vacuum Einstein equation

[Jacques]

Hi

I have a general understanding of general relativity, but when it comes to equation, I have a very vague idea of the math involved and cannot answer by myself some questions.

1- Does Einsten equation include the space-time expension ?

2- Does the vacuum Einstein equation give an expanding space-time or a static space-time ?

Thanks

wiki Einstein equation

Edited December 1, 2010 by Jacques

[ajb]

1- Does Einsten equation include the space-time expension ?

 

The Einstein field equations allow for space-times that are not stationary. That is you cannot find coordinates such that metric is independent of the time coordinate.

 

 

2- Does the vacuum Einstein equation give an expanding space-time or a static space-time ?

 

You can have vacuum solutions that are not stationary. For example the Kasner metric (which is a weird one!)

[Jacques]

Thanks for your answers

The universe is expanding, so the solution to Einstein equation must give an expanding universe...

 

I did some other reading and saw the scale factor

a(t). Is it the space expansion ?

[ajb]

The universe is expanding, so the solution to Einstein equation must give an expanding universe...

 

You don't really need the field equations, from arguments about homogeneity and isotropy you get the FRW cosmologies. The field equations are needed, when you define the matter content of your universe to calculate the scale factor a(t).

 

 

I did some other reading and saw the scale factor

a(t). Is it the space expansion ?

 

Yes, the scale factor scales the space as a function of time. It is related to the Hubble factor as

 

[math]H = \left( \frac{\dot{a}(t)}{a(t)} \right)[/math],

 

where the dot is the time derivative.

 

Just to recap, an expanding or contracting universe is inherent in any theory of gravity that is based on the local geometry. Einstein's field equations only come into this when solving for the scale factor.

 

 

As a note, FWR cosmologies are not vacuum solutions to the field equations. The universe is full of "stuff".

Edited December 2, 2010 by ajb