was changed. Unde the influence of the gavitational field only physical popeties of the ods was changed, but not the space popeties. Let us now tun ou aention to the tie. Accoding to the eq. (4) we have ~ dτ dt dt () whee we assued that d dθ dφ and ~ () If we assue that then clocks with the effective ass will easue diffeent tie than the clocks with the bae ass and d τ < dt. But if we assue that then clocks with the effective ass and the clocks with the bae ass will easue the sae tie d τ dt. The easueent esults ae consistent with GR but thei physical eaning was changed. Unde the influence of the gavitational field only physical popeties of the clocks was changed, but not the tie popeties. Siple two exaples (see below) will illustates ou discussion fo the popeties of the EMT in the Schwazschild etic. Exaple. Let us conside a od of one ete length (d ) with the bae ass. What is the length dr of one ete fo the sae od if we will put it adially in a weak gavitational field fo the? The answe is: dr. ete. In the gavitational field a od with effective ass will show a length geate than the od with the bae ass. Exaple. Let us conside a clock with the bae ass, which easues the tie with an accuacy of the one second (dt s). What is the length d τ of the one second fo the sae clock if we will put it in a weak gavitational field fo the? The answe is: d τ.99 s. In the gavitational field a clock with effective ass uns slowe than the clock with the bae ass. 3

4 III. The EMT in the Schwazschild etic The EMT in the Schwazschild etic has the fo (see eq. ()) sin θ () If we assue that then the 4-diensional EMT has the fo (3) sin θ IV. The EMT anisotopy in the Sola Syste As we known fo the pape [] GM (4) c Many physicists looks fo the (inetial) ass anisotopy []. We will show whee to look fo the ass anisotopy in the Sola Syste if the concept of EMT is tue. The EMT fo the Schwazschild etic has fo GM c GM c sin θ (5) Coponents of the and depends on the distance between the planet and the sta. As we known in the Sola Syste the distance between the Eath and the Sun is changed (the obit is an ellipse). The elative changes in the ass anisotopy fo the coponents and we should obseve duing a yea, easuing fo peihelion (aphelion) to peihelion (aphelion). Coponent of the is the sallest in the peihelion while the coponent the biggest elative changes (fo exaple fo peihelion to peihelion) fo the be easued in the Sola Syste and the estiated value is equal to 4 is the biggest. Annual and fo should

5 δ δ GM 6, 6 c peih aphel ( ) ( ) ( ) ( ) peih aphel peih aphel (6) V. Conclusion In this pape we consideed the concept of the EMT in GR in the aspect of the clocks and ods. We have found that the gavitational field has an ipact on the asses of the clocks and ods located in this field. In the gavitational field all clocks and ods have the effective ass. When thee is no gavitational field all clocks and ods have the bae asses. All clocks with the effective ass uns slowe than the clocks with the bae ass while a ods with the effective ass will show a length geate than the ods with the bae ass. The concept of the EMT is a new physical intepetation of the GR, whee the cuvatue of space-tie has been eplaced by the EMT. We believe that this concept will help bee undestand the gavitational phenoena. Appendix Fo the siila consideation but fo the Newton s etic in the Catesian coodinates we have 3- diensional EMT (A) Coponents of the does not depends on the tie t. If we assue in eq. (A) that then 3- diensional EMT in the Catesian coodinates takes the fo (A) and does not depends on diection in the space. We see that in Catesian coodinates the EMT is the 3-diensional bae ass tenso. This tenso is isotopic and physically epesents the ass isotopy in the 3-diensional space. This ass isotopy is geneally intepeted as a scala. The 3-diensional bae ass tenso is isotopic if and only if it his popeties do not depend on the diection in the space (all coponents have the sae value in all otated coodinate systes). 5

6 The 4-diensional EMT in the Newton s etic in the Catesian coodinates has fo (A3) If we assue that then EMT has the fo (A4) and does not depends on diection in the space-tie. We see that in Catesian coodinates the EMT is the 4-diensional bae ass tenso. This tenso is also isotopic and physically epesents the ass isotopy in the 4-diensional space. So fa, the concept of the ass isotopy was associated only with the 3-diensional space but not with the 4-diensional space-tie. The 4-diensional bae ass tenso it is a new te in ou consideations. The eq. (A4) we can ewite in the fo whee: η is the Minkowski tenso. η (A5) Refeences []. M. J. Kubiak, The Concept of the Effective Mass Tenso in the Geneal Relativity, hp://vixa.og/abs/3.6. []. Coss D. J., Anisotopy of Inetia fo the CMB Anisotopy, hp:// The 4-diensional bae ass tenso is isotopic if and only if it his popeties do not depend on diection in the space-tie. 6