Astroparticle Physics

17.8 Chapter 8 3615. Photon mass m = hνc 2 = E c 2 ;energy loss of a photon in a gravitational potential U: E = mU ;reduced photon energy: E ′ = E − E = hν − hν(ν ′ = ν 1 − U )c 2 ,νν= Uc 2 ;c 2 U ⇒gravitational potential:U = GM R⇒6.νν= GMRc 2 ⇒ GMRc 2is dimensionless;using the Schwarzschild radius R S = 2GMνc 2 one hasν= R S2R .This result is, however, only valid far away from the event horizon. The exact resultfrom general relativity reads, see, e.g., [9]: ν/ν = z = 1/ √ 1 − R S /R − 1.xSunRylight beamAcceleration due to gravity at the solar surface: g = GM/R 2 .Assumption: the deflection takes place essentially over the Sun’s diameter 2R. The photonstravel on a parabola:y = g 2 t2 ,x= ct ⇒ y(x) = GM x 22R 2 c 2 .The deflection δ corresponds to the increase of y(x) at x = 2R,dydx = GMR 2 c 2 x ⇒ y′ (2R) = 2GMRc 2= R SR = δ,R = 6.961 × 10 5 km, M = 1.9884 × 10 30 kg ⇒ δ ≈ 4.24 × 10 −6 ≈ 0.87 arcsec,1arcsec= 1 ′′ .This is the classical result using Newton’s theory. The general theory of relativity givesδ ∗ = 2δ = 1.75 arcsec.The solution to this problem may alternatively be calculated following Problem 13.7,see also Problem 3.4.