Chapter 3: Relativity 17 3.2.4 The trajectory of a photon For the trajectory of a photon (and for each particle with zero restmass) holds ds 2 =0 . Substituting the external Schwarzschild metric results in the following orbital equation: du d ϕ ± d 2 u d ϕ 2 + u-3 mu ² =0 3.2.5 Gravitational waves Starting with the approximation g μ ν = η μ ν + h μ ν for weak gravitational ±elds and the de±nition h ± μ ν = h μ ν-1 2 η μ ν h α α it follows that ± h ± μ ν =0 if the gauge condition ∂ h ± μ ν / ∂ x ν =0 is satis±ed. From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are ± c and for wavelengths ² the size of the system, is given by: dE dt =-G 5 c 5 ³ i,j ± d 3 Q ij dt 3 ² 2 with Q ij = ´ ± ( x i x j-1 3 δ ij r 2 ) d 3 x the mass quadrupole moment. 3.2.6 Cosmology If for the universe as a whole is assumed: 1. There exists a global time coordinate which acts as x0 of a Gaussian coordinate system, 2. The 3-dimensional spaces are isotrope for a certain value of
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.