Time, spacelike foliations and timelike vector fields in GR

[ShayanJ]

Recently I've had some discussions about time in GR. I've always read in different places that people usually want a spacetime to have a hypersurface-orthogonal timelike Killing vector field so that they can assign a time dimension to that spacetime. But Why is this needed?
I can understand it that a hypersurface-orthogonal timelike vector field allows you to define the worldlines of some observers and the hypersurfaces will be their "space" in each instant of "time". So this seems natural. But why do we need it to be a Killing vector field? What's wrong with associating a time to a spacetime that tells you the spacetime is changing with that time?

What condition should a spacetime satisfy to allow a globally hypersurface-orthogonal timelike vector field?

Also, in GR, we work with Lorentzian manifolds, so the signature is always (-+++)(or the other equivalent convention). But sometimes the metric can't be diagonalized. So is it still as easy as checking the sign of the diagonal terms to check the signature? Can we always put a metric in form such that one of the diagonal elements is always negative(or positive in the other convention)?

Another question I have, is that is it always possible to foliate a spacetime into spacelike hypersurfaces? If not, what condition should a spacetime satisfy to allow such a foliation?

Sorry if my questions are too many and too diverse. But I asked them in one thread because I thought there can be a discussion about time in GR which contains the answer of all of the questions above.

Thanks

 

[Orodruin]

But why do we need it to be a Killing vector field?

We do not. The interior of the Schwarzschild metric is an example of a space time which does not allow this.

But sometimes the metric can't be diagonalized.

Do you have an example of this?
You cannot just check the signs on the diagonal - the diagonal entries may even be zero depending on your coordinate system.

 

[ShayanJ]

We do not. The interior of the Schwarzschild metric is an example of a space time which does not allow this.

Yes, but in cases where the metric is diagonal, we can simply call the negative element, the time coordinate. But what about more general spacetimes?

Do you have an example of this?
You cannot just check the signs on the diagonal - the diagonal entries may even be zero depending on your coordinate system.

Surely you can't diagonalize the Kerr metric!

 

[Orodruin]

Surely you can't diagonalize the Kerr metric!

Why not? It is symmetric like any other metric.

 

[ShayanJ]

Why not? It is symmetric like any other metric.

OK...so in principle we can diagonalize any metric. But do you have a reference where someone actually finds a coordinate transformation that diagonalizes the Kerr metric?
Anyway, I can still keep my question. Suppose a physicist is working with a non-diagonal metric but is simply unable to diagonalize it. How can s\he speak of time in a spacetime with that metric?

 

[Orodruin]

But do you have a reference where someone actually finds a coordinate transformation that diagonalizes the Kerr metric?

You do not need to globally diagonalise the metric for it to be diagonalisable at each point in spacetime. For a given point it is just a matter of diagonalising a 4x4 matrix.

Suppose a physicist is working with a non-diagonal metric but is simply unable to diagonalize it.

Again, you do not need to find a global time. In fact, it is not even certain there is a global time (this relates to your other question - it is not always possible to find a spacelike foliation).

Also note that there locally are many (in fact an infinite number of) different time-like directions.

 

[ShayanJ]

You do not need to globally diagonalise the metric for it to be diagonalisable at each point in spacetime. For a given point it is just a matter of diagonalising a 4x4 matrix.Again, you do not need to find a global time. In fact, it is not even certain there is a global time (this relates to your other question - it is not always possible to find a spacelike foliation).

Also note that there locally are many (in fact an infinite number of) different time-like directions.

Thanks.
Do you know any book or paper that has an in-depth discussion about such issues?

 

[PeterDonis]

What's wrong with associating a time to a spacetime that tells you the spacetime is changing with that time?

There's nothing wrong with it (except that it should be "space is changing", not "spacetime is changing" ). Orodruin gave one example of a region of spacetime where space is changing with time (the interior of Schwarzschild spacetime). Another obvious example is FRW spacetime; space is expanding with time with respect to comoving observers.

is it always possible to foliate a spacetime into spacelike hypersurfaces? If not, what condition should a spacetime satisfy to allow such a foliation?

The definitive source for answers to questions like these is Hawking & Ellis. Your question as you state it is probably too general--the answer to it as you state it is, I think, "yes", but that answer isn't going to be very informative since it basically amounts to saying that yes, there are spacelike hypersurfaces in every spacetime.

What I think you mean to ask is, when is it possible to foliate a spacetime with a set of Cauchy surfaces, which are a set of spacelike hypersurfaces each of which intersects every timelike and null worldline exactly once. A spacetime that satisfies this condition is called globally hyperbolic. Hawking & Ellis have a lot of discussion of globally hyperbolic spacetimes and the conditions required to have one, since those spacetimes are the ones that always have a well-posed initial value formulation.