Spacetime we are able to distinguish between different inertial frames

[captain]

A professor told me that in our spacetime we are able to distinguish between different inertial frames because the cosmic microwave background looks different. If this is true then what spacetime do we live in? What would it look like mathematically. Would the det(g) not be negative?

 

[atyy]

Eq 1 and following gives the metric for our spacetime on large scales:
http://nedwww.ipac.caltech.edu/level5/March03/Lineweaver/Lineweaver2.html#2.1

Here they talk about how the rest frame comes about:
http://nedwww.ipac.caltech.edu/level5/March03/Lineweaver/Lineweaver7_5.html

 

[Vanadium 50]

We can also distinguish between different inertial frames because the speed and position of Jupiter looks different in them. There's no difference here.