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phys documents (dragged) 23 - Chapter 3 Relativity 17 3.2.4...

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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 = 0 3.2.5 Gravitational waves Starting with the approximation g μ ν = η μ ν + h μ ν for weak gravitational fields and the definition 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 satisfied. From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are c and for wavelengths the size of the system, is given by: dE dt = - G 5 c 5 i,j d 3 Q ij dt 3 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 x 0 of a Gaussian coordinate system, 2. The 3-dimensional spaces are isotrope for a certain value of x 0 , 3. Each point is equivalent to each other point for a fixed
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