Introduction to
Astrophysics and Cosmology
|
|
General Relativity
There exist three distinct options for the spatial geometry of a spacetime with the given metric, governed by the parameter k such that k=1, k=0, or k=-1.
|
|
A sphere has constant positive curvature.
|
A hyperboloid has constant negative curvature.
To solve Einstein’s field equation, one needs to postulate “stuff” in the spacetime – like matter, radiation vacuum energy, along with the energy momentum tensor 888, whose components are the energy density 888 and pressure p of the “stuff”.
|
|
|
|
The equation will then tell us the curvature of space from the stuff content of the spacetime “stuff”.
|
The three possible values of the parameter k correspond to three distinct and separate curvatures:
k=1 is positive curvature
k=0 is zero curvature
k=-1 is negative curvature
|
|
|
|
k W Topology Time Evolution
1 >1 Closed Space is positively curved and finite, expands from zero size to a maximum size and then shrinks back to zero again
0 =1 Open Space is flat and infinite, and expands forever
-1 <1 Open Space is negatively curved and infinite, and expands forever
|
|
|
|
|
|
|
|
|