Observations demonstrate that the Universe is homogenous and isotropic on the largest scales. This means that the there is no unique center or direction in the universe. In general relativity the shape of space-time is described by a "metric" equation. The general form of the metric equation that satisfies the conditions of homogeneity and isotropy in the universe is the Robertson-Walker metric:
The function f(x) describes the spatial geometry of the universe, it is parameterized by the curvature constant k:
For k < 0 the spatial geometry of the universe is "open", for k = 0 the universe is "flat", and for k > 0 the universe is "closed".
The coordinates x, 
, and 
in the metric equation are "comoving" coordinates. A comoving coordinate system is one which expands with the universe. Therefore, the comoving distant between two point remains constant during the universe's evolution. The physical distance between two points does however change as the universe expands. It is the cosmic scale factor a which relates the comoving coordinates to physical distances, through the relation: d = a x. With the metric defined with comoving coordinates, the time evolution of the universe is described by the time evolution of the cosmic scale factor. If the cosmic scale factor grows in time then the universe expands, if it diminishes with time then the universe collapses.
One of the fundamental ideas in general relativity is that matter and energy act to curve space-time, i.e. they tell the metric equation how to behave. It is Einstein's field equations that describe this mathematically:
Here, g ij represents the metric, R ij is the Ricci tensor (it is essentially derivatives of the metric), R is the trace of the Ricci tensor (it is like the radius of curvature of space-time), 
is the cosmological constant, G is the gravitational constant, and T ij is the stress-energy tensor. The above equation is a tensor equation and thus represents several equations. The stress energy tensor describes the distribution of matter and energy and has diagonal components equal to density and pressure. Thus, given a distribution of matter and energy the field equations govern the form of space-time (i.e. the metric). The inclusion of the cosmological constant term in this equation is of course dictated by whether it is zero or not. One of the field equations can be shown to look like the Newtonian equation for the gravitational potential 
, it however has an extra ingredient:
The added ingredient is that, besides density, pressure also contributes to the gravitational potential. This is a purely general relativistic effect. It is the fact that pressure also contributes to gravity that makes the inclusion of the cosmological constant interesting. If the field equations are rewritten so the cosmological constant appears on the right hand side of the equation, the cosmological constant term can then be associated with a vacuum energy density:
Because the cosmological constant term is proportional to the metric, the pressure associated with the vacuum is then given by the relation:
So the cosmological constant behaves gravitationally like matter and energy except that it has negative pressure. The net effect of a positive cosmological constant is then to create a repulsive gravitational force. This repulsion acts to expand the universe.
The vacuum energy density behaves differently from matter and energy density in another regard. As the universe expands, matter and energy are spread out over more physical space and thus their gravitational attraction is diminished. For the vacuum energy, however, the PdV work done by the vacuum during adiabatic expansion provides exactly the amount of energy to fill the new volume to the same density. Therefore the cosmological constant remains truly constant, and its gravitational repulsion (or attraction) never changes during the universe's evolution.
1.18.99