**1.3. Conservation of momentum and energy?**

Are total momentum and energy conserved in cosmology? This is a nontrivial question because the canonical momentum and Hamiltonian differ from the proper momentum and energy.

Consider first the momentum of a particle in an unperturbed Robertson-Walker
universe. With no perturbations,
= 0 so that
Hamilton's equation for ** p** becomes

The key point is that ** v** =

Energy conservation is more interesting. Let us check whether the
Hamiltonian *H*(** x**,

(1.17) |

Using *H* = *p*^{2} / (2*am*) +
*am*, we
obtain
*dH* / *d* =
- ( /
*a*)(*p*^{2} / 2*am*) + *md*
(*a*) /
*d* which is nonzero
even if
*d* /
*d* = 0. Is this lack
of energy conservation due to the use of non-inertial coordinates? While
the appearance of a Hubble-drag term may suggest this is the case, if we
wish to obtain the total Hamiltonian (or energy) for a system of particles
filling all of space, we have no choice but to use comoving coordinates.

Perhaps the Hamiltonian is not conserved because it is not the proper
energy. To examine this possibility, we use the Hamiltonian for a system
of particles in comoving coordinates, with *H* = *a*(*T*
+ *W*). The proper
kinetic energy (with momenta measured relative to comoving
observers) is

(1.18) |

while the gravitational energy *W* is given in eq. (1.11).
Holding fixed the momenta, we see that *a*^{2}*T* is a
constant, implying
(*aT*) /
=
- *T*. Similarly,
holding fixed the particle positions, we find that
*a* is a
constant, implying
(*aW*) /
= 0. We thus obtain the
Layzer-Irvine equation
(Layzer 1963,
Irvine 1965)

(1.19) |

Total energy (expressed in comoving coordinates) is not conserved in
Newtonian cosmology. (This is also the case in GR - indeed, there is
generally no unique scalar for the total energy in GR.) However, if
almost all of the mass is in virialized systems obeying the classical
virial theorem
2*T* + *W*
0, we recover approximate total energy conservation.