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.However, the velocity and accelerationequations can be obtained directly from the Einstein field equations.Thusthe Einstein equations imply this thermodynamic relationship in the aboveequation.The above equation can also be written asd dR3(R3) +p = 0 (2.2)dt dtand from equation (1.14), 3( + p) =(3 +), it follows thatd(R3+) =0.(2.3)dt13 14 CHAPTER 2.APPLICATIONSIntegrating this we obtainc = (2.4)R3+1where c is a constant.This shows that the density falls as for matter andR31for radiation as expected.R4Later we shall use these equations in a different form as follows.Fromequation (2.1),1 +3( + p) = 0 (2.5)Rwhere primes denote derivatives with respect to R, i.e.x a" dx/dR.Alter-nativelyd(R3) +3pR2 = 0 (2.6)dRso that1 d(R3+) = 0 (2.7)R3+ dRwhich is consistent with equation (2.4)2.2 Age of the UniverseRecent measurements made with the Hubble space telescope [17] have de-termined that the age of the universe is younger than globular clusters.Apossible resolution to this paradox involves the cosmological constant [18].We illustrate this as follows.Writing equation (1.28) as8GX 2 = ( + vac)R2 - k (2.8)3the present day value of k is8G2 2 2k = (0 + 0vac)R0 - H0R0 (2.9)3with H2 a" (X )2.Present day values of quantities have been denoted with aRsubscript 0.Substituting equation (2.9) into equation (2.8) yields8G2 2 2 2X 2 = (R2 - 0R0 + vacR2 - 0vacR0) - H0R0.(2.10)3 2.3.INFLATION 15Integrating gives the expansion ageR0 R0dR dRT0 = =.8G 2 2 2 20 X 0(R2 - 0R0 + vacR2 - 0vacR0) - H0R03(2.11)2For the cosmological constant vac = 0vac and because R2 [ Pobierz całość w formacie PDF ]
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.However, the velocity and accelerationequations can be obtained directly from the Einstein field equations.Thusthe Einstein equations imply this thermodynamic relationship in the aboveequation.The above equation can also be written asd dR3(R3) +p = 0 (2.2)dt dtand from equation (1.14), 3( + p) =(3 +), it follows thatd(R3+) =0.(2.3)dt13 14 CHAPTER 2.APPLICATIONSIntegrating this we obtainc = (2.4)R3+1where c is a constant.This shows that the density falls as for matter andR31for radiation as expected.R4Later we shall use these equations in a different form as follows.Fromequation (2.1),1 +3( + p) = 0 (2.5)Rwhere primes denote derivatives with respect to R, i.e.x a" dx/dR.Alter-nativelyd(R3) +3pR2 = 0 (2.6)dRso that1 d(R3+) = 0 (2.7)R3+ dRwhich is consistent with equation (2.4)2.2 Age of the UniverseRecent measurements made with the Hubble space telescope [17] have de-termined that the age of the universe is younger than globular clusters.Apossible resolution to this paradox involves the cosmological constant [18].We illustrate this as follows.Writing equation (1.28) as8GX 2 = ( + vac)R2 - k (2.8)3the present day value of k is8G2 2 2k = (0 + 0vac)R0 - H0R0 (2.9)3with H2 a" (X )2.Present day values of quantities have been denoted with aRsubscript 0.Substituting equation (2.9) into equation (2.8) yields8G2 2 2 2X 2 = (R2 - 0R0 + vacR2 - 0vacR0) - H0R0.(2.10)3 2.3.INFLATION 15Integrating gives the expansion ageR0 R0dR dRT0 = =.8G 2 2 2 20 X 0(R2 - 0R0 + vacR2 - 0vacR0) - H0R03(2.11)2For the cosmological constant vac = 0vac and because R2 [ Pobierz całość w formacie PDF ]