Foliation of spacetime into spacelike hypersurfaces

[spaghetti3451]

A spacetime can be foliated by spacelike hypersurfaces which are related by an isometry $\partial_t$.1. Why is $\partial_{t}$ called an isometry of the spacetime?

2. How are the spacelike hypersurfaces related by the isometry $\partial_{t}$?

 

[PeterDonis]

A spacetime can be foliated by spacelike hypersurfaces which are related by an isometry $\partial_t$.

Are you saying every spacetime has this property? If so, that's not correct; many spacetimes have no isometries.

Or are you just referring to some particular spacetime? If so, which spacetime?

A source for where you are getting this from would also be helpful; your statement isn't really correct, since an isometry does not "relate" hypersurfaces in a foliation.

 

[spaghetti3451]

The reference is the following: https://arxiv.org/abs/1602.07982.

Look at the sentence below the caption of figure 4 on page 8 of the notes.

 

[PeterDonis]

Look at the sentence below the caption of figure 4 on page 8 of the notes.

Since this is an advanced source, it is assuming that readers already understand in detail the procedure that is being described here; so it doesn't have to explain that procedure in detail and with rigor. You should not be taking this wording to be part of such a detailed rigorous description.

Also, this section is talking about QFT on a Euclidean space, not on a Minkowski spacetime. So the "isometry" it is talking about is not the kind you will see in ordinary Minkowski spacetime. (Note that the word "spacelike", which you used in the OP, does not appear in the sentence you refer to in the paper.)

To illustrate how the foliation concept works in an ordinary (locally Lorentzian) spacetime, consider inertial coordinates on Minkowski spacetime. Each surface of constant coordinate time $t$ is a spacelike hypersurface, and the set of all such hypersurfaces foliates the entire spacetime. Furthermore, the coordinate basis vector $\partial_t$ in these coordinates generates an isometry; heuristically, "isometry" means that each of the spacelike hypersurfaces in the foliation has the same geometry (in this case, each one is just flat Euclidean 3-space). We say that $\partial_t$ "generates" this isometry because integral curves of this vector fields, which are just the worldlines of inertial observers at rest in these coordinates, form a timelike congruence (a family of worldlines that never intersect each other and which cover the entire spacetime), and an observer moving along any of these worldlines sees an unchanging spacetime geometry in his vicinity.

For another example, consider Schwarzschild spacetime outside the event horizon. Here the spacelike hypersurfaces that foliate the spacetime (more precisely, this region of the spacetime) are the surfaces of constant Schwarzschild coordinate time $t$; and the timelike vector field that generates the isometry is again the coordinate basis vector $\partial_t$. Again, each spacelike hypersurface has the same geometry (this time it's the Flamm paraboloid), and observers moving along integral curves of the isometry (call these "static" observers) see an unchanging spacetime geometry in their vicinity. The difference in this case, vs. the Minkowski case above, is that the spacetime geometry is different in the vicinity of different static observers (more precisely, static observers at different radius).

So the isometry doesn't "relate" the different spacelike hypersurfaces; it tells us which observers (which worldlines) see unchanging spacetime geometry in their vicinity. The spacelike hypersurfaces all have the same geometry; there's nothing to "relate".

 

[physAWAY221]

This is interesting. I understand that you are talking about the wordlines for a set of static observers and the hypersurfaces seen by them which are orthogonal to these worldlines. Isn't the isometry there because you are talking about a static solution for the metric in both cases?

I am just getting into the technicalities of GR. Could you point me towards the references where I will be able to learn the following with rigor.

integral curves of this vector fields

a timelike congruence

the timelike vector field that generates the isometry

 

[PeterDonis]

Isn't the isometry there because you are talking about a static solution for the metric in both cases?

Yes, that is what "static" means: that there is a timelike Killing vector field (isometry) which is hypersurface orthogonal (the spacetime can be foliated by a family of spacelike hypersurfaces that are everywhere orthogonal to the integral curves of the isometry).

Could you point me towards the references where I will be able to learn the following with rigor.

The classic GR textbook that goes into this in some detail is Wald (1984). The more advanced classic source is Hawking & Ellis (1973).

 

[physAWAY221]

Thanks. I will go look it up. :)