Introduction to Cosmology

Weak Field Limit 63

The left-hand side of Equation (3.42) is not covariant, but it does contain second
derivatives of the metric, albeit of only one component. Thus it is already related to
curvature. The next step would be to replace∇^2 푔 00 by a tensor matching the properties
of푇휇휈on the right-hand side.

(i) It should be of rank two.

(ii) It should be related to the Riemann curvature tensor푅훼훽훾 휎.Wehavealready

found a candidate in the Ricci tensor푅휇휈in Equation (3.17).

(iii) It should be symmetric in the two indices. This is true for the Ricci tensor.

(iv) It should be divergence-free in the sense of covariant differentiation. This is not

true for the Ricci tensor, but a divergence-free combination can be formed with

the Ricci scalar푅in Equation (3.18).

The Einstein tensor퐺휇휈contains only terms which are either quadratic in the first
derivatives of the metric tensor or linear in the second derivatives.
For weak stationary fields produced by nonrelativistic matter,퐺 00 indeed reduces
to∇^2 푔 00. The Einstein tensor vanishes for flat space-time and in the absence of matter
and pressure, as it should. Thus the problems encountered by Newtonian mechanics
and discussed at the end of Section 1.7 have been resolved in Einstein’s theory. The
recession velocities of distant galaxies do not exceed the speed of light, and effects of
gravitational potentials are not felt instantly, because the theory is relativistic. The dis-
continuity of homogeneity and isotropy at the boundary of the Newtonian universe
has also disappeared because four-space is unbounded, and because space-time in
general relativity is generated by matter and pressure. Thus space-time itself ceases
to exist where matter does not exist, so there cannot be any boundary between a homo-
geneous universe and a void outside space-time.

Problems

1. Derive the Taylor expansions quoted below Equation (3.15).


2. Derive Newton’s second law in the generally covariant form Equation (3.12).


3. Show that the Ricci tensor푅훽훾is symmetric.


4. Show that퐺휇휈in Equation (3.28) is divergence-free in the sense of covariant
   differentiation.

References

[1] Kenyon, I. R. 1990General relativity. Oxford University Press, Oxford.
[2] Shore, G. M. 2002Nuclear Phys.B 633 , 271.
[3] Peebles, P. J. E. 1993Principles of physical cosmology. Princeton University Press,
Princeton, NJ.
[4] Peacock, J. A. 1999Cosmological physics. Cambridge University Press, Cambridge.