Introduction to Cosmology

Relativistic Distance Measures 41 We then insert푎(푡)from Equation (2.54) to lowest order in푡 0 −푡, obtaining 푑P≈푐 ∫ 푡 0 푡 [ 1 +퐻 ...

42 Special Relativity Redshift and Luminosity Distance. Consider an astronomical object emitting pho- tons isotropically with po ...

Relativistic Distance Measures 43 temperature–luminosity plot. Temperature can be determined from color—note that astronomers de ...

44 Special Relativity and they are bright enough to be seen out to about 300kpc. A very important class of standard candles are ...

Tests of Special Relativity 45 The Tully–Fisher relation for spiral galaxies is calibrated by nearby spiral galax- ies having Ce ...

46 Special Relativity Variable speed of light is in conflict with Lorentz invariance and with Einstein’s time dilation formula i ...

Tests of Special Relativity 47 Calculate whether the following space-time intervals from the origin are space- like, timelike o ...

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3 General Relativity Newton’s law of gravitation, Equation (1.28), runs into serious conflict with special relativity in three d ...

50 General Relativity This is most conveniently done using tensor notation, briefly presented in Section 3.2, which has the adva ...

The Principle of Equivalence 51 that the lift is accelerating upwards, tests inside the lift reveal that the objects acquire an ...

52 General Relativity F Figure 3.2Tidal force퐹acting between two test bodies falling freely towards the surface of a gravitating ...

The Principle of Equivalence 53 A g C B Figure 3.3A pocket lamp at ‘A’ in the Einstein lift is shining horizontally on a point ‘ ...

54 General Relativity Equation (3.1) clearly shows that light emerging from a star with mass푀is redshifted in proportion to푀. Th ...

The Principle of Covariance 55 In general, tensors can have several contravariant and covariant indices running over the dimensi ...

56 General Relativity described by the Robertson–Walker metric the approach to general covariance is to find appropriate invaria ...

The Principle of Covariance 57 The path of a test body in free fall follows from Equation (3.12) by requiring that no forces act ...

58 General Relativity to Einstein’s system of ten gravitational equations as we shall see later. Finally, we may sum over the tw ...

The Einstein Equation 59 The action principle then requires the variation of this action with respect to the inverse metric푔휇휈to ...

60 General Relativity matter are compactly described by thestress–energy tensor푇휇휈with the following components. The time–time ...