What can be said about S-T global properties from the EFE?

[loislane]

In another thread I was arguing that leaving aside reasonable physical conditions that are added independently from the math of GR, [wich I consider basically the EFE, the EP, general covariance and the metric and curvature tensors in the neighbourhood of points that solve the EFE in the context of background independence from any fixed geometry that might be inferred], there is no grounds strictly to discuss about global properties of spacetimes like singularities other than as informed speculations based on what subjectively one might consider to be reasonable physically or more pleasing aesthetically or more convenient under certain particular coordinates but certainly not as something derived from the math of GR by the inherent locality of the EFE solutions determined by the absence of an absolute spacetime of constant curvature like in the Minkowski case in SR and the fact that the symmetries in GR are determined by the Diif group of GR, i.e. invariance under arbitrary local changes of coordinates .

Any commnets to the points above?(please be specific)
see for instance http://physics.stackexchange.com/questions/111670/global-properties-of-spacetime-manifolds for background

 

[martinbn]

What exactly is your agument?

It seems that you are saying that most of Hawking and Ellis' book is just informed speculations based, any any of the global results since the 1960's.

 

[PeterDonis]

he math of GR, [wich I consider basically the EFE, the EP, general covariance and the metric and curvature tensors in the neighbourhood of points that solve the EFE in the context of background independence from any fixed geometry that might be inferred]

This is too narrow a definition, at least the way GR is actually done. The way GR is actually done certainly includes global methods, such as those in Hawking & Ellis, as martinbn mentioned. That reference goes into excruciating detail about singularities, precisely because the global properties of solutions that have are of great interest (since those solutions include the FRW solutions of cosmology and the black hole solutions). So I don't see the point of limiting "the math of GR" to purely local properties; that's not all we use GR for.

 

[PeterDonis]

there is no grounds strictly to discuss about global properties of spacetimes like singularities other than as informed speculations based on what subjectively one might consider to be reasonable physically or more pleasing aesthetically or more convenient under certain particular coordinates but certainly not as something derived from the math of GR

The singularity theorems of Hawking & Ellis, which are certainly not "informed speculations" but rigorous mathematical results, are, as I said in my last post, certainly considered part of "the math of GR" by workers in the field. I suggest taking a look at Hawking & Ellis and their definition of what it means for a spacetime to contain a "singularity". The basic point (which has been made in multiple prior threads on this topic) is that singularities are not defined as "places where curvature blows up" or something like that; they are defined in terms of geodesic incompleteness--the existence of geodesics in the spacetime that cannot be extended to or beyond some finite value of their affine parameter. A common reason for geodesic incompleteness is that some invariant quantity evaluated along the geodesic diverges in the limit as some finite value of the affine parameter is approached. However, it is geodesic incompleteness, not the divergence of any particular quantity, that defines the presence of a singularity.

 

[loislane]

This is too narrow a definition, at least the way GR is actually done. The way GR is actually done certainly includes global methods, such as those in Hawking & Ellis, as martinbn mentioned. That reference goes into excruciating detail about singularities, precisely because the global properties of solutions that have are of great interest (since those solutions include the FRW solutions of cosmology and the black hole solutions). So I don't see the point of limiting "the math of GR" to purely local properties; that's not all we use GR for.

The singularity theorems of Hawking & Ellis, which are certainly not "informed speculations" but rigorous mathematical results, are, as I said in my last post, certainly considered part of "the math of GR" by workers in the field. I suggest taking a look at Hawking & Ellis and their definition of what it means for a spacetime to contain a "singularity". The basic point (which has been made in multiple prior threads on this topic) is that singularities are not defined as "places where curvature blows up" or something like that; they are defined in terms of geodesic incompleteness--the existence of geodesics in the spacetime that cannot be extended to or beyond some finite value of their affine parameter. A common reason for geodesic incompleteness is that some invariant quantity evaluated along the geodesic diverges in the limit as some finite value of the affine parameter is approached. However, it is geodesic incompleteness, not the divergence of any particular quantity, that defines the presence of a singularity.

So Hawking and Ellis book is more than 40 years old and it seems even their definition of spacetime(for curved Lorentzian manifolds that is) has been modified by slightly more recent books like "Global Lorentzian geometry" by Beem, Ehrlich and Easley. It might be a good time to update the bibliography a bit.
In H&E definition any pair M, g with g a Lorentzian metric is considered a spacetime but it seems to me that just with that basic starting point it is not possible to prove things like the initial value proble being well posed and the existence of a globally hyperbolic manifold in GR. So nowadays one needs to assume that a Lorentzian manifold must be time-oriented to be called a spacetime.
Now, I would like to get a better grasp of the math requirements in order to assume that a general Lorentzian manifold is time-oriented, that is for admitting a continuous, nowhere vanishing timelike vector field in the presence of curvature..

 

[PeterDonis]

So Hawking and Ellis book is more than 40 years old and it seems even their definition of spacetime(for curved Lorentzian manifolds that is) has been modified by slightly more recent books like "Global Lorentzian geometry" by Beem, Ehrlich and Easley.

Really? Can you give specifics? I wasn't aware that anyone's "definition of spacetime" had been modified.

In H&E definition any pair M, g with g a Lorentzian metric is considered a spacetime but it seems to me that just with that basic starting point it is not possible to prove things like the initial value proble being well posed and the existence of a globally hyperbolic manifold in GR

Of course not. Those proofs require additional assumptions, and H&E are quite clear about what those additional assumptions are.

So nowadays one needs to assume that a Lorentzian manifold must be time-oriented to be called a spacetime.

Just as a note, time orientability is not a sufficient condition for the other properties you named. As is discussed, IIRC, in H&E.

In any case, AFAIK restricting the term "spacetime" to time orientable manifolds only is not a mainstream use of language.

Now, I would like to get a better grasp of the math requirements in order to assume that a general Lorentzian manifold is time-oriented, that is for admitting a continuous, nowhere vanishing timelike vector field in the presence of curvature..

IIRC, H&E go into exactly this in quite some detail.