Metrics: Stationary & Rotating - Can They Coexist?

[Emilie.Jung]

Can metrics be stationary and rotating at the same time? Doesn't stationary here means that the metric is time-independent. Thus, if a metric is time-indepedent how could it be rotating?

 

[Dale]

As long as it is rotating the same today as it was yesterday. This can be described in terms of Killing vector fields.

 

[Emilie.Jung]

So there isn't a contradiction there? If something is rotating, say earth, the east hemisphere could be at point x in space and after a few hours it will be at point x'. What do you say?? @Dale

 

[Orodruin]

Generally a stationary state does not exclude motion. This is true here as well as in other areas of physics. The state being stationary only implies that it does not change with time. Things can move, but the state will look the same after that movement.

 

[PeterDonis]

if a metric is time-indepedent how could it be rotating?

Because there are rotating metrics in which none of the metric coefficients depend on time. For example, the Kerr metric describes a rotating black hole; none of its metric coefficients depend on time. (A more rigorous definition would be in terms of Killing vector fields, as Dale says.)

the east hemisphere could be at point x in space and after a few hours it will be at point x'

That doesn't change the metric--the geometry of spacetime. It just changes the locations of particular points on the Earth. The definition of "stationary", as you said in the OP, is that the metric is independent of time. It is not that the positions of all objects are independent of time.

 

[Orodruin]

Just to take a completely unrelated example. Consider the heat conduction through a homogeneous wall in winter (it is cold outside and we wish to keep it warm inside). If the inside and outside temperatures are held constant and different, there temperature in the wall will eventually reach a stationary state. This does not mean that heat is not flowing through the wall - we definitely still need to keep the radiators on to keep it warm inside.

The same idea applies here. Whereas things may be moving, this does not affect the metric, which can be stationary in the same way as the temperature distribution can be stationary in the wall even though heat was flowing.

 

[Dale]

So there isn't a contradiction there? If something is rotating, say earth, the east hemisphere could be at point x in space and after a few hours it will be at point x'. What do you say?? @Dale

That doesn't change the metric. Stationary refers to the metric, not the matter.

Edit: I see the others have explained this in more detail