I The name of the 3+1 metric where time is normal to space?

[gnnmartin]

I am interested in looking at the metric where time is everywhere normal to space, so g^ta=0 everywhere, where t is the time coordinate and 'a' is any of the space coordinates. I'm finding it hard to look up in the literature: does it have a name that I can search for?

My main interest is in seeing which space/times can be described by it, and which (if any) can't. I think I remember that any chart can be re-expressed in this form, but I can't find that in my textbooks.

 

[fresh_42]

Did you mean the Minkowski metric?

https://en.wikipedia.org/wiki/Minkowski_space
https://en.wikipedia.org/wiki/Four-dimensional_space

 

[gnnmartin]

Did you mean the Minkowski metric?

https://en.wikipedia.org/wiki/Minkowski_space
https://en.wikipedia.org/wiki/Four-dimensional_space

No, Minkowski is a special case. The metric I want has no restrictions on g^ab where a and b are space dimensions (Minkowski by contrast has g^ab=0 where a≠b).

In the metric I want, only g^ta and its derivatives are zero.

In particular, I am interested in whether given a spatial 3 space which is part of a time slice of a space/time taken so that time is normal to the 3 surface, is it always possible to extend that throughout the space time? The fact that there are only 6 constrained derivatives (g^ta,b where b is a space coordinate normal to the edge of the surface bounding the 3 space), suggests it is possible.

I'm then interested in knowing if one can evolve this 3 space forward in time. The fact that there are then only 3 constrained derivatives again suggests it is possible, and if it is not I would like to understand how it becomes impossible.

 

[robphy]

Possibly useful:
https://en.wikipedia.org/wiki/Stationary_spacetime
https://en.wikipedia.org/wiki/Static_spacetime
http://grwiki.physics.ncsu.edu/wiki/Splitting_Spacetime

 

[PeterDonis]

In the metric I want, only g_ta and its derivatives are zero.

I don't know that there's a general term for a spacetime geometry that meets this condition--that one can find a coordinate chart in which the timelike basis vector $\partial_t$ is everywhere orthogonal to spacelike surfaces of constant time. The term I would use is "hypersurface orthogonal", but that term usually has a more restricted application, to spacetimes with time translation symmetry (a timelike Killing vector field). However, for example, FRW spacetime meets your condition, but it does not have time translation symmetry (it describes an expanding universe).

I am interested in whether given a spatial 3 space which is part of a time slice of a space/time taken so that time is normal to the 3 surface, is it always possible to extend that throughout the space time?

No, it isn't. In fact it's not even possible to find a coordinate chart on every spacetime in which any spacelike surface of constant coordinate time is orthogonal to the timelike basis vector $\partial_t$ at more than one event. For an example, look at Kerr spacetime. Matt Visser's paper is a good resource:

https://arxiv.org/pdf/0706.0622.pdf

He discusses a number of possible coordinate charts on Kerr spacetime, but none of them meet your requirement, and I'm pretty sure it's been proven that it is impossible to find one that does on this spacetime.

Possibly useful:

A static spacetime meets the OP's condition, yes, but as above, so does FRW spacetime, which is not static or even stationary. So the condition the OP is interested in is more general.

 

[gnnmartin]

Peter, thanks.