Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

approaches zero. This distance R s is the Schwarzschild radius encountered
before, representing the horizon of a black hole.
Finally, under the assumptions that the universe is isotropic and homogeneous,
the FLRW interval result for general relativity is
[
ds 2 = (c dt) 2 − R(t) 2 dr 2
]
1 − Kr + 2 r2 dθ 2 + r 2 sin 2 θdφ 2 , (21)
where R(t) is the scale parameter and K represents the type of spacetime
curvature. The equations can be re-scaled so that K conventionally can take
on one of three discrete values:
• K = 0. Flat spacetime. The 2-d analog is a flat sheet.
• K = +1. Positive curvature. The surface of a balloon in 2-d.
• K = −1. Negative curvature. A saddle point in 2-d.
The scale parameter R(t) represents the fractional expansion of the distance
between two points due to cosmological expansion (see next sections).
So if r o is a distance between two objects, then at a later time, the true coordinate
distance between the objects has become r(t) = R(t) r o . (r o is known
as the comoving coordinate distance.) Note that R(t) is only a function of
time.
5.2 Hubble’s Observation and its interpretation
The redshift z measures the fractional shift of the observed wavelength λ ′ of
a spectral feature, e.g. an emission line, relative to the wavelength λ of the
feature measured in a proper frame. So z = ∆λ/λ = (λ ′ − λ)/λ, or
λ ′ = λ(1 + z) (22)
Hubble observed redshifts for tens of relatively nearby galaxies for which
their distance was independently known. The relativistic Doppler shift provides
an explanation for a shift of wavelength due to the motion v of the
source with respect to the observer:
λ ′ /λ =
[ ] 1/2
1 + (v/c)
≈ 1 + v 1 − (v/c) c ,
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