12.4 Summary
201
geometry on the left side of the field equations, or as dark energy on the right side
of the equations, is an important question in that it could determine our mindset in
developing future theories.
Exercises
12.1 Carefully apply Newton’s second law to an element of a fluid and derive the
basic fluid equation (12.8).
12.2 Estimate very roughly the relation of pressure to density for the present-day
cosmological gas of galaxies by looking up the approximate random motion
velocity of a typical galaxy and using simple kinetic theory. Is the neglect of
pressure in the present-day universe justified?
12.3 Suppose that the dominant ingredients of the present-day universe are not really
the visible galaxies and gas and dust that we observe but instead are unseen
very low mass and fast-moving particles such as neutrinos. What happens to
your answers to Exercise 12.2? This is called the hot dark matter universe.
12.4 We have called the cosmological constant. Could it instead be a variable
depending on time, (t)? Assume that energy-momentum is conserved and
show that must be constant by taking the divergence of both sides of the
field (12.18).
12.5 Suppose the contrary to Exercise 12.4, that the cosmological term does vary
slowly with time. What would that specifically imply for conservation of
energy in the universe and what experiment or observation could test for it?
12.6 Could it be that some special type of energy-momentum does not couple to
gravity? Show that this would not be consistent with conservation of energymomentum.
12.7 As we have indicated the cosmological term can be thought of as part of
the geometry on the left side of the field equations or as some sort of dark
energy stuff on the right side—that is “geometry versus stuff.” What is your
preference—based purely on esthetic and philosophical grounds? There is, of
course, no right or wrong answer to this question.
12.8 Consider a cylinder filled with a material whose total energy is proportional to
its volume. Using standard energy arguments as in thermodynamic textbooks
show that the equation of state must be p = −ρc
2 as in (12.21); that is the
pressure must be negative.
201
geometry on the left side of the field equations, or as dark energy on the right side
of the equations, is an important question in that it could determine our mindset in
developing future theories.
Exercises
12.1 Carefully apply Newton’s second law to an element of a fluid and derive the
basic fluid equation (12.8).
12.2 Estimate very roughly the relation of pressure to density for the present-day
cosmological gas of galaxies by looking up the approximate random motion
velocity of a typical galaxy and using simple kinetic theory. Is the neglect of
pressure in the present-day universe justified?
12.3 Suppose that the dominant ingredients of the present-day universe are not really
the visible galaxies and gas and dust that we observe but instead are unseen
very low mass and fast-moving particles such as neutrinos. What happens to
your answers to Exercise 12.2? This is called the hot dark matter universe.
12.4 We have called the cosmological constant. Could it instead be a variable
depending on time, (t)? Assume that energy-momentum is conserved and
show that must be constant by taking the divergence of both sides of the
field (12.18).
12.5 Suppose the contrary to Exercise 12.4, that the cosmological term does vary
slowly with time. What would that specifically imply for conservation of
energy in the universe and what experiment or observation could test for it?
12.6 Could it be that some special type of energy-momentum does not couple to
gravity? Show that this would not be consistent with conservation of energymomentum.
12.7 As we have indicated the cosmological term can be thought of as part of
the geometry on the left side of the field equations or as some sort of dark
energy stuff on the right side—that is “geometry versus stuff.” What is your
preference—based purely on esthetic and philosophical grounds? There is, of
course, no right or wrong answer to this question.
12.8 Consider a cylinder filled with a material whose total energy is proportional to
its volume. Using standard energy arguments as in thermodynamic textbooks
show that the equation of state must be p = −ρc
2 as in (12.21); that is the
pressure must be negative.