108 Cosmological Models
and훤휎휈휇gives a system of second-order differential equations. This will of course give
rise to somewhat different physics than the Einstein–Hilbert variation, so in due time
we will probably have data permitting us to decide between them.
Another degree of freedom in the action [Equation (5.85)] is the dimensionality푛
of space-time. At this point in my treatise there is no obvious reason to assume that
푛is larger than four. However, the gravitational interaction is so much weaker than
all the standard particle interactions, that one could imagine that it propagates in a
different space-time. We shall come back to models in later chapters which explore
this freedom.
Problems
- On the solar surface the acceleration caused by the repulsion of a non-vanishing
cosmological constant휆must be much inferior to the Newtonian attraction.
Derive a limiting value of휆from this condition. - In Newtonian mechanics, the cosmological constant휆can be incorporated by
adding to gravity an outward radial force on a body of mass푚,adistance푟
from the origin, of퐹=+푚휆푟∕6. Assuming that휆=− 10 −^20 yr−^2 ,andthat퐹is the
only force acting, estimate the maximum speed a body will attain if its orbit is
comparable in size with the Solar System (0.5 light day [11]). - de Sitter’s static universe has푎∝eHt, zero curvature of its co-moving coordinates
(푘=0), and a proper density of all objects that is constant in time. Show that
the co moving volume out to red-shift푧is푉(푧)=^43 휋(cz∕퐻)^3 , and hence that the
number-count slope for objects at typical red-shift푧becomes[( 3 +훼)ln푧]−^1 for
푧≫1, where훼is the spectral index for the objects [12]. - Starting from Equation (5.55) with the parameters훺 0 =1,훺r=0, show that the
age of the Universe can be written
푡 0 =
2
3 퐻 0
tanh−^1√
훺휆
√
훺휆
.
- A galaxy at푧= 0 .9 contains a quasar showing red-shift푧= 1 .0. Supposing that
this additional red-shift of the quasar is caused by its proximity to a black hole,
how many Schwarzschild radii away from the black hole does the light emitted
by the quasar originate? - Estimate the gravitational red-shift푧of light escaping from a galaxy of mass
109 푀⊙after being emitted from a nearby star at a radial distance of 1kpc from
the center of the galaxy. (Assume that all matter in the galaxy is contained within
that distance [11].) - Light is emitted horizontallyin vacuonear the Earth’s surface, and falls freely
under the action of gravity. Through what vertical distances has it fallen after
travelling 1km? Calculate the radial coordinate (expressed in Schwarzschild
radii) at which light travels in a circular path around a body of mass푀[11].